Structural Design Optimization Considering Uncertainties

Structural design optimization considering uncertainties

Structures and Infrastructures Series ISSN 1747-7735

Book Series Editor:

Dan M. Frangopol Professor of Civil Engineering and Fazlur R. Khan Endowed Chair of Structural Engineering and Architecture Department of Civil and Environmental Engineering Center for Advanced Technology for Large Structural Systems (ATLSS Center) Lehigh University Bethlehem, PA, USA

Volume 1

Structural design optimization considering uncertainties

Edit ed by

Yiannis Tsompanakis1, Nikos D. Lagaros2 & Manolis Papadrakakis3 1

Department of Applied Sciences, Technical University of Crete, University Campus, Chania, Crete, Greece 2,3 Institute of Structural Analysis & Seismic Research, Faculty of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens, Greece

LONDON / LEIDEN / NEW YORK / PHILADELPHIA / SINGAPORE

Colophon Book Series Editor : Dan M. Frangopol Volume Editors: Yiannis Tsompanakis, Nikos D. Lagaros and Manolis Papadrakakis Cover illustration: Objective space of the M-3OU multi-criteria optimization problem Nikos D. Lagaros September 2007 This edition published in the Taylor & Francis e-Library, 2008. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Taylor & Francis is an imprint of the Taylor & Francis Group, an informa business ©2008 Taylor & Francis Group, London, UK All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Structural design optimization considering uncertainties / Edited by Yiannis Tsompanakis, Nikos D. Lagaros & Manolis Papadrakakis. p. cm. – (Structures and infrastructures series ; 1747-7735) Includes bibliographical references and index. ISBN 978-0-415-45260-1 (hardcover : alk. paper) ISBN 978-0-203-93852-2 (e-book) 1. Structural optimization. I. Tsompanakis, Yiannis. 1969– II. Lagaros, Nikos D. 1970– III. Papadrakakis, Manolis. 1948– TA658.8.S73 2007 624.1
7713–dc22 2007040343 Published by: Taylor & Francis/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.balkema.nl, www.taylorandfrancis.co.uk, www.crcpress.com ISBN 0-203-93852-6 Master e-book ISBN

ISBN13 978-0-415-45260-1(Hbk) ISBN13 978-0-203-93852-2(eBook) Structures and Infrastructures Series: ISSN 1747-7735 Volume 1

Table of Contents

Editorial About the Book Series Editor Foreword Preface Brief Curriculum Vitae of the Editors List of Contributors Author Data

IX XI XIII XV XXI XXIII XXV

PART 1

Reliability-Based Design Optimization (RBDO) 1

2

Principles of reliability-based design optimization Alaa Chateauneuf , University Blaise Pascal, France Reliability-based optimization of engineering structures

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31

John D. Sørensen, Aalborg University, Aalborg, Denmark 3

4

Reliability analysis and reliability-based design optimization using moment methods Sang Hoon Lee, Northwestern University, Evanston, IL, USA Byung Man Kwak, Korea Advanced Institute of Science and Technology, Daejeon, Korea Jae Sung Huh, Korea Aerospace Research Institute, Daejeon, Korea Efficient approaches for system reliability-based design optimization Efstratios Nikolaidis, University of Toledo, Toledo, OH, USA Zissimos P. Mourelatos, Oakland University, Rochester, MI, USA Jinghong Liang, Oakland University, Rochester, MI, USA

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5

6

Contents

Nondeterministic formulations of analytical target cascading for decomposition-based design optimization under uncertainty Michael Kokkolaras, University of Michigan, Ann Arbor, MI, USA Panos Y. Papalambros, University of Michigan, Ann Arbor, MI, USA Design optimization of stochastic dynamic systems by algebraic reduced order models Gary Weickum, University of Colorado at Boulder, Boulder, CO, USA Matt Allen, University of Colorado at Boulder, Boulder, CO, USA Kurt Maute, University of Colorado at Boulder, Boulder, CO, USA Dan M. Frangopol, Lehigh University, Bethlehem, PA, USA

7

Stochastic system design optimization using stochastic simulation Alexandros A. Taflanidis, California Institute of Technology, CA, USA James L. Beck, California Institute of Technology, CA, USA

8

Numerical and semi-numerical methods for reliability-based design optimization Ghias Kharmanda, Aleppo University, Aleppo, Syria

9

Advances in solution methods for reliability-based design optimization Alaa Chateauneuf , University Blaise Pascal, France Younes Aoues, University Blaise Pascal, France

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Non-probabilistic design optimization with insufficient data using possibility and evidence theories Zissimos P. Mourelatos, Oakland University, Rochester, MI, USA Jun Zhou, Oakland University, Rochester, MI, USA

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155

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217

247

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A decoupled approach to reliability-based topology optimization for structural synthesis 281 Neal M. Patel, University of Notre Dame, Notre Dame, IN, USA John E. Renaud, University of Notre Dame, Notre Dame, IN, USA Donald Tillotson, University of Notre Dame, Notre Dame, IN, USA Harish Agarwal, General Electric Global Research, Niskayuna, NY, USA Andrés Tovar, National University of Colombia, Bogota, Colombia

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Sample average approximations in reliability-based structural optimization: Theory and applications Johannes O. Royset, Naval Postgraduate School, Monterey, CA, USA Elijah Polak, University of California, Berkeley, CA, USA

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Cost-benefit optimization for maintained structures

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Rüdiger Rackwitz, Technical University of Munich, Munich, Germany Andreas E. Joanni, Technical University of Munich, Munich, Germany 14

A reliability-based maintenance optimization methodology Wu Y.-T., Applied Research Associates Inc., Raleigh, NC, USA

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C o n t e n t s VII

15

Overview of reliability analysis and design capabilities in DAKOTA with Application to shape optimization of MEMS Michael S. Eldred, Sandia National Laboratories, Albuquerque, NM, USA Barron J. Bichon, Vanderbilt University, Nashville, TN, USA Brian M. Adams, Sandia National Laboratories, Albuquerque, NM, USA Sankaran Mahadevan, Vanderbilt University, Nashville, TN, USA

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PART 2

Robust Design Optimization (RDO) 16

Structural robustness and its relationship to reliability Jorge E. Hurtado, National University of Colombia, Manizales, Colombia

17

Maximum robustness design of trusses via semidefinite programming Yoshihiro Kanno, University of Tokyo, Tokyo, Japan Izuru Takewaki, Kyoto University, Kyoto, Japan

18

19

20

21

Design optimization and robustness of structures against uncertainties based on Taylor series expansion Ioannis Doltsinis, University of Stuttgart, Stuttgart, Germany Info-gap robust design of passively controlled structures with load and model uncertainties Izuru Takewaki, Kyoto University, Kyoto, Japan Yakov Ben-Haim, Technion, Haifa, Israel Genetic algorithms in structural optimum design using convex models of uncertainty Sara Ganzerli, Gonzaga University, Spokane, WA, USA Paul De Palma, Gonzaga University, Spokane, WA, USA Metamodel-based computational techniques for solving structural optimization problems considering uncertainties Nikos D. Lagaros, National Technical University of Athens, Athens, Greece Yiannis Tsompanakis, Technical University of Crete, Chania, Greece Michalis Fragiadakis, University of Thessaly, Volos, Greece Vagelis Plevris, National Technical University of Athens, Athens, Greece Manolis Papadrakakis, National Technical University of Athens, Athens, Greece

References Author index Subject index

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599 631 633

Editorial

Welcome to the New Book Series Structures and Infrastructures. Our knowledge to model, analyze, design, maintain, manage and predict the lifecycle performance of structures and infrastructures is continually growing. However, the complexity of these systems continues to increase and an integrated approach is necessary to understand the effect of technological, environmental, economical, social and political interactions on the life-cycle performance of engineering structures and infrastructures. In order to accomplish this, methods have to be developed to systematically analyze structure and infrastructure systems, and models have to be formulated for evaluating and comparing the risks and benefits associated with various alternatives. We must maximize the life-cycle benefits of these systems to serve the needs of our society by selecting the best balance of the safety, economy and sustainability requirements despite imperfect information and knowledge. In recognition of the need for such methods and models, the aim of this Book Series is to present research, developments, and applications written by experts on the most advanced technologies for analyzing, predicting and optimizing the performance of structures and infrastructures such as buildings, bridges, dams, underground construction, offshore platforms, pipelines, naval vessels, ocean structures, nuclear power plants, and also airplanes, aerospace and automotive structures. The scope of this Book Series covers the entire spectrum of structures and infrastructures. Thus it includes, but is not restricted to, mathematical modeling, computer and experimental methods, practical applications in the areas of assessment and evaluation, construction and design for durability, decision making, deterioration modeling and aging, failure analysis, field testing, structural health monitoring, financial planning, inspection and diagnostics, life-cycle analysis and prediction, loads, maintenance strategies, management systems, nondestructive testing, optimization of maintenance and management, specifications and codes, structural safety and reliability, system analysis, time-dependent performance, rehabilitation, repair, replacement, reliability and risk management, service life prediction, strengthening and whole life costing. This Book Series is intended for an audience of researchers, practitioners, and students world-wide with a background in civil, aerospace, mechanical, marine and automotive engineering, as well as people working in infrastructure maintenance, monitoring, management and cost analysis of structures and infrastructures. Some volumes are monographs defining the current state of the art and/or practice in the field, and some are textbooks to be used in undergraduate (mostly seniors), graduate and

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postgraduate courses. This Book Series is affiliated to Structure and Infrastructure Engineering (http://www.informaworld.com/sie), an international peer-reviewed journal which is included in the Science Citation Index. If you like to contribute to this Book Series as an author or editor, please contact the Book Series Editor ([email protected]) or the Publisher ([email protected]). A book proposal form can be downloaded at www.balkema.nl. It is now up to you, authors, editors, and readers, to make Structures and Infrastructures a success. Dan M. Frangopol Book Series Editor

About the Book Series Editor

Dr. Dan M. Frangopol is the first holder of the Fazlur R. Khan Endowed Chair of Structural Engineering and Architecture at Lehigh University, Bethlehem, Pennsylvania, USA, and a Professor in the Department of Civil and Environmental Engineering at Lehigh University. He is also an Emeritus Professor of Civil Engineering at the University of Colorado at Boulder, USA, where he taught for more than two decades (1983–2006). Before joining the University of Colorado, he worked for four years (1979–1983) in structural design with A. Lipski Consulting Engineers in Brussels, Belgium. In 1976, he received his doctorate in Applied Sciences from the University of Liège, Belgium, and holds an honorary doctorate degree (Doctor Honoris Causa) and a B.S. degree from the Technical University of Civil Engineering in Bucharest, Romania. He is a Fellow of the American Society of Civil Engineers (ASCE), American Concrete Institute (ACI), and International Association for Bridge and Structural Engineering (IABSE). He is also an Honorary Member of both the Romanian Academy of Technical Sciences and the Portuguese Association for Bridge Maintenance and Safety. He is the initiator and organizer of the Fazlur R. Khan Lecture Series (www.lehigh.edu/frkseries) at Lehigh University. Dan Frangopol is an experienced researcher and consultant to industry and government agencies, both nationally and abroad. His main areas of expertise are structural reliability, structural optimization, bridge engineering, and life-cycle analysis, design, maintenance, monitoring, and management of structures and infrastructures. He is the Founding President of the International Association for Bridge Maintenance and Safety (IABMAS, www.iabmas.org) and of the International Association for Life-Cycle Civil Engineering (IALCCE, www.ialcce.org), and Past Director of the Consortium on Advanced Life-Cycle Engineering for Sustainable Civil Environments (COALESCE). He is also the Chair of the Executive Board of the International Association for Structural Safety and Reliability (IASSAR, www.columbia.edu/cu/civileng/iassar) and the Vice-President of the International Society for Health Monitoring of Intelligent Infrastructures (ISHMII, www.ishmii.org). Dan Frangopol is the recipient of several prestigious awards including the 2007 ASCE Ernest Howard Award, the 2006 IABSE OPAC Award, the 2006 Elsevier Munro Prize, the 2006 T. Y. Lin Medal, the 2005 ASCE Nathan M. Newmark Medal, the 2004 Kajima Research Award, the 2003

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ASCE Moisseiff Award, the 2002 JSPS Fellowship Award for Research in Japan, the 2001 ASCE J. James R. Croes Medal, the 2001 IASSAR Research Prize, the 1998 and 2004 ASCE State-of-the-Art of Civil Engineering Award, and the 1996 Distinguished Probabilistic Methods Educator Award of the Society of Automotive Engineers (SAE). Dan Frangopol is the Founding Editor-in-Chief of Structure and Infrastructure Engineering (Taylor & Francis, www.informaworld.com/sie) an international peerreviewed journal, which is included in the Science Citation Index. This journal is dedicated to recent advances in maintenance, management, and life-cycle performance of a wide range of structures and infrastructures. He is the author or co-author of over 400 refereed publications, and co-author, editor or co-editor of more than 20 books published by ASCE, Balkema, CIMNE, CRC Press, Elsevier, McGraw-Hill, Taylor & Francis, and Thomas Telford and an editorial board member of several international journals. Additionally, he has chaired and organized several national and international structural engineering conferences and workshops. Dan Frangopol has supervised over 70 Ph.D. and M.Sc. students. Many of his former students are professors at major universities in the United States, Asia, Europe, and South America, and several are prominent in professional practice and research laboratories. For additional information on Dan M. Frangopol’s activities, please visit www.lehigh.edu/∼dmf206/

Foreword

The aim of structural optimization is to achieve the best possible design by maximizing benefits under conflicting criteria. Uncertainties are unavoidable in the structural optimization process. Therefore, a realistic optimal design process should definitely consider uncertainties. Two broad types of uncertainty have to be considered: (a) uncertainty associated with randomness, the so-called aleatory uncertainty, and (b) uncertainty associated with imperfect modeling, the so-called epistemic uncertainty. It has been clearly demonstrated that both aleatory and epistemic uncertainties can be treated, separately or combined, and analyzed using the principles of probability and statistics. Structural reliability theory has been developed during the past decades to handle problems considering such uncertainties. This continuous development has had considerable impact in recent years on structural optimization. The purpose of this book is to present the latest research findings in the field of structural optimization considering uncertainties. A wide variety of topics are covered by leading researchers. The first part (Chapters 1 to 15) is devoted to reliability-based design optimization, and the second part (Chapters 16 to 21) deals with robust design optimization. To provide the reader with a good overview of pertinent literature, all cited papers and additional references on the topics discussed, are collected in a comprehensive list of references. The Book Series Editor would like to express his appreciation to the Editors and all Authors who contributed to this book. It is his hope that this first volume in the Structures and Infrastructures Book Series will generate a lot of interest and help engineers to design the best structural systems under uncertainty.

Dan M. Frangopol Book Series Editor Bethlehem, Pennsylvania November 2, 2007

Preface

Uncertainties are inherent in engineering problems and the scatter of structural parameters from their nominal ideal values is unavoidable. The response of structural systems can sometimes be very sensitive to uncertainties encountered in the material properties, manufacturing conditions, external loading conditions and analytical and/or numerical modelling. In recent years, probabilistic-based formulations of optimization problems have been developed to account for uncertainties through stochastic simulation and probabilistic analysis. Stochastic analysis methods have been developed significantly over the last two decades and have stimulated the interest for the probabilistic optimum design of structures. There are mainly two design formulations that account for probabilistic systems response: Reliability-Based Design Optimization (RBDO) and Robust Design Optimization (RDO). The main goal of RBDO methods is to design for safety with respect to extreme events. RDO methods primarily seek to minimize the influence of stochastic variations on the mean design. The selected contributions of this book deal with the use of probabilistic methods for optimal design of different types of structures and various considerations of uncertainties. This volume is a collective book of twenty-one self-contained chapters, which present state-of-the-art theoretical advances and applications in various fields of probabilistic computational mechanics. The first fifteen chapters of the book are focused on RBDO theory and applications, while the rest of the chapters deal with advances in RDO and combined RBDO-RDO theory and applications. Apart from the reference list that is given separately for each chapter, a complete list of references is also provided for the reader. In order to obtain contributions that cover a wide spectrum of engineering problems, the problem of optimum design is considered in a broad sense. The probabilistic framework allows for a consistent treatment of both cost and safety. In what follows a short description of the book content is presented. In the introductory chapter by Chateauneuf, the fundamental theoretical and computational issues related to RBDO are described and the advantages of RBDO compared to conventional deterministic optimization approaches are outlined. This chapter emphasizes the role of uncertainties in deriving a “true’’ optimal solution, defined as the best compromise between cost minimization and safety assurance. The presented RBDO formulations cover various important probabilistic issues (theoretical, computational and practical), such as multi-component reliability analysis, safety factor calibration, multi-objective applications, as well as a great variety of engineering applications, such as topology, maintenance and time-variant problems.

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The theoretical basis for reliability-based structural optimization is described by Sørensen within the framework of Bayesian statistical decision theory. This contribution presents the latest findings in RBDO with respect to three major types of decision problems with increased degree of complexity and uncertainty: a) decisions with given information (e.g. planning of new structures), b) decisions when new information is provided (e.g. for re-assessment and retrofitting of existing structures), c) decisions involving planning of experiments/inspections to obtain new information (e.g. for inspection planning). Furthermore, RBDO issues related to decisions with systematic reconstruction are also discussed. Reliability-based, cost-benefit problems are formulated and exemplified with structural optimization. Illustrative examples are presented including a simple introductory example, a decision problem related to bridge re-assessment and a reliability-based decision problem for offshore wind turbines. Lee, Kwak and Huh deal with reliability analysis and reliability-based design optimization using moment methods. By using this approach, a finite number of statistical moments of a system response function are calculated and the probability density function (PDF) of the system response is identified by empirical distribution systems, such as the Pearson or the Johnson system. In this chapter, a full factorial moment method (FFMM) procedure is introduced for reliability analysis calculations. A response surface augmented moment method (RSMM) is developed to construct a series of approximate response surface for enhancing the efficiency of FFMM. The probability of failure is calculated using an empirical distribution system and the first four statistical moments of system’s performance function are calculated from appropriate design simulations. The design sensitivity of the probability of failure, required during RBDO process, is calculated in a semi-analytic way using moment methods. As stated in the chapter by Nikolaidis, Mourelatos and Liang, a designer faces many challenges when applying RBDO to engineering systems. The high computational cost required for RBDO and the efficient computation of the system failure probability are the two principal challenges. As a result, most RBDO studies are restricted to the safety levels of the individual failure modes. In order to overcome this deficiency, two efficient approaches for RBDO are presented in this chapter. Both approaches apportion optimally the system reliability among the failure modes by considering the target values of the failure probabilities of the modes as design variables. The first approach uses a sequential optimization and reliability assessment (SORA) approach, while the second system RBDO approach uses a single-loop method where the searches for the optimum design and for the most probable failure points proceed simultaneously. The two approaches are illustrated and compared on characteristic design examples. Moreover, it is shown that the single-loop approach, enhanced with an active set strategy, is considerably more efficient than the SORA approach. According to the work of Kokkolaras and Papalambros, design subproblems are formulated and solved so that their solution can be integrated to represent the optimal design of the decomposed system. This approach requires appropriate problem formulation and coordination of the distributed, multilevel system design problem. The presented analytical target cascading (ATC) is a methodology suitable for multilevel optimal design problems. Design targets are cascaded to lower levels using the modelbased, hierarchical decomposition of the original design problem. An optimization problem is posed and solved for each design subproblem to minimize deviations from

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propagated targets. Solving the subproblems and using an appropriate coordination strategy the overall system compatibility is preserved. The required computational effort motivated Weickum, Allen, Maute and Frangopol to address the need for developing efficient numerical probabilistic techniques for the reliability analysis and design optimization of stochastic dynamic systems. This work seeks to alleviate the computational costs for optimizing dynamic systems by employing reduced order models. The key to utilize reduced order models in stochastic analysis and optimization lies in making them adaptable to design changes and variations of the random parameters. For this purpose, an extended reduced order model (EROM) method, which is a reduced order model accounting for parameter changes, is integrated into stochastic analysis and design optimization. The application of the proposed EROM is tested both for deterministic and probabilistic optimization of the characteristic connecting rod example. Taflanidis and Beck consider a two stage framework for efficient implementation of RBDO of dynamical systems under stochastic excitation (e.g. earthquake, wind or wave loading), where uncertainties are assumed for both the excitation characteristics and the structural model adopted. In the first stage a novel approach, the so called stochastic subset optimization (SSO), is implemented for iteratively identifying a subset of the original design space that has high probability of containing the optimal design variables. The second stage adopts a stochastic optimization algorithm to pinpoint, if needed, the optimal design variables within that subset. Topics related to the combination of the two different stages, in order to enhance the overall efficiency of the presented methodology, are also discussed. An illustrative example for the seismic retrofitting, via viscous dampers, is presented. The minimization of the expected lifecycle cost is adopted as the design objective, in which the cost associated with damage caused by future earthquakes is calculated by stochastic simulation via a realistic probabilistic model for the structure and the ground motion that involves the formulation of an effective loss function model. Kharmanda discusses in his contribution issues related to RBDO formulation and solution procedures. The RBDO formulation is defined as a nonlinear mathematical programming problem in which the mean values of uncertain system parameters are used as design variables and its weight or cost is optimized subjected to prescribed probabilistic constraints. In this chapter, recent developments for the efficient RBDO problem solving using semi-numerical and numerical techniques are presented. Following a detailed description of the proposed methods, their efficiency is demonstrated in computationally demanding dynamic applications. The obtained results as well as the computational implications of the methods are compared and their advantages and disadvantages are highlighted in a comprehensive manner. In the contribution by Chateauneuf and Aoues, the main objective is to apply appropriate numerical methods in order to solve RBDO problems more efficiently. A comprehensive description of the most commonly used RBDO formulations and the corresponding numerical methods is provided. A good RBDO algorithm should satisfy the conditions of efficiency (computation time), precision (accuracy of finding the optimum), generality (capability to deal with different kinds of problems) and robustness (stability of the convergence for any admissible initial point, local or global convergence criteria, etc). All these aspects are discussed in detail, and effective solutions are proposed via characteristic test examples.

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In the chapter by Mourelatos and Zhou, evidence theories are used to account for uncertainty in structural design with incomplete and/or fuzzy information. A sequential possibility-based design optimization (SPDO) method is presented which decouples the design loop and the reliability assessment of each constraint and is also capable of handling both random and possibilistic design variables. Furthermore, a computationally efficient optimum design formulation using evidence theory is presented, which can handle a mixture of epistemic and aleatory uncertainties. Numerical examples demonstrate the application of possibility and evidence theories in probabilistic optimum design and highlight the trade-offs among reliability-based, possibility-based and evidence-based design approaches. In the chapter by Patel, Renaud, Tillotson, Agarwal, and Tovar, the mode of failure is considered to be the maximum deflection of the structure in reliability-based topology optimization (RBTO). A decoupled approach is employed in which the topology optimization stage is separate from the reliability analysis. The proposed decoupled reliability-based design optimization methodology is an approximate technique to obtain consistent reliable designs at lower computational expense. An efficient nongradient Hybrid Cellular Automaton (HCA) method has been implemented in the proposed decoupled approach for evaluating density changes, while the strain energy for every new design is evaluated via finite element structural analyses. The chapter by Royset and Polak presents recent advances in combining Monte Carlo sampling and nonlinear programming algorithms for RBDO problems utilizing effective approximation techniques that can lead to the reduction of the excessive computational cost. More specifically, they present an approach where the reliability term in the problem formulation is replaced by a statistical estimate of the reliability obtained by means of Monte Carlo sampling. The authors emphasize on the calculation of “adaptive optimal’’ sample size which is achieved using sample-adjustment rules by solving auxiliary optimization tasks during the evolution of RBDO. The efficiency of the methods is verified in a number of numerical examples arising in design of various types of structures having a single or multiple limit-state functions, in which reliability terms are included in both objective and constraint functions. Rackwitz and Joanni describe theoretical and practical issues leading to cost-efficient optimization formulations for existing aging structures. In order to establish an efficient methodology for optimizing maintenance, an elaborate model, based on renewal theory that uses systematic reconstruction or repair schemes after suitable inspection, is formulated in which life-cycle cost perspective is used. The presented implementation shows the impact of the choice of the objective function, the risk acceptability and the transient behaviour of the failure rate. The emphasis is given on concrete structures, but the described methodology can be applied to any material and any type of engineering structures. In particular, minimal age-dependent block repairs and maintenance by inspection and repair have been studied via an illustrative example. Wu describes in his contribution a reliability-based damage tolerance (RBDT) approach that employs a systematic approach for probabilistic fracture-mechanics damage tolerance analysis with maintenance planning under various uncertainties. Moreover, he presents the successful integration of RBDT in the proposed reliabilitybased maintenance optimization (RBMO) methodology, focusing on efficient sampling and other computational strategies for handling the uncertainties related to structural maintenance issues (fatigue, failure, inspection, repair, etc). A comparison of different

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versions of the proposed RBMO for analytical benchmark examples as well as for realistic test cases is presented. Eldred, Bichon, Adams and Mahadevan present an overview of recent research related to first and second-order reliability methods. They outline both the forward reliability analysis of computing probabilities for specified response levels (using the so-called RIA, i.e. the reliability index approach) and the inverse reliability analysis of computing response levels for specified probabilities (the performance measure approach or PMA). A number of algorithmic variations are described and the effect of different limit state approximations, probability integrations, warm starting, most probable point search algorithms, and Hessian approximations is discussed. Relative performance of these reliability analysis and design algorithms is presented for several benchmark test problems as well as for real-world applications related to the probabilistic analysis and design of micro-electro-mechanical systems (MEMS) using the DAKOTA software. Hurtado aims at exploiting the complementary nature of RDO and RBDO probabilistic optimization approaches, using effective expansion techniques. Under this viewpoint, an efficient approximate methodology that integrates RDO and RBDO is proposed, in an effort to allow the designer to foresee the implications of adopting RDO or RBDO in the optimization process of probabilistic applications and to combine them in an optimum manner. On this basis, the concept of “robustness assurance’’ in structural design is introduced, in a similar manner to the “quality assurance’’ in the construction phase. For this purpose, a practical method for robust optimal design interpreted as entropy minimization is presented. Illustrative examples are presented to elucidate the advantages of the proposed approach. The robustness function is a measure of the performance of structural systems and expresses the greatest level of non-probabilistic uncertainty at which any constraint on structural performance cannot be violated. Kanno and Takewaki propose an efficient scheme for robust design optimization of trusses under various uncertainties. The structural optimization problem is formulated in the framework of an info-gap decision theory, aiming at maximizing the robustness function and is solved using semi-definite programming methods. Characteristic truss examples are used to demonstrate the efficiency of the proposed methodology. In his chapter, Doltsinis advocates the importance of an elaborate consideration of random scatter in industrial engineering with regard to reliability, and for securing standards of operation performance (robustness). For this purpose, synthetic Monte Carlo sampling and analytic Taylor series expansion that offer alternatives of stochastic analysis and design improvement are described. The robust optimum design problem is formulated as a two-criteria task that involves minimization of mean value and standard deviation of the objective function, while randomness of the constraints is also considered. Numerical applications justify the efficiency of the proposed approach are presented with linear and nonlinear structural response. Takewaki and Ben-Haim present a robust design concept, capable of incorporating uncertainties for both demand (loads) and capacity (various structural design parameters) of a dynamically loaded structure. Since uncertainties are prevalent in many cases, it is necessary to satisfy critical performance requirements, rather than to optimize performance, and to maximize the robustness to uncertainty. In the proposed implementation, the so called, “info-gap models of uncertainty’’ are used to represent

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uncertainties in the Fourier amplitude spectrum of the dynamic loading and the basic structural parameters related the vibration model of the structure. Furthermore, earthquake input energy is introduced as a new measure of structural performance for passively controlled structures and uncertainties of damping coefficients of control devices are also considered. Ganzerli and De Palma focus on the use of convex models of uncertainty with genetic algorithms for optimal structural design. Convex models theory together with probability and fuzzy sets, convex models can be considered part of the so-called “uncertainty triangle’’. Following, a literature review on convex models and their applications a description of convex models theory as an efficient alternative way to deal with problems having severe structural uncertainties, is presented. Subsequently, applications including the use of convex models of uncertainty combined with genetic algorithms for optimal structural design of trusses are demonstrated, and directions for further research in this area are given. In the last chapter, Lagaros, Tsompanakis, Fragiadakis, Plevris and Papadrakakis present efficient methodologies for performing standard RBDO and combined reliability-based and robust design optimization (RRDO) of stochastic structural systems in a multi-objective optimization framework. The proposed methodologies incorporate computationally efficient structural optimization and probabilistic analysis procedures. The optimization part is performed with evolutionary methods while the probabilistic analysis is carried out with the Monte Carlo Simulation (MCS) method with the Latin Hypercube Sampling (LHS) technique for the reduction of the sample size. In order to reduce the excessive computational cost and make the whole procedure feasible for real-world engineering problems the use of Neural Networks (NN) based metamodels is incorporated in the proposed methodology. The use of NN is motivated by the time-consuming repeated FE solutions required in the reliability analysis phase and by the evolutionary optimization algorithm during the optimization process. The editors of this book would like to express their deep gratitude to all the contributors for their most valuable support during the preparation of this volume and for their time and effort devoted to the completion of their contributions. In addition, we are most appreciative to the Book Series Editor, Professor Dan M. Frangopol, for his kind invitation to edit this volume, for preparing the foreword of this book, and for his constructive comments and suggestions offered during the publication process. Finally, the editors would like to thank all the personnel of Taylor and Francis Publishers, especially Germaine Seijger, Richard Gundel, Lukas Goosen, Tessa Halm, Maartje Kuipers and Janjaap Blom, for their most valuable support for the publication of this book.

Yiannis Tsompanakis Nikos D. Lagaros Manolis Papadrakakis September 2007

Brief Curriculum Vitae of the Editors

Yiannis Tsompanakis is Assistant Professor in the Department of Applied Sciences of the Technical University of Crete, Greece, where he teaches structural and computational mechanics as well as earthquake engineering lessons. His scientific interests include computational methods in structural and geotechnical earthquake engineering, structural optimization, probabilistic mechanics, structural assessment and the application of artificial intelligence methods in engineering. Dr. Tsompanakis has published many scientific papers and is the co-editor of several books in computational mechanics. He is involved in the organization of minisymposia and special sessions in international conferences as well as special issues of scientific journals as guest editor. He serves as a board member in various conferences, organized the COMPDYN-2007 conference together with the other editors of this book and acts as a co-editor of the resulting selected papers volume.

Nikos D. Lagaros is Lecturer of structural dynamics and computational mechanics in the School of Civil Engineering of the National Technical University of Athens, Greece. His research activity is focused on the development and the application of novel computational methods and information technology to structural and earthquake engineering analysis and design. In addition, Dr. Lagaros has provided consulting and expert-witness services to private companies and federal government agencies in Greece. He also serves as a member of the editorial board and reviewer of various international scientific journals. He has published numerous scientific papers, and is the co-editor of a number of forthcoming books, one of which is dealing with innovative soft computing applications in earthquake engineering. Nikos Lagaros is co-organizer of COMPDYN 2007 and co-editor of its selected papers volume.

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Brief Curriculum Vitae of the Editors

Manolis Papadrakakis is Professor of Computational Structural Mechanics in the School of Civil Engineering at the National Technical University of Athens, Greece. His main fields of interest are: large-scale, stochastic and adaptive finite element applications, nonlinear dynamics, structural optimization, soil-fluid-structure interaction and soft computing applications in structural engineering. He is co-Editor-in-chief of the Computer Methods in Applied Mechanics and Engineering Journal, an Honorary Editor of the International Journal of Computational Methods, and an Editorial Board member of a number of international scientific journals. He is also a member of both the Executive and the General Council of the International Association for Computational Mechanics, Chairman of the European Committee on Computational Solid and Structural Mechanics and Vice President of the John Argyris Foundation. Professor Papadrakakis has chaired many international conferences and presented numerous invited lectures. He has written and edited various books and published a large variety of scientific articles in refereed journals and book chapters.

List of Contributors

Adams, B.M., Sandia National Laboratories, Albuquerque, NM, USA Agarwal, H., General Electric Global Research, Niskayuna, NY, USA Allen, M., University of Colorado at Boulder, Boulder, CO, USA Aoues, Y., University Blaise Pascal, France Beck, J.L., California Institute of Technology, CA, USA Ben-Haim, Y., Technion, Haifa, Israel Bichon, B.J., Vanderbilt University, Nashville, TN, USA Chateauneuf, A., University Blaise Pascal, France De Palma, P., Gonzaga University, Spokane, WA, USA Doltsinis, I., University of Stuttgart, Stuttgart, Germany Eldred, M.S., Sandia National Laboratories, Albuquerque, NM, USA Fragiadakis, M., University of Thessaly, Volos, Greece Frangopol, D.M., Lehigh University, Bethlehem, PA, USA Ganzerli, S., Gonzaga University, Spokane, WA, USA Huh, J.S., Korea Aerospace Research Institute, Daejeon, Korea Hurtado, J.E., National University of Colombia, Manizales, Colombia Joanni, A.E., Technical University of Munich, Munich, Germany Kanno, Y., University of Tokyo, Tokyo, Japan Kharmanda, G., Aleppo University, Aleppo, Syria Kokkolaras, M., University of Michigan, Ann Arbor, MI, USA Kwak, B.M., Korea Advanced Institute of Science and Technology, Daejeon, Korea Lagaros, N.D., National Technical University of Athens, Athens, Greece Lee, S.H., Northwestern University, Evanston, IL, USA Liang, J., Oakland University, Rochester, MI, USA Mahadevan, S., Vanderbilt University, Nashville, TN, USA Maute, K., University of Colorado at Boulder, Boulder, CO, USA Mourelatos, Z.P., Oakland University, Rochester, MI, USA Nikolaidis, E., University of Toledo, Toledo, OH, USA Papadrakakis, M., National Technical University of Athens, Athens, Greece Papalambros, P.Y., University of Michigan, Ann Arbor, MI, USA Patel, N.M., University of Notre Dame, Notre Dame, IN, USA Plevris, V., National Technical University of Athens, Athens, Greece Polak, E., University of California, Berkeley, CA, USA Rackwitz, R., Technical University of Munich, Munich, Germany

XXIV

List of Contributors

Renaud, J.E., University of Notre Dame, Notre Dame, IN, USA Royset, J.O., Naval Postgraduate School, Monterey, CA, USA Sørensen, J.D., Aalborg University, Aalborg, Denmark Taflanidis, A.A., California Institute of Technology, CA, USA Takewaki, I., Kyoto University, Kyoto, Japan Tillotson, D., University of Notre Dame, Notre Dame, IN, USA Tovar, A., National University of Colombia, Bogota, Colombia Tsompanakis, Y., Technical University of Crete, Chania, Greece Weickum, G., University of Colorado at Boulder, Boulder, CO, USA Wu, Y.-T., Applied Research Associates Inc., Raleigh, NC, USA Zhou, J., Oakland University, Rochester, MI, USA

Author Data

Adams, B.M. Sandia National Laboratories PO Box 5800, MS 1318 Albuquerque, NM 87185-1318 USA Phone: (505)284-8845 Fax: (505)284-2518 Email: [email protected] Agarwal, H. General Electric Global Research Niskayuna, New York, 12309 USA Phone: (574) 631-9052 Fax: (574) 631-8341 Email: [email protected] Allen, M. Research Assistant Center for Aerospace Structures Department of Aerospace Engineering Sciences University of Colorado at Boulder Boulder, CO 80309-0429, USA Phone: (303) 492 0619 Fax: (303) 492 4990 Email: [email protected] Aoues, Y. Laboratory of Civil Engineering University Blaise Pascal Complexe Universitaire des Cézeaux, BP 206 63174 Aubière Cedex, France Phone: +33(0)473407532 Fax: +33(0)473407494 Email: [email protected]

XXVI A u t h o r D a t a

Beck, J.L. Professor Engineering and Applied Science Division California Institute of Technology Pasadena, CA 91125 USA Phone: (626) 395-4139 Fax: (626) 568-2719 Email: [email protected] Ben-Haim, Y. Professor Faculty of Mechanical Engineering Technion – Israel Institute of Technology Haifa 32000, Israel Phone: 972-4-829-3262 Fax: 972-4-829-5711 Email: [email protected] Bichon, B.J. PhD Student Civil and Environmental Engineering Vanderbilt University VU Station B 351831 Nashville, TN 37235 USA Phone: 615-322-3040 Fax: 615-322-3365 Email: [email protected] Chateauneuf, A. Professor Polytech’Clermont-Ferrand Department of Civil Engineering University Blaise Pascal Complexe Universitaire des Cézeaux, BP 206 63174 Aubière Cedex, France Phone: +33(0)473407526 Fax: +33(0)473407494 Email: [email protected] De Palma, P. Professor Department of Computer Science School of Engineering and Applied Science Gonzaga University Spokane, WA 99258-0026 USA

Author Data

Phone: 509-323-3908 Email: [email protected] Doltsinis, I. Professor Institute for Statics and Dynamics of Aerospace Structures Faculty of Aerospace Engineering and Geodesy University of Stuttgart Pfaffenwaldring 27 D-70569 Stuttgart, Germany Phone: 0711-685-67788 Fax: 0711-685-63644 Email: [email protected] Eldred, M.S. Sandia National Laboratories P.O. Box 5800, Mail Stop 1318 Albuquerque, NM 87185-1318 USA Phone: (505)844-6479 Fax: (505)284-2518 Email: [email protected] Fragiadakis, M. Lecturer Faculty of Civil Engineering University of Thessaly Pedion Areos, Volos 383 34, Greece Phone: +30-210-748 9191 Fax: +30-210-772 1693 Email: [email protected] Frangopol, D.M. Professor of Civil Engineering and Fazlur R. Khan Endowed Chair of Structural Engineering and Architecture Department of Civil and Environmental Engineering Center for Advanced Technology for Large Structural Systems (ATLSS Center) Lehigh University 117 ATLSS Drive, Imbt Labs Bethlehem, PA 18015-4729, USA Phone: 610-758-6103 or 610-758-6123 Fax: 610-758-4115 or 610-758-5553 Email: [email protected] Ganzerli, S. Associate Professor Department of Civil Engineering School of Engineering

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XXVIII A u t h o r D a t a

Gonzaga University Spokane, WA 99258-0026 USA Phone: 509-323-3533 Fax: 509-323-5871 Email: [email protected] Huh, J.S. Senior Researcher Engine Department/KHP Development Division Korea Aerospace Research Institute 45 Eoeun-Dong, Yuseong-Gu Daejeon 305-330, Republic of Korea Phone: +82-42-860-2334 Fax: +82-42-860-2626 Email: [email protected] Hurtado, J.E. Professor Universidad Nacional de Colombia Apartado 127 Manizales Colombia Phone: +57-68863990 Fax: +57-68863220 Email: [email protected] Joanni, A.E. Research Engineer Institute for Materials and Design Technical Univerisity of Munich D-80290 München, Germany Phone: +49 89 289-25038 Fax: +49 89 289-23096 Email: [email protected] Kanno, Y. Assistant Professor Department of Mathematical Informatics Graduate School of Information Science and Technology University of Tokyo, Tokyo 113-8656, Japan Phone & Fax: +81-3-5841-6906 Email: [email protected] Kharmanda, G. Dr Eng Faculty of Mechanical Engineering

Author Data

University of Aleppo Aleppo – Syria Phone: +963-21-5112 319 Fax: +963-21-3313 910 Email: [email protected] Kokkolaras, M. Associate Research Scientist, Research Fellow Optimal Design (ODE) Laboratory Mechanical Engineering Department University of Michigan 2250 G.G. Brown Bldg. 2350 Hayward Ann Arbor, MI 48109-2125, USA Phone: (734) 615-8991 Fax: (734) 647-8403 Email: [email protected] Kwak, B.M. Samsung Chair Professor Center for Concurrent Engineering Design Department of Mechanical Engineering Korea Advanced Institute of Science and Technology 373-1 Guseong-dong, Yuseong-gu Daejeon 305-701 Republic of Korea Phone: +82-42-869-3011 Fax: +82-42-869-8270 Email: [email protected] Lagaros, N.D. Lecturer Institute of Structural Analysis & Seismic Research Faculty of Civil Engineering National Technical University of Athens Zografou Campus Athens 157 80, Greece Phone: +30-210-772 2625 Fax: +30-210-772 1693 Email: [email protected] Lee, S.H. Postdoctoral Research Fellow Department of Mechanical Engineering Northwestern University 2145 Sheridan Road Tech B224 Evanston IL 60201, USA Phone: +1-847-491-5066

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XXX A u t h o r D a t a

Fax: +1-847-491-3915 Email: [email protected] Liang, J. Graduate Research Assistant Department of Mechanical Engineering Oakland University Rochester, MI 48309-4478 USA Phone: (248) 370-4185 Fax: (248) 370-4416 Email: [email protected] Mahadevan, S. Professor Civil and Environmental Engineering Vanderbilt University VU Station B 351831 Nashville, TN 37235, USA Phone: 615-322-3040 Fax: 615-322-3365 Email: [email protected] Maute, K. Associate Professor Center for Aerospace Structures Department of Aerospace Engineering Sciences University of Colorado at Boulder Room ECAE 183, Campus Box 429 Boulder, Colorado 80309-0429, USA Phone: (303) 735 2103 Fax: (303) 492 4990 Email: [email protected] Mourelatos, Z.P. Professor Department of Mechanical Engineering Oakland University Rochester, MI 48309-4478 USA Phone: (248) 370-2686 Fax: (248) 370-4416 Email: [email protected] Nikolaidis, E. Professor Mechanical Industrial and Manufacturing Engineering Department

Author Data

4035 Nitschke Hall The University of Toledo Toledo, OH 43606 USA Phone: (419) 530-8216 Fax: (419) 530-8206 Email: [email protected] Papadrakakis, M. Professor Institute of Structural Analysis & Seismic Research Faculty of Civil Engineering National Technical University of Athens Zografou Campus Athens 157 80, Greece Phone: +30-210-772 1692 & 4 Fax: +30-210-772 1693 Email: [email protected] Papalambros, P.Y. Professor Director, Optimal Design (ODE) Laboratory University of Michigan 2250 GG Brown Building Ann Arbor, Michigan 48104-2125 USA Phone: (734) 647-8401 Fax: (734) 647-8403 Email: [email protected] Patel, N.M. Graduate Research Assistant Design Automation Laboratory Aerospace and Mechanical Engineering 365 Fitzpatrick Hall of Engineering University of Notre Dame Notre Dame, Indiana 46556-5637 USA Phone: (574) 631-9052 Fax: (574) 631-8341 Email: [email protected] Plevris, V. PhD Candidate Institute of Structural Analysis & Seismic Research Faculty of Civil Engineering National Technical University of Athens

XXXI

XXXII A u t h o r D a t a

Zografou Campus Athens 157 80, Greece Phone: +30-210-772-2625 Fax: +30-210-772-1693 Email: [email protected] Polak, E. Professor Emeritus, Professor in the Graduate School Department of Electrical Engineering and Computer Sciences University of California at Berkeley 255M Cory Hall 94720-1770 Berkeley, CA USA Phone: 510-642-2644 Fax: 510-841-4546 Email: [email protected] Rackwitz, R. Professor Institute for Materials and Design Technical Univerisity of Munich D-80290 München, Germany Phone: +49 89 289-23050 Fax: +49 89 289-23096 Email: [email protected] Renaud, J.E. Professor Design Automation Laboratory Aerospace and Mechanical Engineering 365 Fitzpatrick Hall of Engineering University of Notre Dame Notre Dame, Indiana 46556-5637 USA Phone: (574) 631-8616 Fax: (574) 631-8341 Email: [email protected] Royset, J.O. Assistant Professor Operations Research Department Naval Postgraduate School Monterey, California 93943 USA Phone: 1-831-656-2578 Fax: 1-831-656-2595 Email: [email protected]

Author Data

Sørensen, J.D. Professor Department of Civil Engineering Aalborg University Sohngardsholmsvej 57 9000 Aalborg, Denmark Phone: +45 9635 8581 Fax: +45 9814 8243 Email: [email protected] Taflanidis, A.A. Ph.D Candidate Engineering and Applied Science Division California Institute of Technology Pasadena, CA 91125 USA Phone: (626) 379-3570 Fax: (626) 568-2719 Email: [email protected] Takewaki, I. Professor Department of Urban and Environmental Engineering Graduate School of Engineering Kyoto University Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8540 Japan Phone: +81-75-383-3294 Fax: +81-75-383-3297 Email: [email protected] Tillotson, D. Research Assistant Design Automation Laboratory Aerospace and Mechanical Engineering 365 Fitzpatrick Hall of Engineering University of Notre Dame Notre Dame, Indiana 46556-5637 USA Phone: (574) 631-8616 Fax: (574) 631-8341 Email: [email protected] Tovar, A. Assistant Professor Department of Mechanical and Mechatronic Engineering Universidad Nacional de Colombia

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XXXIV A u t h o r D a t a

Cr. 30 45-03, Of. 453-401 Bogota, Colombia Phone: +57-3165320 - 3165000 ext. 14062 Fax: +57-3165333 - 3165000 ext. 14065 Email: [email protected] Tsompanakis, Y. Assistant Professor Department of Applied Sciences Technical University of Crete University Campus Chania 73100, Crete, Greece Phone: +30 28210 37 634 Fax: +30 28210 37 843 Email: [email protected] Weickum, G. Graduate Research Assistant Center for Aerospace Structures Department of Aerospace Engineering Sciences University of Colorado at Boulder Room ECAE 188, Campus Box 429 Boulder, Colorado 80309-0429 USA Phone: (303) 492 0619 Fax: (303) 492 4990 Email: [email protected] Wu, Y.-T. Fellow, Applied Research Associates, Inc. 8540 Colonnade Center Dr., Ste 301 Raleigh, NC 27615 USA Phone: 919-582-3335 or 919-810-1788 Email: [email protected] Zhou, J. Graduate Research Assistant Department of Mechanical Engineering Oakland University Rochester, MI 48309-4478 USA Phone: (248) 370-4185 Fax: (248) 370-4416 Email: [email protected]

Part 1

Reliability-Based Design Optimization (RBDO)

Chapter 1

Principles of reliability-based design optimization Alaa Chateauneuf University Blaise Pascal, France

ABSTRACT: Reliability-Based Design Optimization (RBDO) aims at searching for the best compromise between cost reduction and safety assurance, by controlling the structural uncertainties allover the design process, which cannot be achieved by deterministic optimization. This chapter describes the fundamental concepts in RBDO. It aims to explain the role of uncertainties in deriving the optimal solution, where emphasis is put on the comparison with conventional deterministic optimization. The interest of RBDO formulation can also be extended to cover different design aspects, such as multi-component reliability analysis, safety factor calibration, multi-objective applications and time-variant problems.

1 Introduction The design of structures must fulfill a number of different criteria, such as cost, safety, performance and durability, leading to conflicting requirements to be simultaneously considered by the engineer. Therefore, the challenge in the design process is how to define the best compromise between contradictory design requirements. Moreover, the complexity of the design process does not allow for simultaneous optimization of all the design criteria with respect to all the parameters. Traditionally, this complexity is reduced by dividing the process into simpler sub-processes where each requirement can be handled separately. The designer can hence concentrate his effort on only one goal, generally the cost, and then checks if the other requirements can be, more or less, fulfilled. If necessary, further adjustments are introduced in order to improve the obtained solution. However, this procedure cannot assure performance-based optimal design. In structural engineering, the deterministic optimization procedures have been successfully applied to systematically reduce the structure cost and to improve the performance. However, uncertainties related to design, construction and loading, lead to structural behavior which does not correspond to the expected optimal performance. The gap between expected and obtained performances is even larger when the structure is optimized, as the remaining margins are reduced to their lower bounds; in other terms, the optimal structure is usually sensitive to uncertainties. In deterministic design, the propagation of uncertainties is usually hidden by the use of the well-known “safety factors’’, without direct connection with reliability specifications. Traditionally, the optimal cost is looked for by iterative search procedures, while the required reliability level is assumed to be ensured by the applied safety factors, as described by the design codes of practice. As a matter of fact, these safety factors are calibrated for average

4

Structural design optimization considering uncertainties

design situations and cannot ensure consistent reliability levels for specific design conditions. They may even lead to poor design, as the optimization procedure will search for the weakest region in the domain covered by the code of practice. This weakest region often presents not only the lowest cost but also the lowest safety. The deterministic optimal design is pushed to the admissible domain boundaries, leaving very little space for safety margins in design, manufacturing and operating processes. Moreover, the optimization process leads to a redistribution of the roles of uncertainties which can only be controlled by reliability assessment on the basis of the sensitivity measures. For these reasons, the Deterministic Design Optimization (DDO) cannot ensure appropriate reliability levels. If the DDO solution is more reliable than required, the losses can be avoided in construction and manufacturing costs; however, if the reliability is lower than required, the economic solution is not really achieved, because of the increase of the failure rate, leading to failure losses higher than the expected money saving. In this sense, the Reliability-Based Design Optimization (RBDO) becomes a very powerful tool for robust and cost-effective designs (Frangopol 1995). The RBDO aims to find a balanced design by reducing the expected total cost, which is defined in terms of the initial cost (i.e. including design, manufacturing, transport and construction costs), the failure cost, the operation cost and the maintenance costs. In addition, the RBDO takes the benefit of driving the search procedure by the wellcontrolled variables having great impact on the total cost. On the other side, the variables with high uncertainties are penalized independently of their mechanical role. In this sense, the system robustness is achieved as the role of highly uncertain and fluctuating variables is diminished during the optimization process. Contrary to the DDO, the solution does not lie in the weakest domain of the design code of practice, but a better compromise is defined by satisfying the target reliability levels. The RBDO can also be applied for robust design purposes, where the mean values of random variables are used as nominal design parameters, and the cost is minimized under a prescribed probability. Therefore, the solution of RBDO provides not only an improved design but also a higher level of confidence in the design. From the practical point of view, solving the RBDO problems is a heavy task because of the nested nonlinear procedures: optimization procedure, reliability analysis and numerical simulation of structural systems. Several methods have been developed for solving efficiently this problem, in order to allow for complex industrial applications; this topic will be discussed in a subsequent chapter by Chateauneuf and Aoues. This chapter aims at describing the RBDO principles, in order to give a clear vision of the links between classical deterministic approach and the reliability-based one. It emphasizes the fact that the deterministic optimization, based on safety factor considerations, is not anymore sufficient for safety control and assurance. The Reliability-Based Design Optimization has the advantage of ensuring a minimum cost without affecting the target safety level. At the end of the present chapter, the use of the RBDO in different kinds of engineering problems is briefly discussed in order to show how large can be the application spectrum.

2 Historical background Since the beginning of the twentieth century, the need for rational way to consider structural safety motivated a number of researchers, such as Forsell (1924), Wierzbicki

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

5

(1936) and Lévi (1948). In the conference on structural safety, held in Liège 1948, by the Association Internationale des Ponts et Charpentes, Torroja stated, probably for the first time, that the reduction of the total cost, have to include not only the construction cost, but also the expected failure cost. CT = CI + CF

(1)

where CT is the expected total cost, CI is the initial cost (i.e. design and construction cost) and CF is the expected failure cost. This expression has been easily approved, as the increase of construction cost should lead to higher safety margin and so decreasing the failure probability. Even that the formulation of the RBDO is known since 1948 (and even earlier), the direct application was impossible because of the difficulties related to the failure probability computation for realistic structures. With the development of the reliability theory starting in the 1950s, the solution procedures became available in 1970s and improved in the 1980s, in order to allow for the analysis of practical engineering structures. However, till now, the difficulty to estimate the failure cost is still remain a main problem, especially when dealing with human lives and environmental deterioration. On the basis of the target reliability index, the RBDO is really born in the second half of 1980s and developed along 1990s. Nowadays, the industrial applications of RBDO still face many difficulties due to the very high computational effort required to solve large-scale systems. Most of practical applications of structural optimization requires at least three conflicting goals (Kuschel and Rackwitz 1997): – – –

Low structural cost, including or not the expected failure cost. High reliability levels for components and systems. Good structural performance under various operating conditions.

Actually, the new trend is to include the inspection, maintenance, repair and operating costs in the definition of the expected total cost CT , in order to reach a performance-based design on the basis of multi-criteria considerations (Frangopol 2000). A comprehensive overview of these approaches is given by Frangopol and Maute (2003).

3 Reliability analysis The design of structures requires the verification of a certain number of rules resulting from the knowledge of physics and mechanics, combined with the experience of designers and constructors. These rules come from the necessity to limit the loading effects such as stresses and displacements. Each rule represents an elementary event and the occurrence of several events leads to a failure scenario. In addition to the deterministic variables d to be used in the system control and optimization, the uncertainties are modeled by stochastic variables affecting the failure scenario. The knowledge of these variables is not, at best, more than statistical information and we admit a representation in the form of random variables X, whose realizations are noted x. For a given design rule, the basic random variables are defined by their probability distribution with some statistical parameters (generally, the mean and the standard deviation).

6

Structural design optimization considering uncertainties

Safety domain

Joint distribution

x2

G>0

Pf

Failure domain

G 0 is known as the limit state surface G(x, d) = 0. Having the performance function G(x, d), known also as the limit state function or the safety margin, it is possible to evaluate the probability of failure by integrating the joint probability density over the failure domain (Figure 1.1):
Pf (d) = fX (x, d) dx (2) G(x,d)≤0

It is to be noted that the joint density function fX (x, d) depends on the design parameters d only when the distribution parameters belong to the design variables; this is especially the case when the mean value is considered as a design variable in the optimization process. There is a special case when the performance function is simply written by the margin between the resistance R and the load effect S, where both variables are independent normal random variables. The performance function and the failure probability are simply given by: G(X, d) = R − S Pf (d) =
(−β(d))

with:

m R − mS β(d) =  σR2 + σS2

(3)

where
( · ) is the standard gaussian cumulated distribution function, β(d) is the reliability index, mR , mS , σR and σS are respectively the means and standard deviations of the resistance and load effect. For this simple configuration, the optimization variable could be the mean design strength, and probably, in some cases, the mean load effect.

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

7

It to note that also standard deviations can be taken as optimization variables if the relationship between the quality control and the structural cost can be established. In practice, the performance function cannot be written in a simple linear form of normal variables, and equation 3 can rarely be applied. It is thus necessary to evaluate, more or less precisely, the failure probability as given in equation 2. Direct integration is impossible even for small structures due to: 1) the high-required precision, 2) the computation cost of the mechanical response, and 3) the multidimensional space. Numerical methods have to be applied to give an approximation of the failure probability. Three methods are commonly used for this purpose: –

–

–

Monte Carlo simulations allowing to estimate the failure probability for any general problem. It has two main advantages: 1) the possibility to deal with practically any mechanical or physical model (linear, nonlinear, continuous, discrete, . . .) and 2) the simple implementation without any modification of the mechanical model (e.g. finite element software) which is considered as a blackbox. However, the two main drawbacks are: 1) the very high computational time, especially for realistic structures with low failure probability and 2) the numerical noise due to random sampling, leading to non-monotonic estimates during simulations, and consequently, it becomes impossible to get accurate and stable evaluation of the response gradient. Although computation time can be reduced by using importance sampling and other variance reduction techniques, the numerical noise still remains a serious difficulty for practical applications in RBDO. First- and Second-Order Reliability Methods, known as FORM/SORM, which are based on the approximation of the performance function in the standard gaussian space by using polynomial series. An optimization algorithm is applied to search for the design point, called also the most probable failure point or β-point, which is the nearest failure point to the origin in the normal space. Then, linear (FORM) or quadratic (SORM) approximations are adopted for the performance function in order to get an asymptotic approximation of the failure probability. It is approved that FORM is usually sufficient for the majority of practical engineering systems. In RBDO context, FORM/SORM techniques have the advantages of: 1) high numerical efficiency; and 2) direct computation of the gradients of the reliability index, and consequently of the failure probability. The main drawbacks are: 1) the limited precision and convergence problems in some cases, especially for highly nonlinear limit states; and 2) the computation time for large number of random variables. Response Surface Methods (RSM), which are commonly used to approximate the mechanical response of the structure, by building what is called a meta-model. Quadratic polynomials are shown to be suitable for localized approximation of structural systems. The large part of the computational cost lies in the evaluation of the polynomial coefficients. Then, the failure probability can be simply evaluated by using the response surface which is an analytical expression, instead of the mechanical model itself (generally, complex finite element model). The advantages are mainly: 1) the reduction of the computation time for moderate number of random variables; and 2) the possibility of coupling reliability and optimization algorithms to achieve high efficiency. The most common drawback lies in the large number of mechanical calls for moderate and high number of variables.

8

Structural design optimization considering uncertainties

x2

u2

Physical space

Normalized space Failure domain Gu(u, d)  0 P* MPP

Failure domain G(x, d)  0 mX

2

Safe domain mX

G(x, d)
0

b x1

1

u*2

a

Gu(u, d)
0

u*1

u1

Figure 1.2 Reliability index and the Most Probable Failure Point (MPP).

In First Order Reliability Method, the failure probability Pf is approximated in terms of the reliability index β according to the expression: Pf (d) = Pr[G(X, d) ≤ 0] ≈
(−β(d))

(4)

where Pr[·] is the probability operator and
( · ) is the standard Gaussian cumulated function. The invariant reliability index β, introduced by Hasofer and Lind (1974), is evaluated by solving the constrained optimization problem (Figure 1.2):

β = min

u =



(Ti (x))2

i

(5)

under the constraint: G(T(x), d) ≤ 0 where u is the distance between the median point (corresponding to the space origin) and the failure subspace in the normalized space u and T(x) is an appropriate probabilistic transformation: i.e. ui = Ti (x). The image of the performance function G(x) in the normalized space is noted: Gu (u, d) = G(T(x), d). The solution of this problem is called the Most Probable Failure Point, the design point or the β-point; it is noted P∗ , or either x* or u*, whether physical or normalized space is considered, respectively. At this point, the following relationship holds: β = u. For the case of two random variables, Figure 1.3 illustrates the important points involved in structural design: the mean point represents the average stress and strength at operation, the characteristic values are loading and resistance values that can be guaranteed in the design process (they correspond to small probability to find higher loading level or to find lower strengths; percentiles of 95% or 5% are commonly adopted) and finally the Most Probable Failure Point (MPP) where the failure configuration has the highest joint probability density. While the reliability analysis aims at finding the Most Probable failure Point, the design procedure aims at setting the characteristic and mean values of strength and dimensions, according to economical considerations.

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

s

Strength R

fS(s), fR(r) Load effect S

9

P*

Limit state

s*

sk

xk G(x, d) = 0

mS

mX r

mS

sk

rk mR s* = r*

s, r

r* rk

mR

Figure 1.3 Mean, characteristic and design points.

Alternatively to equation 2, the reliability level of a structure can also be characterized by the performance function Pp defined as:
Pp (d, p) = fX (x, d) dx (6) G(x,d)≤p

where the subscript p is the performance measure (in standard reliability, p is set to zero). This formulation can be useful for specific RBDO formulations (see chapter by Chateauneuf and Aoues). 3.1

S ystem reliability analys is

Due to optimization, the structural components are strongly stretched close to the limit state, and their contribution in the overall safety becomes significant. That is why structural reliability cannot be correctly computed unless the complete system is considered, by taking into consideration the contributions of all the failure modes through appropriate modeling of system configurations, material behavior, load variability, strength uncertainty and statistical correlation. As structures are made of the assembly of several members, the overall ultimate capacity is highly conditioned by the redundancy degree. For many structures, several components can reach their ultimate capacity much before reaching the overall structural failure load. On the other hand, the structure could contain a number of critical members, leading to the overall failure if any one of them fails. In this context, the system reliability can be quite different from the reliability of its components. In the last decades, many research works have been dedicated to compute the system reliability, especially for series and parallel systems. A series system, representing a “weakest-link’’ chain, fails if any link fails; superstructures and building foundations are generally good examples of a series system. A parallel system implies that each component contributes more or less in the structural good-standing; the system failure takes place if all components fail.

10

Structural design optimization considering uncertainties

Practical expressions for system reliability include lower and upper bounds for both series and parallel systems, some of these bounds consider the correlation between pairs of potential failure modes. Also, more complex system models involving mixed series-parallel systems can be used (Ditlevsen and Madsen 1996). For series and parallel systems, the first order approximation of the failure probabilities can be computed as following: 



Pf = Pr

Gj (X, d) ≤ 0 ≈ 1 −
m (β(d), ρ)

j

 Pf = Pr





for series system



(7)

Gj (X, d) ≤ 0 ≈
m (−β(d), ρ)

for parallel system

j

where
m (β(d), ρ) is the multi-dimensional standard normal distribution, β(d) is the vector of the reliability indices for the different modes and ρ is the matrix of correlations between the failure modes. For practical RBDO analysis, the failure probability can be estimated by Ditlevsen bounds (Ditlevsen 1979), which is written for series system as:

Pf1 +

m  j=2

⎡ max ⎣Pfj −

j−1  k=1

⎤ Pfjk , 0⎦ ≤ Pfs ≤

m  j=1

Pfj −

m  j=2

max Pfjk k<j

(8)

where m is the number of dominant failure modes, Pfj is the failure probability of mode j and Pfjk is the probability of the intersection of modes j and k; in this expression, the failure probabilities follow a decreasing order.

4 Formulation of Reliability-Based Design Optimization The Reliability-Based Design Optimization (RBDO) aims at finding the optimal solution that fulfills the prescribed reliability requirements. The fluctuation of loads, the variability of material properties and the uncertainties regarding the analysis models, contribute to make the performance of the optimal design different from the expected one. In this sense, the optimization process has a large effect on the structural reliability. It is today well recognized that the safety factor approach cannot ensure the required safety levels, as they do not explicitly consider the probability of failure regarding some performance criteria. In other words, the optimal design resulting from deterministic optimization procedures does not necessarily meet the reliability targets. The RBDO allows us to consider the safety margin evolution, leading to the settlement of the best compromise between the life-cycle cost and the required reliability. This task is rather complicated due to the inherent non-deterministic nature of the input information. For this reason, many analysis methods have been developed to deal with the statistical nature of data. The process efficiency is mandatory to deal with realistic engineering problems (Kharmanda et al. 2002). The solution based on reliability concepts is rather robust, as the uncertain parameters are penalized during the design process, compared to a greater commitment of the well-controlled parameters.

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4.1

11

Insuf f iciency of Determinis tic Des ig n O p ti mi zati o n

In Deterministic Design Optimization (DDO), it is aimed to reduce the initial structural cost CI (d) under a number of constraints gj (d, γ), j = 1, 2, . . . , ng ; where d is the vector of design parameters and γ is the vector of partial safety factors. The optimization problem is thus written: min CI (d) d

subject to

gj (d, γ) ≤ 0 for

(9)

j = 1, 2, . . . , ng

In this problem, the structural safety is assumed to be ensured by the introduction of the safety factors within the constraint equations. A typical constraint for stress can be written as: g = σ − fy /γ where σ is the applied stress, fy is the yield stress and γ is the safety factor. Usually the strength fy is defined either by the mean value or by the characteristic value; the former is common in mechanical engineering and the latter is common in civil engineering. In the DDO, it is assumed that the safety factors are appropriate whatever the chosen optimal configuration. For most of systems, it can be shown that the safety level is not independent of the selected optimal design parameters. Figure 1.4a illustrates how the deterministic optimal design is defined on a constraint which is simply described by shifting the limit state. It can be said that the failure limit state g(d) is transformed to a safe limit state g(d, γ), by introducing the safety factor to take account for uncertainties. Starting from the initial point x0 , the DDO is based on the use of classical optimization algorithms to find the optimal design d∗ , which is generally located on the boundary of the reduced design space, including the safety factor. The Reliability-Based Design Optimization aims at finding the optimal solution, such that the failure limit state is kept sufficiently far from the operating point. In other words, the failure surface must lie on the iso-reliability level corresponding to the prescribed safety target (Figure 1.4b). It is clear that even for simple cases, the solution can be quite different from the deterministic optimization where homothetic Safe design constraint

s

Limit state

s

g(d )
0

Optimum design

γ

Optimum design Iso-reliability contours

Limit state n

ctio

g(d) = 0

d*

s*

s

Co

g(d, γ) = 0 x0 s

Co

(a)

d*

s* x0

n

du t re

r*

du t re

ctio

r

r (b)

r*

Figure 1.4 Comparison of optimal points corresponding to DDO and RBDO.

12

Structural design optimization considering uncertainties

fG(g)

Safety factor g1

1

g

g

Figure 1.5 Distribution of the global safety factor.

reduction of the design space is applied. In this sense, the RBDO can really ensure optimal cost, without compromising the structural safety. As a matter of fact, the uncertainties related to structural geometry, material properties and loading lead to stochastic cost, strength and stress in the structure, which automatically leads to random safety factors. When the optimal design is defined, it can be possible to see the effect of uncertainties on the global safety factor. This can be carried out by plotting the distribution of the strength-stress ratio, as illustrated in figure 1.5. In this case, the structural failure is observed when the safety factor realizations become less than unity. The failure probability can thus be computed by evaluating either Pr [γ ≤ 1] or Pr [G(d∗ , X) ≤ 0]. It is also to emphasize that the system uncertainties may lead to random total cost, which can be considered in one of the two following ways: –

–

If the optimal configuration is specified, the structure realization involves random variations in material properties, geometrical parameters, material unit cost, construction costs and failure costs. These viabilities and fluctuations produce a random total cost, where the probabilistic distribution depends on the inherent random variables. The goal of RBDO is usually to minimize the expected total cost. If the structural realizations are considered for cost optimization, a scatter of the optimal solutions is obtained, as different optimum is found for each structural realization. In other words, even if random variables are not involved in the cost, the optimal deterministic solution changes according to structure and loading fluctuations. Therefore, the total cost becomes random as solutions differ in terms of input uncertainties. This leads to what can be seen as a lack of robustness.

Deterministic optimization is even worst when multi-constraints or multicomponents are considered. The difficulty lies in the way to set the safety factors in order to ensure simultaneously safe and optimal design. As illustrated in figure 1.6, while the deterministic optimum leads to uncontrolled safety levels with respect to various limit states (due to the application of either the same safety factor or an inconsistent set of safety factors), the reliability-based design optimization looks for the situation where the safety levels can be simultaneously controlled for all the limit states. In this case, the optimum design is oriented such that safety requirements are fulfilled with respect to the uncertainty degrees; practically, greater margins are taken for largely scattered variables, while small margins are considered for well controlled variables.

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Deterministic optimum

x2

Limit state with safety factors Real limit state

Iso-reliability contours

Reliability-based optimum

x1

Figure 1.6 Comparison between DDO and RBDO solutions.

P

1

y

S1

v

2 S2

600 mm

u

x 800 mm

450 mm

Figure 1.7 Two-link structure under vertical force.

In other words, the design is driven by the variables with small uncertainties. That is why the Reliability-Based Design Optimization (RBDO) aims at searching for the best compromise between cost reduction and reliability assurance, by taking the system uncertainties into consideration; therefore, the RBDO ensures economical and safe design. It offers a good alternative to the safety factor approach, which is based on deterministic considerations and cannot take account for reduction of safety margins during the optimization procedure. In order to illustrate this idea, let us consider the two-bar system shown in figure 1.7. The system is supported at the end nodes and a vertical load P is applied at the internal node. The bar cross-sections are noted S1 and S2 for bars 1 and 2, respectively. The

14

Structural design optimization considering uncertainties

design criteria are related to member strengths and nodal displacements; buckling is assumed to be neglected. Under stress and deflection constraints, the deterministic optimization problem is written: min V = 1000 S1 + 750 S2 S1 ,S2

subject to g1 = F1 −

fY S1 ≤ 0 γσ (10)

fY S2 ≤ 0 γσ vL ≤0 g3 = v − γv

g2 = F2 − and with: F1 = 0.6P;

F2 = 0.8P;

P v= E

480 360 + S1 S2

 (11)

where E is the Young’s modulus, fY is the yield stress, vL is the limit displacement, γσ is the load safety factor and γv is the displacement safety factor. They are calculated by: 

fY S1 fY S2 ; γσ = min 0.6P 0.8P

 (12)

E S1 S2 vL γv = P (480 S1 + 360 S2 )

For deterministic optimization, these safety factors are taken as γσ = 1.5 and γv = 1.2. Considering only the two resistance limit states, Figure 1.8 shows the

S2

G1(d) ⴝ 0

G1(d, γ) ⴝ 0

Deterministic optimum 0.9P/fY

Iso-reliability contours G2(d, γ) ⴝ 0

P1*

0.6P/fY

Safety design domain G1(d, γ)  0 and G2(d, γ )  0

P*2

G2(d) ⴝ 0

Failure domain S1 0.8P/fY

1.2P/fY

Figure 1.8 Deterministic design and inconsistent reliability levels.

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

15

deterministic optimum solution in the space of the design variables S1 and S2 . This optimum is obtained at the intersection of the shifted limit states, due to the application of the safety factor of 1.5. If the uncertainties on the cross-sections are considered by using independent normal variables with the same standard deviation, we get directly the iso-reliability contours in this superposed design/random space. It is clearly observed that the Most probable failure points P1∗ and P2∗ do not lie on the same reliability level. As the system represents a series combination, the bar 2 reduces the overall structural reliability and the target safety is not met. A more rational approach implies that the two MPP should be at the same reliability contour; for more complex systems, this should take into account the failure costs and system reliability models for effective setting of the MPP locations. It has to be stressed that even for this simple problem, the task is not that easy, it can be imagined how complex is to set appropriate safety factors for nonlinear limit states (e.g. the displacement constraint) with non-gaussian variables, including statistical correlations. For this reason, the DDO approach cannot clearly give convenient solution with consistent uncertainty considerations.

4.2

Reliability-bas ed des ign optimizatio n

Although the total cost includes all costs from construction till destruction and recycling, including the in-service costs, the high complexity of engineering systems leads to difficulties in dealing with both aspects: design and maintenance uncertainties. A common procedure consists in separating the design into two steps. In the first step, the structure is designed in order to avoid “failure’’ configurations, with respect to the limit states (ultimate, serviceability, fatigue, . . .). At this design stage, the optimization is applied to assure the structural survival (or good-standing) with the lowest cost. In the second step, the maintenance planning is optimized for the structure designed in the first step. In this way, the design optimization is first carried out to define the optimal structural configuration for which Reliability-Based Maintenance Optimization (RBMO) is performed to define the maintenance-inspection-replacement policy. In this sense, the total cost minimization is carried out in two times: minimizing the initial and failure costs at the first time and then minimizing the maintenance cost at the second time. Of course, this implies an approximation, as some design variables can play different roles in the cost components. For some engineering systems, the decoupling of design and maintenance costs may not lead to globally optimal costs, due to interaction between design decisions and deterioration rates and time-dependent failure probability. In other words, the variation of some variables may increase failure cost and decrease maintenance costs, and vice-versa. However, this approach is widely accepted in engineering practice. It has also a practical advantage, as the optimal design becomes independent of the maintenance policy, where the operating conditions (loading, environment, deterioration rates, costs, . . .) may strongly varies over the structure lifespan. The total cost CT depends on two kinds of variables (Kharmanda et al. 2002b): –

Design variables, noted d, which are the deterministic control parameters. They should be optimized for cost reduction. They can be mechanical parameters

16

Structural design optimization considering uncertainties

Cost

Expected total cost CT Expected failure cost CF
Cf Pf

Minimum expected total cost

Initial construction cost CI Failure probability Optimum reliability level

Pf

Figure 1.9 Evolution of the costs in function of the failure probability.

–

(e.g. geometrical dimensions, material properties) or probabilistic parameters (e.g. means of random distributions). Random variables, noted X, whose realizations are x, representing the uncertainties and the fluctuations in the system configuration. Each of the random variables is defined by a probabilistic distribution. They usually represent geometrical, material or loading uncertainties.

Basically, the RBDO aims at minimizing the total expected cost CT (Figure 1.9) which is given in terms of initial cost CI (including design, manufacturing, transport and construction costs) and direct failure cost Cf (Torroja 1948, Ditlevsen and Madsen 1996). min CT (d) = CI (d) + Cf Pf (d) d

subject to gj (d) ≤ 0

(13)

A more rigorous mathematical notation consists in writing E[CT (d, X)] instead of CT (d), as what is optimized is the expectation, not the cost itself (which is a random function); however, for simplicity, the notation CT (d) is maintained to indicate the expectation. This problem can also be written in terms of the utility function U(d) as following (Frangopol 1995): max U(d) = B(d) − CI (d) − L(d) x

subject to

gj (d) ≤ 0

(14)

where B is the benefit derived from the system operation, C I is the initial construction cost and L is the expected loss due to inspection, maintenance and failure. This total cost expression indicates that the possible increase of initial cost should be balanced by a decrease in the risk C F (i.e. product: Cf Pf ). The minimization is carried out for the design parameters such as member sizes, structural configuration

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

17

and material parameters. These design parameters may correspond to probabilistic distribution parameters: cost is related to the mean value when it represents the nominal design value and to the standard deviation when it represents the quality control and the dispersion reduction aspects. Usually, the cost of consequences is taken as fixed, but in fact it should be a function of the failure probability. This means that the failure cost Cf (e.g. reconstruction cost, direct damage cost and pollution) is independent of the failure probability Pf , and consequently, the expected failure cost can be written: CF = Cf × Pf . This expression holds as long as the failure rate remains below a commonly accepted level. However, with abnormal failure rates, the failure cost Cf becomes a function of the failure probability according to indirect damage (propaganda, market losses, public opinion on company/authority, accelerated effects, . . .). For example in automotive industry, if a defect is observed for few cars, the failure cost for each car can be assumed equal to the repair of that car (added to some indemnity for the car owner). But if the defect is observed for a large number of cars, the company should repair the whole produced cars, beside the social damage to the company itself which can be traduced by significant selling losses. In other domains, such as nuclear energy, the failure cost can be a jump function as only one accident in a nuclear power plant leads to very high economic, social and political consequences. To take account for this failure rate dependence, it could be appropriate to estimate the failure cost by nonlinear functions, such as: CF = E[Cf ] ≈ Cf (Pf ) × Pf

(15)

where Cf (Pf ) may take one of the following forms: Polynomial Exponential Sigmoidal

Cf (Pf ) = Cf0 (1 + Pfα )  Cf0 for Pf ≤ Pf0 Cf (Pf ) = Cf0 exp(µ(Pf − Pf0 )α ) for Pf > Pf0 Cf 1 Cf (Pf ) = Cf0 + 1 + exp (−µ(Pf − Pf0 ))

where Cf0 and Cf1 are respectively the basic and the extra failure costs, Pf0 is the probability threshold, and α, µ are parameters to be estimated in terms of failure consequences. More generally, the expected total cost CT can be expressed in terms of all the costs involved in the structural system, from birth to death. It thus includes inspection, maintenance, repair and operating costs (Frangopol 2003), leading to: CT = CI + CF + CM + CS + CR + CU + CD

(16)

where CI is the initial construction cost, CF is the expected failure cost, usually defined as: CF = Cf × Pf , CM is the expected preventive maintenance cost, CS is the expected inspection cost, CR is the expected repair cost, CU is the expected use cost and CD is the expected recycling and destruction cost, which is particularly important for sensitive structures, such as nuclear powerplants.

18

Structural design optimization considering uncertainties

In practice, the design objective of only minimizing the expected total cost is not yet applicable, and is somehow dangerous from human point of view. For example, if the designer underestimates the failure consequences with respect to the initial cost, the optimal solution will allow for high failure rates, leading to accept the use of low-reliable structures. The extrapolation to rich and poor countries or cities, leads also to low reliability levels in poor countries (or cities) because of the lower failure costs, as human lives and constructions have statistically lower monetary values. One can imagine the political consequences of such a strategy. At least theoretically, the correct estimation of the failure cost should lead to coherent results. The problem of cost estimation is even more complicated when talking about the whole lifetime management, because the failure cost may change along the structure lifetime due to socio-economical considerations (e.g. life quality of the society). In all cases, special care is strongly required when minimizing the expected total cost, even when other reliability constraints are considered. Due to difficulties in estimating the failure cost Cf (especially when dealing with human lives and environmental deterioration, political consequences, . . .), the direct use of the above equation is not that easy. For design purpose, an alternative to the expect total cost formulation is usually to minimize the initial cost under a prescribed reliability constraint (Moses 1977): min CI (d) d

subject to Pf (d) ≤ Pft d ≤d≤d L

(17) U

where dL and dU are respectively the lower and upper bounds of the design variables and Pft is the admissible failure probability, which is set on the basis of engineering state-of-knowledge and experience. An equivalent formulation is defined in terms of the target reliability index βt : min C(d) d

subject to

β(d) ≥ βt d ≤d≤d L

(18) U

This formulation has the advantage of avoiding the failure cost computation. Nevertheless, the failure consequences can be indirectly included by selecting suitable target safety levels. In civil engineering, it is common to use an admissible failure probabilities of 10−4 for the ultimate limit state and of 10−2 for the serviceability limit state. More refined target values are given in the Eurocodes, in terms of the economical gravity and the number of exposed persons. In principle, the target system reliability should be determined by social and economical considerations. There is no general rule, so far, to select the target value of the system-reliability index. Furthermore, the designer’s experience and preferences still play an important role in the process. A reasonable choice consists in taking the reliability of old design codes as a target for the new codes. Nevertheless, the choice of the target value is still very important in system reliability-based optimization, because it is the regulator of the reliability indexes of the failure modes.

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19

The above formulation represents two embedded optimization problems (Enevoldsen and Sørensen 1994; Enevoldsen 1994). The outer one concerns the search for optimal design variables to minimize the cost and the inner one concerns the evaluation of the reliability index in the space of random variables. The coupling between the optimization and reliability problems is a complex task and leads to a very high calculation cost. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a particular optimization procedure. In the random variable space, the reliability analysis implies a large number of mechanical calls, where in the design variable space, the search procedure modifies the structural configuration and hence requires the re-evaluation of the reliability level at each iteration. For this reason, the solution of these two problems (optimization and reliability) requires very large computation resources that seriously reduces the applicability of this approach. This topic will be intensively discussed later on in the chapter by Chateauneuf and Aoues. In general, the RBDO can be formulated according to one of the following forms: –

RBDO1: Minimize the design cost under reliability and structural constraints: min: CI (d) d

subject to: β(d) ≥ βt and: gj (d) ≤ 0

min: CI (d) or

d

subject to: Pf (d) ≤ Pft and: gj (d) ≤ 0

where βt is the target reliability index and Pft is the maximum allowable failure probability. When first order approximation is applied, the relationship between these two forms is given by: Pf =
(−β) or β = −
−1 (Pf ). –

RBDO2: Maximize the reliability under cost and structural constraints: max: β(d)

min: Pf (d)

d

subject to: CI (d) ≤ CIt and: gj (d) ≤ 0 –

d

or

subject to: CI (d) ≤ CIt and: gj (d) ≤ 0

RBDO3: Maximize the reliability per unit cost under structural constraints: max: β(d)/CI (d) d

subject to: gj (d) ≤ 0

or

max: 1/Pf (d)/CI (d) d

subject to: gj (d) ≤ 0

which is equivalent to minimize the ratio cost/reliability: min: CI (d)/β(d) d

subject to: gj (d) ≤ 0

or

min: CI (d) · Pf (d) d

subject to: gj (d) ≤ 0

This kind of formulation is particularly useful when there is no limitation on the total cost in RBDO2.

20

Structural design optimization considering uncertainties

P

h

mR

A

L

Figure 1.10 Perforated beam subjected to uniform load.

–

RBDO4: Minimize the total expected cost under reliability and structural constraints: min: CI (d) + Cf Pf (d) d

subject to: β(d) ≥ βt and: gj (d) ≤ 0

min: CI (d) + Cf Pf (d) or

d

subject to: Pf (d) ≤ Pft and: gj (d) ≤ 0

These formulations are considered as the basic forms of reliability-based design optimization, where the goal is to better redistribute the material within the structure by taking into account the effects of uncertainties and fluctuations. 4.3

Il l ustrati o n o n pe r fo r at e d s imple be a m

A simply supported beam, with length L = 2 m and height h = 0.3 m, is perforated by 5 holes of mean radius mR . The beam is subjected to uniformly distributed load P with mean value 1 MN/m and coefficient of variation of 15%. The maximum stress is located at point A in Figure 1.10. Under the effect of geometrical uncertainties, the nominal hole radius mR has to be designed according to the RBDO basis. In Figure 1.11, the initial, failure and total costs are plotted in function of the mean hole radius. The minimum cost corresponds to mR = 7.5 cm, and to the failure probability of 1.07 × 10−4 . Figure 1.12 shows the expected total cost for different values of consequence severity. It is observed that the hole radius should be decreased with higher consequence costs, in order to reduce the probability of failure and therefore the risk. The optimal solutions are found with respect to each failure cost case: Low: mR = 7.9 cm (Pf = 3.4 × 10−3 ), Moderate: mR = 7.5 cm (Pf = 1.1 × 10−4 ), High: mR = 7.1 cm (Pf = 4.3 × 10−6 ) and Very High: mR = 6.7 cm (Pf = 3.7 × 10−7 ). It can be observed that the failure probability levels are very sensitive to the failure consequences, showing that special care should be considered in estimating these consequences, as they changes drastically the optimal solution.

5 Multi-component RBDO In practical structural systems, the overall failure is generally dependent on a certain number of components where each one may have several failure modes, arranged in

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

21

Expected costs (Euros)

4000 3500

Initial cost Failure cost Total cost

3000 2500 2000 1500 1000 500 0 6,500

7,000 7,500 Mean hole radius (cm)

8,000

Figure 1.11 Initial, failure and total costs of the perforated beam.

Expected costs (Euros)

4,00E03 3,50E03 3,00E03 2,50E03 2,00E03 1,50E03 1,00E03 5,00E02 0,00E00 6,500

Low failure cost Moderate failure cost High failure cost Very high failure cost 7,000

7,500

8,000

Mean hole radius (cm)

Figure 1.12 Expected total costs in function of the failure consequence costs.

series and/or parallel systems. During the optimization of redundant structures, the contribution of various members is highly redistributed and the prediction of the most important components is not easy. Some insignificant components at the beginning of the RBDO procedure can become very important in the neighborhood of the optimal point. That is why structural reliability cannot be correctly computed without the whole system consideration, by taking account for all the failure modes. In this case, the constraint on system reliability becomes a computational challenge because of the different levels of embedded optimization loops. So, the system RBDO has common limitations due to system reliability computation and the necessity to make some approximations in practical cases (e.g. bounds, reduction of failure paths, . . .). This is probably the main reason why the system approach is less popular than the component approach. Another difficulty arises from the fact that the component assembly is rather a logical combination (i.e. union and intersection of events) than just algebraic operation, which is hard to deal with in system optimization, as sensitivity computation is not easy for

22

Structural design optimization considering uncertainties

logical operators. For example, the derivation of the union of two events is not simple to handle when one of them is totally or partially included inside the other, as the derivative operator can only capture the dominant event sensitivity. This difficulty is emphasized by the fact that failure mode combination is strongly related to the few significant failure modes at a given instance of the computing process. However, as the design variable values change in each iteration, the significant failure modes are not always the same, which greatly influences the convergence of the optimization procedure. Fortunately, in practice the significant failure modes identified in the system reliability analysis tend to be stabilized after few iterations. 5.1 Sy stem RB DO fo r mulat io n The system RBDO can be formulated either at the component level or at the system level (Enevoldsen 1994). At the component level, the RBDO can be written by specifying the target reliability for each one of the structural components, leading to: min CI (d) x

subject to βi (d) ≥ βti

and

gj (d) ≤ 0

(19)

where βi (d) and βti are respectively the reliability index and the target index for the ith component. Each one of the component reliability constraints includes a minimum reliability requirement for a specific failure mode at a specific location in the structure. For example, a member has several critical cross-sections which may fail according to several modes, such as yielding, cracking and excessive deformation, in addition to member buckling failure and structural instability. At the system level, the RBDO is formulated by only specifying the target system reliability for the whole structure: min CI (d) d

subject to βsys (d) ≥ βt

and

gj (d) ≤ 0

(20)

where βsys (d) and βt are respectively the reliability index and the target index for the whole system. The system reliability is generally evaluated by the use of upper and lower bounds. Some authors combined the constraints on component and system reliabilities, but this approach could lead to either redundant or inconsistent constraints. Aoues and Chateauneuf (2007) proposed a scheme for consistent RBDO of structural systems. The basic idea consists in updating the component target safety levels in order to fulfill the overall system target. In the main optimization loop, the cost function is minimized under the constraints that component reliability indexes must satisfy the updated target values. min C(d) subject to

Updated

βj (d) ≥ βtj

d ≤d≤d L

Updated

(21)

U

where βtj is the updated target reliability index for the jth failure mode and βj (d) is the reliability index for the considered design configuration. In the updating procedure,

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

P

q

23

P

q q/8

d2

d2 d1

Lc
3 m

Lc
3 m

L
8m M1

M2 M2
M1/8

M1

M2

Bending moment diagram

Figure 1.13 Overhanged beam with variable cantilever depth.

the target indexes are adjusted to meet the system reliability requirement. This can be performed by solving the problem: min

Updated j

βt

mp 

Updated

(βtj

− β j )2

(22)

i=1

subjected to

Updated βsys (βtj , ρjk )

≥ βt Updated

which is solved for the updated target indexes βtj 5.2

.

Overhanged reinforced concrete be am

In order to show the interest of system analysis, an overhanged beam with variable cantilever depth is considered, as shown in Figure 1.13 (Aoues and Chateauneuf 2007). With a constant breadth of 20 cm, the beam is defined by the middle-span depth d1 and the cantilever end depth d2 . The span is L = 8 m and the cantilever length is Lc = 3 m. The beam is subjected to uniformly distributed loads q and q/8 as illustrated in Figure 1.13. In order to reduce the negative moments, two tension rods are acting at the cantilever ends, modeled by the tensile force P. The concrete strength is taken as fcu = 25 MPa and the steel yield strength is fY = 200 MPa. An extreme loading case is considered where q = 40 kN/m and P = 30 kN; leading to the maximum moments M(x = 0.75) = 11.25 kNm and M(x = 3) = −90 kNm. The considered random variables are the applied loads and the effective depth of RC cross-sections, which are considered as normally distributed to allow for easy graphical illustrations. For a given cross-section, the design equation is written by:

Gi = fY Asi

f Y As i di − 2(0.85fcu b)

 − Mi

(23)

24

Structural design optimization considering uncertainties Table 1.1 Statistical data for random variables. Random variable

Mean

St-deviation

Middle span depth d1 Cantilever end depth d2 Reference moment M

md1 md2 mM = 90 kNm

σd1 = 5 cm σd2 = 2.5 cm σM = 18 kNm

The reinforcement is chosen as As1 =12 cm2 and As2 = 6 cm2 , leading to the limit states: G1 = 0.24(d1 − 0.02824) − M1 G2 = 0.12(d2 − 0.01412) − M2

(24)

which can be written in the normalized space by probabilistic transformation: H1 = 0.24(md1 + σd1 ud1 − 0.02824) − (mM + σM uM )  2 H2 = 0.12(md2 + ρσd2 ud1 + 1 − ρ σd2 ud2 − 0.01412) −0.125(mM + σM uM )

(25)

where M is a reference moment (equal to M1 ), ui are the normalized variables and ρ is the correlation between d1 and d2 . The distribution parameters are given in Table 1.1. The correlation between d1 and d2 is taken as ρ = −0.6. As this situation is considered as extreme one, the allowable failure probability of the system is set to: Pf _system = 0.05 (naturally, this is a conditional probability as it assumes that extreme situation occurs). The reliability solution leads to the direction cosines: αd1 = 0.55 and αM = −0.83 for the limit state H1 , and αd1 = −0.48, αd2 = 0.64 and αM = −0.60 for the limit state H2 . Thus, the correlation between these two limit states is equal to 0.233. The overall RC volume in this beam is computed by V = 0.2(11 d1 + 3 d2 ). To take account for the workmanship in the cost calculation, the depths are set to the power 3. The final cost of RC is estimated by 150a/m3 . The system RBDO is applied to the structure, by adopting two considerations: 1) the target reliability index is the same for all the limit states, and 2) the target reliability indexes are adapted to find a better solution, under the satisfaction of the system target. In the first case, the target system failure probability of 0.05 is reached when both components have reliability indexes of 1.943, knowing the correlation of 0.233. In the second case, the target of 0.05 is searched, where the cost is to be set as low as possible. The interest of the adaptive target strategy is shown by comparing these two RBDO formulations, as indicated in Table 1.2. For the same system reliability level, the adaptive target methodology allows us to significantly decrease the structural cost, by better distributing the material within the structure. Figure 1.14 compares the failure domains for both solutions (the 2D graph is given for the limit states projected on the plane uM = 0). As the system failure probability is the same for both formulations, the decrease of the margin for H1 implies the increase of the margin for H2 . For the same system reliability, the adaptive approach allows us to reach a cost reduction of 12.4%. Figure 1.14 shows also the beam profile obtained

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

25

Table 1.2 RBDO formulation and solutions. Considered aspect

Component-based formulation

System-based formulation

Formulation

Minimize: 300(11m3d1 + 3m3d2 ) under: β1 ≥ 1.9434 and: β2 ≥ 1.9434

Minimize: 300(11m3d1 + 3m3d2 ) under: βsys ≥ 1.64485

Failure point U ∗ :

u∗d1 = −1.07; u∗d2 = 0; u∗M = 1.61 ∗ ∗ ud1 = 0.93; ud2 = −1.24; u∗M = 1.17

u∗d1 = −0.91; u∗d2 = 0; u∗M = 1.37 ∗ ∗ ud1 = −1.58; ud2 = −2.11; u∗M = 1.98

Reliability levels at optimum:

β1 = 1.9434 β2 = 1.9434 Pfsys = 0.05

β1 = 1.6487 β2 = 3.2959 Pfsys = 0.05

Optimum design:

m∗d1 = 57.8 cm m∗d2 = 16.9 cm CT = 64.3a

m∗d1 = 55.2 cm m∗d2 = 21.1 cm CT = 56.3a

H1

ud

2

H2 ud

1

Limit states for identical component reliabilities

Limit states for adaptive targets

16.9 cm 21.1 cm 55.2 cm 57.8 cm Design with identical component reliabilities

Design with adaptive targets

Figure 1.14 Failure domains and optimum design for identical and adaptive formulations.

by the two approaches. It is clear that the adaptive target approach tries to decrease the depth where cost is widely involved, without decreasing the overall system safety.

6 RBDO issues The interest of RBDO is not limited to the design of new structures, but it also offers a powerful tool to solve a large class of structural problems. The RBDO is applied to various levels of reliability assessment, design, maintenance and codification. Some of these issues are briefly presented in this section. 6.1

Multicriteria approach for RBDO

As a matter of fact, the RBDO is a multicriteria optimization problem where the objective is to minimize the costs and to maximize the safety (Kuschel and Rackwitz 1997). It is generally acceptable that reliability and economy have conflicting requirements

26

Structural design optimization considering uncertainties

which must be considered simultaneously in the optimization process. The usual formulations aims either to combine these two objectives in only one weighted objective or to deal with one of these objectives as an optimization constraint. A more rational formulation can be stated as real multicriteria problem where the designer can get the Pareto optimal configurations in order to make consistent choices in the design process. As an example, Frangopol (2003) proposed a four-objective vector for bridge structures: f(d, x) = [V(d), PfCOL (d, x), Pf YLD (d, x), Pf DFM (d, x)]

(26)

where V(d) is the material volume, Pf COL (d, x) is the collapse probability, Pf YLD (d, x) is the first yield probability and Pf DFM (d, x) is the excessive deformation probability. This problem can be solved by any general multi-criteria technique. 6.2

C o d e c al i b r at io n b y R B DO

The design codes of practice must fit a certain objective for the whole applicable domain. Many actual design codes derive from a reliability-based calibration procedure to determine the partial safety factors to be applied in design (Sørensen et al. 1994). The objective of these codes is generally to keep the structural reliability above the specified target level (Ang and De Leon 1997). The problem of defining the safety factors is solved by the minimization of a penalization function for all the design situations covered by the design code (Gayton et al. 2004); the optimization problem is thus (Ditlevsen & Madsen 1996): min f (γi ) = γi

L 

W(ωj , βj (γi ), βt )

(27)

j=1

where W( · ) is a penalty function, γi are the partial safety factors, βj (γi ) is the safety index for the j-th situation and βt is the target reliability. Several kinds of penalty function have been proposed in the literature. The simplest one is defined by the weighted least square function: W1 (γi ) = ωj (βj (γi ) − βt )2

(28)

This function has the advantage of being very simple and the solution of the optimization problem (equation 27) can be greatly simplified if βj (γi ) has a simple explicit expression. Nevertheless, this function is symmetrical with respect to βt , i.e. it only depends on the difference βj − βt , and structures with a reliability index smaller than the target are not more penalized than structures with higher reliability index. Another function can take the following form (Lind 1977): W2 (γi ) = ωj (k(βj (γi ) − βt ) + exp(−k(βj (γi ) − βt )) − 1)

(29)

where k > 0 is the curvature parameter. This function penalizes the reliability indexes which are smaller than the target, compared to those higher than the target. When the parameter k increases this function becomes more penalizing for βj < βt than the least square function. For large values of k, the penalty goes to infinity for βj < βt , and so

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

27

reliability indexes lower than the target become forbidden. Other penalty functions can be proposed on the basis of socio-economic measures of the gap between the code and its objective. In such a case, the relationship between cost and target reliability index must be known. Classically, the goal of the design codes is to minimize the expression (27) for the whole spectrum of the design situations. Nevertheless, new evolutions of some codes of practice tend to homogenize the risk (i.e. product of failure probability by the consequences) instead of the reliability level (or failure probability). As an example, the RBDO calibration could take the form: min CT (d, γ) = d,γ

subject to

L 

L 

CTi (d, γ)

i=1

W(ωj , βj (d, γ), βt ) ≤ ε

(30)

j=1

gj (d) ≤ 0 dL ≤ d ≤ dU where ε is an acceptable tolerance for target fitting. 6.3 Topology bas ed RBDO Knowing that the RBDO concerns mostly shape optimization, the application to topology optimization is a new research field (Kharmanda et al. 2004). The basic idea concerns the use of uncertainties as a control parameter for topology selection. In fact, the reliability constraint allows us to get a robust structural topology. Figure 1.15 illustrates the fact that different topologies can be suitable for the same ground structure. Usually, the comparison in deterministic topology optimization is only related to minimized mean compliance, without observing the solution dispersion. The principle of reliability-based topology robustness consists in defining the topology which is less sensitive to system uncertainties. The main difficulty in dealing with topology lies in the fact that topology optimization is a qualitative approach, while the reliability-based design is a quantitative

Robust topology Compliance

Large dispersion

Optimization procedure iterations

Figure 1.15 RBTO and principle of reliability-based topology robustness.

28

Structural design optimization considering uncertainties

approach. The coupling of the two methods requires special developments to overcome formulation and efficiency problems. 6.4 T i m e-v a ri ant R B DO Every designer knows well that system information are not perfect and their validity is limited under system aging. In fact, most of phenomena involved in the total cost function are time-variant. It can be mentioned, for example, the loading fluctuation over the structural lifetime, the deterioration of material properties with time, the variation of operating and maintenance costs, and the monetary fluctuation of failure costs. All these time-variant phenomena lead to time-variant optimal solution. However, the designer must take decisions in a given stage of the project (largely before the construction or the manufacturing of the system), in function of the available data at that stage. The resulting solution is optimal only in the first part of the structure lifetime, as it does not account for aging and long-term exposure. In time-variant RBDO (Kuschel and Rackwitz 1998), the ideal scheme consists in designing the system for best optimal solution, considering the whole lifetime of the system. In this case, the utility function takes the form: max x

U(p, d, T) = B(d, t) − CI (d) − L(p, d, T)

subject to

gj (d) ≤ 0

(31)

with
T B(d, T) =

b(t) d(t) (1 − Pf (p, d, t))dt 0

(32)


T L(p, d, x, T) =

Cf (p, d) f (p, d, t) d(t)dt 0

where b(t) is the benefit derived from the existence of the system, Cf is the failure cost, f (p, d, t) is the probability density of the time to failure, d(t) is the discount function (or capitalization function) and T is the system age. 6.5

C o upl ed r e liab ilit y-b as e d d es ig n an d m a i n t e n a n ce p l a n n i n g

Although in design practice and due to system complexity, the maintenance planning is often considered as an independent step, the Reliability-based optimization can also be applied to a coupled set of design and maintenance parameters. In this case, the problem is formulated as: min x

CT (d) = CI (d) + CF (d) + CM (d)

subject to gj (d) ≤ 0

(33)

At the design stage, the maintenance cost is minimized by selecting the best set of parameters. At this stage, there is no available site information (as the system is

P r i n c i p l e s o f r e l i a b i l i t y-b a s e d d e s i g n o p t i m i z a t i o n

29

not constructed yet) and a priori hypotheses have to be formulated. Generally, regular maintenance intervals are chosen at this stage. The maintenance cost is usually a function of the type of inspection method mS , the number of inspections in the remaining lifetime nS , and the time for different inspections t. The maintenance cost can be described by (Enevoldsen and Sørensen 1994): CM (d, p) = CPM (d, p) + CINS (d, p) + CREP (d, p)

(34)

where CM is the expected maintenance cost, CPM is the preventive maintenance cost, CINS is the expected inspection cost, CREP is the expected cost of repairs, and p is the vector of maintenance parameters. Enevoldsen and Sørensen (1994) suggested to use the following expressions to evaluate inspection and repair costs: CINS (d, mS , nS , t) =

nS  i=1

CREP (d, nS , t) =

nS  i=1

CSi (mS )(1 − Pf (d, ti ))

1 (1 + r)ti

1 CRi (x)PRi (d, ti ) (1 + r)ti

(35)

where CSi is the ith inspection cost, Pf is the failure probability in the time interval [0, ti ], r is the discount rate, CRi is the cost of a repair at the ith inspection and PRi is the probability of performing a repair after the ith inspection for surviving components.

7 Conclusions RBDO is a powerful tool for robust design of structural systems. The explicit consideration of safety level allows us to optimize the total cost where the solution becomes less sensitive to system uncertainties. Contrary to traditional deterministic design optimization, the RBDO allows us to modulate the safety margins in function of the uncertainty effects for each variable, in order to reach economic, safe, efficient and robust design. In this sense, the safety factors are optimally defined within the system, compared to deterministic design where the safety factors are set before undergoing the optimization process. RBDO is still an active research field in order to extend the possibilities for new applications. Design, topology and time-variant reliability-based optimizations are very interesting field to reach performance-based design for cost-effective, durability and lifetime management of structural systems.

References Ang, A.H.-S. & De Leon, D. 1997. Determination of optimal target reliabilities for design and upgrading of structures. Structural Safety 19:91–103. Aoues, Y. & Chateauneuf, A. 2007. Reliability-based optimization of structural systems by adaptive target safety application to RC frames. Structural Safety. Article in Press. Ditlevsen, O. 1979. Narrow reliability bounds of structural systems. Journal of Structural Mechanics 7:435–451. Ditlevsen, O. & Madsen, H. 1996. Structural Reliability Methods. John Wiley & Sons.

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Structural design optimization considering uncertainties

Enevoldsen, I. 1994. Reliability-based optimization as an information tool. Mech. Struct. & Mach. 22:117–135. Enevoldsen, I. & Sørensen, J.D. 1994. Reliability-based optimization in structural engineering. Structural Safety 15:169–196. Frangopol, D.M. 1995. Reliability-based optimum structural design. In: Probabilistic structural mechanics handbook, edited by C. Raj Sundararajan, Chapman Hall, USA, 352–387. Frangopol, D.M. 1999. Life-cycle cost analysis for bridges. In: Bridge safety and reliability. ASCE, Reston, Virginia, 210–236. Frangopol, D.M. 2000. Advances in life-cycle reliability-based technology for design an maintenance of structural systems. In: Computational mechanics for the twenty-first century. Edinburgh: Saxe-Coburg Publishers, 257–270. Frangopol, D.M. & Maute K. 2003. Life-cycle reliability-based optimization of civil and aerospace structures. Computers & Structures 81:397–410. Gayton, N., Mohamed-Chateauneuf, A., Sørensen, J.D., Pendola, M. & Lemaire, M. 2004. Calibration methods for reliability-based design codes. Structural Safety 26(1):91–121. Hasofer, A.M. & Lind, N.C. 1974. An Exact and Invariant First Order Reliability Format. J. Eng. Mech., ASCE, 100, EM1:11–121. Kharmanda, G., Mohamed-Chateauneuf, A. & Lemaire, M. 2002. Efficient reliability-based design optimization using a hybrid space with application to finite element analysis. Journal of Structural and Multidisciplinary Optimization 24(3):233–245. Kharmanda, G., Mohamed-Chateauneuf, A. & Lemaire, M. 2002. CAROD: Computer-Aided Reliable and Optimal Design as a concurrent system for real structures. Journal of Computer Aided Design and Computer Aided Manufacturing CAD/CAM 1(1):1–12. Kharmanda, G., Olhoff, N., Mohamed-Chateauneuf, A. & Lemaire, M. 2004. Reliability-based topology optimization. Struct. Multidisc. Optim. 26:295–307. Kuschel, N. & Rackwitz, R. 1997. Two basic problems in reliability-based structural optimization. Mathematical Methods of Operations Research 46:309–333. Kuschel, N. & Rackwitz, R. 1998. Structural optimization under time-variant reliability constraints. Proceeding of the eighth IFIP WG 7.5 Working conference on Reliability and Optimization of Structural Systems, edited by Nowak, University of Michigan, Ann Arbor, Michigan, USA, 27–38. Kuschel, N. & Rackwitz, R. 2000. A new approach for structural optimization of series system. In: R.E. Melchers & M.G. Stewart (eds). Proceedings of the 8th International conference on applications of statistics and probability (ICASP) in Civil engineering reliability and risk analysis, Sydney, Australia, December 1999, Vol. 2. pp. 987–994. Lemaire, M., in collaboration with Chateauneuf, A. & Mitteau, J.C. 2006. Structural reliability. ISTE, UK. Lind, N.C. 1977. Reliability based structural codes, practical calibration. Safety of structures under dynamic loading, Trondheim, Norway, 149–160. Madsen, H.O. & Friis Hansen, P. 1991. Comparison of some algorithms for reliabilitybased structural optimization and sensitivity analysis. In: C.A. Brebbia & S.A. Orszag (eds): Reliability and Optimization of Structural Systems, Springer-Verlag, Germany, 443–451. Moses, F. 1977. Structural System Reliability and Optimization. Comput. Struct. 7:283–290. Moses, F. 1997. Problems and prospects of reliability based optimization. Engineering Structures 19(4):293–301. Rackwitz, R. 2001. Reliability analysis, overview and some perspectives. Structural Safety 23:366–395. Sørensen, J.D., Kroon, I.B. & Faber, M.H. 1994. Optimal reliability-based code calibration. Structural Safety 15:197–208.

Chapter 2

Reliability-based optimization of engineering structures John D. Sørensen Aalborg University, Aalborg, Denmark

ABSTRACT: The theoretical basis for reliability-based structural optimization within the framework of Bayesian statistical decision theory is briefly described. Reliability-based cost benefit problems are formulated and exemplified with structural optimization. The basic reliability-based optimization problems are generalized to the following extensions: interactive optimization, inspection and repair costs, systematic reconstruction, re-assessment of existing structures. Illustrative examples are presented including a simple introductory example, a decision problem related to bridge re-assessment and a reliability-based decision problem for offshore wind turbines.

1 Introduction The theoretical basis for reliability-based structural optimization can be formulated within the framework of Bayesian statistical decision theory mainly developed and described in the period 1940–60, see for example (Raiffa & Schlaifer 1961), (Aitchison & Dunsmore 1975), (Benjamin & Cornell 1970) and (Ang & Tang 1975). By statistical decision theory it is possible to solve a large number of decision problems where some of the parameters are modeled as uncertain. The uncertain parameters are modeled by stochastic variables or stochastic processes. Uncertain costs and benefits can thus be accounted for in a rational way. A large number of “simple’’ examples for application of statistical decision theory within structural and civil engineering are given in e.g. (Benjamin & Cornell 1070), (Rosenbleuth & Mendoza 1971) and (Ang & Tang 1975). During the last decades significant achievements have been obtained in development of efficient numerical techniques which can be used in solving problems formulated by statistical decision theory. Especially the development of FORM (First Order Reliability Methods), SORM (Second Order Reliability Methods) and simulation techniques to evaluate the reliability of components and systems has been important, see e.g. (Madsen et al. 1986). In the same period efficient methods to solve non-linear optimization problems have also been developed, e.g. the sequential quadratic optimization algorithms (Schittkowski 1986) and (Powell 1982). These developments have made it possible to solve problems formulated in a decision theoretical framework. Examples are: •

Reliability-based inspection and repair planning for offshore structures and concrete structures, formulated as a preposterior decision problem, see e.g. (Kroon 1994), (Engelund 1997), (Skjong 1985), (Thoft-Christensen & Sørensen 1987),

32

•

Structural design optimization considering uncertainties

(Fujita et al. 1989), (Madsen et al. 1989), (Madsen & Sørensen 1990), (Fujimoto et al. 1989), (Sørensen & Thoft-Christensen 1988) and (Faber et al. 2000). Reliability-based structural optimization problems and associated techniques for sensitivity analysis and numerical solution. Basic formulations of reliability-based structural optimization are given in e.g. (Murotsu et al. 1984), (Frangopol 1985), (Sørensen & Thoft-Christensen 1985) and (Enevoldsen & Sørensen 1994). System aspects are considered in e.g. (Enevoldsen & Sørensen 1993), interactive reliabilitybased optimization in (Sørensen et al. 1995) and optimization with time-variant reliability in e.g. (Kuschel & Rackwitz 1998). Further it is noted that a one-level approach for reliability-based optimization is described in (Streicher & Rackwitz 2002) based on an idea in (Madsen & Hansen 1992).

In section 2 a short description of Bayesian decision theory for engineering decisions is given and in section 3 reliability-based structural optimization problems are formulated. Only time-invariant reliability problems are considered. Three levels of decision problems with increasing degree of complexity can be identified: (1) decisions with given information (e.g. for new structures), (2) decisions with given new information (e.g. for existing structures), (3) decisions involving planning of experiments/inspections to obtain new information (e.g. for inspection planning). Further, interactive optimization aspects are discussed. In order to solve reliability-based optimization problems it is important to have accurate and numerically effective methods to evaluate probabilities of different events and of expectations. In section 4 some probabilistic methods, such as FORM/SORM, are briefly mentioned. Also techniques are described for sensitivity analyses to be used in numerical solution of the optimization problems using general optimization algorithms. In section 5 illustrative examples are presented, including applications with re-assessment of a concrete bridge and with reliability-based design of support structure for wind turbines.

2 Decision theory for engineering decisions Engineers are often in the situation to take decisions on design of a new structure, on repair/maintenance of existing structures where statistical information is available. In the following it is shown how Bayesian statistical decision theory can be used for making such decisions in a rational way, see (Raiffa & Schlaifer 1961) and (Benjamin & Cornell 1970) for a detailed description. An important difficulty in Bayesian statistical decision theory when applied in civil and structural engineering is that it can be difficult to assign values to cost of failure, or not acceptable behavior, especially when loss of human lives is involved. One solution is to calibrate the cost models to existing structures or to base the decisions on comparisons with alternative solutions. Further, organizational factors can have a rather significant influence in the decision process. These factors often have an influence, which is not rational from a cost-benefit point of view. Examples are the influence of the organizational structure, personal preferences and organizational culture. The first problem to consider is that of making rational decisions when some of the parameters defining the model are uncertain, but a statistical description of the

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33

Cost C(z, X) Design decision z

State of nature X

Figure 2.1 Decisions with given information.

parameters is available, i.e. the statistical information is given. The uncertain parameters are modeled by n stochastic variables X = (X1 , X2 , . . . , Xn ). The density function of the stochastic variables is fX (x, θ) where θ are statistical parameters, for example mean values, standard deviations and correlation coefficients. Further, it is assumed that a decision has to be taken between a number of alternatives which can be modeled by design/decision variables z = (z1 , z2 , . . . , zN ). In figure 2.1 a decision model with one discretized variable z is shown. The decision is taken before the realization by nature of the stochastic variables is known. Besides the decision variables z and the uncertain variables X also a cost function C(z, X) is introduced in the decision model in figure 2.1. When a decision z has been taken and a realization x of the stochastic variables appears then the value obtained is denoted C(z, X) and represents a numerical measure of the consequences of the decision and the realization obtained. C(z, X) is assumed to be related to money and represents in general costs minus benefits, if relevant. As an example the design parameters z could be the geometrical parameters of a structural system (cross-sectional dimensions and topology), the stochastic variables X could be loads and material strengths and objective function C could be the cost of the structure. In some decision problems it can be difficult to specify the cost function, especially if the consequences not directly measurable in money are involved, for example personal preferences. However, as described in von (von Neumann & Morgenstern 1943) rational decisions can be taken if the cost function is made such that the expected value of the cost function is consistent with the personal preferences. Thus, if the decisionmaker wants to act rationally the strategy z, which minimizes the expected cost, has to be chosen as
C ∗ = min EX [C(z, X)] = C(z, X)fX (x) dx (1) z

EX [−] is the expectation with respect to the joint density function of the stochastic variables X is the minimum cost corresponding to the optimal decision z∗ . The optimization problem can be generalized to include benefits B(z) such that the total expected benefits minus costs, Z are maximized. (1) is then written
(2) Z∗ = max Z(z) = B(z) − EX [C(z, X)] = B(z) − C(z, X)fX (x) dx z

where it is assumed that the benefits are not dependent on the stochastic variables X.

34

Structural design optimization considering uncertainties

3 Reliability-based structural optimization The formulations given above can be used in a number of cases related to design of structures. As mentioned in section 2 they can e.g. be used in a design situation where z models the design variables (size and shape variables in a structural system), X models uncertain loads and material parameters, B models the benefits and C models the total expected costs to design and possible failure. As mentioned only time-invariant reliability problems are considered.

3.1 Ba si c rel i ab ilit y-b as e d o pt imizat io n f o r m u l a t i o n s First, it is assumed that • •

There is no systematic reconstruction of the structure in case of failure Discounting can be ignored The total expected cost-benefits can then be written Z(z) = B(z) − C(z) = B(z) − CI (z) − Cf PF (z)

(3)

where CI (z) and Cf model the costs due to construction and failure, B(z) models the benefits and PF (z) is the probability of failure. Failure/no failure should here be considered in a general sense as satisfactory/not satisfactory behavior. The optimal design z∗ is obtained from the optimization problem: max Z(z) = max {B(z) − CI (z) − Cf PF (z)} z

z

(4)

(4) can equivalently be formulated as a reliability-constrained optimization problem max B(z) − CI (z) z

(5)

subject to β(z) ≥ βmin where the generalized reliability index is defined by β(z) = −
−1 (PF (z))

(6)


is the standard normal distribution function. βmin can be a code specified minimum acceptable reliability level related to annual or lifetime reference time intervals. Other design constraints can be added to (5) if needed. (4) and (5) give the same optimal decision if βmin is chosen as the reliability level corresponding to the optimal solution z∗ of (4): βmin = β(z∗ ), i.e. there is a close connection between βmin and Cf /CI . This can easily be seen considering the Kuhn-Tucker optimality conditions for (4) and (5). (5) is a two-level optimization problem, sine the calculation of the reliability index β by FORM requires an optimization problem to be solved, see section 4.

R e l i a b i l i t y-b a s e d o p t i m i z a t i o n o f e n g i n e e r i n g s t r u c t u r e s

35

The optimization problem in (5) can be generalised to the following element reliability-based structural optimization problem: m  mP D   max Z(z) = B(z) − C(z) = B(z) − CIi Vi (z) + Cfi
(−βi (z)) z

subject to

i=1

βi (z) ≥ BI,i (z) ≥ 0, BE,i (z) = 0, z1i ≤ zi ≤ ziu , βimin ,

i=1

i = 1, . . . , M i = 1, . . . , mI i = 1, . . . , mE i = 1, . . . , N

(7)

where z = (z1 , . . . , zN ) are the design (or optimization) variables. The optimization variables are assumed to be related to parameters defining the geometry of the structure (for example diameter and thickness of tubular elements) and coordinates (or related parameters) defining the geometry (shape) of the structural system. The objective function C consists of a deterministic and a probabilistic part with mD and mP terms, respectively. Vi is e.g. a volume in the ith deterministic term and Vi is the cost per volume of the ith term modelling the construction costs. Vi is assumed to be deterministic. If stochastic variables influence Vi then design values, see below, are assumed to be used to calculate Vi . In the probabilistic part Cfi is the cost due to failure of failure mode i. βi , i = 1, . . . , mP are reliability indices for the mP failure modes. The general formulation of (7) allows the objective function to model both the structural weight and the total expected costs of construction and failure. The constraints in (7) are based on the reliability indices βi , i = 1, . . . , M for M failure modes. βimin , i = 1, . . . , M are the corresponding lower limits on the reliabilities. BI,i , i = 1, . . . , mI and BE,i , i = 1, . . . , mE define the deterministic inequality and equality constraints in (7) which can ensure that response characteristics such as displacements and stresses do not exceed codified critical values. Determination of the inequality constraints usually includes finite element analyses of the structural system. The inequality constraints can also include general design requirements for the design variables. Finally also simple bounds are included as constraints. The variables (parameters) used to model the structure are characterized as stochastic or deterministic if the variable can be modelled as stochastic or deterministic and design or fixed if the variable is a design (optimization) variable or a fixed constant. The optimization problem in (5) can further be generalised to the following system reliability-based structural optimization problem: m  mP D  CIi Vi (z) + Cfi
(−βi (z)) max Z(z) = B(z) − C(z) = B(z) − z

subject to

βS (z) ≥ βmin , BI,i (z) ≥ 0, BE,i (z) = 0, z1i ≤ zi ≤ ziu ,

i=1

i = 1, . . . , mI i = 1, . . . , mE i = 1, . . . , N

i=1

(8)

where βS is the system reliability index. If failure of the structure can be modelled as by a series/parallel system then βS can be obtained from: βS (z) = −
−1 (Pf (z))

(9)

36

Structural design optimization considering uncertainties

where Pf (z) is the probability of failure of the system, e.g. obtained by FORM/SORM techniques. 3.2

Intera c ti v e o pt imizat io n

In practical solution of an optimization problem it will often be very relevant to be able to make different types of interaction between the user and the numerical formulation/ solution of the design problem. The basic types of interactive optimization which influences the formulation of the optimization problems are, see (Haftka & Kamat 1985) and (Sørensen et al. 1995): • • • •

include (delete) a design (optimization) variable include (delete) a constraint modify a constraint or modify (change) the objective function.

In order to investigate the effect of interactive optimization on the optimality criteria, (9) is restated as the following general optimization problem: min C(z) z

subject to

ci (z) = 0, ci (z) = 0,

i = 1, . . . , mE i = mE + 1, . . . , m

(10)

First order necessary conditions that have to be satisfied at a (local) optimum point z∗ are given by the Kuhn-Tucker conditions. If the optimization process has almost converged, a good guess on the optimal design is available. A modification of the optimization problem is then specified by the user. In (Sørensen et al. 1995) the details are described. Figure 2.2 illustrates the data flow in interactive structural optimization. The modules used are: • • • • • 3.3

User interface OPT: general optimization algorithm REL: module for reliability analysis, e.g. FORM, incl. optimization FEA: finite element program module DSA: module for calculating design sensitivity coefficients. General i z a t io n: inc lud e ins pec t io n a n d r e p a i r co s t s

The basic decision problems considered in section 2 can as mentioned be generalized to be used in reliability-based experiment and inspection planning, see figure 2.3. If e model the inspection times and qualities, and d models the repair decision given uncertain inspection result S, the optimization problem can be written: 0 (e) + CR0 (e, d)PR (e, d) + Cf0 PF (e, d)} max Z(e, d) = B0 − {CIN e,d

(11)

0 where B0 models the benefits, CIN models the inspection costs, CR0 models the repair costs, PR is the probability of repair and PF is the probability of failure, both obtained using stochastic models for S and X.

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37

CARBOS: User interface

OPT

DSA

Modifications ? N Final design

Y

Reliability analysis

FEA

Sensitivity analysis

DSA

FEA

Interactive optimization Optimization (probabilistic) Optimization (deterministic) Sensitivity analysis

REL

CARBOS: Modify variables, constraints and obj. function

Figure 2.2 Data flow interactive optimization, from (Sørensen et al. 1995).

Inspection plan e

Inspection result S

Repair decision d

State of nature X

Cost-benefit Z(e, S, d, X)

Figure 2.3 Decisions for with given information.

(11) can be further generalised if the total expected costs are divided into construction, inspection, repair and failure costs and a constraint related to a maximum annual (or accumulated) failure probability PFmax is added. If the inspections performed at times T1 , T2 , . . . , TN are part of e the optimization problem can be written max Z(e, d) = B(e, d) − {CI (e, d) + CIN (e, d) + CR (e, d) + CF (z, e)} e,d

subject to

ei1 ≤ ei ≤ eiu ,

i = 1, . . . , N

Pt (t, e, d) ≤ PFmax ,

t = 1, 2, . . . , TL

(12)

where, B is the expected benefits, CI is the initial costs, CIN is the expected inspection costs, CR is the expected costs of repair and CF is the expected failure costs. The annual probability of failure in year t is PF,t . The N inspections are assumed performed at times 0 ≤ T1 ≤ T2 ≤ · · · ≤ TN ≤ TL .

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Structural design optimization considering uncertainties

The total capitalised benefits are written B(e, d) =

N 

Bi (1 − PF (Ti ))

i=1

1 (1 + r)Ti

(13)

The ith term represents the capitalized benefits in year i given that failure has not occurred earlier, Bi is the benefits in year i, PF (Ti ) is the probability of failure in the time interval [0, Ti ] and r is the real rate of interest. The total capitalised expected inspection costs are written CIN (e, d) =

N 

CIN,i (e)(1 − PF (Ti ))

i=1

1 (1 + r)Ti

(14)

The ith term represents the capitalized inspection costs at the ith inspection when failure has not occurred earlier, CIN,i is the inspection cost of the ith inspection, PF (Ti ) is the probability of failure in the time interval [0, Ti ] and r is the real rate of interest. The total capitalised expected repair costs are CR (e, d) =

N 

CR,i PRi (e, d)

i=1

1 (1 + r)Ti

(15)

CR,i is the cost of a repair at the ith inspection and PRi is the probability of performing a repair after the ith inspection when failure has not occurred earlier and no earlier repair has been performed. The total capitalised expected costs due to failure are estimated from CF (e, d) =

TL 

CF (t) PF,t PCOL|FAT

t=1

1 (1 + r)t

(16)

where CF (t) is the cost of failure at the time t. PCOL|FAT is the conditional probability of collapse of the structure given failure of the considered component. 3.4

G en eral i z a t io n: inc lud e s y s t emat ic r e co n s t r u ct i o n

The following assumptions are made: (1) the structure is assumed to be systematically rebuild in case of failure, (2) only initial costs, CI (z) and direct failure costs, CF are included, (3) the benefits per year are b and (4) failure events are assumed to be modeled by a Poisson process with rate λ. The probability of failure is PF (z). The optimal design is determined from the following optimization problem, see e.g. (Rackwitz 2001): 

CI (z) CF λPF (z) b Ci (z) − − + max Z(z) = z rC0 C0 C0 C0 r + λPF (z) (17) subject to zl ≤ z ≤ zu , i = 1, . . . , N i

i

PF (z) ≤

i

PFmax

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39

where zl and zu are lower and upper bounds on the design variables. PFmax is the maximum acceptable probability of failure e.g. with a reference time of one year. This type of constraint is typically required by regulators. The optimal design z∗ is determined by solution of (17). If the constraint on the maximum acceptable probability of failure is omitted, then the corresponding value PF (z∗ ) can be considered as the optimal probability of failure related to the failure event and the actual cost-benefit ratios used. The failure rate λ and probability of failure can be estimated for the considered failure event, if a limit state equation, g(X1 , . . . , Xn , z) and a stochastic model for the stochastic variables, (X1 , . . . , Xn ) are established. If more than one failure event is critical, then a series-parallel system model of the relevant failure modes can be used. 3.5

Generalisation: optimal re-as s es s me nt o f e xi s ti ng s truc ture s

In re-assessment of structures and engineering systems, engineers are often in the situation to be involved in decisions on repair and/or strengthening of an existing system/structure where some statistical information is available. In the following it is shown how Bayesian statistical decision theory can be used for making such decisions in a rational way. The theoretical basis is detailed described in e.g. (Raiffa & Schlaifer 1961) and (Benjamin & Cornell 1970). It is assumed that the decision is taken on behalf of the owner of the structure, and that a cost-benefit approach is used with constraints related to minimum safety requirements specified by national/international codes of practice and/or the society. The same principles can be applied in case of other decision makers. It is noted that the optimal solution from the cost-benefit problem should be used as one input to the decision process. The decision problem on possible repair and/or strengthening in a re-assessment situation is illustrated in figure 2.4. It is assumed that the design variables in the initial design situation are denoted z. After the initial design information about the uncertain variables influencing the behaviour of the structure is collected, and are denoted S. Often this information will be collected in connection with the re-assessment. The decision variables at the time TR of re-assessment are denoted d. The uncertain variables describing the state of nature are denoted X.

Time TR

Design decision z

Information S

Repair/re-design decision d

State of nature X

Cost-benefit Z(z, S, d, X)

Figure 2.4 Decisions in re-assessment with given information. The vertical line illustrates the time of re-assessment.

40

Structural design optimization considering uncertainties

The decision is taken before the realization by nature of the stochastic variables is known. Besides the decision variables d and the uncertain variables X also a costbenefit function Z(z, S, d, X) is introduced in the decision model. When a decision d in the re-assessment problem has been taken and a realisation x of the stochastic variables appears then the value obtained is denoted Z(z, S, d, x) and represents a numerical measure of the consequences of the re-assessment decision and the realisation obtained. Z(z, S, d, x) is assumed to be measured in monetary units and represents in general costs minus benefits, if relevant. Illustrative examples of the decision variables z and d, and the stochastic variables S and X are: • • • •

z: design parameters, e.g. geometrical parameters of a structural system (crosssectional dimensions and topology). The design parameters are already chosen at the initial design, and are therefore fixed at the time of re-assessment. S: information collected, e.g. concrete compression strengths obtained from samples taken from the structure, measured wave heights, non-failure of the structure, no-find of defects by an inspection. d: design parameters in the re-assessment, e.g. geometrical parameters of a repair (cross-sectional dimensions and topology). X : stochastic variables, representing e.g. loads and material strengths.

In some decision problems it can be difficult to specify the cost function, especially if the consequences not directly measurable in money are involved, for example personal preferences. However, as described in (von Neumann, J. and Morgenstern 1943) rational decisions can be taken if the cost function is made such that the expected value of the cost function is consistent with the personal preferences. If the information S is related the stochastic variables X then a predictive density function (updated density function) fX

(x|s) of the stochastic variables X taking into account a realization s can be obtained using Bayesian statistical theory, see (Lindley 1976) and (Aitchison & Dunsmore 1975). If the decision-maker wants to act rationally, taking into account the information s the strategy d, which maximizes the expected cost-benefits, has to be chosen from Z∗ = max E

X|s [Z(z, s, d, X)] d

(18)

E

X|s [−] is the expectation with respect to the predictive (updated) density function fX

(x|s). In the following the initial design variables z are not written explicitly. Z∗ is the maximum cost-benefit corresponding to the optimal decision. If the benefits are not dependent on the stochastic variables then the optimization problem can be written: Z∗ = max Z(d) = max {B(d) − E

X|s [C(s, d, X)]} d

d

(19)

where the future benefits are denoted B and the future costs are denoted C. Both benefits and costs should be discounted to the time of the re-assessment. The optimization formulation can also be generalised to include decision variables related to experiment planning.

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41

In the following time-invariant reliability problems are considered. It is assumed that there is no systematic reconstruction of the structure in case of failure and discounting can be ignored. The total expected cost-benefits can then be written Z(d) = B(d) − C(d) = B(d) − CS (d) − Cf Pf

(d)

(20)

where CS (d) and Cf models the costs due to repair/strengthening after the re-assessment and due to failure, B(d) models the benefits and Pf

(d) is the probability of failure updated with the information s. Failure/no failure should here be considered in a general sense as satisfactory/not satisfactory behaviour. In the case the information S models (one or more) events modelled by an event margin {h(d, X) ≤ 0}, and failure is modelled by a limit state function g(d, X), the updated probability of failure is obtained from: Pf

(d) = P(g(d, X) ≤ 0|h(d, X) ≤ 0)

(21)

In the case the information S is related to the measurements of the stochastic variables X then the (updated) density function fX

(x|s) is used. The optimal design d∗ is obtained from the optimization problem max Z(d) = max {B(d) − CS (d) − Cf Pf

(d)} d

d

(22)

(22) can equivalently be formulated as a reliability-constrained optimization problem max B(d) − CS (d), d

subject to

β

(d) ≥ βmin

(23)

where the generalised reliability index is defined by β

(d) = −
−1 (Pf

(d)). βmin is a code specified minimum acceptable reliability level related to annual or lifetime reference time intervals. Other design constraints can be added to (23) if needed. (22) and (23) give the same optimal decision if βmin is chosen as the reliability level corresponding to the optimal solution d∗ of (22): βmin = β

(d∗ ), i.e. there is a close connection between βmin and Cf /CS . This can easily be seen considering the Kuhn-Tucker optimality conditions for (22) and (23). The basic decision problems considered above can be generalized to be used in reliability-based experiment and inspection planning as described in section 3.3. 3.6

Numeric al s olution of decis ion pro bl e ms

Numerical solution of the decision problems requires solution of one or more optimization problems. Since the optimization problems formulated are generally continuous with continuous derivatives sequential quadratic optimization algorithms such as (Schittkowski 1986) and (Powell 1982) can be expected to be the most effective, see (Gill et al. 1981). These algorithms require that values of the objective function and the constraints be evaluated together with gradients with respect to the decision variables. The probabilities in the optimization problems can be solved using FORM techniques, see (Madsen et al. 1986). Associated with the FORM estimates of the

42

Structural design optimization considering uncertainties

probabilities also sensitivities with respect to parameters are obtained. If the decision problem includes analysis of a structural system the finite element method in combination with sensitivity analyses can be used. The sensitivity analyses can be based on the direct or adjoint load method in combination with the discrete quasi-analytical method or with the continuum method.

4 Reliability analysis and sensitivity analysis As mentioned in the previous section the evaluation of the probability of failure events is an integral part of decision analysis and reliability-based structural optimization problems. Further, the decision analysis involves the evaluation of expected values of the costs. Both the relevant failure probabilities and expected values can be determined using modern reliability analysis techniques. If all variables in the reliability problem can be modelled as time-invariant random variables, the failure probability, PF (z), for a given limit state equation, g(x, z) can be evaluated as
PF (z) = P(g(X, z) ≤ 0) = fX (x, z) dx (24) g(x,z)≤0

where fX (x, z) is the joint density function of the stochastic variables X. The integral in (24) plays a central role in the reliability analysis and has therefore been devoted special attention over the last decades. As the integral in general has no analytical solution it is easily realised that its solution or numerical approximation becomes a major task for integral dimension larger than say 6 and for small probabilities. Sufficiently accurate approximations have been developed which are based on asymptotic integral expansions. These FORM/SORM methods are standard in reliability analysis and commercial software, see e.g. (Madsen et al. 1986). Also simulation methods can in many cases be very effective alternatives to FORM/SORM methods. By FORM analysis the failure surface is approximated by its tangent at the design point. On the basis of the linearised failure surface the probability of failure can be approximated by, see (9): PF (z) ≈
(−β(z))

(25)

Most optimization algorithms for solution of the reliability-based optimization problems formulated in section 3 require that the sensitivities with respect to objective functions and reliability estimates can be determined efficiently. By a FORM analysis these derivatives can be computed numerically by the finite difference method. However, it is more efficient to use a semi-analytical expression. For an element analysis the derivative of the first order reliability index, β, with respect to a parameter p, which may be a design variable z, is ∂β ∂ g(u∗ ; p) 1  = ∇u g(u∗ ; p) ∂p ∂p

(26)

If a gradient-based algorithm is used in order to locate the design point the gradient vector ∇u g(u∗ ; p) is already available and it is only necessary to determine the derivative

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43

of the failure function with respect to the parameter p. The derivative of the first order estimate of the probability of failure with respect to p is ∂Pf ∂β = −ϕ(−β) ∂p ∂p

(27)

where ϕ denotes the density function of a standard normally distributed variable. Also for series and parallel systems semi-analytical expressions for the derivatives of the first order reliability index can be derived. The following optimization problem corresponding to the general optimization problems defined in section 3, is considered.  min CI (z, p) = C0 (z, p) + Cj (z, p)Pj (z, p) z j (28) subject to Pf (z, p) ≤ Pfmax where z are decision/design variables, p are quantities defining the costs and/or the stochastic model. Pj denotes a probability (failure or repair), Pf denotes a failure probability and Pfmax is the maximum accepted failure probability. The sensitivity of the total expected costs C with respect to the elements in p is obtained from, see (Haftka & Kamat 1985) and (Enevoldsen 1994)  dPj dPf dC = Cj +λ dpi dpi dpi

(29)

j

where λ is the Lagrangian multiplier associated with the constraint in (25). The sensitivity of the decision variables z with respect to pi can be calculated using the formulas given below which are obtained from a sensitivity analysis of the Kuhn-Tucker conditions related to the optimization problem defined in (28). dz/dpi is obtained from ⎡

⎤ dz ⎡ ⎤ A B ⎢ C ⎥ dp i ⎥=⎣ ⎦ ⎣ ⎦⎢ ⎣ ⎦ dλ 0 BT 0 dpi ⎡

⎤

The elements in the matrix A and the vectors B and C are   ∂2 Cj ∂2 Pf ∂Pj ∂Cj ∂2 Pj ∂ 2 C0 Ars = Pj + +2 + Cj + λ ∂zr ∂zs ∂zr ∂zs ∂zr ∂zs ∂zr ∂zs ∂zr ∂zs

(30)

(31)

j

Br =

∂Pf ∂zr

Cr = −

(32)

  ∂2 Cj ∂Pj ∂Cj ∂ 2 C0 − + ∂zr ∂pi ∂zr ∂pi ∂zr ∂pi

(33)

j

It is seen that the sensitivity of the objective function (the total expected cost) with respect to some parameters can be determined on the basis of the first order sensitivity

44

Structural design optimization considering uncertainties

coefficients of the probabilities and of the cost functions, see (29). However, calculation of the sensitivities of the decision parameters is much more complicated because it involves estimation of the second order sensitivity coefficients of the probabilities, see e.g. (Enevoldsen 1994).

5 Examples 5.1 Ex am pl e 1 – S imple c o s t-b e nefit an a l ys i s In this section a simple, introductory example is presented. A structural component is considered. It is assumed to have strength R and load S, which for simplicity both are Normal distributed: Load S: expected value µS = 20 kN and Coefficient of Variation = 25% Strength R: expected value µR = 50 kN/m2 and Coefficient of Variation = 10% The design variable z represents the cross-sectional area. The limit state equation is written: g = zR − S

(34)

In the initial design situationz = z0 = 1 m2 is chosen. The corresponding reliability index is β = (1 · 50 − 20)/ (1 · 5)2 + 52 = 4.24 and the probability of failure Pf =
(−4.24) = 1.1 · 10−5 . The benefits and cost of failure are B0 = 10 and CF = 107 . New information has been collected. It consists of n = 5 tests with samples of similar components with the following results: 51, 53, 56, 57 and 58 kN/m2 . The mean value of the test results is X = 55 kN/m2 . For updating Bayesian statistics is used. It is assumed that the strength has a known standard deviation σR = 4 kN/m2 . The expected value is assumed to have a prior which is Normal distributed with expected value µ0 = 50 kN/m2 and standard deviation σ0 = 3 kN/m2 . It is noted that these assumptions are consistent with the initial model for the strength (µR = 50 kN/m2 and COV = 10%). The (updated) posterior for the expected value becomes Normal distributed with (nXσ02 + µ0 σR2 )/(nσ02 + σR2 ) = 53.7 kN/m2 and expected value of µR equal to µ

= 

standard deviation of µR equal to σ

= (σ02 σR2 )/(nσ02 + σR2 ) = 1.5 kN/m2 . The predictive (updated) distribution for the strength becomes Normal distributed with expected  value of R equal to µ


= µ

= 53.7 kN/m2 and standard deviation of R

equal to σ


= σ 2 + (σ02 σR2 )/(nσ02 + σR2 ) = 4.3 kN/m2 .  The updated reliability index and probability of failure becomes β = (1 · 53.7 − 20)/ (1 · 4.3)2 + 52 = 5.12 and the probability of failure Pf = 1.56 · 10−7 . At time TR the following two alternatives re-design situations are considered: 1)

continue with existing design The cost-benefits becomes: Z = B0 − CF Pf

= 10 − 107 · 1.5610−7 = 8.44

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45

9.0

Z(z)

8.5

8.0

7.5

7.0 1.00

1.05

1.10

1.15

1.20

z

Figure 2.5 Cost-benefit as function of design variable z.

2)

use a modified design with increased benefits The design variable is chosen to be z = 1.1 m2 . The benefits are assumed to be changed to: B
(z) = B0 + (z − z0 ) · 0.5. The cost of the design change is assumed to be: CI (z) = 1 + (z − z0 ) · 2. The updated reliability index and probability of failure becomes:  β = (1.1 · 53.7 − 20)/ (1.1 · 4.3)2 + 52 = 5.68 and the probability of failure Pf = 6.60 · 10−9 . The cost-benefits become: Z = B0 + (z − z0 ) · 0.5 − (1 + (z − z0 ) · 2) − CF Pf

= 10 + (1.1 − 1) · 0.5 − (1 + (1.1 − 1) · 2) − 107 · 6.6010−9 = 8.78

Since the cost-benefits are larger for the modified design than continuing with the existing design, the modified design should be chosen. In figure 2.5 the cost-benefits are shown as function of z. It is seen that the optimal decision is to chose a modified design with z = 1.12. It is noted that the known information also could be in the form of an event, e.g. an inspection, and that there could be many more decision alternatives. 5.2 Example2 – Repair decis ion for conc re te bri dg e A road bridge with concrete columns is considered. The total expected lifetime is assumed to be TL . The concrete columns are exposed to chloride ingress due to spread of de-icing salts on and below the bridge. There are some indications that

46

Structural design optimization considering uncertainties

chloride has penetrated the concrete and that corrosion of the reinforcement could be expected within the next few years. Therefore a re-assessment is performed at time TR as illustrated in figure 2.4. Chloride ingress is one of the most common destructive mechanisms for this type of structures. The most typical type of chloride initiated corrosion is pitting corrosion which may locally cause a substantial reduction of the cross-sectional area and cause maintenance and repair actions which can be very costly. Further, the corrosion may make the reinforcement brittle, implying that failure of the structure might occur without warning. The probabilistic analysis of the time to initiation of corrosion in concrete structures is in this example based on models described in (Engelund & Sørensen 1998). At the time of re-assessment it is assumed that chloride profiles are taken from representative parts of the concrete columns. The estimation of the time to initiation of corrosion is based on these chloride profiles combined with prior knowledge. A chloride profile consists of a number of measurements of the chloride concentration as a function of the distance to the surface, y. Using the chloride profiles, the surface concentration and the diffusion coefficient can be estimated. It is assumed that diffusion (transportation) of chlorides into the concrete can be described by a one-dimensional diffusion model where C(y, t) is the content of chloride at time t in the depth y, D(y, t) is the coefficient of diffusion (transportation) at time t in the depth y, CS is the surface concentration and Cinit is the initial chloride concentration. It is assumed that the diffusion coefficients can be written:

a t0 (35) D(y, t) = D0 (y) t where D0 (y) is the reference diffusion coefficient at the reference time t0 and a is an age coefficient (0 < a < 1). Models for the diffusion coefficient can include different diffusion coefficients in different depths. Based on n measurements in one chloride profile the surface concentration cS , the coefficient of diffusion D0 and the age coefficient a can be estimated using the Maximum Likelihood method, see (Engelund & Sørensen 1998). Next using Bayesian statistics a predictive (updated) distribution for the stochastic variables X can be obtained. On the basis of the available information described above the decision maker has to decide which repair/maintenance strategy should be applied. As an example, three different strategies are described below based on the models in (Engelund & Sørensen 1998). All the costs given below are in some monetary unit. It is assumed that the repair is carried out before the probability of any critical event such as total collapse of the bridge. Therefore, in the following the optimization problem is solved without any restriction on the probability of some critical event. Strategy 1: consists of a cathodic protection. This strategy is implemented when corrosion has been initiated at some point. In order to determine when corrosion is initiated, inspections are carried out each year, beginning five years before the expected time of initiation of corrosion. The cost of these inspections is 25 each year except for the last year before expected initiation of corrosion where the cost is 100. The cost of

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47

the cathodic protection is 1000 and the cost of running the cathodic protection is 20 each year. Strategy 2: is implemented when 5% of the surface of the bridge columns shows minor signs of corrosion, e.g. small cracks and discolouring of the surface. The repair consists of repairing the minor damages and applying a cathodic protection. As for strategy 1 the costs related to this strategy are the costs of the repair and the costs of an extended inspection programme which starts three years before the expected time of repair. However, by this strategy, also the costs related to running the cathodic protection must be taken into account. The cost of repair is 2000, the cost of inspection for three years before the repair is 100 each year and the cost of running the cathodic protection is 30 each year. Strategy 3: repair is performed as a complete exchange of concrete and reinforcement in the corroded areas. The strategy is implemented when 30% of the surface at the bridge columns shows distinct signs of corrosion, such as cracking and spalling of the cover. The cost related to this strategy are the cost of the repair and the cost of an extended inspection programme which starts three years before the expected time of repair. The cost of repair is 3000 and the cost of inspection in the three years before repair is 200 each year. Traffic restrictions in the year of repair he bridge decrease the benefits with 1000. The total expected costs for maintenance/repair is determined from CS (z1 , z2 , z3 ) =

TL 

Pi (z)Ci (z)

(36)

i=TR

where z = (z1 , z2 , z3 ) is the three repair/maintenance options, Pi (z) is the probability that repair/maintenance is performed in year i and Ci (z) is the total costs of the repair strategy if the repair is performed in year i: Ci (z) =

TL  j=TR

Ci,j (z)

1 (1 + r)j−TR

(37)

Ci,j (z) is the repair/maintenance cost in year j if the repair is performed in year i. These costs can be found in the descriptions of the repair strategies. The costs are discounted to the time of re-assessment TR using the real rate of interest r. The expected benefits in the remaining lifetime are determined from B(z) =

TL  i=TR

TL  1 B0 − Pi (z)Bi (z) (1 + r)i−TR

(38)

i=TR

where Bi (z) =

TL  j=TR

Bi,j (z)

1 (1 + r)j−TR

(39)

B0 is the basic annual benefit from use of the bridge and Bi,j (z) is the loss of benefits in year j due to repair in year i, e.g. due to traffic restrictions.

48

Structural design optimization considering uncertainties

The optimal repair strategy is obtained solving the optimization problem max B(z) − CS (z) z

(40)

The expected costs are determined using the predictive stochastic model for the surface concentration cS , the coefficient of diffusion D0 and the age coefficient a obtained using the available information. 5.3

Ex am pl e 3 – O pt imal d es ig n o f o ffsh o r e w i n d t u r b i n e s

Wind turbines for electricity production are increasing drastically these years both in production capability and in size. Offshore wind turbines with an electricity production of 2–5 MW are now being produced. The main failure modes are fatigue failure of wings, hub, shaft and main tower, local buckling of main tower, and failure of the foundation. This example considers reliability-based optimization of the tower and foundation, see (Sørensen & Tarp-Johansen 2005a) and (Sørensen & Tarp-Johansen 2005b). 5.3.1 F ormu l a tio n o f r e lia b ilit y-b a s e d o p t im i z at i o n pr o bl e ms f or w in d tu r b in e s Reliability based optimization problems can be formulated in different ways, e.g. with or without systematic reconstruction. In this example it is assumed that the control system is performing as expected, one single wind turbine is considered and the wind turbine is systematically reconstructed in case of failure. It is noted that it is assumed that the probability of loss of human lives is negligible. The the main design variables are denoted z = (z1 , . . . , zN ), e.g. diameter and thickness of tower and main dimension of wings. The initial (building) costs are CI (z), the direct failure costs are CF , the benefits per year are b and the real rate of interest is γ. Failure events are assumed to be modelled by a Poisson process with rate λ. The probability of failure is PF (z). The optimal design can thus be determined from the following optimization problem, see section 3.4:

 CI (z) CF λPF (z) b CI (z) − − + max W(z) = z γC0 C0 C0 C0 γ (41) l u subject to zi ≤ zi ≤ zi , i = 1, . . . , N PF (z) ≤ PFmax where zl and zu are lower and upper bounds on the design variables. C0 is the reference initial cost of corresponding to a reference design z0 . PFmax is the maximum acceptable probability of failure e.g. with a reference time of one year. This type of constraint is typically required by regulators. The optimal design z∗ is determined by solution of (41). If the constraint on the maximum acceptable probability of failure is omitted, then the corresponding value PF (z∗ ) can be considered as the optimal probability of failure related to the failure event and the actual cost-benefit ratios used. The failure rate λ and probability of failure can be estimated for the considered failure event, if a limit state equation, g(X1 , . . . , Xn , z) and a stochastic model for the

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49

DT

t1

H/3

t2

H/3

t3

H/3

H

hw d tP

HP

D DP

Figure 2.6 Design variables in wind turbine example (not in scale).

stochastic variables, (X1 , . . . , Xn ) are established. If more than one failure event is critical, then a series-parallel system model of the relevant failure modes can be used. An offshore 2 MW wind turbine with monopile foundation is considered, see figure 2.6. The wind turbine tower has height h = 63 m and a diameter which h increases linearly from D at bottom to DT at the top. The tower is divided in three sections each with height h/3 and each with the same thickness: t1 in top section, t2 in middle and t3 in bottom section. Diameter and thickness of monopile are constant: DP and tP . Tower and monopile are made of structural steel. The distance from bottom of the tower to the water surface is hw = 7 m and the distance from the water surface to the sea bed (the water depth) is d = 9 m. Wind and wave loads on the tower itself are neglected. The following failure modes are included: (a) yielding in cross sections in tower just above and below changes in thickness, (b) local stability in cross sections in tower just above and below changes in thickness, (c) fatigue in cross sections just above and below changes in thickness, and (d) yielding in monopile in cross-section with maximum bending moment. The stochastic model for the extreme loading at the top of the tower is described in (Sørensen & Tarp-Johansen 2005a) and (Sørensen & Tarp-Johansen 2005b). For the failure mode yielding of cross-section the limit state function is written: σ=

M N + ≥ Fy A W

(42)

where the cross-sectional forces in the cross-section is the normal force N, a shear force Q and a bending moment M. Further A is the cross-sectional area (= πt(D − t)), W is the cross-sectional section modulus and Fy is the yield stress.

50

Structural design optimization considering uncertainties

The cross-sectional forces are calculated from the stochastic variables HT , MT , and NT . The yield stress, Fy , is modelled as a LogNormal variable with coefficient of variation (COV) = 0.05 and characteristic values (5 percentile) equal to 235 MPa and 340 MPa for the tower and the mono-pile, respectively. For the failure mode local buckling of cross-section the limit state function is written: σ=

N M + ≥ Fyc A W

(43)

where the local buckling strength is estimated by the model in (ISO 19902 2001). The cross-sectional forces are calculated from the stochastic variables HT and MT . The yield stress, Fy is modelled as for yielding failure. Model uncertainty is introduced through a factor XB multiplied to Fyc . XB is assumed LogNormal distributed with expected value 1 and COV = 0.10. For the failure mode fatigue failure SN-curves and linear damage accumulation by the Miner rule are used. It is assumed that the SN-curve is bilinear and can be described by: N = K1 ( s)−m1

for N ≤ NC

(44)

N = K2 ( s)−m2

for N > NC

(45)

where s is the stress range, N is the number of cycles to failure, K1 , m1 are the material parameters for N ≤ NC , K2 , m2 are the material parameters for N > NC , sC is the stress range corresponding to NC . Further it is assumed that the total number of stress ranges for a given fatigue critical detail can be grouped in nσ groups/bins such that the number of stress ranges in group i is ni per year. In a deterministic design check the design equation can be written: 

ni TF

si ≥ sC

1 K1C s−m i

+



ni TF

si < sC

2 K2C s−m i

≥1

(46)

where si = Mi /z is the stress range in group i, Mi is the bending moment range z is a design parameter, KiC is the characteristic value of Ki (log KiC mean of log Ki minus two standard deviations of log Ki ), TF = FDF TL is the fatigue life time, TL is the service life and FDF is the Fatigue Design Factor which can be considered as a fatigue safety factor. In a reliability analysis the reliability index (or the probability of failure) is calculated using the limit state function associated with (46). This limit state equation can be written: g =1−



ni TL

si ≥ sC

1 K1 s−m i

−



ni TL

si < sC

2 K2 s−m i

(47)

where si = XS Mi /p is the stress range in group i, XS is a stochastic variable modelling model uncertainty related to the fatigue wind load and to calculation of the relevant fatigue stresses with given wind load. XS is assumed LogNormal distributed with

R e l i a b i l i t y-b a s e d o p t i m i z a t i o n o f e n g i n e e r i n g s t r u c t u r e s

51

Table 2.1 Stochastic model. D: Deterministic; N: Normal; LN: LogNormal. Variable

Distribution

Expected value

Standard deviation

X stress X wind TL m1 log K 1 m2 log K 2

LN LN D D N D N

1 1 T F /FDF 3 12.151 + 2 · 0.20 5 15.786 + 2 · 0.25

COV stress = 0.05 COV wind = 0.15 20 years 0.20 0.25

log K 1 and log K 2 are fully correlated.

 2 2 mean value = 1 and COV = COVwind + COVstress . log Ki is modelled by a Normal distributed stochastic variable according to a specific SN-curve. Representative statistical parameters are shown in Table 2.1. The basic SN curve used correspond to the SN 90 curve in (EC 3 2003). The optimal design is determined from the following optimization problem: b CI (z) max W(z) = − − z rC0 C0 subject to

pli ≤ pi ≤ pui , PF (z) ≤

CI (z) CF + C0 C0



λPF (z) γ

i = 1, . . . , N

(48)

PFmax

ω1 (z) ≥ ωL where PFmax is the maximum acceptable annual probability of failure. ω1 is the lowest natural frequency of the wind turbine structure and ωL is a minimum acceptable eigen frequency.  The probability of failure is estimated by the simple upper bound: PF ≈ N i=1
(−βi ) where βi is the annual reliability index in failure element i of the N failure elements/failure modes. The following design/optimization variables related to the tower and pile model are used: DT is the diameter at tower top, D is the diameter of tower at bottom, t1 , t2 and t3 are thickness of tower sections, DP is the diameter of the monopile, tP is the thickness of monopile, HP is the length of the monopile. The initial costs is modelled by:

 1 1 Vmono + CI = C0,foundation 2 2 Vmono,0

 1 3 Vtower + CI,blades + CI,powertrain + CI,others + C0,tower + 4 4 Vtower,0    turbine

(49)

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Structural design optimization considering uncertainties

where Vmono,0 and Vtower,0 are reference cross-sectional areas for the mono-pile foundation and the tower, respectively. Thus, the model is a linear model that gives the initial costs for designs that deviate from a given reference. The term CI,others accounts for initial costs connected to external and internal grid connections that are of course independent of the extreme load. Because, in current practice, the design of the blades and the power train are driven by fatigue and operation loads respectively, the dependence of the initial costs of these main parts of the turbine on the extreme load is assumed negligible in this model. The following model is used for the normalised initial costs at the considered site  

 CI 1 1 1 3 Vtower 1 1 1 1 1 Vmono 1 + + + + + (50) = + C0 6 2 2 Vmono,0 2 3 4 4 Vtower,0 3 3 3    turbine The ratios appearing in this formula will be site specific. For a far off offshore site the grid connection will become a larger part of the total costs. Likewise the foundation costs will depend on water depth. For other sites the cost ratios may e.g. be: 1 5 , , and 13 for the foundation, the turbine, and the other costs, respectively. For the 4 12 reference turbine Vmono,0 = 25.5 m3 and Vtower,0 = 14.0 m3 , which have been derived from the following reference values: h = 63 m, hw = 7 m, DT = 2.43 m, tT = 17 mm, DB = 3.90 m, tB = 29 mm, hP = 41 m, tp = 49.5 mm, and DP = 4.1 m. Thus 1 Vmono 1 Vtower CI 1 1 1 1 + + = + + + C0 12 12 14.0 m3 24 8 23.2 m3 3 3    turbine

(51)

It is noted that out of the total initial costs only a minor part depends on the loads because the study is restricted to the support structure. For a gravity foundation the normalised failure costs are estimated to be: CF,foundation 1 = C0,foundation 6

(52)

Compared to this the failure costs for the turbine are negligible. The turbine failure costs could be virtually zero if one just leaves the turbine at the bottom of the sea like a shipwreck, a solution that may hardly be accepted by environmentalists. It is noted that, at least in Denmark, it is for aesthetic reasons not accepted to rebuild the turbine a little away from the collapsed turbine, whereby the failure costs could otherwise practically vanish. Indeed Danish building licences demand that a new turbine, which replaces a collapsed turbine, must be situated at the exact same spot. That is, the space cannot even be left unused. Assuming that the damage to the grid is small the failure costs become: CF 1 = C0 36

(53)

For the considered site and turbine, and Assumption: Given site-i.e. climate (A = 10.8, k = 2.4), specified rated power (2 MW) and turbine height and rotor

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53

Table 2.2 Optimal values of design variables, objective function and natural frequency. γ

0.03

0.05

0.10

0.05

0.05

b/C 0 C F /C 0 DT D t1 t2 t3 DP tP HP W ω1

1/8 1/36 2.92 m 4.00 m 20 mm 28 mm 35 mm 5.41 m 21 mm 34.7 m 3.264 2.71

1/8 1/36 2.89 m 4.00 m 20 mm 29 mm 33 mm 5.40 m 20 mm 34.7 m 1.602 2.67

1/8 1/36 2.81 m 4.00 m 20 mm 25 mm 32 mm 4.93 m 20 mm 34.7 m 0.359 2.43

1/10 1/36 2.89 m 4.00 m 20 mm 29 mm 33 mm 5.40 m 20 mm 34.7 m 1.102 2.67

1/8 1/360 2.77 m 4.00 m 20 mm 28 mm 33 mm 5.31 m 20 mm 34.7 m 1.603 2.63

diameter. The average power is 1095 kW which with an assumption of 2% down time the annual average production may be computed. In the Danish community subsidising currently ensures that the market price for 1 kWh wind turbine generated electric power is 0.43 DKK/kWh. From this, one should subtract, as a lifetime average, 0.1 DKK/kWh for operation and maintenance expenses. The normalised average benefits per year becomes approximately b 1 = C0 8

(54)

The real rate of interest r is assumed to be 5% because, as argued, a purely monetary reliability optimization is considered. Assuming a lower tower frequency of 0.33 Hz a frequency constraint becomes ω1 ≥ 2π 0.33 Hz = 2.07 s−1 . The optimal design is determined from the optimization problem (5.9). The following bounds on the design variables are used: Thicknesses: 20 mm and 50 mm Diameter tower: 2m and 4 m Diameter monopile: 2 m and 6 m The optimal values of the design variables are shown in Table 2.2, including cases where the real rate of interest r is 3%, 5% and 10%, b/C0 is 1/8 and 1/10, and CF /C0 is 1/36 and 1/360. In Table 2.3 reliability indices for the different failure modes and for the system are shown. It is seen that • •

For increasing rate γ the dimensions and the value of the objective function as expected decreases. Further also the corresponding system reliability indices and eigenfrequencies decrease slightly. The optimal dimensions are not influenced by a change in the benefits – only the value of the objective function decreases with decreasing benefits per year.

54

Structural design optimization considering uncertainties

Table 2.3 Optimal values reliability indices for failure modes and system – first value is for local buckling/yielding and second value is for fatigue. γ

0.03

0.05

0.10

0.05

0.05

b/C 0 C F /C 0 Top section

1/8 1/36 13.6/4.90 5.63/4.25 7.96/5.62 5.12/3.67 6.37/4.32 5.08/3.60 7.09/4.09 3.41

1/8 1/36 13.4/4.79 5.55/4.20 8.02/5.64 5.22/3.72 6.08/4.09 4.85/3.49 7.09/3.98 3.34

1/8 1/36 12.9/4.52 5.37/4.08 6.91/5.01 4.35/3.50 5.79/3.95 4.66/3.26 7.09/3.47 3.06

1/10 1/36 13.4/4.79 5.55/4.20 8.02/5.64 5.22/3.72 6.08/4.09 4.85/3.49 7.09/3.98 3.34

1/8 1/360 12.7/4.52 5.28/4.01 7.39/5.20 4.83/3.54 5.95/4.01 4.86/3.49 7.09/3.86 3.26

Middle section Bottom section

Top Bottom Top Bottom Top Bottom

Pile System

• • • •

For decreasing failure costs the optimal dimensions, the objective function, the system reliability level and the eigenfrequency decrease slightly. The system reliability index β is 3.1–3.4. In this example the fatigue failure mode has the smallest reliability indices (largest probabilities of failure). The frequency constraint is not active.

The example shows that the optimal reliability level related to structural failure of offshore wind turbines is of the order of a probability per year equal to 2 · 10−4 − 10−3 corresponding to an annual reliability index equal to 3.1–3.4. This reliability level is significantly lower than for civil engineering structures in general.

6 Conclusions The theoretical basis for reliability-based structural optimization within the framework of Bayesian statistical decision theory is briefly described. Reliability-based cost benefit problems are formulated and exemplified with structural optimization. The basic reliability-based optimization problems are generalized to the following extensions: interactive optimization, inspection and repair costs, systematic reconstruction, re-assessment of existing structures. Illustrative examples are presented including a simple introductory example, a decision problem related to bridge re-assessment and a reliability-based decision problem for offshore wind turbines.

References Aitchison, J. & Dunsmore, I.R. 1975. Statistical Prediction Analysis. Cambridge University Press, Cambridge. Ang, H.-S.A. & Tang, W.H. 1975. Probabilistic concepts in engineering planning and design, Vol. I and II, Wiley.

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55

Benjamin, J.R. & Cornell, C.A. 1970. Probability, Statistics and Decision for Civil Engineers. Mc-Graw-Hill. EN 1993-1-9 2003. Eurocode 3: Design of steel structures – Part 1–9: Fatigue. Enevoldsen, I. & Sørensen, J.D. 1993. Reliability-Based Optimization of Series Systems of Parallel Systems. ASCE Journal of Structural Engineering, Vol. 119, No. 4, pp. 1069–1084. Enevoldsen, I. 1994. Sensitivity Analysis of a Reliability-Based Optimal Solution. ASCE, Journal of Engineering Mechanics. Enevoldsen, I. & Sørensen, J.D. 1994. Reliability-based optimization in structural engineering. Structural Safety, Vol. 15, pp. 169–196. Enevoldsen, S. & Sørensen, J.D. 1998. A Probabilistic Model for Chloride-Ingress and Initiation of Corrosion in Reinforced Concrete Structures. Structural Safety, Vol. 20, pp. 69–89. Engelund, S. 1997. Probabilistic models and computational methods for chloride ingress in concrete. Ph.D. thesis, Department of Building Technology and Structural Engineering, Aalborg University. Faber, M.H., Engelund, S. Sørensen, J.D. & Bloch, A. 1989. Simplified and generic risk based inspection planning. Proc. OMAE2000, New Orleans. Frangopol, D.M. 1985. Sensitivity of reliability-based optimum design. ASCE, Journal of Structural Engineering, Vol. 111, No. 8, pp. 1703–1721. Fujimoto, Y., Itagaki, H., Itoh, S., Asada, H. & Shinozuka, M. 1989. Bayesian Reliability Analysis of Structures with Multiple Components. Proceedings ICOSSAR 89, pp. 2143–2146. Fujita, M., Schall, G. & Rackwitz, R. 1989. Adaptive Reliability Based Inspection Strategies for Structures Subject to Fatigue. Proceedings ICOSSAR 89, pp. 1619–1626. Gill, P.E., Murray, E.W. & Wright, M.H. 1981. Practical Optimization. Academic Press. Haftka, R.T. & Kamat, M.P. 1985. Elements of Structural Optimization. Martinus Nijhoff, The Hague. ISO 19902 2001. Petroleum and natural gas industries – Fixed steel offshore structures. Kroon, I.B. 1994. Decision Theory Applied to Structural Engineering Problems. Ph.D. thesis, Department of Building Technology and Structural Engineering, Aalborg University. Kuschel, N. & Rackwitz, R. 1998. Structural optimization under time-variant reliability constraints. Proc. 8th IFIP WG 7.5 Conf. On Reliability and optimization of structural systems, University of Ann Arbor, pp. 27–38. Lindley, D.V. 1976. Introduction to Probability and Statistics from a Bayesian Viewpoint, Vol. 1 + 2. Cambridge University Press, Cambridge. Madsen, H.O. & Friis-Hansen, P. 1992. A comparison of some algorithms for reliabilitybased structural optimization and sensitivity analysis. Proc. IFIP WG7.5 Workshop, Munich, Springer-Verlag, pp. 443–451. Madsen, H.O. & Sørensen, J.D. 1990. Probability-Based Optimization of Fatigue Design Inspection and Maintenance. Presented at Int. Symp. On Offshore Structures, University of Glasgow. Madsen, H.O., Krenk, S. & Lind, N.C. 1986. Methods of Structural Safety. Prentice-Hall. Murotsu, Y., Kishi, M., Okada, H., Yonezawa, M. & Taguchi, K. 1984. Probabilistically optimum design of frame structures. Proc. 11th IFIP Conf. On System modeling and optimization, Springer-Verlag, pp. 545–554. Madsen, H.O., Sørensen, J.D. & Olesen, R. 1989. Optimal Inspection Planning for Fatigue Damage of Offshore Structures. Proceedings ICOSSAR 89, pp. 2099–2106. Powell, M.J.D. 1982. VMCWD: A FORTRAN Subroutine for Constrained Optimization. Report DAMTP 1982/NA4, Cambridge University, England. Rackwitz, R. 2001. Risk control and optimization for structural facilities. Proc. 20th IFIP TC7 Conf. On System modeling and optimization, Trier, Germany. Raiffa, H. & Schlaifer, R. 1961. Applied Statistical Decision Theory. Harward University Press, Cambridge, Mass.

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Rosenblueth, E. & Mendoza, E. 1971. Reliability optimization in isostatic structures. J. Eng. Mech. Div. ASCE, pp. 1625–1642. Schittkowski, K. 1986. NLPQL: A FORTRAN Subroutine Solving Non-Linear Programming Problems. Annals of Operations Research. Skjong, R. 1985. Reliability-Based Optimization of Inspection Strategies. Proc. ICOSSAR’85 Vol. III. pp. 614–618. Streicher, H. & Rackwitz, R. 2002. Structural optimization – a one level approach. Proc. Workshop on Reliability-based Design and Optimization – rbo02, IPPT, Warsaw. Sørensen, J.D. & Thoft-Christensen, P. 1985. Structural optimization with reliability constraints. Proc. 12th IFIP Conf. On System modeling and optimization, Springer-Verlag, pp. 876–885. Sørensen, J.D. & Thoft-Christensen, P. 1988. Inspection Strategies for Concrete Bridges. Proc. IFIP WG 7.5, Springer Verlag, Vol. 48, pp. 325–335. Sørensen, J.D., Thoft-Christensen, P., Siemaszko, A., Cardoso, J.M.B. & Santos, J.L.T. 1995. Interactive reliability-based optimal design. Proc. 6th IFIP WG 7.5 Conf. On Reliability and Optimization of Structural Systems, Chapman & Hall, pp. 249–256. Sørensen, J.D. & Tarp-Johansen, N.J. 2005a. Reliability-based optimization and optimal reliability level of offshore wind turbines. International Journal of Offshore and Polar Engineering (IJOPE), Vol. 15, No. 2, pp. 1–6. Sørensen, J.D. & Tarp-Johansen, N.J. 2005b. Optimal Structural Reliability of Offshore Wind Turbines. CD-rom Proc. ICOSSAR’2005, Rome. Thoft-Christensen, P. & Sørensen, J.D. 1987. Optimal Strategies for Inspection and Repair of Structural Systems. Civil Engineering Systems, Vol. 4, pp. 94–100. von Neumann, J. & Morgenstern, O. 1943. Theory of Games and Economical Behavior. Princeton University Press.

Chapter 3

Reliability analysis and reliabilitybased design optimization using moment methods Sang Hoon Lee Northwestern University, Evanston, IL, USA

Byung Man Kwak Korea Advanced Institute of Science and Technology, Daejeon, Korea

Jae Sung Huh Korea Aerospace Research Institute, Daejeon, Korea

ABSTRACT: Reliability analysis methods using the design of experiments (DOE) are introduced and integrated into reliability based design optimization (RBDO) frame work with a semi-analytic design sensitivity analysis (DSA) for the reliability measure. A procedure using the full factorial DOE with optimal levels and weighs is introduced and named as full factorial moment method (FFMM) for reliability analysis. The probability of failure is calculated using an empirical distribution system and the first four statistical moments of system performance function calculated from DOE. To enhance the efficiency of FFMM, a response surface augmented moment method (RSMM) is developed to construct a series of approximate response surface approaching to that of FFMM. A semi-analytic design sensitivity analysis for the probability of failure is proposed in combination with FFMM and RSMM. It is shown that the proposed methods are accurate and effective especially when the inputs are non-normal.

1 Introduction One of the fundamental problems in the structural reliability theory is the calculation of the probability of failure which is defined as a multifold probability integral of the joint probability density function of random variables over the domain of structural failure. Because the analytic calculation of this integral is practically impossible, many approximate methods and simulation methods are developed so far (Madsen et al. 1986, Kiureghian 1996, Bjerager 1991). Among the methods, the first order reliability method (FORM) (Hasofer & Lind 1974, Rackwitz & Fiessler 1978) is considered to be one of the most efficient computational methods and over the past three decades, contributions from numerous studies have made FORM the most popular reliability method. The reliability based design approaches (Lee & Kwak 1987–1988, Enevoldsen & Sørensen 1994, Frangopol & Corotis 1996, Tu et al. 1999, Youn et al. 2003) have adopted FORM as their main analysis tool for reliability due to its efficiency. The difficulties in FORM such as numerical difficulty in finding the most probable failure point (MPFP), errors involved in the nonlinear failure surface including the possibility of multiple design points (Kiureghian & Dakessian 1998), and errors caused

58

Structural design optimization considering uncertainties

by non-normality of variables (Hohenbichler & Rackwitz 1981) are well recognized and efforts to overcome these are also made. They include the second order reliability method (SORM) (Fiessler et al. 1979, Breitung 1984, Koyluoglu & Nielsen 1994, Kiureghian et al. 1987), the advanced Monte Carlo simulation (MCS) such as importance sampling (Bucher 1988, Mori & Ellingwood 1993, Melchers 1989), directional sampling (Bjerager 1988, Nie & Ellingwood 2005) and the response surface based approaches (Faravelli 1989, Bucher & Bourgund 1990, Rajashekhar & Ellingwood 1993). However, finding MPFP is a still numerically difficult task in FORM and often the error involved degrades the accuracy of final results. In this chapter, we investigate another route for structural reliability, the moment method. The moment method calculates the probability of failure by computing the statistical moments of the performance function and fitting the moments with some empirical distribution systems such as the Pearson system, Johnson system, Gram-Charlier series, and so on (Johnson et al. 1995). For this purpose, the performance function must be computed for a set of well-designed calculation points, often called quadrature points or designed experimental points. Compared with FORM, the moment method has advantages that it dose not involve the difficulties of searching for the MPFP and the information of cumulative distribution function (CDF) is readily available. Not so many attempts are reported about the reliability analysis using the moment method. For statistical moment estimation, Evans (1972) proposed a quadrature formula which uses 2n2 + 1 nodes and weights for a system with n random variables and applied it to tolerance analysis problems. Li & Lumb (1985) adopted Evans’ quadrature formula in structural reliability analysis in combination with the Pearson system. Rosenblueth (1981) devised a 2n point estimate method and (Hong 1996) proposed a nonlinear system of equations for point estimate of probability in combination with the Johnson distribution system and Gram-Charlier series. Zhao & Ono (2001) proposed a point estimate method using Rosenblatt transformation and kn point concentration where k is the number of quadrature points for each random variable. Taguchi (1978) proposed a design of experiment (DOE) technique which uses three level experiments for each random variable to calculate the mean and standard deviation of performance function for tolerance design. Taguchi’s method was improved by (D’Errico & Zaino 1988). These methods can treat only normally distributed random variables and the DOE becomes a 3n full factorial design when n random variables are under consideration. Actually, the levels and weights proposed by D’Errico & Zaino are equivalent to the nodes and weights in Gauss-Hermite quadrature formula (Abramowitz & Stegun 1972, Engels 1980). Seo & Kwak (2002) extended D’Errico & Zaino’s method to treat non-normal distributions by deriving an explicit formula of three levels and weights for general distributions. In addition to the strong points of moment method mentioned above, the moment method using DOE has several good aspects. It is very easy and simple to use and does not involve any deterioration of accuracy or additional efforts for treating non-normal random variables. However, the common problem of the moment based methods is the numerical efficiency. The methods often tend to become very expensive as the number of random variables increases. To overcome this shortcoming, (Lee & Kwak 2006) developed a new moment method integrating the response surface method with the 3n full factorial DOE. (Huh et al. 2006) developed a response surface approximation scheme based on the moment method and applied it to the design study of a precision nano-positioning system.

Reliability analysis and optimization using moment methods

59

In this chapter, we present our previous developments of moment methods which utilize DOE for statistical moment estimation and propose a RBDO framework with a semi-analytic design sensitivity analysis in combination with the moment methods. In section 2, the full factorial moment method (FFMM) is introduced with an explanation on the selection of optimal DOE. In section 3, the response surface augmented moment method (RSMM) is introduced and the accuracy and efficiency of RSMM are compared with other methods via several examples. In section 4, a RBDO procedure is proposed using FFMM and RSMM with a semi-analytic design sensitivity analysis. Section 5 provides some discussions on moment methods and the proposed RBDO procedure and concluding remarks.

2 Reliability analysis using full factorial moment method The probability of failure of a system is defined by a multifold probability integral as
Pf = Pr [g(X) < 0] =

fX (x)dx

(1)

g(x) 0, replace I by I + 1, and set CI = {K + 1}. Then, F(x) satisfies the assumption about a bounded safe domain. This is equivalent to enlarging the failure domain. The probability associated with the enlarged failure domain is slightly larger than the one associated with the original failure domain. The difference, however, is no

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Structural design optimization considering uncertainties

2 greater than 1 − χm (ρ2 ) and therefore negligible for sufficiently large ρ. Consequently, this boundedness assumption is not restrictive in practice. From the above derivation we see that the failure probability is differentiable with a continuous gradient given by (16), i.e., the failure probability satisfies the required smoothness assumption for nonlinear optimization. However, for this to have any practical value, we also need to be able to compute the failure probability and its gradient, i.e., we need the computability assumption to be satisfied. Clearly, (10) and (16) cannot, in general, be evaluated analytically, but must be estimated by Monte Carlo sampling. Let w1 , w2 , . . . , wN be a set of N sample points, each generated by independent sampling from the uniform distribution on the m-dimensional unit hypersphere. Given this sample, we define the estimate of (10):

pN (x) =

N 

φ(x, wj )/N

(17)

j=1

Since W corresponds to a direction, this type of Monte Carlo simulation is referred to as directional sampling (Bjerager 1988). It is well-known (see, e.g., (Rubinstein and Shapiro 1993) for a proof) that pN (x) converges to p(x) uniformly over any closed and bounded set, as N → ∞. Hence, at least in principle, we can obtain an accurate estimate of the failure probability by computing (17) with a large N. (Of course, a large sample size may be prohibitive computationally.) We now consider an estimate of the gradient (16). Since φ(x, w) is not differentiable with respect to x, we see that pN (x) is generally not differentiable either. However, since φ(x, w) has a subgradient, see (15), it can be shown that pN (x) also has a subgradient denoted by ∂pN (x), see (Royset and Polak 2007). This subgradient is given by ∂pN (x) =

N 

∂φ(x, wj )/N

(18)

j=1

see (15) for the expression for ∂φ(x, wj ). It is shown in (Royset and Polak 2007) that the subgradient ∂pN (x) converges (shrinks) to ∇p(x) uniformly over any closed and bounded set, as N → ∞. We note that there is typically no need to estimate the subgradient ∂pN (x), but only one of its elements. To generate such an element, proceed as follows: (i) obtain N sample points w1 , w2 , . . . , wN , (ii) for each sample point wj deterˆ mine one active limit-state function, i.e., find one element in K(x, wj ), and compute the numerical value of the vector within the brackets of (15) for the active limit-state function, and (iii) average the numerical values over all the sample points.

4 Algorithm based on sample average approximations In this section, we follow (Royset and Polak 2007; Polak and Royset 2007) and present an algorithm that utilizes the sample average estimates of the failure probability and its gradient derived in the previous section, see (17) and (18). The algorithm carries out nonlinear optimization iterations on a sequence of sample average approximations for

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the original problem P. Given the sample points w1 , w2 , . . . , wN , we define the sample average approximation of P as the following optimization problem: PN : min{c(x)|pN (x) ≤ q, x ∈ X} x

(19)

It is noted that the only difference between P and PN is that p(x) has been replaced by its sample average. Intuitively, PN becomes a better approximation to P as N increases. In fact, under weak assumptions, a global minimum of PN converges to a global minimum of P, as N → ∞, see (Royset and Polak 2007) and more generally Chapter 6 of (Ruszczynski and Shapiro 2003) and references therein. Since we can evaluate pN (x) for a given sample, PN satisfies our computability assumption. However, PN does not satisfy our smoothness assumption since (17) is generally not differentiable – it only has a subgradient (18). Hence, standard nonlinear programming algorithms may perform poorly when applied to PN . As seen in Subsection 5.1 below, we are able to overcome this difficulty by utilizing the fact that P satisfies the smoothness assumption. In this section, we proceed under the assumption that there is some optimization algorithm that can effectively be applied to PN . As discussed in Section 1, the simplest scheme for approximately solving P would be to select some sample size N and apply some optimization algorithm to PN for a number of iterations. The obtained design would be an estimate of the optimal design of P. However, this may be a poor estimate if the sample size is small, and if the sample size is large, the computational cost may be prohibitive. In (Royset and Polak 2007; Polak and Royset 2007), the following adaptive scheme is proposed. Conceptual Algorithm for Solving P. Step 0. Select an initial design x0 , an initial sample size N, and sample w1 , w2 , . . . , wN . Set iteration counter j = 0. Step 1. Consider the sample average approximation PN and compute a new design xj+1 by carrying out one iteration of some optimization algorithm applied to PN . This iteration is initialized by the current design xj . Step 2. Use some sample-adjustment rule and determine if the sample size should be augmented. If the sample size should be augmented, replace N by some larger N and generate additional sample points to complement the existing sample points. Step 3. Replace j by j + 1, and go to Step 1. The conceptual algorithm describes an adaptive scheme, but does not specify how Steps 1 and 2 can be implemented. What optimization algorithm can be used in Step 1? What sample-adjustment rule should be used in Step 2? At first glance, the first question appears easier. However, as discussed above, PN may not satisfy the smoothness assumption and standard nonlinear programming algorithms may perform poorly. In fact, as we will see in Subsection 5.1 below, care must be taken when selecting the optimization algorithm in Step 1 to ensure convergence of the overall algorithm. The second question appears to be difficult and embodies the following fundamental trade-off. A rapid increase in sample size may result in many iterations with large N and hence high computational cost. As we see in Subsection 5.1 below, there is also a theoretical concern; a rapid increase in sample size may prevent convergence to an optimal design.

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On the contrary, a slow increase in sample size may lead to unnecessarily many iterations on coarse sample average approximations. The next section discusses two approaches for implementing Step 2. We also briefly discuss the implementation of Step 1. There is also a third question that is not addressed in the conceptual algorithm: when to stop the calculations? As in all nonlinear programming, this is a fundamentally difficult questions that is substantially aggravated by the presence of sample averages. A simple approach would be to augment the sample size until it reaches a “sufficiently large’’ number, e.g., an N that results in a coefficient of variation for pN (x) of less than 5%. Then, keep that sample size for a number of iterations until the optimization algorithm in Step 1 ceases to make substantial progress from iteration to iteration. Another approach is to simply run the algorithm until the dedicated time is consumed. Techniques for checking whether a given design is close to optimal includes statistical testing, see e.g. Section 6.4 of (Ruszczynski and Shapiro 2003). A further discussion of stopping criteria and solution quality is beyond the scope of this chapter.

5 Selection of sample sizes The conceptual algorithm presented in the previous section needs a sample-adjustment rule (see Step 2). There are two main concerns when constructing a sample-adjustment rule: (i) theoretical convergence and (ii) computational efficiency. This section presents two different rules. The first rule satisfies (i), but its efficiency is sensitive to input parameters. The second rule has weaker convergence properties, but allocates samples optimally in some sense. 5.1 F eed b ac k r ule The first sample-adjustment rule for Step 2 of the conceptual algorithm augments the sample size when the progress of the optimization algorithm in Step 1 is sufficiently small. This rule is motivated by the following observation: when the optimization algorithm in Step 1 is making small progress towards an optimal design of the current sample average approximation PN , the current design is probably near that optimal design. Hence, there is little to be gained from computing even better designs for PN ; it is better to increase the sample size N and start to calculate with a more accurate sample average approximation. In (Royset and Polak 2007), the progress of the optimization algorithm in Step 1 is measured in terms of a function FN (x
, x

) defined by FN (x
, x

) = max{c(x

) − c(x
) − γψN (x
)+ , ψN (x

) − ψN (x
)+ }

(20)

where

  ψN (x) = max pN (x) − q, max fj (x) j∈J

(21)

ψN (x)+ = max{0, ψN (x)}, and the parameter γ > 0. The function FN (x
, x

) measures how much “better’’ the design x

is compared to design x
. Suppose that x
is a feasible design for PN . Then, ψN (x
) ≤ 0 and ψN (x
)+ = 0 and, hence, FN (x
, x

) = max{c(x

) − c(x
), ψN (x

}

(22)

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We see that if FN (x
, x

) ≤ −ω, with ω being some positive number, then the objective function in PN for design x

is reduced with at least the amount ω compared to the value for design x
. Additionally, x

is feasible for PN because ψN (x

) ≤ −ω. Suppose that x
is not a feasible design for PN . Then, ψN (x
) > 0. When FN (x
, x

) ≤ −ω, the constraint violation for PN at x

is reduced with at least the amount ω compared to the value at x
because ψN (x

) − ψN (x
) ≤ −ω. The above observation leads to the following sample-adjustment rule: If FN (xj , xj+1 ) is no larger than a threshold, then the progress is sufficient and the current sample size is kept. (Note that FN (xj , xj+1 ) is a negative number and that it measures the decrease in cost or constraint violation. Hence, a large negative number corresponds to a large progress towards an optimal design.) If FN (xj , xj+1 ) is larger than the threshold, then the progress is too small and the sample size is increased. The challenge with this rule is to determine an appropriate threshold. In (Royset and Polak 2007), we find a sample-size dependent threshold that results in the following sample-adjustment rule: If

 τ FN (xj , xj+1 ) > −η ( log log N)/N

(23)

then augment the sample size. Otherwise, keep the current sample size in the next iteration. Here, η is a positive parameter and τ is a parameter strictly between 0 and 1. Since the threshold is increasing (approaches zero from below) with increasing sample size, the rule becomes successively more stringent. For large N, the sample size is only increased if the optimization algorithm in Step 1 of the conceptual algorithm makes a tiny progress (FN (xj , xj+1 ) is close to zero). This means that for large N, it is necessary to solve the sample average approximation to near optimality before the sample size is increased. On the other hand, for small N, the sample size is increased even if the optimization algorithm in Step 1 is making a relatively large progress. Hence, the rule avoids having to solve low-precision sample average approximations to high accuracy before switching to a larger sample size. But, the rule eventually forces the algorithm to solve high-precision sample average approximations to high accuracy. The double logarithmic form of the threshold in (23) relates to the Law of the Iterated Logarithm (see (Royset and Polak 2007) and references therein). It is shown in (Royset and Polak 2007) that this exact form of the sample-adjustment rule guarantees convergence of the conceptual algorithm when implemented with a specific optimization algorithm in Step 1. This specific optimization algorithm is motivated by the Polak-He algorithm (see Section 2.6 of (Polak 1997)) and takes the following form. For any current design xj and current sample size N, the next iteration xj+1 = xj + λN (xj , d)hN (xj , d)

(24)

where d is any element in the subgradient ∂pN (xj ), see (18) and its subsequent paragraph, and the stepsize λN (xj , d) is given by Armijo’s rule: λN (xj , d) =

max {βk |FN (xj , xj + βk hN (xj , d)) ≤ βk αθN (xj , d)}

k∈{0,1,2,...,}

(25)

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Structural design optimization considering uncertainties

Here, α ∈ (0, 1] and β ∈ (0, 1) are parameters, and θN (x, d) = − min{zT bN (x) + zT BN (x, d)T BN (x, d)z/(2δ)} z∈Z

with parameter δ > 0, the J + 2-dimensional unit simplex Z given by ⎫ ⎧  J+2 ⎨  ⎬ Z = z  zj = 1, zj ≥ 0, ∀j ⎭ ⎩  j=1

(26)

(27)

the (J + 2)-dimensional vector (γ as in (20)) bN (x) = (γ ψN (x)+ , ψN (x)+ − pN (x) + q, ψN (x)+ − f1 (x), . . . , ψN (x)+ − fJ (x))T (28) and the n × (J + 2)-matrix BN (x, d) = (∇c(x), d, ∇f1 (x), . . . , ∇fJ (x))

(29)

Finally, the search direction hN (xj , d) = −BN (xj , d)ˆz/δ

(30)

where zˆ is any optimal solution of (26). The problem in (26) is quadratic and can be solved in a finite number of iterations by a standard QP-solver (e.g. Quadprog (Mathworks, Inc. 2004)). Usually, the one-dimensional root finding problems in the evaluation of rk (x, w), needed in (15), cannot be solved exactly in finite computing time. One possibility is to introduce a precision parameter that ensures a gradually better accuracy in the root finding as the algorithm progresses. Alternatively, we can prescribe a rule saying that the root finding algorithm should terminate after CN iterations, with C being some constant. For simplicity, we have not discussed the issue of root finding. In fact, this issue is not problematic in practice. The root finding problems can be solved in a few iterations with high accuracy using standard algorithms. Hence, the root finding problems are solved with fixed precision for all iterations in the algorithm giving a negligible error. The feedback rule (23) requires the user to determine the parameters η and τ as well as the amount of sample size increase. To avoid a quick increase in sample size and corresponding high computational costs, the parameter τ is typically set to 0.9999. However, it is nontrivial to determine an efficient value for the parameter η. If η is large, then the sample size tends to be augmented frequently. Hence, η should be small to avoid costly sample average approximations in the early iterations. However, too small η may result in an excessive number of iterations for each sample average approximation. Overall, in Section 6 we see empirically that the numerical value of η may influence computing times significantly. Furthermore, neither the conceptual algorithm nor the feedback rule specify how much the sample size should be increased – only when to increase it. Typically, the user specifies a rule of the form: replace N by ξN, with ξ > 1, whenever the sample size needs to be increased. Naturally, the

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computationally efficiency may vary with the amount increased each time. We note that (Royset and Polak 2007) proves that the conceptual algorithm with the sampleadjustment rule (23) and the optimization algorithm (24) is guaranteed to converge to a solution for any τ ∈ (0, 1), η > 0, and sample size increase. Hence, the above discussion only relates to how fast the algorithm will converge. As indicated in the previous paragraph, it can be difficult to select efficient values for the parameter η as well as an efficient sample size increase every time the algorithm is prompted by the sample-adjustment rule. Typically, some numerical experimentation and parameter tuning for the problem at hand is needed. In the next subsection, we describe an alternative, more complex sample-adjustment rule that avoids such tuning. 5.2

Ef f icient s cheme

In this subsection, we present the sample-adjustment scheme given in (Polak and Royset 2007), which modifies a methodology originally developed in (He and Polak 1990). Instead of having a simple sample-adjustment rule as in Subsection 5.1 for Step 2 of the conceptual algorithm, the scheme in (Polak and Royset 2007) consists of a precalculation step that determines the “optimized’’ sample size for subsequent iterations. In the pre-calculation step, the user selects a required accuracy of the final design (e.g., a feasible design with cost within 5% of the minimum cost) and solves an auxiliary optimization problem that determines the sample size for each iteration (e.g., 100, 100, 100, 200, 200, 300, etc., sample points, for iterations 1, 2, 3, 4, 5, 6, etc., respectively). Hence, whenever the conceptual algorithm reaches Step 2, it simply looks up the prescribed sample size from the output of the auxiliary optimization problem. The objective function of the auxiliary optimization problem, to be derived below, is the total computational work needed to obtain a solution of required accuracy, and the constraint is that the required cost reduction be achieved. Let a stage be a number of iterations carried out by the conceptual algorithm for a constant sample size. The decision variables in the auxiliary problem are (i) the number of stages, s, to be used, (ii) the sample size Ni to be used in stage i, i = 1, 2, . . . , s, and (iii) the number of iterations ni to be carried out in stage i. For example, 100, 100, 100, 200, 200, and 300 sample points, for iterations 1, 2, 3, 4, 5, and 6 respectively, correspond to three stages, with stage 1 consisting of three iterations (n1 = 3) and sample size 100 (N1 = 100), stage 2 consisting of two iterations (n2 = 2) and sample size 200 (N2 = 200), and stage 3 consisting of one iteration (n3 = 1) and sample size 300 (N3 = 300). While the number of stages s has to be treated as an integer variable, the variables Ni and ni can be treated as continuous variables and rounded at the end of their optimization. In practice, it turns out that the optimal number of stages s∗ hardly ever exceeds 10, with 3–7 being a most likely range for s∗ . Incidentally, if one assigns the number of stages to be s > s∗ , and then solves the reduced auxiliary optimization problem for the Ni and ni , the optimal solution will consist of several Ni being equal, so that the total number of distinct stages is s∗ . The auxiliary problem depends on a sampling-error bound, on the initial distance to the optimal value, and on the rate of convergence of the optimization algorithm applied in Step 1 of the conceptual algorithm. All of these may have to be estimated. As a result, it may be presumptuous to call the solution of the auxiliary optimization

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problem an “optimal strategy,’’ and hence we will call it an “efficient strategy.’’ As we will see from our numerical results, despite the use of estimated quantities, the efficient strategy is considerably more effective than the obvious alternatives. 5.2.1 Au xi li a ry o p t im iza t io n p r o b le m We begin by deriving the auxiliary optimization problem. First we penalize the constraint in P to convert it into an equivalent, unconstrained min-max problem. This simplifies the derivation since it avoids distinguishing between feasible and infeasible design. For a given parameter π > 0, we define c˜ (x) = c(x) + π max{0, p(x) − q, f1 (x), f2 (x), . . . , fJ (x)}

(31)

c˜ N (x) = c(x) + π max{0, pN (x) − q, f1 (x), f2 (x), . . . , fJ (x)}

(32)

and the unconstrained problem P˜ : min c˜ (x)

(33)

x

We refer to π as a penalty since it adds a positive number to the objective functions c(x) and cN (x) for any infeasible design x. If P is calm (see, e.g., (Burke 1991; Clarke 1983)) and π is sufficiently large, then the design x is a local minimizer of P˜ if and only if it is a local minimizer of P. Similarly, the unconstrained problem P˜ N : min c˜ N (x)

(34)

x

is equivalent to PN for sufficiently large π. An appropriate penalty π can be selected using well-known techniques such as the one in Section 2.7.3 of (Polak 1997). The implementation of such techniques is beyond the scope of this chapter and we assume in the following that a sufficiently large penalty π > 0 has been determined so that optimal solutions of P˜ and P˜ N are feasible for P and PN , respectively. As above, we assume that each sample point is independently generated and that sample points are reused at later stages, i.e., for all stages i = 2, 3, . . . , s, the sample at stage i consists of the Ni−1 sample points at stage i − 1 and of Ni − Ni−1 new, independent sample points. To construct an auxiliary optimization model for determining the number of stages, the sample size at each stage, and the number of iterations to be performed at each stage, we introduce the following assumptions. Suppose that the optimization algorithm in Step 1 of the conceptual algorithm is linearly convergent with a rate of convergence coefficient independent of the sample size in the sample average approximations. That i is, for any stage i and iteration j, the costs of the design at the next iteration, xj+1 , and i ∗ the current design, x , relate to the cost of the optimal design x of P˜ N as follows: j

i ∗ ∗ c˜ Ni (xj+1 ) − c˜ Ni (xN ) ≤ θ(˜cNi (xji ) − c˜ Ni (xN )) i i

Ni

i

(35)

where θ ∈ (0, 1) is the rate of convergence coefficient. Hence, every iteration of the optimization algorithm reduces the remaining distance to the optimal value by a factor θ.

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Many optimization algorithms including the Pshenichnyi-Pironneau-Polak Min-Max Algorithm (see Section 2.4.1 of (Polak 1997)) are linearly convergent. Next, we assume that for any design x the sampling error is given by |˜cN (x) − c˜ (x)| ≤ (N)

(36)

where (N) is a strictly decreasing positive function with (N) → 0, as N → ∞. We return to the form of (N) below, but for now we only assume that such a function exists. To simplify the notation, we deviate from the numbering scheme of the conceptual algorithm and let j note the iteration number of the current stage (and not from the beginning). Then, xji is the design at iteration j of the i-th stage. Hence, we plan to compute the designs x01 , x11 , . . . , xn11 on stage 1, x02 , x12 , . . . , xn22 on stage 2, . . . , and , i.e., designs x0s , x1s , . . . , xns s on stage s. To make use of “warm’’ starts, we set x0i = xni−1 i−1 the last design of the current stage is taken as the initial design of the next stage. ∗ Let x∗ and xN be optimal designs for P˜ and P˜ N , respectively. Then, in view of (36) we have that ∗ ∗ c˜ (x∗ ) ≤ c˜ (xN ) ≤ c˜ N (xN ) + (N) ∗ ) c˜ N (xN

∗

∗

≤ c˜ N (x ) ≤ c˜ (x ) + (N)

(37) (38)

We refer to the distance between the cost c˜ (x) of some design x and the cost c˜ (x∗ ) of an optimal design x∗ for P˜ as the cost error of design x. Here, the term “error’’ refers to the discrepancy between x and x∗ . For any stage i = 1, 2, . . . , s, we define the cost error after the last iteration of the i-th stage by ei = c˜ (xni i ) − c˜ (x∗ )

(39)

Also let e0 = c˜ (x01 ) − c˜ (x∗ ). Using (36)–(38) and (35), we obtain that for all i = 1, 2, . . . , s, ∗ ei ≤ c˜ Ni (xni i ) − c˜ Ni (xN ) + 2 (Ni ) i ∗ )] + 2 (Ni ) ≤ θ ni [˜cNi (x0i ) − c˜ Ni (xN i

≤θ

ni

[˜c(xni−1 ) i−1

∗

− c˜ (x )] + 4 (Ni )

≤ θ ni ei−1 + 4 (Ni )

(40) (41) (42) (43)

Hence, es ≤ e0 θ k0 (s) + 4

s 

θ ki (s) (Ni )

(44)

i=1

 where ki (s) = sl=i+1 nl if i < s and ki (s) = 0 if i = s. We observe that (44) gives an upper bound on the cost error after completing s stages with ni iterations and Ni sample points at stage i. As shown in (Polak and Royset 2007), the cost error is guarantee to vanish as the number of stages s increases to infinity. This shows that such gradual sample

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size increase can lead to asymptotic convergence. This is a valuable result, but in this subsection we aim to determine efficient sample-adjustment schemes, i.e., schemes that minimize the computing time to reach a specific reduction in cost error from an initial value. To be able to construct efficient sample-adjustment schemes we need to quantify the computational effort associated with one iteration of the optimization algorithm used in Step 1 of the conceptual algorithm as a function of the sample size N. Suppose that this computational effort is given by the positive function w(N) for any design x. We are now ready to present the auxiliary optimization problem. Given an initial cost error e0 > 0 and a required fractional reduction in cost error  ∈ (0, 1), we seek to determine the number of stages s as well as sample sizes Ni and numbers of iterations ni at each stage i, i = 1, 2, . . . , s, such that the computational effort to reach a cost error of e0 is minimized. We note that the cost error is the discrepancy between the cost of the current design and the cost of the optimal design ˜ In view of (44), this optimization problem takes the following form of P. min

s,ni ,Ni

 s  i=1

s    ni w(Ni )e0 θ k0 (s) + 4 θ ki (s) (Ni ) ≤ e0 i=1

Ni+1 ≥ Ni ,

i = 1, 2, . . . , s − 1

s, ni , Ni integer,



(45)

i = 1, 2, . . . , s

The objective function in D(e0 , ) represents the total computational effort needed to carry out the planned iterations. The first constraint ensures that the cost error has at least been reduced to the required level e0 and the second set of constraints ensures that the sample size is nondecreasing. The estimation of the parameters defining problem D(e0 , ) is discussed in the next section. 5.2.2 Im p lem en t a t io n o f a u x ilia r y o p t im iza t i o n pr o bl e m The auxiliary optimization problem D(e0 , ) involves the work and sampling-error functions w(N) and (N) as well as the rate of convergence parameter θ and the initial cost error e0 = c˜ (x01 ) − c˜ (x∗ ). All these quantities must be determined before D(e0 , ) can be solved. We deal with these issues one at a time. In view of (17) and (18), the computing effort required to evaluate pN (x) and an element of the subgradient grows linearly in N. Hence, the work associated with one iteration of the optimization algorithm used in Step 1 of the conceptual algorithm is proportional to N and we set the work function w(N) = N. The (almost sure) sampling error (N) can be determined  using the Law of the Iterated Logarithm, see (Royset and Polak 2007). However, ( log log N)/N is a pessimistic estimate of the sampling error “typically’’ experienced. Since our goal is to determine efficient number of stages, sample sizes, and numbers of iterations, it appears √ to be more reasonable to assume that the sampling error is proportional to 1/ N as proposed by classical estimation theory: For a given design x, it follows under weak assumption from the Central Limit Theorem that pN (x) is approximately normally

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distributed with mean p(x) and variance σ(x)2 /N for large N, where σ(x)2 = Var[φ(x, W)]. Hence, for sufficiently large N, √ P[|pN (x) − p(x)| ≤ 1.96σ(x)/ N] ≥ 0.95 (46) However, we are primarily interested in the difference between c˜ N (x) and c˜ (x). Since the max-function in (32) only makes the variance less, it follows that P[|˜cN (x) − c˜ (x)| ≤ (N)] ≥ 0.95 (47) √ when (N) = 1.96πσ(x)/ N. This error expression appears to be appropriate for our auxiliary optimization problem, and we set √ (48)

(N) = 1.96π max σ(x)/ N where the maximization is over all designs examined in a preliminary calculation described below. We determine σ(x), θ, and e0 in an estimation phase consisting of n0 iterations of the optimization algorithm in Step 1 of the conceptual algorithm applied to P˜ N0 , with 0 N0 being a small sample size. Let {xj0 }nj=0 be the iterates computed in this estimation phase. Each time pN0 (x) is computed, the corresponding variance σ(x)2 is estimated by σ(x) = 2

N0 

(φ(x, wj ) − pN0 (x))2 /(N0 − 1)

(49)

j=1

We always retain the largest σ(x)-value computed and use that in the calculation of

(N), see (48). The rate of convergence parameter θ is estimated by the solution of the following ∗ least-squares problem, where the optimal value c˜ N0 (xN ) of P˜ N0 is also estimated: 0 min θˆ ,ˆc

n0 

[(ˆc + (˜cN0 (x00 ) − cˆ )θˆ j ) − c˜ N0 (xj0 )]2

(50)

j=0

This least-square problem minimizes the squared error between the calculated cost at each iteration c˜ N0 (xj0 ) and the nonlinear model cˆ + (˜cN0 (x00 ) − cˆ )θˆ j . The nonlinear model estimates that the cost of the design at iteration j is the optimal cost cˆ plus the initial cost error c˜ N0 (x00 ) − cˆ reduced by a factor. The factor is simply the rate of convergence coefficient raised to the power of the number of iterations. Using the results of the ∗ ) by cˆ . Finally, we (coarsly) least square calculations, we estimate θ by θˆ and c˜ N0 (xN 0 1 ∗ estimate the initial cost error e0 = c˜ (x0 ) − c˜ (x ) by eˆ 0 = c˜ N0 (x00 ) − cˆ . We have now established procedures for estimating all the unknown quantities in D(e0 , ). D(e0 , ) is a nonlinear integer program that appears difficult to solve directly, but this fact can be circumvented by the following observations. First, the restriction of D(e0 , ) obtained by fixing s to a number in the range 5–10 tends to be insignificant since more than 5–10 stages is rarely advantageous and fewer than 5–10 stages is still effectively allowed in the model by setting Ni = Ni+1 for some i. Second, Ni , and to some extent also ni , tend to be large integers. Hence, a continuous relaxation

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with rounding of the optimal solutions to the nearest integers is justified. In view of these observations, D(e0 , ) can be solved approximately using a standard nonlinear programming algorithm. 5.2.3 Overa ll a lg o r it h m w it h e f f icie n t s a m p l e-adj us t me nt s c he me We now summarize our approach and discuss how the auxiliary optimization problem can be integrated in an algorithm for solving P. As indicated above, the process of solving the auxiliary optimization problem must be preceded by an estimation phase where parameters are determined. This leads to the following overall algorithm for solving P approximately. Algorithm with Efficient Sample-Adjustment Scheme. Parameters. Number of iterations in estimation phase n0 , sample size in estimation phase N0 , maximum number of stages s, and constraint penalty π > 0. Data. Required fractional reduction in cost error  > 0, initial design x00 , and independent sample points w1 , w2 , . . . . Step 0. Compute variance estimate σ(x00 )2 using (49). Step 1. For j = 0 to n0 − 1, perform: 0 by starting from xj0 and carrying out Sub-step 1.1. Compute the next design xj+1 one iteration of some optimization algorithm applied to PN0 . 0 Sub-step 1.2. Compute the variance estimate σ(xj+1 )2 using (49). Step 2. Set σˆ equal to the largest variance estimate encountered in Steps 0 and 1. Step 3. Determine θˆ and cˆ as√the optimal solution of (50). ˆ Step 4. Set (N) = 1.96πσ/ ˆ N, and determine ni and Ni by solving  s s     ˆ i ) ≤ ˆe0 min θˆ ki (s) (N ni Ni ˆe0 θˆ k0 (s) + 4 ni ,Ni

i=1

i=1

Ni+1 ≥ Ni ,

i = 1, 2, . . . , s − 1 

ni , Ni ≥ 1,

i = 1, 2, . . . , s

(51)

Step 5. For i = 1 to s, perform: Sub-step 5.1. Set the first design of the current stage equal to the last design of the previous stage, i.e., x0i = xni−1 . i−1 i Sub-step 5.2. For j = 0 to ni − 1, compute the next design xj+1 by starting from xji and carrying out one iteration of some optimization algorithm applied to PNi . We note that the optimization algorithm used in Sub-Step 1.1 should be identical to the one used in Sub-Step 5.2 since the former sub-step is used to estimate the behavior of the latter. However, any nonlinear programming algorithm can be used in Steps 3 and 4. The proposed algorithm consists of three phases: estimation of parameters (Steps 0–3), solution of auxiliary optimization problem (Step 4), and main iterations (Step 5). This represents the simplest implementation of our idea. Alternatively, we can adopt a moving-horizon approach, where Step 5 is completed only for i = 1, followed by Step 4, then by Step 5 for i = 1 again, followed by Step 4, etc. Hence,

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the sample-adjustment plan is re-optimized after each stage, which may lead to an improved plan. With re-optimization, it is also possible to re-compute σ, ˆ using all previous iterates, as well as θˆ and cˆ . Other implementations can also be imagined. In the following numerical study, we adopt the simple implementation described above.

6 Numerical examples We illustrate our sample-adjustment approaches using three numerical examples. The examples are implemented in Matlab 7.0 (Mathworks, Inc. 2004) on a 2.8 GHz PC running Microsoft Windows 2000. 6.1

Feedbac k rule and efficient s chem e

This subsection presents a comparative study of the two sample-adjustment approaches given in Section 5. The numerical results of this subsection were reported in (Polak and Royset 2007). Ex a mple 1 The first example arises in the optimal design of a short structural column with a rectangular cross section of dimensions x1 × x2 . Hence, x = (x1 , x2 ) is the design vector. The column is subjected to bi-axial bending moments V1 and V2 , which, together with the yield strength V3 of the material, are considered to be independent, lognormally distributed random variables. The column is also subject to a deterministic axial force af . This gives rise to a failure probability p(x) = P[{G(x, V) ≤ 0}]

(52)

where the random vector V = (V1 , V2 , V3 ) and G(x, V) is a limit-state function defined by G(x, V) = 1 −

4V1 4V2 − 2 − 2 x1 x2 V3 x1 x2 V3

af x1 x2 V3

2 (53)

As discussed in Section 2, this limit-state function can be transformed into one give in terms of a standard normal vector U. Let g1 (x, U) be this transformed limit-state function. Since the resulting safe domain is not bounded, we introduce an auxiliary limit-state function g2 (x, U) = ρ − U, where ρ = 6.5 in this example. (This introduces negligible error.) Then, we redefine the failure probability of the structure as p(x) = P[{g1 (x, U) ≤ 0} ∪ {g2 (x, U) ≤ 0}]

(54)

which is in the form considered in this chapter. We seek a design of the column which satisfies the constraints defined by f1 (x) = −x1 , f2 (x) = −x2 , f3 (x) = x1 /x2 − 2, f4 (x) = 0.5 − x1 /x2 , f5 (x) = x1 x2 − 0.175, and minimize p(x). This is problem (1) with c0 (x) = 0, c(x) = 1, and J = 5. As discussed above, pN (x) does not satisfy the smoothness assumption. Hence, care must be taken when selecting an optimization algorithm for Step 1 in the conceptual algorithm or in Sub-Steps 1.1 and 5.2 in the algorithm with efficient sample-adjustment scheme. For simplicity in

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Structural design optimization considering uncertainties

these numerical tests, we ignore the fact that the smoothness assumption may be violated and use the Pshenichnyi-Pironneau-Polak Min-Max Algorithm (see Section 2.4.1 of (Polak 1997)) as the optimization algorithm for solving PN . No detrimental behavior of the Pshenichnyi-Pironneau-Polak Min-Max Algorithm was observed because of this simplification. (Note that since p(x) is smooth, pN (x) is, for practical purposes, effectively smooth for large N.) The parameters for the algorithm with efficient sample-adjustment scheme were selected to be n0 = 25, N0 = 50, s = 5, and π = 2. We note that π = 2 suffices to ensure feasibility. Finally, the required fractional reduction in cost error was  = 0.01 and the √ √ initial point was chosen to be x00 = ( 0.175, 0.175). The auxiliary optimization problem yielded a sample-adjustment strategy of three stages with 25, 8, and 8 iterations, with sample sizes 50, 251, and 1621, respectively, which was executed in 458 seconds. Note that this computing time includes the estimation phase (30 seconds) and the solution time of the auxiliary optimization problem (3 seconds). For comparison, we also solve the problem using the feedback rule of Subsection 5.1 to adjust the sample size. We experiment with the thresholds −η

τ

 ( log log N)/N

(55)

and √ −η/ N

(56)

for determining if the progress is “small’’ in Step 1 of the conceptual algorithm. We note that (55) is the same as in (23). This threshold formula guarantees convergence as proven in (Royset and Polak 2007). The threshold in (56) leads to a heuristic algorithm, but offers the advantage that the threshold tends to zero faster for increasing N as compared to (55). In the numerical tests, we set τ = 0.9999. As mentioned above, it is difficult to select and effective value of η, so we experiment with a range of values. Furthermore, we must determine how much the sample size should increase when prompted by the sample-adjustment rule. In this example, we selected five stages with sample sizes equally spaced between the minimum and maximum sample sizes given by the auxiliary optimization problem, i.e., 50, 443, 836, 1228, and 1621. We used the same random seed in both algorithms. We ran the algorithm with the feedback rule until c˜ 1621 ( · ) was equal to the cost achieved in the last iteration of the algorithm with the efficient scheme. We did not augment the sample size beyond 1621, but continued computing iterates at that stage until the target costvalue was achieved. This is a somewhat favorable stopping criterion for the algorithm with the feedback rule because this algorithm might augment, prematurely, the sample size beyond 1621 resulting in long computing times. The computing times for the algorithm with the feedback rule are summarized in Table 12.1 for various values of the parameter η and for the two threshold formulae (55) and (56). In Table 12.1, the row with η = ∞ gives the computing time for a fixed sample size equal to the largest sample size 1621 for all iterations.

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Table 12.1 Computing times [seconds] for the algorithm with feedback rule for sample adjustment as applied to Example 1. The algorithm with efficient sample-adjustment scheme computes the same design in 458 seconds. η

∞ 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9

Threshold (55)

(56)

980 1044 1084 678 675 682 476 574 603 898

980 1036 654 675 677 676 477 554 601 901

As seen from Table 12.1, a fixed sample size can result in poor computing times compared to an adaptive scheme using a feedback rule. However, in the adaptive schemes there is a trade-off between solving the approximating problems accurately at an early stage (i.e., using small η), potentially wasting time, and solving the early approximations too coarsely (i.e., using large η), leading to many iterations at stages with high computational cost. In the efficient sample-adjustment scheme of Sub-Section 5.2, the trade-off is balanced by solving the auxiliary optimization problem. In the feedback rule, the user needs to consider the trade-off manually by selecting a value for the parameter η. If the right balance is found, i.e., a good η, then the feedback rule can be efficient. In fact, the feedback rule with η = 10−6 is only marginally slower than the efficient scheme. Of course, it is difficult to select η a priori. To illustrate this difficulty, we repeated the example for the higher accuracy  = 0.005. Then, the efficient scheme increased the sample size up to 6473 and solved the problem in 1461 seconds. From Table 12.1 it appears that η = 10−6 is a good choice. We selected this value and re-solved the problem using the feedback rule with five stages equally spaced in the range [50, 6473] as above. The computing time turned out to be 4729 seconds. Hence, η = 10−6 was not efficient in this case. Exa mple 2 The second example considers the design of a simply supported reinforced concrete T-girder for minimum cost according to the specifications in (American Association of State Highway and Transportation Officials 1992), using the nine design variables x = (As , b, hf , bw , hw , Av , S1 , S2 , S3 ), where As is the area of the tension steel reinforcement, b is the width of the flange, hf is the thickness of the flange, bw is the width of the web, hw is the height of the web, Av is the area of the shear reinforcement (twice the cross-section area of a stirrup), and S1 , S2 and S3 are the spacings of

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Structural design optimization considering uncertainties Table 12.2 Computing times [seconds] for the algorithm with feedback rule for sample adjustment as applied to Example 2. The algorithm with efficient sample-adjustment computes the same design in 1001 seconds. η

∞ 10−2 10−3 10−4 10−5 10−6 10−7

Threshold (55)

(56)

>36000 >12600 2004 2256 6721 1209 11108

>36000 7416 1990 2342 2327 1608 >7200

shear reinforcements in the high, medium, and low shear force zones of the girder, respectively. We model uncertainty using eight independent random variables collected in a vector V. We assumed that the girder can fail in four different modes corresponding to bending stress in mid-span and shear stress in the high, medium, and low shear force zones. Structural failure occurs if any of the four failure modes occur. This gives rise to four nonlinear, smooth limit-state functions Gk (x, V), k = 1, 2, 3, 4, whose exact form is rather complicated and is given in (Royset et al. 2006). This results in a failure probability p(x) = P[ ∪4k=1 {Gk (x, V) ≤ 0}]. As above, these limit-state functions can be transformed into ones given in terms of a standard normal vector U. Let gk (x, U) be these transformed limit-state functions. Since the resulting safe domain is not bounded, we introduce an auxiliary limit stage function g5 (x, U) = ρ − U, where ρ = 10 in this example. (This introduces negligible error.) Then, we redefine the failure probability of the structure as   5 ' p(x) = P {gk (x, U) ≤ 0} (57) k=1

which is in the form considered in this chapter. We also imposed 24 deterministic, nonlinear constraints as described in (Royset et al. 2006). Algorithm parameters were selected to be n0 = 50, N0 = 50, s = 5, and π = 1. Finally, the required fractional reduction in cost error  = 0.0001 and the initial point x00 = (0.01, 0.5, 0.5, 0.5, 0.5, 0.0005, 0.5, 0.5, 0.5) were chosen. The algorithm with the efficient sample-adjustment scheme gave three stages with 65, 20, and 20 iterations, with sample sizes 50, 373, and 2545, respectively. The total computing time was 1001 seconds. Again we compared this result with that obtained using the algorithm with the feedback rule. Here, we use five stages of equally spaced sample sizes between 50 and 2545. Using the same stopping criterion as for the first example, we obtained the computing times in Table 12.2. We observe that the computing times using the feedback rule can be significantly longer than those achieved using the efficient scheme. We also

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4

3

6

5

8.66 m

7

2

1 L 10 m

10 m

Figure 12.1 Truss for Example 3.

note that an approach with a fixed sample size of 2545 for all iterations takes more than 10 hours (see the first row in Table 12.2). 6.2 Alternative objective functions We conclude this chapter by demonstrating how our solution methodology can also solve other problems than P (and (1) and (3)). Typically, engineers need to account for not only quantitative factors such as cost and reliability, but also esthetic, social, and political requirements. Most esthetic, social, and political requirements are qualitative in nature and cannot easily be incorporated into numerical models. Even quantitative factors may not fully represent reality due to imprecise models and lack of data. In this subsection, we show how multiple optimization models can be formulated and solved to account for this situation. We adopt an approach originally proposed in (Brill Jr. 1979) for public sector planning: determine a small set of design alternatives that satisfy the stated requirements, are “good’’ with respect to the stated objective, and are also dispersed in the design space. Instead of searching for one optimal design or an efficient frontier, as in singleand multi-objective objective optimization, respectively, this approach seeks several design alternatives (e.g., 3–12) that the engineer and the decision maker can further assess using qualitative objectives. As pointed out in (Brill Jr. 1979), the best design from the perspective of the decision maker may not be located on the efficient frontier, as assumed by a multi-objective optimization formulation, due to the fact that not all objectives are included in the multi-objective formulation. Furthermore, by seeking a dispersed set of design alternatives, the engineer and decision maker are presented with a wide range of alternatives which may stimulate new considerations and ideas about designs, objectives, and constraints. See also (White 1996; Drezner and Erkut 1995) for similar approaches. We illustrate this approach with an example. Ex am ple 3 Consider the simply supported truss in Figure 12.1. The truss is subject to a random load L in its mid-span. L is lognormally distributed with mean 1000 kN and standard

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Structural design optimization considering uncertainties

deviation 400 kN. Let Sk be the yield stress of member k. Members 1 and 2 have lognormally distributed yield stresses with mean 100 N/mm2 and standard deviation 20 N/mm2 . The other members have lognormally distributed yield stresses with mean 200 N/mm2 and standard deviation 40 N/mm2 . The yield stresses of members 1 and 2 are correlated with correlation coefficients 0.8. However, their correlation coefficients with the other yield stresses are 0.5. Similarly, the yield stresses of members 3–7 are correlated with correlation coefficients 0.8, but their correlation coefficients with the yield stresses of members 1 and 2 are 0.5. The load L is independent of the yield stresses. Let V = (S1 , S2 , . . . , S7 , L). The design vector x = (x1 , x2 , . . . , x7 ), where xk is the cross-section area (in 1000 mm2 ) of member k. The truss fails if any of the members exceed their yield stress. (We ignore the possibility of buckling.) This gives rise to seven limit state functions: Gk (x, V) = Sk xk − L/ζk ,

k = 1, 2, . . . , 7

(58)

where ζk is factor given by √ the geometry and loading of√the truss. From Figure 12.1, we determine that ζk = 1/(2 3) for k = 1, 2, and ζk = 1/ 3 for k = 3, 4, . . . , 7. Using a Nataf distribution (see (Ditlevsen and Madsen 1996), Section 7.2), we transform these limit-state functions into limit-state functions given in terms of a standard normal random vector U. Let gk (x, U) denote these transformed limit-state functions. Since the resulting safe domain is not bounded, we introduce an auxiliary limit state function g8 (x, U) = ρ − U, where ρ = 20 in this example. (This introduces negligible error.) Then, we redefine the failure probability of the structure as  p(x) = P

8 '

 {gk (x, U) ≤ 0}

(59)

k=1

which is in the form considered in this chapter. We impose the constraint that the failure probability should be no larger than 0.001350, i.e., p(x) ≤ q = 0.001350. We also impose the 14 deterministic constraints 0.5 ≤ xk ≤ 2, k = 1, 2, . . . , 7, that limit the allowable area of each member to be between 500 mm2 and 2000 mm2 . We initially seek a design of the truss that minimizes the cost of  the truss, i.e., we aim to solve P. Since all members are equally long, the cost c(x) = 7k=1 xk . We use the conceptual algorithm implemented with the feedback rule (23) for sample-adjustment, with parameters η = 0.002 and τ = 0.9999, and optimization algorithm (24) for Step 1, with parameters α = 0.5, β = 0.8, γ = 2, and δ = 1. The sample size is initially 375 and is increased by a factor of 4 every time it is prompted by the sample-adjustment rule. However, the sample size is not increased beyond 24000. We start the calculations with initial design x0 = (1.000, 1.000, . . . , 1.000) and stop when a feasible solution for P24000 is found. The resulting design is given in the first row of Table 12.3. With the motivation that a decision maker may want to be presented with a small set of good designs, from which he or she may select, we formulate an optimization model that generates substantially different designs. Specifically, suppose that we have a set of existing design alternatives xd , d ∈ D. Let cˆ be the smallest cost over all existing design alternatives, i.e., cˆ = mind∈D c(xd ). Then, the following optimization model provides a

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Table 12.3 Alternative designs for Example 3. The first row gives the optimal design, but the subsequent rows are at most 10% more costly. Design

Dispersion

x1

x2

x3

x4

x5

x6

x7

1.138 1.169 1.982 1.124 1.121 1.123 1.122 1.123 1.087

1.156 2.000 1.164 1.146 1.145 1.147 1.146 1.146 1.595

1.118 1.089 1.100 1.110 1.113 1.107 1.944 1.106 1.536

1.107 1.096 1.100 1.946 1.108 1.109 1.109 1.108 1.104

1.119 1.096 1.102 1.109 1.109 1.947 1.109 1.110 1.107

1.113 1.103 1.104 1.111 1.949 1.111 1.110 1.110 1.119

1.108 1.091 1.092 1.100 1.100 1.101 1.104 1.941 1.098

– 0.8451 0.8449 0.8393 0.8367 0.8286 0.8269 0.8331 0.6085

design that is no more costly than aˆc, with a > 1, and that is as “different’’ compared to the existing designs xd as possible: max{x0 |p(x) ≤ q, x ∈ X, c(x) ≤ aˆc, x − xd  ≥ x0 , d ∈ D} x0 ,x

(60)

Here, x0 is an auxiliary design variable that we seek to maximize. The last set of constraints in (60) ensures that the difference (measured in the Euclidean distance) between the new design x and the existing designs xd are all no smaller than x0 . Hence, (60) maximizes the smallest difference between a new design and the existing designs, while ensuring that the new design is feasible and no more costly than aˆc. We note that (60) is in the form P (after redefining the cost and constraint functions) and, hence, it can be solved by the conceptual algorithm described in Section 4. Using the same algorithm parameters as in the beginning of this example, we obtain the designs reported in Table 12.3. In this table, the first row reports the optimal design. The second row is obtained by solving (60) with a = 1.1 and D consisting only of the design in the first row. We observe that the design in the second row is substantially different than the one in the first row, even though it is no more than 10% more costly. The last column of Table 12.3 shows that the second design lies 0.8451 “away’’ from the first design measured in the Euclidean distance. The remaining rows in Table 12.3 are computed in a similar manner, but with D now consisting of all the designs in the rows above. We note that all the designs cost no more than 10% more than the minimum cost. It is seen from Table 12.3 that the minimum cost design (row 1) distributes the material evenly between the different members. However, good designs can also be achieved by selecting one of the members to have cross-section area close to 2 (rows 2–8). Moreover, good designs can be found by setting two members to approximately 1.5 (last row). Naturally, it becomes harder and harder to find a “different’’ design as the set of existing designs D grows, i.e., the last column of Table 12.3 tends to decrease for later designs. Hence, after some solutions of (60) with steadily increasing D, the designs we generate will not be substantially different

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compared to the ones already computed. This is an interactive process, which should be ended whenever a useful set of designs have been generated and further calculations will provide only limited insight.

7 Conclusions We have presented an approach for solving reliability-based optimal structural design problems using Monte Carlo sampling and nonlinear programming. The approach replaces failure probabilities in the problems by Monte Carlo estimates with increasing sample sizes, and solves the resulting approximate problems with increasing precision. We have also described rules for adjusting the sample sizes, which ensure theoretical convergence and computational efficiency. The numerical examples show empirically that the sample-adjustment rules can reduce computing times substantially compared with an implementation using a fixed sample size. The approach in this chapter is directed towards reliability-based structural optimization problems where the design variables are not restricted to be integers and the relevant limit-state functions are differentiable with continuous gradients. Furthermore, the approach requires many limit-state function evaluations, which (currently) prevent its application to problems involving, e.g., computationally intensive finite element analysis. We note, however, that the sample-adjustment rules described in this chapter dramatically reduce the number of limit-state function evaluations compared to an approach with a fixed sample size. Consequently, the results of this chapter open the possibility for solving, to high accuracy, many previously intractable reliability-based structural optimization problems.

References Akgul, F. & Frangopol, D.M. 2003. Probabilistic analysis of bridge networks based on system reliability and Monte Carlo simulation. In A. Der Kiureghian, S. Madanat & J.M. Pestana (eds), Applications of Statistics and Probability in Civil Engineering, Rotterdam, Netherlands, pp. 1633–1637. Millpress. American Association of State Highway and Transportation Officials (1992). Standard specifications for highway bridges. Washington, D.C.: American Association of State Highway and Transportation Officials. 15th edition. Beck, J.L., Chan, E., Irfanoglu, A. & Papadimitriou, C. 1999. Multi-criteria optimal structural design under uncertainty. Earthquake Engineering & Structural Dynamics 28(7):741–761. Bjerager, P. 1988. Probability integration by directional simulation. Journal of Engineering Mechanics 114(8):1288–1302. Brill Jr., E.D. 1979. The use of optimization models in public-sector planning. Management Science 25(5):413–422. Burke, J.V. 1991. Calmness and exact penalization. SIAM J. Control and Optimization 29(2):493–497. Clarke, F. 1983. Optimization and nonsmooth analysis. New York, New York: Wiley. Deak, I. 1980. Three digit accurate multiple normal probabilities. Numerische Mathematik 35:369–380. Ditlevsen, O. & Madsen, H.O. 1996. Structural reliability methods. New York, New York: Wiley.

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Ditlevsen, O., Oleson, R. & Mohr, G. 1987. Solution of a class of load combination problems by directional simulation. Structural Safety 4:95–109. Drezner, Z. & Erkut, E. 1995. Solving the continuous p-dispersion problem using nonlinear programming. The Journal of the Operational Research Society 46(4):516–520. Eldred, M.S., Giunta, A.A., Wojtkiewicz, S.F. & Trucano, T.G. 2002. Formulations for surrogate-based optimization under uncertainty. In Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Paper AIAA-2002-5585, Atlanta, Georgia. Enevoldsen, I. & Sørensen, J.D. 1994. Reliability-based optimization in structural engineering. Structural Safety 15(3):169–196. Gasser, M. & Schuëller, G.I. 1998. Some basic principles in reliability-based optimization (RBO) of structures and mechanical components. In Stochastic programming methods and technical applications, K. Marti & P. Kall (eds), Lecture Notes in Economics and Mathematical Systems 458, Springer-Verlag, Berlin, Germany. He, L. & Polak, E. 1990. Effective diagonalization strategies for the solution of a class of optimal design problems. IEEE Transactions on Automatic Control 35(3):258–267. Holicky, M. & Markova, J. 2003. Reliability analysis of impacts due to road vehicles. In A. Der Kiureghian, S. Madanat & J.M. Pestana (eds), Applications of Statistics and Probability in Civil Engineering, Rotterdam, Netherlands, pp. 1645–1650. Millpress. Igusa, T. & Wan, Z. 2003. Response surface methods for optimization under uncertainty. In Proceedings of the 9th International Conference on Application of Statistics and Probability, A. Der Kiureghian, S. Madanat & J. Pestana (eds), San Francisco, California. Itoh, Y. & Liu, C. 1999. Multiobjective optimization of bridge deck maintenance. In Case Studies in Optimal Design and Maintenance Planning if Civil Infrastructure Systems, D.M. Frangopol (ed.), ASCE, Reston, Virginia. Kuschel, N. & Rackwitz, R. 2000. Optimal design under time-variant reliability constraints. Structural Safety 22(2):113–127. Liu, P.-L. & Kuo, C.-Y. 2003. Safety evaluation of the upper structure of bridge based on concrete nondestructive tests. In A. Der Kiureghian, S. Madanat & J.M. Pestana (eds), Applications of Statistics and Probability in Civil Engineering, Rotterdam, Netherlands, pp. 1683–1688. Millpress. Madsen, H.O. & Friis Hansen, P. 1992. A comparison of some algorithms for reliability-based structural optimization and sensitivity analysis. In Reliability and Optimization of Structural Systems, Proceedings IFIP WG 7.5, R. Rackwitz & P. Thoft-Christensen (eds), SpringerVerlag, Berlin, Germany. Marti, K. 1996. Differentiation formulas for probability functions: the transformation method. Mathematical Programming 75:201–220. Marti, K. 2005. Stochastic Optimization Methods. Berlin: Springer. Mathworks, Inc. 2004. Matlab reference manual, Version 7.0. Natick, Massachusetts: Mathworks, Inc. Nakamura, H., Miyamoto, A. & Kawamura, K. 2000. Optimization of bridge maintenance strategies using GA and IA techniques. In Reliability and Optimization of Structural Systems, Proceedings IFIP WG 7.5, A.S. Nowak & M.M. Szerszen (eds), Ann Arbor, Michigan. Polak, E. 1997. Optimization. Algorithms and consistent approximations. New York, New York: Springer-Verlag. Polak, E. & Royset, J.O. 2007. Efficient sample sizes in stochastic nonlinear programming. J. Computational and Applied Mathematics. To appear. Royset, J.O., Der Kiureghian, A. & Polak, E. 2006. Optimal design with probabilistic objective and constraints. J. Engineering Mechanics 132(1):107–118. Royset, J.O. & Polak, E. 2004a. Implementable algorithm for stochastic programs using sample average approximations. J. Optimization. Theory and Application 122(1):157–184.

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Royset, J.O. & Polak, E. 2004b. Reliability-based optimal design using sample average approximations. J. Probabilistic Engineering Mechanics 19(4):331–343. Royset, J.O. & Polak, E. 2007. Extensions of stochastic optimization results from problems with simple to problems with complex failure probability functions. J. Optimization. Theory and Application 133(1):1–18. Rubinstein, R. & Shapiro, A. 1993. Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. New York, NY: Wiley. Ruszczynski, A. & Shapiro, A. 2003. Stochastic Programming. New York, New York: Elsevier. Torczon, V. & Trosset, M.W. 1998. Using approximations to accelerate engineering design optimization. In Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symp. on Multidisciplinary Analysis and Optimization, AIAA Paper 98-4800, St. Louis, Missouri. Tretiakov, G. 2002. Stochastic quasi-gradient algorithms for maximization of the probability function. A new formula for the gradient of the probability function. In Stochastic Optimization Techniques, New York, pp. 117–142. Springer. Uryasev, S. 1995. Derivatives of probability functions and some applications. Annals of Operations Research 56:287–311. White, D.J. 1996. A heuristic approach to a weighted maxmin dispersion problem. IMA Journal of Mathematics Applied in Business and Industry 7:219–231.

Chapter 13

Cost-benefit optimization for maintained structures Rüdiger Rackwitz & Andreas E. Joanni Technical University of Munich, Munich, Germany

ABSTRACT: In this chapter the theoretical and practical issues for setting up effective costbenefit optimization formulations for existing aging structures are presented. These formulations include deterioration and failure models as well as inspection and repair models. An elaborate optimization methodology, based on renewal theory that uses systematic reconstruction or repair schemes after suitable inspection is formulated, in which life-cycle cost perspectives are used is implemented for maintained concrete structures.

1 Introduction Many civil engineering structures are exposed not only to loads but also to the technical or natural environment. They are aging because of wear, corrosion, fatigue and other phenomena. At a certain age they need to be inspected and, possibly, repaired or replaced. Many aging phenomena are rather complex and all but fully understood in their physical and chemical context. For concrete structures the most important aging phenomena in temperate climates are corrosion due to carbonation and/or chloride attack, for steel structures it is rusting and fatigue. Moreover, the concepts for costbenefit optimization of such structures are not very well developed, although it is known that the cost for maintenance can be considerable and, in the long term, can even exceed the cost of the initial investment. It should be clear that only a rigorous lifecycle consideration can fully account for all cost involved, and that design rules and maintenance strongly interact. While the techniques for design optimization appear sufficiently developed, no clear concepts exist for optimizing maintenance. In this contribution suitable failure models for physically based deterioration phenomena are first reviewed. Their computation is essentially based on FORM/SORM (see, for example, (Rackwitz 2001)) which can be shown to be accurate enough for the purpose under discussion. Several schemes for computing first passage time distributions are discussed. Failure time models for series systems are also given. This is followed by some remarks about classical renewal theory, Bayesian updating, inspection and repair models. Then, the well-known renewal theory (Rosenblueth and Mendoza 1971; Rackwitz 2000) for cost-benefit optimization of structures is outlined. It is extended and generalized to optimal and integrated inspection and maintenance strategies. When setting up suitable maintenance strategies we follow closely the concepts developed in classical reliability theory as described, for example, in (Barlow and Proschan 1965; Barlow and Proschan 1975) which we find still very valid and which, to our knowledge,

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have not been applied to structures so far (see, however, (Van Noortwijk 2001)). In particular, we study minimal, age-dependent and block repairs and maintenance by inspection and repair. The models are generalized for maintenance optimization of series systems. Some special optimization techniques are briefly reviewed. An example illustrates aspects of the theory. Clearly, the considerations are no more valid if other than economic reasons exist to repair and/or retrofit an existing structure.

2 Preliminaries 2.1 F ai l ure m od e ls wit ho ut d e t er io r at i o n As a matter of fact, there are very few exact, time-variant failure models available which are amenable to practicable computation. In some cases consideration of (stationary or non-stationary) time-variant actions and time-variant structural state function is necessary. Let G(X(t), t) be the structural state function such that G(X(t), t) ≤ 0 denotes failure states and X(t) a random process. Examples of such processes are the Gaussian and related processes and the rectangular wave renewal processes. But X(t) can also include simple random variables. Then, the failure time distribution can be computed numerically by the outcrossing approach. A well-known upper bound is


t

F(t) ≤

ν(τ)dτ ≤ 1

(1)

0

with the outcrossing rate (more specifically, the downcrossing rate) 1 P({G(X(t), t) > 0} ∩ {G(X(t + ), t + ) ≤ 0})

→0

ν(τ) = lim

(2)

This upper bound is only tight for small probabilities. Frequently, an asymptotic result is used (Cramér and Leadbetter 1967) 
t F(t) ≈ 1 − exp − ν(τ)dτ (3) 0

with 
t f (t) ≈ ν(t) exp − ν(τ)dτ

(4)

0

Equation (3) implies a non-homogeneous Poisson process of failure events with intensity ν(t). For stationary failure processes Equation (3) reduces to a homogeneous Poisson process and simplifies somewhat. In general, computations are done by first transforming the original process and/or random variables into the so-called standard space of uncorrelated standard normal variates (Hohenbichler and Rackwitz 1981) which enables to use FORM/SORM (see, for example, (Rackwitz 2001)) provided that the dependence structure of the two events {G(X(t), t) > 0} and {G(X(t + ), t + ) ≤ 0} can be determined in terms of correlations coefficients. Some computational details are given in (Streicher and Rackwitz 2004). However, the relevant conditions must be fulfilled, i.e. the outcrossing events must become independent

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and rare asymptotically. For example, the independence property is lost if X(t) contains not only (mixing) random processes but also simple random variables. Therefore, in many cases this approach yields only crude approximations. An alternative approach will be discussed in the next subsection. 2.2

Failure mo dels for deterioration

Obviously, the outcrossing approach can also be applied if there is deterioration. It appears as if it performs better if the outcrossing rate is increasing with time. For aging structures a closed-form failure time (first passage time) distribution is hardly available except for some special, usually oversimplifying cases. The log-normal, inverse Gaussian or Weibull distribution function with a suitable deterioration mechanism for the mean (or other parameters) has been used. They, at most, can serve as approximations. Realistic failure models must be derived from physical multi-variable deterioration models (cumulative wear, corrosion, fatigue, etc.). For (monotonically and continuously) deteriorating structures a widely used failure model is as follows. Let G(X, t) = g(U, t) be the (differentiable) structural state function of a structural component with G(X, t) = g(U, t) ≤ 0 the failure domain. X is a vector of random variables and time t is a parameter. Transition from X to U denotes the usual probability transformation from the original into the standard space of variables (Hohenbichler and Rackwitz 1981). Within FORM/SORM the probability of the time to first failure is F(t) = P(T ≤ t) = P(g(U, t) ≤ 0) ≈
(−β(t))C(t)

(5)

for t ≥ 0 and the failure density is ∂F(t) ∂β(t) ∂C(t) ≈ −ϕ(β(t)) C(t) +
(−β(t)) ∂t ∂t ∂t ) * − ∂t∂ g(u∗ ,t) ∂C(t) = −ϕ(β(t)) C(t) +
(−β(t)) ∇u g(u∗ , t) ∂t

f (t) =

(6)

T is the time to first entrance into a failure state.
( · ) and ϕ( · ) denote the univariate standard normal distribution function and corresponding density, respectively. β(t) is the (geometrical) reliability index. C(t) is a correction factor evaluated according to SORM and/or importance sampling which can be neglected in many cases. In Equation (6) it frequently can be assumed that C(t) does not vary with t. Clearly, this model does not take account of the randomness in the deterioration process caused by a (large) number of small disturbances which, however, is small to negligible for cumulative deterioration phenomena, at least for larger t. A numerical computation scheme for first-passage time distributions under less restrictive conditions than the outcrossing approach can also be given. It is based on the following lower bound formula F(t) = P(T ≤ t) ≥ P

) n ' i=1

* P(G(X(ti ), ti ) ≤ 0)

(7)

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with t = tn and ti < t denoting a narrow not necessarily regular time spacing of the interval [0, t]. As demonstrated by examples in (Au and Beck 2001), the lower bound F(t) = P(T ≤ t) = 1 − P(G(X(θ), θ) > 0 for all θ in [0, t]) ) n * ) n * ' ' ≥ P {g(U(θi ), θi ) ≤ 0} ≈ P {α(θi )T U(θi ) + β(θi ) ≤ 0} i=0

)

= 1−P

*

n :

i=0

{Zi ≤ β(θi )} = 1 −
n+1 (β; R)

(8)

i=0

to the first-passage time distribution turns out to be surprisingly accurate for all values of F(t), if the time-spacing τ = θi − θi−1 is chosen sufficiently close and where θi = iτ and t = θn . Here again, a probability distribution transformation from the original space into the standard space is performed and the boundaries of each failure domain are linearized. The last line represents a first order approximation (Hohenbichler and Rackwitz 1983) where
n (·; ·) is the n-dimensional standard normal integral with β = {β(θi )} the vector of reliability indices of the various components in the union and the dependence structure of the events is determined in terms of correlation coefficients R = {ρij = α(θi )T α(θj )}. Suitable computation schemes for the multinormal integral even for high dimensions and arbitrary probability levels have been proposed, for example in (Hohenbichler and Rackwitz 1983; Gollwitzer and Rackwitz 1988; Pandey 1998; Ambartzumian et al. 1998; Genz 1992). It would appear that slight improvements can be achieved if the probabilities for the individual events are determined by SORM (or any other suitable improvement) and an equivalent value of βe (θ) is computed from βe (θ) = −
−1 (
(−β(θ))CSORM ). This computation scheme is approximate but quite general if the correlation structure of the state functions in the different points in time can be established. In (Au and Back 2001) a Monte Carlo method is used to compute Equation (7) which can be recommended if high accuracy requirements are imposed – at the expense of in part considerable numerical effort. The special case of equi-dependent (equi-correlated) components is worth mentioning. In this case we simply have (see, for example (Dunnett and Sobel 1955))
Fe (t) = 1 −

∞ −∞

ϕ(τ)

t  i=1




√  βi − ρτ dτ √ 1−ρ

(9)

For equi-reliable components (no variation of resistance quantities with time) this result simplifies further. The corresponding values of the density function needed when taking Laplace transforms as required later are most easily calculated by f (θi ) = (F(θi ) − F(θi−1 ))/τ or a higher order differentiation rule. For equi-reliable components Equation (9) has a decreasing risk function.

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The results obtained so far carry over to systems without any further conceptual difficulty. Only the numerical computations become more involved. Any system can be reduced to a minimal cut set system so that its failure probability is represented as ⎛ ⎞ mi s : ' Pf (t) = P(T ≤ t) = F(t) = P ⎝ {Tij ≤ t}⎠

(10)

i=1 j=1

Assume that the failure times of the parallel systems can be determined which, in general, can involve quite some numerical effort. The remaining series system then is computed as ) s * ) s * s ' :  Pf (t) = P(T ≤ t) = F(t) = P {Ti ≤ t} = 1 − P {Ti > t} ≤ P(Ti ≤ t) i=1 i=1 i=1 (11) where usually the failure and survival events are dependent. The upper bound in Equation (11) is less useful for larger, low reliability systems. Equation (8) can be combined with Equation (11), especially if the parallel systems can be represented sufficiently well by equivalent, linearly bounded failure domains of the components (Gollwitzer and Rackwitz 1983). Some specific results for the computation of series systems are given in (Streicher and Rackwitz 2004). The failure densities are obtained by differentiation. Note that, by definition, a series system fails if any of its components fails. In passing it is also noted that the formulation in Equation (11) also includes failure due to extreme disturbances. And it should be clear that the series system model must be applied if several hazards are present. Deterioration of structural resistance is frequently preceded by an initiation phase. In this phase failure is dominated by normal (extreme-value) failure. Structural resistance is virtually unaffected. Only in the succeeding phase resistances degrade. Examples are crack initiation and crack propagation or chloride penetration into concrete up to the reinforcement and subsequent reduction of the reinforcement cross-section by corrosion and, similarly, for initial carbonation and subsequent corrosion. In many cases the initiation phase is much longer than the actual degradation phase. Let Ti denote the random time of initiation, Te the random time to normal (first-passage extreme-value) failure and Td the random time from the end of the initiation phase to deterioration failure with degraded resistance. Then, F(t) = P(T ≤ t) = P[({Ti > t} ∩ {Te ≤ t}) ∪ ({Ti ≤ t} ∩ {Te < Ti }) ∪({Ti ≤ t} ∩ {Te > Ti } ∩ {Ti + Td ≤ t})]

(12)

= P[{Ti > t} ∩ {Te ≤ t}] + P[{Ti ≤ t} ∩ {Te < Ti }] + P[{Te > Ti } ∩ {Ti + Td ≤ t}] Note, extreme-value failure during the initiation phase and failure in the deterioration phase are mutually exclusive. Assume that Ti is independent of the other two variables.

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If the variables Te and Td can also be assumed independent, the following formula can be used
t F(t) = Fe (t)F i (t) + fi (τ)[Fe (τ) + (1 − Fe (τ))Fd (t − τ)]dτ (13) 0

2.3 T h e renew al mo d el A sufficiently general setting is to assume that the structure fails at a random time in the future. After failure or serious deterioration it is systematically renewed by reconstruction or retrofit/repair. Reconstruction, repair or retrofit reestablish all (stochastic) structural properties. The times between failure (renewal) events have identical distribution functions F(t), t ≥ 0 with probability densities f (t) and are independent. The sequence of failures and renewals then forms an ordinary renewal process. Renewal theory allows for a useful refinement which will be found to be important for the problem under discussion, namely the distribution of the time to the first event can have distribution function F1 (t) = F(t), t ≥ 0 (see (Cox 1962) for details). The process of renewals is then denoted by modified or delayed renewal process. The independence assumption between failure times needs to be verified carefully. In particular, one has to assume that loads and resistances in the system are independent for consecutive renewal periods and there is no change in the design rules after the first and all subsequent failures (renewals). Even if designs change failure time distributions must remain the same. But the model allows for a different design rule for the initial design which can be one of the reasons for F1 (t) = F(t). Throughout the chapter the point process of renewals is an orderly point process, that is multiple occurrences of renewals in a small time interval are excluded (Cox and Isham 1980). The renewal function for a modified renewal process which will be used extensively later on is (Cox 1962) E[N(t)] = M1 (t) =

∞ 

np(N(t) = n) =

n=1

= F1 (t) +

∞
 n=1

∞ 

n(Fn (t) − Fn+1 (t)) =

n=1 t

Fn (t)

n=1




t

Fn (t − u)dF(u) = F1 (t) +

0

∞ 

M1 (t − u)dF(u)

(14)

0

with N(t) the random number of renewals and Fn (t) = P(N(t) ≥ n) = P(Tn ≤ t) the distribution function of the time to the n-th renewal. The renewal intensity (or, if applied to failure processes, the unconditional failure rate) is obtained upon differentiation ∞

dM1 (t)  P(one renewal in [t, t + dt]) = = fn (t) m1 (t) = lim dt dt dt→0

(15)

n=1

For ordinary processes the index ‘1’ is omitted. The last expression in Equation (14) is called ‘renewal equation’. As pointed out in (Cox 1962), m(t) (or m1 (t)) has a limit m(t → ∞) = lim m(t) = t→∞

1 E[Tf ]

(16)

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for f (t) → 0 if t → ∞. In approaching the limit m(t) can be strictly increasing, strictly decreasing or oscillate in a damped manner around 1/E[Tf ]. Ordinary renewal processes then tend to be large around t = E[Tf ], 2E[Tf ], . . . and small around t = 0, 32 E[Tf ], 53 E[Tf ], . . .. For a Poisson process with parameter λ it is constant, i.e. m(t) = λ. If there are oscillations they die out more rapidly for larger dispersions of the failure time distribution. In many examples oscillations have been found when the risk function is increasing. Also, in many cases the failure rate is increasing for small t. Only for some special models, especially those with very large coefficient of variation of failures times, m(t) is decreasing. The transient behavior of m(t) will later be of interest. Unfortunately, Equation (14) has closed-form solutions for only very few special mathematical failure models (see (Streicher et al. 2006) for a list of relevant references) and otherwise can be computed directly only with extreme numerical effort. In general, Equation (14) or Equation (15) have to be determined numerically. A particularly suitable numerical method is proposed in (Ayhan et al. 1999). It makes use of the upper and lower sum in Riemann-Stieltjes integration for the discrete version of Equation (14). ( t Because M(t) is non-decreasing, we have the following bounds for M(t) = F(t) + 0 M(t − s)dF(s)

MLB (kτ) = F(kτ) +

k 

MLB ((k − i)τ) F(iτ)

i=1

≤ M(kτ) ≤ F(kτ) +

k 

MUB ((k − i + 1)τ) F(iτ) = MUB (kτ)

(17)

i=1

for equal partitions of length τ in [0, t] with F(iτ) = F(iτ) − F((i − 1)τ) and nτ = t. The resulting system of linear equations is solved easily. If the first failure time distribution is different from the others one obtains by one additional convolution


t

M1 (t) = F1 (t) +

F1 (t − s)dM(s)

(18)

0

which, in turn, is bounded by

M1,LB = F1 (t) +

k  i=1

inf

(i−1)τ≤x≤iτ

≤ M1,UB ≤ F1 (t) +

F1 (t − x)(MLB (iτ) − MLB ((i − 1)τ)) ≤ M1 (kτ)

k 

sup

i=1 (i−1)τ≤x≤iτ

F1 (t − x)(MUB (iτ) − MUB ((i − 1)τ)) (19)

m1 (t) is obtained by numerical differentiation. The computation methods in Equations (17) and (19) are useful whenever interest lies in the (unconditional) failure rate or risk acceptance questions. Other approximation methods have also been proposed.

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For aging components with increasing risk function the following bounds on the renewal function are given in (Barlow and Proschan 1965, p. 54) t tF(t) t t − 1 ≤ M(t) ≤ ( t ≤ −1 ≤ (t E[Tf ] E[T f] 0 (1 − F(τ))dτ 0 (1 − F(τ))dτ

(20)

The sharper upper bound in Equation (20) turns out to be remarkably close to the exact result for small t. Under suitable conditions one also has m(t) =

d d tF(t) M(t) ≤ (t dt dt 0 (1 − F(τ))dτ

(21)

Again, the upper bound for Equation (21) is found to be very close to the exact result up to approximately E[T]. It approaches the limit 1/E[T] for large t. The lower bound obtainable from Equation (20) by differentiation is generally less useful. Equation (21) can be used with advantage in Sections 4.4 and 4.5. 2.4

U pd a ti ng t he pr o b ab ilis t ic mo d el

There are many types of updating of a probabilistic model depending on the type of information collected during the experimental and numerical investigations. In general, one can distinguish between variable updating and event updating. In a Bayesian context information is collected about a variable by taking (independent) samples and testing them. This leads to an improved estimate of the parameters of the distribution of a variable. Let xn be values of a sample of size n and θ a parameter (vector), then an improved posterior distribution is






f (θ | xn ) = (

L(xn | θ)f (θ)
L(x n | θ)f (θ)dθ θ

(22)


where L(xn | θ) is the likelihood function and f (θ) the prior density. The Bayesian or predictive density function is


f (x | xn ) = f (x | θ)f (θ | xn )dθ (23) θ

For many important distributions analytical results are available (Aitchison and Dunsmore 1975). Updating by events is generally more difficult. We  show this for the model from Equation (5) and previous informative events B = i=1 Bi . For example, such events could be the knowledge about the maximum load in the past, some measured damage indicator or just the knowledge that the structure has survived up to the present time. Then, we have two types of observations, namely equalities and inequalities which  require different treatment. For B = i=1 bi (X, t0 ) ≤ 0 it is F(t | B) =

 P({g(X, t) ≤ 0} ∩ i=1 {bi (X, t0 ) ≤ 0})  P( i=1 bi (X, t0 ) ≤ 0)

(24)

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It is assumed that the observation events B can always be written in the form given. In most cases the observation and decision point is t0 = 0. Within FORM one can write for one observation event F(t | B) =


2 (−βg (t), −βb (t0 ), ρ)
(−βb (t0 ))

(25)

where
2 (x, y, ρ) is the two-dimensional normal integral and ρ = αTg αb with αg , αb the two normalized gradients of the limit state functions. This scheme applies analogously if more than one event has to be considered. For B = {b(X, t0 ) = 0} we have F(t | B) =

∂ ∂βb

( βb

−∞

P(Zg ≤ βg | Zb (t0 ) = z)ϕ(z)dz ϕ(βb (t0 ))

)

−βg (t) + ρ(t, t0 )βb (t0 ) =
 1 − ρ(t, t0 )2

*

(26)

3 Cost-benefit optimization 3.1

G eneral

It is generally accepted that the ultimate target to be achieved in structural design including proper maintenance is to maximize the net benefit derived from the structure over its lifetime, subject to constraints related to safety and serviceability. For technical facilities the following objective has been proposed by (Rosenblueth and Mendoza 1971) based on earlier proposals in economics for cost benefit analysis: Z(p) = B(p) − C(p) − D(p)

(27)

A facility is financially optimal if Equation (27) is maximized. It is assumed that all quantities in Equation (27) can be measured in monetary units. p is the vector of all safety relevant parameters. B(p) is the (expected) benefit derived from the existence of the facility, C(p) is the cost of design and construction and D(p) is the (expected) cost in case of failure. Later we will also include all expenses for maintenance in D(p). Statistical decision theory dictates that expected values are to be taken. In the following it is assumed that C(p) and D(p) are differentiable in each component of p. The facility has to be optimized during design and construction at the decision point which is taken as t = 0. Now it is a well-established principle of cost-benefit analysis that future costs and benefits must be discounted, using a compound interest formula. A continuous discounting function is assumed for analytical convenience which is accurate enough for all practical purposes. δ(t) = exp [−γt]

(28)

γ is a time-independent, time-averaged interest rate. In most cost-benefit analyses a tax and inflation-free discount rate should be taken. If a discrete discount rate γ
is given, one converts with γ = ln (1 + γ
). The principles of choosing appropriate discount rates are thoroughly discussed in (Rackwitz et al. 2005).

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Cost and benefits may differ for the different parties involved having different economic objectives, e.g. the owner, the builder, the user and society. Also, the discount rate may vary among the different parties in their cost-benefit analysis. A facility makes sense only if Z(p) is positive within certain parameter ranges for all parties involved. 3.2 D eri v ati o ns A complete cost-benefit analysis must include not only the direct and indirect cost for possible failure and for maintenance of the structure to be built, but also the cost for all future realizations if the concepts of sustainability are applied (Rackwitz et al. 2005). But this is just the situation for the application of renewal theory. It is assumed that structures will be systematically reconstructed after failure and/or maintained. This rebuilding strategy is in agreement with the principles of life cycle engineering and also fulfills the demand for sustainability (Rackwitz et al. 2005). Clearly, it rests on the assumption that future preferences are the same as the present preferences. For regular renewal processes some objective functions based on the renewal model are already derived in (Rosenblueth and Mendoza 1971; Rackwitz 2000; Streicher and Rackwitz 2004) and elsewhere. For existing structures the time to first failure is generally different from the other failure times due to additional experimental and numerical investigations and subsequent updating of the structural state and/or due to repair or retrofit of the existing structure. But there can also be other reasons for assuming f1 (t, p) = f (t, p). Therefore, we derive our model for cost-benefit optimization in full generality. The objective function is given by Equation (27). The expected damage cost D(p) are derived as follows. The discrete cost associated with failure including the reconstruction or repair cost are denoted as CV,1 at the first renewal and CV = CV,n at subsequent renewals. Let θi = ti − ti−1 be the times between renewals with density f (t, p) whereas θ1 = t1 has density f1 (t, p). The time to the n-th renewal is Tn = ni=1 θi . Systematic reconstruction is assumed. The discounted expected damage cost are then ∞   n   D(p) = E CV,n exp −γ θk n=1

k=1



= E[CV,1 exp[−γ θ1 ]] + E  = E[CV,1 exp[−γ θ1 ]] + E

∞  n=2 ∞ 

CV,n

n 

∞ 

exp[−γ θk ]

k=1

CV,n exp[−γ θ1 ]

n=2

= E[CV,1 exp[−γ θ1 ]] +



n−1 

 exp[−γ θk ] exp[−γ θn ]

k=2

E[ exp[−γ θ1 ]]E[ exp(−γ θ)]n−2 E[CV,n exp[−γ θn ]]

n=2

= E[CV,1 exp[−γ θ1 ]] + E[ exp[−γ θ1 ]] =

E[CV exp[−γ θ]] 1 − E[ exp[−γ θ]]

CV,1 E[exp[−γ θ1 ]] 1 − E[exp[−γ θ]] −CV,1 E[exp[−γ θ1 ]]E[exp[−γ θ]] + E[exp[−γ θ1 ]]CV E[exp[−γ θ]] + (29) 1 − E[exp[−γ θ]]

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 a n−k where we have made use of the relation s = ∞ = 1−q for k < ∞. n=k aq (∞ (∞ ∗ E[exp[−γ θ1 ]] = 0 exp[−γt]f1 (t, p)dt = f1 (t, p) and E[exp[−γ θ]] = 0 exp[−γt] f (t, p)dt = f ∗ (t, p) is also denoted as Laplace transform of f1 (t, p) and f (t, p). If f (t, p) is a probability density it is f ∗ (0, p) = 1 and 0 < f ∗ (γ, p) ≤ 1 for all γ ≥ 0. Equation (27) can be rewritten in case of systematic reconstruction after failure with CV,1 = (C1 (p) + L) as well as CV = (C(p) + L) as Z(p) = B(p) − C(p) − +

(C1 (p) + L)f1∗ (γ, p) 1 − f ∗ (γ, p)

(C1 (p) + L)f ∗ (γ, p)f1∗ (γ, p) − (C(p) + L)f1∗ (γ, p)f ∗ (γ, p) 1 − f ∗ (γ, p)

(30)

for the modified renewal process. L is the monetary loss in case of failure including direct failure cost, loss of business and, possibly, the cost to reduce the risk to human life and health (or, better, the compensation cost). If only C1 (p) = C(p) the two terms in the numerator of the forth term cancel. This is usually the case for existing and systematically renewed structures and, therefore Z(p) = B(p) − Cini (p) − (C(p) + L)

f1∗ (γ, p) 1 − f ∗ (γ, p)

(31)

It has to be mentioned that the design parameters p can be different after the first renewal compared to the initial design. Also, the cost for the initial design Cini (p) can be different from the reconstruction cost C(p). The term m∗1 (γ, p) =

f1∗ (γ, p) 1 − f ∗ (γ, p)

(32)

is also denoted by the Laplace transform of the renewal intensity. If f1 (t, p) = f (t, p), f1∗ (t, p) in Equation (31) must be replaced by f ∗ (t, p). The benefit B(p) is also discounted down to the decision point. For a benefit rate b(t) unaffected by possible renewals and negligibly short times of reconstruction (retrofitting) one finds
∞ B= b(t) exp[−γt]dt (33) 0

Clearly, the integral must converge imposing some restriction on the form of b(t). If the benefit rate b = b(t) is constant one can integrate to obtain


∞

B= 0

b exp[−γt]dt =

b γ

(34)

The upper integration limit is extended to infinity because the full sequence of life cycle benefits is considered. A model which represents realistically the observation that with increasing age of a component its suitability for use diminishes according to b(t) has been established in (Hasofer and Rackwitz 2000). Decreasing benefit was associated with obsolescence in

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Structural design optimization considering uncertainties

this reference. But b(t) can have any form. At each renewal the benefit rate starts again at b(0) for systematic reconstruction. The total benefit is already given in (Streicher 2004) and is repeated here in full generality.  B(p) = E

∞ 

)

 exp −γ

i=1







θi

θk

* exp[−γ τ]b(τ)dτ

0

k=1

θ1

=E

i−1 

exp[−γ τ]b(τ)dτ

0

 + E exp[−γ θ1 ]

∞  i−1 


exp[−γ θk ]




θ1

 exp[−γ τ]b(τ)dτ

0

i=2 k=2

=E

θi

exp[−γ τ]b(τ)dτ + E[exp[−γ θ1 ]]

0

∞ 

E[exp[−γ θ]]i−2

i=2




θ

×E

exp[−γ τ]b(τ)dτ

0




θ1

= 0

(θ E[exp[−γ θ1 ]]E[ 0 exp[−γ τ]b(τ)dτ] exp[−γ τ]b(τ)dτ + 1 − E[ exp[−γ θ]]

(35)

Equation (35) can be simplified for the case of systematic reconstruction after failure to
∞
t B(p) = exp[−γτ]b(τ)dτ f1 (t, p)dt 0

0

f1∗ (γ, p) 1 − f ∗ (γ, p)

+


∞

=


∞


t

exp[−γτ]b(τ)dτ f (t, p)dt 0

0

BD (t)f1 (t, p)dt +

0

f1∗ (γ, p) 1 − f ∗ (γ, p)




∞

BD (t)f (t, p)dt

(36)

0

with


t

BD (t) =

exp[−γτ]b(τ)dτ

(37)

0

For f1 (t, p) = f (t, p) Equation (36) simplifies to: 1 B(p) = 1 − f ∗ (γ, p)




∞

BD (t)f (t, p)dt

(38)

0

For completeness, the objective function is also given for the case where the component is given up after failure or a finite service time ts
Z(p) = 0

ts




ts

BD (t)f1 (t, p)dt − C(p) − L

exp[−γt]f1 (t, p)dt 0

(39)

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347

Because the failure densities, in general, are known only numerically and pointwise, the corresponding Laplace transforms have to be taken numerically. Suitable techniques are presented in (Streicher and Rackwitz 2004) and Section 5. The formulae are easily extended for systems with several components and/or multiple failure modes in series as demonstrated by Equation (11) (see also (Streicher and Rackwitz 2004)). In particular, one component of the system can model replacement due to obsolescence. Non-constant discounting is discussed in (Rackwitz et al. 2005). Optimization of Equation (31) with respect to the design parameter p can be performed by one of the available algorithms (see Section 5). Application to existing, aging but maintained structures requires a few more remarks. It is assumed that the structure is already in use for some time. At a special point in time it will be decided to inspect and possibly repair or retrofit the structure. The cost which occur at this decision point are CR (p). Clearly, all cost incurred before that point are irrelevant if the decision is to keep the structure rather than demolishing and rebuilding it. The value of CR (p) can be zero if the structure is left as is but the probabilistic model for the time to first failure f1 (t, p) possibly is updated. Then, renewal of the structure is a question as to when the possibly updated failure rate is no more acceptable. The modified density f1 (t, p) of the time to first failure has to be determined depending on the repair/retrofitting actions and the information collected about the actual state of the structure. CR (p) generally differs from C(p), the reconstruction cost after failure, or even exceeds it if retrofitting is more expensive than reconstruction. A maintenance plan for the existing structure has to be designed. After the first renewal due to future failure the regular failure time density f (t, p) is valid.

3.3 Applicatio n to s tationary Pois s onia n di s turbanc e s Unfortunately, analytic Laplace transforms are available only for a few analytic failure models, for instance the exponential, uniform, gamma, normal and inverse normal distribution. The important exponential distribution with parameter λ corresponding to a Poisson process has f1∗ (γ) = f ∗ (γ) = λ/γ + λ and, therefore, m∗ (γ) = λ/γ. A very useful generalization is when a modified renewal process models disturbance (loading) events (Hasofer 1974; Rosenblueth 1976). Such disturbances generally are extreme events like shocks, explosions, earthquakes, storms or floods. The distribution functions between events are G1 (t) and G(t), respectively. If such an event occurs the failure probability is Pf (p). By definition, the occurrence of disturbance events and the failure events are independent. The density function of the time to the first failure event then is f1 (t, p) =

∞ 

gn (t)Pf (p)Rf (p)n−1

(40)

n=1

i.e. the first failure event can occur after the first, second, third, . . . disturbance event and where Rf (p) = 1 − Pf (p). The density of the n-th event can be obtained by recursive convolution so that in terms of Laplace transforms ∗ gn∗ (γ, p) = gn−1 (γ, p)g ∗ (γ, p) = g1∗ (γ, p)[g ∗ (γ, p)]n−1

(41)

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Structural design optimization considering uncertainties

Application to the renewal intensity yields m∗1 (γ, p) =

∞ 

∗ g1∗ (γ)gn−1 (γ)Pf (p)Rf (p)n−1

n=1

=

∞ 

g1∗ (γ)[g ∗ (γ)]n−1 Pf (p)Rf (p)n−1 =

n=1

Pf (p)g1∗ (γ) 1 − Rf (p)g ∗ (γ)

(42)

For the regular renewal process m∗1 (γ, p) has to be replaced by m∗ (γ, p). Let reconstruction and damage cost be C(p) and L, respectively. Also, as a special case, let the times between disturbances be the (exponential failure time distributions with (failure) rate ∞ λPf (p). Therefore, E(e−γt ) = −∞ e−γt (λPf (p))e−λPf (p)t dt = λPf (p)/γ + λPf (p). Then, if only failures due to such disturbances are considered it is (Rackwitz 2000) Z(p) = B − Cini (p) − (C(p) + L)

λPf (p) γ

(43)

  For a series system it is Pf (p) = P( sk=1 P(Fk (p))) = 1 − P( sj=1 F j (p)) in Equation (11) where Fk (p) is the failure event in the k-th mode and F k (p) its complement. Then, the following generalization is possible ⎡ ⎛ ⎞⎤ s : λ Z(p) = B − Cini (p) − (C(p) + L) ⎣1 − P ⎝ F j (p)⎠⎦ (44) γ j=1

The benefit B and the initial cost Cini (p) as well as the damage cost are related to the whole system. If there are n different, independent hazards each with rate λi one derives ⎡ ⎛ ⎞⎤ n s : λi Z(p) = B − Cini (p) − i=1 (Ci (p) + Li ) ⎣1 − P ⎝ F ij (p)⎠⎦ (45) γ j=1

These generalizations also apply analogously for the more complicated cases discussed below.

4 Preventive maintenance 4.1 Ma i n tenanc e s t r at eg ie s Repair after failure is but the simplest maintenance strategy. For aging components, i.e. components with increasing risk function (conditional failure rate) r(t) = f (t)/1 − F(t), i.e. r
(t) > 0, the risk of failure with potentially large consequences increases with age and alternative maintenance strategies have been proposed in order to reduce expected failure consequences. The most important alternative is called preventive maintenance at random or fixed times. Preventive maintenance actions can be replacements or

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

349

(perfect) repairs. Preventive repairs occur only if corrective renewals have not occurred before due to failure or obsolescence. Note that preventive maintenance is usually suboptimal for non-aging components, i.e. with constant or decreasing risk function. A first strategy repairs a system (component) at age a or after failure, whichever comes first. In (Barlow and Proschan 1965) this strategy is denoted by age replacement. It requires knowledge of the age a of a component. (Barlow and Proschan 1965) also investigate so-called block repairs. In this maintenance strategy the components in a system are repaired either after failure or all at once at a given time d irrespective of their actual age. It is clear for increasing risk functions and, in fact, is shown in (Barlow and Proschan 1965) that the total number of repairs is smaller for age repairs than for block repairs. However, the number of failures (with large consequences) is larger in the first strategy and so, possibly, the total cost. Block repairs also may be organizationally easier. Sometimes they are necessary, i.e. whenever a single repair of a component prevents the whole system from functioning. While knowledge about the actual deterioration state of a component is irrelevant for the block repair strategy, this may be vital for the age repair strategy. An improvement is when repairs are only performed if inspections indicate that they are necessary. Otherwise further inspections and possible repairs are postponed to a later time. A strategy where repairs are preceded by inspections is also denoted as condition-based strategy. In practice, mixtures of these maintenance strategies will also be found. 4.2

Inspections

Inspections should determine the actual state of a component in order to decide on repair or leave it as is. But inspections can rarely be perfect. A decision about repair can only be reached with certain probability depending on the inspection method used. The repair probability depends on the magnitude of one or more suitable damage indicators (chloride penetration depth, crack length, abrasion depth, etc.) measured during inspection. For cumulative damage phenomena the damage indicators increase with time and so does the repair probability PR (t). The parameter t is the time elapsed since the beginning of the deterioration process. For example, the repair probability may be presented as PR (t) = P(S(t, X) > sc ) = P(sc − S(t, X) ≤ 0)

(46)

with S(t, X) a suitable, monotonically increasing damage indicator, X a random vector taking into account of all uncertainties during inspection and sc a given threshold level. If this is exceeded a decision for repair is taken. The vector X usually also includes a random variable modeling the measurement error. Frequently, the damage indicator function S(t, X) reflects the damage progression and has a similar form as the failure function. It involves, at least in part, the same random variables. In this case failure and no repair/repair events become dependent events. It is, of course, possible to consider multidimensional damage indicators and derive repair decisions from an arbitrary combination thereof. A discussion of the details of the efficiency of various inspection methods and the corresponding repair probabilities is beyond the scope of this chapter. They depend on the particular deterioration phenomenon under consideration.

350

Structural design optimization considering uncertainties

4.3 Repai r m od el After failure of a system or component it is repaired unless it is given up after failure or it is repaired systematically in the age-dependent maintenance strategy or it is repaired after an indicative inspection in the condition-based maintenance strategy. The name repair is used synonymously for renewal, replacement or reconstruction. Repairs, if undertaken, restore the properties of a component to its original (stochastic) state, i.e. repairs are equivalent to renewals (AGAN = As Good As New) so that the life time distribution of the repaired component is again F(t). The repair times can either be assumed negligibly short or have finite length. The model is a somewhat idealized model. It rests on a number of assumptions the most important of which is probably that repairs fully restore the (stochastic) properties of the component. Imperfect repairs cannot be handled because the renewal argument repeatedly used in the following breaks down. In the literature several models for imperfect repairs are discussed which only partially reflect the situations met in the structures area. An important case is when minimal repairs not essentially changing the initial lifetime are done right after an inspection. If one generalizes this model to a model where a renewal (perfect repair) occurs with probability π but minimal repair with probability 1−π, one has essentially the model proposed in (Brown and Proschan 1983). This model, in fact, resembles the one studied herein with π = PR (t). Negligibly short times of inspection and repair are most often only a more or less good approximation. Consideration of finite, random renewal times in the age-repair strategy appears possible but is complicated because inspections and probably also failures cannot occur during repairs. No benefit can be earned during repair times. Another important case is when repairs are delayed, for example due to budget restrictions. It appears possible to handle this case by adding a random delay time to the random repair time. During a delay time the component can still degrade or fail while this is unlikely to happen during repair. Finite renewal times are not considered in this chapter. Some more but still first results are given in (Joanni and Rackwitz 2006). It turns out that for realistic repair times their influence is very small. Inspection/repair at strictly regular time intervals as assumed below is also not very realistic. However, as will be shown in the examples, the objective function is rather flat in the vicinity of the optimal value so that small variations will not noticeably change the results. Repair operations necessarily lead to discontinuities (drops) in the risk function, and similarly in the renewal intensity. They can substantially reduce the number of failures and, thus, corrective renewals. In an effective maintenance scheme the majority of renewals will, in fact, be preventive renewals. 4.4 Ag e-d epend ent r e pair s It is convenient to start with the general case of replacements (repairs, renewals) at random times Tr with distribution Fr (t) or after failure at random times Tf with distribution Ff (t, p). The renewal time then is the minimum of these times with distribution function F(t, p) = 1 − (1 − Ff (t, p))(1 − Fr (t)) = 1 − F f (t, p)F r (t)

(47)

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

351

for independent times Tf and Tr with density f (t, p) = ff (t, p)F r (t) + fr (t)F f (t, p)

(48)

and where the notation F(x) = 1 − F(x) is used. Application of Equation (29) then gives for the damage term of an ordinary renewal process D(p) =

(C(p) + L)fF∗ (γ, p) + R(p)fF∗ (γ, p) r

1−

(fF∗ (γ, p) f

f

(49)

+ fF∗ (γ, p)) r

and, similarly, for the benefit term with the model in Equation (35) (∞ B(p) =

0

(∞ BD (t)ff (t, p)F r (t)dt + 0 BD (t)fr (t)F f (t, p)dt 1 − (fF∗ (γ, p) + fF∗ (γ, p)) f

where fF∗ (γ, p) = r

(∞ 0

(50)

r

exp[−γt]ff (t, p)F r (t)dt and fF∗ (γ, p) =

(∞

f

0

exp[−γt]fr (t)F f (t, p)dt

are the modified complete Laplace transforms of ff (t, p)F r (t) and fr (t)F f (t, p), respectively. R(p) is the cost of repair and BD (t) is as in Equation (37). The case of random maintenance actions has hardly any practical application except if there is continuous monitoring of the system state. Then, the time until intervention by repair is random and can be defined as the first passage time of a given threshold by the continuous observation process. Alternatively, assume maintenance actions at (almost) fixed intervals a, 2a, 3a, . . . so that fr (t) = δe (a) and Fr (t) = He (a) (δe (x) = Dirac’s delta function, He (a) = Heavyside’s unit step function. Equation (49) then specializes to DM (p,a) =

(C(p) + L)f ∗∗ (γ, p,a) + R(p) exp[−γa]F(p,a) 1 − (f ∗∗ (γ, p,a) + exp[−γa]F(p,a))

(51)

and similarly Equation (50) to (a BM (p,a) =

0

BD (t)f (t, p)dt + BD (a)F(p,a)

(52) 1 − (f ∗∗ (γ, p,a) + exp[−γa]F(p,a)) (a with f ∗∗ (γ, p,a) = 0 exp[−γt]f (t, p)dt the incomplete Laplace transform of f (t, p) and F(p,a) the probability of survival up to a. The quantity BD (t) is given in Equation (37). Note that the Laplace transform of a deterministic repair time fr (t) = δe (a) is f ∗ (γ) = exp[−γa]. The repair cost R(p) should be substantially smaller than C(p) + L so that it is worth making preventive repairs and, thus, avoiding the large failure and reconstruction cost in case of failure. Equation (51) goes back to some early work in (Cox 1962; Barlow and Proschan 1965; Fox 1966). In (Van Noortwijk 2001) parallel results are developed for discrete failure models and discrete discounting schemes. Next, assume that the structure is already in use. At a special decision point in time inspection, retrofit or repair at cost CR (p) takes place. Depending on the state of the structure and the action which is done, an updated failure time density f1 (t, p) for

352

Structural design optimization considering uncertainties

the time to the first renewal is calculated. Therefore, a new cost benefit optimization is necessary in order to find optimal replacement intervals and design variables. The first replacement interval a1 with f1 (t, p) is different from the subsequent intervals a with ordinary failure time density f (t, p). It will further be assumed that for the first renewal the optimized parameter p, which is also valid for all subsequent renewals, is calculated without having regard to the special parameters realized in the existing structure. If the structure undergoes a complete renewal at the decision point it is even possible to introduce the design variables p already in that structure. Then, the existing design variables p have to be augmented by the additional variables. The expected damage cost are then determined according to Equation (49) as DMa1−a (p, a1 , a = (C(p) + L)f1∗∗ (γ, p, a1 ) + R(p) exp[−γa1 ]F 1 (p, a1 ) +

f1∗∗ (γ, p, a1 ) + exp[−γa1 ]F 1 (p, a1 ) 1 − (f ∗∗ (γ, p, a) + exp [−γa]F(p, a))

× ((C(p) + L)f ∗∗ (γ, p, a)

+ R(p) exp[−γa]F(p, a))

(53)

For constant benefit rate b(t) = b the benefit is as in Equation (34). The expected benefit for a non-constant rate b(t) as in Equation (35) is (Streicher 2004)
a1 BMa1−a (p, a1 , a) = BD (t)f1 (t, p)dt + BD (a1 )F 1 (p, a1 ) 0

+

f1∗∗ (γ, p, a1 ) + exp[−γa1 ]F 1 (p, a1 )

1 − (f ∗∗ (γ, p, a) + exp[−γa]F(p, a))


a  × BD (t)f (t, p)dt + BD (a)F(p, a)

(54)

0

with BD (t) from Equation (37). The cost for continuous monitoring and/or maintenance could alternatively also be taken into account in the benefit term by replacing b(t) with b(t) − c(t). The objective function then is ZMa1−a (p, a1 , a) = BMa1−a (p, a1 , a) − CR (p) − DMa1−a (p, a1 , a)

(55)

Repair is interpreted as preventive renewal (replacement of an aging component after a finite time of use a). Renewal after failure is called corrective renewal. Equation (55) can be subject to optimization not only with respect to the design parameter p but also with respect to the inspection/repair intervals a1 and a, respectively. Optimal inspection/ repair intervals do not always exist, as pointed out already in (Fox 1966). They exist for failure models with increasing risk function (Fox 1966). If they do not exist, then it is preferable to wait with renewal until failure unless the failure rate exceeds a given value. When optimizing Equation (55) it is, of course, important that the owner, builder or other party does not only enjoy the benefits but also carries the cost of construction, the cost of failures and the cost for preventive maintenance. Only then, a joint optimization of design and maintenance makes sense. If one is only interested in optimal maintenance it is still possible to optimize the cost for preventive and corrective repairs with respect to the repair intervals keeping the design parameter p constant.

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

4.5

353

Bloc k repairs

The damage cost for block repairs are composed of the (discounted) cost of planned systematic renewals at time d (or d1 for the first interval, where the time to the first failure has the updated failure time density f1 (t, p)) plus the (discounted) cost of failure(s) before d (or d1 ). Therefore, DB (p, d1 , d) = R(p)e−γd1 + (C(p) + L)[f1∗∗ (γ, p, d1 ) + m∗∗ 1 (γ, p, d1 )] +

e−γd1 [R(p)e−γd + (C(p) + L)f ∗∗ (γ, p, d)[1 + m∗∗ (γ, p, d)]] 1 − e−γd (56)

(d ( d1 −γt ∗∗ (γ, p, d1 ) = 0 (1) e−γt f(1) (t, p)dt, m∗∗ m(1) (t, p)dt with where f(1) (1) (γ, p, d1 ) = 0 e m1 (t, p) for the updated failure rate to the first renewal until d1 and m(t, p) in subsequent intervals until d as the renewal (Cox 1962). m1 (t, p) intensities in Equation (15) ∞ f (t, p) and m(t, p) = and m(t, p) are given by m1 (t, p) = ∞ n=1 1,n n=1 fn (t, p), respectively (see Equation (15)). Here and in the following the notation x(1) means either x or x1 whatever is relevant. Remember, integration of m(1) (t) is simply the mean number of renewals in [0, d(1) ] but here discounting is introduced additionally. The computation of m(1) (t) is the numerically expensive part (see Equation (17), Equation (19) or Equation (21)). Note that all components are repaired at time d(1) with certainty and cost R(p) but some components are already renewed earlier because they failed. For f1 (t, p) = f (t, p) and d1 = d Equation (56) simplifies to DB (p, d) =

R(p)e−γd + (C(p) + L)f ∗∗ (γ, p, d)[1 + m∗∗ (γ, p, d)] 1 − e−γd

(57)

For benefit rates unaffected by renewals one simply has the results in Equation (34) or (33). The benefit term for the case in Equation (35) is for finite integration intervals [0, d].
BB (p, d1 , d) =

d1

BD (t)f1 (t, p)dt +

0

e−γd1 + 1 − e−γd




d

m∗∗ 1 (γ, p, d1 )




d1

BD (t)f (t, p)dt 0 ∗∗




BD (t)f (t, p)dt + m (γ, p, d)

0



d

BD (t)f (t, p)dt 0

(58) with BD (t) in Equation (35). For f1 (t, p) = f (t, p) and d1 = d Equation (58) simplifies to (d BB (p, d) =

0

BD (t)f (t, p)dt + m∗∗ (γ, p, d) 1 − e−γd

(d 0

BD (t)f (t, p)dt

(59)

The length d (and/or d1 ) of a replacement interval can also be subject to optimization with respect to benefits and cost. In general, there is little difference between agedependent and block repairs unless the failure cost are very large.

354

Structural design optimization considering uncertainties

4.6 Inspec ti o n and r epair In the structures and many other areas any expensive maintenance operation is preceded by inspections involving cost I if damage progression and/or changes in system performance are observable. We understand that the inspections are essential inspections leading eventually to decisions about repair or no repair. If there are inspections at times a(1) , 2a(1) , 3a(1) , . . . there is not necessarily a repair because aging processes and inspections are uncertain or the signs of deterioration are vague. Repairs occur only with a certain probability PR (t), for example according to Equation (46). For cumulative damage phenomena this probability should increase with time as in Equation (46) and should depend on the actual observed damage state. As mentioned before the same (physical or chemical) damage process determines an (observable) damage state but also failure. For this reason inspection results and thus repair events and failure events are dependent. In fact, only if inspections address the same damage process, specifically the same realization, can we expect to make reasonable decisions about repair or no repair. Such dependencies makes an analytical and numerical treatment complicated but still computationally manageable. The objective is ZIR (p, a1 , a) = BIR (p, a1 , a) − CR (p) − DIR (p, a1 , a)

(60)

where in generalizing Equation (53) DIR (p, a1 , a) = N1 +

N2 N3 D

(61)

with:

N1 = (C(p) + L)

n=1

+I

∞ 

⎛ ⎞ n−1 d ⎝: exp[−γt] P {R(ja1 )} ∩ {T1 ≤ θ}⎠ dθ (n−1)a1

∞


na1

j=0

⎛ exp[−γ(na1 )]P⎝{R(na1 )} ∩

n=1

+ (I + R(p))

∞ 

n−1 :

exp[−γna1 ]P⎝{R(na1 )} ∩

n=1

+

⎞

n−1 :

{R(ja1 )} ∩ {T1 > na1 }⎠ (62a)

j=0

⎛ ⎞ n−1 d ⎝: exp[−γt] P {R(ja1 )} ∩ {T1 ≤ θ}⎠ dθ (n−1)a1

∞


∞  n=1

|θ=t

{R(ja1 )} ∩ {T1 > na1 }⎠

j=0

⎛

n=1

N2 =

⎞

dt

na1

j=0

⎛ exp[−γna1 ]P⎝{R(na1 )} ∩

|θ=t

n−1 :

dt ⎞

{R(ja1 )} ∩ {T1 > na1 }⎠

j=0

(62b)

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

355

⎛ ⎞ n−1 d ⎝: N3 = (C(p) + L) exp[−γt] P {R(ja)} ∩ {T ≤ θ}⎠ dt dθ n=1 (n−1)a j=0 |θ=t ⎛ ⎞ ∞ n−1  : exp[−γ(na)]P⎝{R(na)} ∩ {R(ja)} ∩ {T > na}⎠ +I ∞


na

n=1

+ (I + R(p))

∞ 

⎛

j=0

exp[−γna]P⎝{R(na)} ∩

n=1

⎞

n−1 :

{R(ja)} ∩ {T > na}⎠

⎛ ⎞ n−1 d ⎝: exp[−γt] P {R(ja)} ∩ {T ≤ θ}⎠ dt D=1− dθ n=1 (n−1)a j=0 |θ=t ⎛ ⎞ ∞ n−1  : exp[−γna]P⎝{R(na)} ∩ − {R(ja)} ∩ {T > na}⎠ ∞


(62c)

j=0

na

n=1

(62d)

j=0

and CR (p) = cost of investigating and/or retrofitting an existing structure C(p) = reconstruction cost after failure L = direct damage cost after failure R(na(1) ) = repair event at the j-th inspection R(ja(1) ) = no-repair event at the j-th inspection PR (ja(1) ) = probability of repair after the j-th inspection PR (ja(1) ) = (1 − PR (ja(1) )) = probability of no repair after the j-th inspection a(1) = deterministic inspection interval I = cost per inspection R(p) = repair cost for preventive maintenance. The first term N1 in Equation (61) is the replacement cost after first failure or repair, N3 the replacement cost for subsequent renewal cycles. In both cases the replacement cost include the cost of failure, the cost of inspections given that no failure and no repairs have occurred before and the third term accounts for the cost of inspection and repair given that no failure occurred before. Here, one has to extend the renewal interval to 2a(1) , 3a(1) , . . . if an inspection is not followed by repair and no failure occurred. Since PR (a(1) ) < 1 it is usually sufficient to consider only a few terms in the sums. The higher order terms vanish for PR (a(1) ) → 1 and are significant only for relatively small a(1) . As concerns numerical computions consider the fractional Laplace transform of the failure density given dependencies between no repair and failure events, that is (Joanni and Rackwitz 2006). ∗∗∗ f(1) (γ, p, (n − 1)a(1) ≤ t ≤ na(1) ) ⎛ ⎞
na(1) n−1 d ⎝: = exp[−γt] P {R(ja(1) )} ∩ {T(1) ≤ θ}⎠ dθ (n−1)a(1) j=0

|θ=t

dt

(63)

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Structural design optimization considering uncertainties

where T(1) is the random time to failure. Here again, the intersection probabilities can be determined by FORM/SORM but alternative methods such as Monte Carlo n−1 simulation can also be used. Remember that a typical intersection event j=0 {R(ja(1) )} ∩ {T(1) ≤ t} after the probability distribution transformation into  standard space is given by n−1 j=0 {sc − S(ja(1) , UR ) > 0} ∩ {g(1) (UF , t) ≤ 0} according to Equations (5) and (46), for example. UR and UF denote the variables in the random vector defining the damage indicator (including measurement error) and the variables defining failure, respectively. Because UR and UF have some components in common the events are dependent. Within FORM/SORM the event boundaries are now linearized in the most likely failure point(s) and the correlation coefficients between the respective state functions are computed. The dependencies can be taken into account by evaluating the corresponding multivariate normal integrals. The differentiation under the integral that is necessary for evaluation of Equation (63) is best done numerically, but can also be performed analytically under certain conditions. For F1 (t) = F(t) the damage term in Equation (61) simplifies to DIR =

N3 . D

(64)

The benefit is given by Equation (33) or (34) if it is unaffected by renewals. It has a similar structure as Equation (61). Generalizing Equation (54) for the model in Equation (35) one obtains BIR (p, a1 , a) = B1 +

B2 B3 D

(65)

and B1 =

n=1

B2 =

⎡

∞


na1 (n−1)a1

d ⎣: P P({R(ja1 )} ∩ {T1 ≤ θ})|θ=t dt dθ j=0 ⎛ ⎞⎤ ∞ n−1  : + B∗D (na1 )P ⎝ {R(ja1 )} ∩ {T1 > na1 }⎠⎦ (66a)

⎡

∞
 n=1

B∗D (t)

na1 (n−1)a1

n−1

n=1

j=0

d ⎣: P P({R(ja1 )} ∩ {T1 ≤ θ})|θ=t dt + exp[−γ(na1 )] dθ j=0 ⎛ ⎞⎤ n−1 : {R(ja1 )} ∩ {T1 > na1 }⎠⎦ × P⎝{R(na1 )} ∩ n−1

(66b)

j=0

B3 =

⎛ ⎞ n−1 : d B∗D (t) P⎝ {R(ja)} ∩ {T ≤ t}⎠ dθ (n−1)a

∞
 n=1

+

∞  n=1

na

⎛ B∗D (na)P⎝

j=0

n−1 : j=0

⎞

{R(ja)} ∩ {T > na}⎠

dt |θ=t

(66c)

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

where BD (t) is given in Equation (37) and
t ∗ BD (t) = exp [−γ τ]b(τ)dτ.

357

(67)

(n−1)a(1)

For F1 = F(t) an analogous simplification as in Equation (64) is possible. For independent repair and failure events the intersection signs must simply be replaced by product signs simplifying the numerical computations considerably. The question is when the independence assumption becomes at least approximately true. This must depend on the case under consideration. Dependencies become weaker for larger measurement errors during inspections and for smaller dependencies between damage indicators and failure processes. 4.7

Preventive maintenance for s eries s y s te ms

By definition, a series system fails if any of its components fails. Consequently, all of its components have to renewed. This requires only a few modifications of the theory developed in Section 4.6. For a system with s components we have ⎛ ⎞ ∞
na1 n−1 s  : dP ⎝ : N1s = (C(p) + L) exp[−γt] × (−1) {R(ja1 )} ∩ {Tm1 > θ}⎠ dt dθ (n−1)a1 n=1

+I

∞ 

j=1

⎛

exp[−γ(na1 )]P⎝{R(na1 )} ∩

n=1

+ (I + R(p))

∞ 

n−1 :

∞
 n=1

+

exp[−γt] × (−1)

s :

{R(ja1 )} ∩

N3s = (C(p) + L)

n=1 ∞ 

m=1

+ (I + R(p))

∞  n=1

m=1

n−1 :

s :

j=0

m=1

{R(ja1 )} ∩

(68a) dt

θ=t

⎞

{Tm1 > na1 }⎠

(68b)

⎞ ⎛ n−1 s : dP ⎝ : exp[−γt] × (−1) {R(ja)} ∩ {Tm > θ}⎠ dθ (n−1)a na

j=1

⎛

exp[−γ(na)]P⎝{R(na)} ∩

n=1

{Tm1 > na1 }⎠ ⎞

j=1

exp[−γna1 ]P⎝{R(na1 )} ∩ ∞


⎞

n−1 s : dP ⎝ : {R(ja1 )} ∩ {Tm1 > θ}⎠ dθ

⎛

n=1

+I

n−1 : j=0

(n−1)a

∞ 

{Tm1 > na1 }⎠

⎛

na

θ=t

⎞

m=1

exp[−γna1 ]P⎝{R(na1 )} ∩

n=1

N2s =

s :

{R(ja1 )} ∩

j=0

⎛

m=1

⎛

m=1

n−1 :

s :

j=0

m=1

{R(ja)} ∩

exp[−γna]P⎝{R(na)} ∩

n−1 : j=0

dt

θ=t

⎞

{Tm1 > na1 }⎠

{R(ja)} ∩

s :

⎞ {Tm > na}⎠

m=1

(68c)

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Structural design optimization considering uncertainties

Ds = 1 −

∞
 n=1

+

∞ 

⎞ ⎛ n−1 s : dP ⎝ : exp [−γt] × (−1) {R(ja)} ∩ {Tm > θ}⎠ dθ (n−1)a na

j=1

⎛

exp[−γna]P ⎝R(na) ∩

n=1

n−1 :

s :

R(ja) ∩

j=0

m=1

⎞

dt θ=t

{Tm > na}⎠

(68d)

m=1

in Equation (61) with N1 , N2 , N3 and Dd replaced by N1s , N2s , N3s and Ds , respectively. Similar modifications have to made for the benefit term.

B1s =

⎛ ⎞ n−1 s : : dP ⎝ {R(ja1 )} ∩ B∗D (t) × (−1) {Tm1 > θ}⎠ dθ (n−1)a1

∞
 n=1

+

∞ 

na1

j=1

⎛ B∗D (na1 )P ⎝{R(na1 )} ∩

n=1

B2s =

+

n−1 :

s :

j=0

m=1

{R(ja1 )} ∩

+

(69a)

⎛

exp [−γ(na1 )]P ⎝{R(na1 )} ∩

n−1 :

s :

j=0

m=1

{R(ja1 )} ∩

dt θ=t

⎞

{Tm1 > na1 }⎠

(69b)

⎞ ⎛ n−1 s : : dP ⎝ {R(ja)} ∩ B∗D (t) × (−1) {Tm > θ}⎠ dθ (n−1)a

∞
 n=1

{Tm1 > na1 }⎠

m=1

n=1

B3s =

θ=t

na1

j=1

∞ 

dt

⎞

⎛ ⎞ n−1 s : dP ⎝ : ×(−1) {R(ja1 )} ∩ {Tm1 > θ}⎠ dθ (n−1)a1

∞
 n=1

m=1

∞  n=1

na

⎛ B∗D (na)P⎝{R(na)} ∩

j=1

m=1

n−1 :

s :

j=0

m=1

{R(ja)} ∩

⎞

{Tm > na}⎠

θ=t

(69c)

  Here, we have used again P( s=1 E ) = 1 − P( sm=1 Em ) in order to retain operations solely on intersections. The series system is a realistic assumption for many but not for all civil engineering systems. For example, if one bridge of several bridges in a road connection between A and B fails or a river dam breaks at a certain point the infrastructure or flood protection system fails but only the failed bridge or dam section must be restored in order to make the system functional again. This may require certain modifications in the models outlined so far. If the block maintenance regime is chosen, all components in the systems will be restored. But if the age-dependent regime with inspection and repair is chosen any repair action may also be associated to a specific component. An analytical treatment will then become rather difficult and complex because the components in the system will have different ages. More complex systems can involve considerable numerical effort.

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359

5 Some remarks about suitable optimization methods 5.1

G eneral

It is necessary to speak a little about the technical aspects of optimization. When designing and applying appropriate optimization techniques to the objective functions derived in the foregoing sections one faces the problem that, in fact, two optimization tasks have to be solved: (i) Optimization with respect to the design parameter p and (ii) Optimization with respect to the standard vector u to find the (local) reliability index, at least if FORM/SORM methods are applied. More specifically, the reliability optimization has to be solved for each step in the design parameter optimization. Even if one assumes differentiability of the objective and in the stochastic model as well as in the structural state function and uniqueness of the solution point(s), overall optimization still is a formidable task requiring quite some numerical effort. In the recent literature one distinguishes between one-level and bi-level optimization methods. For the bi-level method one optimizer solves the cost benefit optimization and another, possibly different optimizer solves the reliability optimization. In the one-level approach both optimization tasks are solved simultaneously by a suitable optimizer. In the following we shall briefly comment on both concepts. Both usually work and it is a matter of taste to select one or the other. If the abovementioned smoothness properties do not hold, then other optimization procedures are in order. 5.2

Bi-level optimization

In order to obtain the set of parameters for which the objective function Z(p) becomes optimal, the so-called bi-level approach can be chosen. Here, the optimization task in standard normal space for computation of the required reliability statistics corresponding to a fixed parameter set p is carried out separately using one of the sequential quadratic programming or similar methods. The results, in turn, serve as input to the main optimization loop for the parameter set p for which any of the available optimization methods can be employed. Alternatively, a direct search optimization method developed by (Powell 1994) can be applied. It does not require derivatives. This approach proved to be robust and reliable and only slightly more expensive than other methods. For the main optimization loop, lower and upper bounds should be imposed on the parameters, and it usually turns out to be advantageous to scale the optimization domain such that its shape becomes a hypercube. 5.3

One-level optimization

Let p be a parameter vector which enters in both the cost function and the limit state function g(u, p) = 0. Benefit, construction and damage function as well as the limit state function(s) are differentiable in p and u. The conditions for the application of FORM/SORM hold. In the so-called β-point u∗ the optimality conditions (Kuhn-Tucker conditions) are (Kuschel and Rackwitz 1997): g(u, p) = 0 u ∇u g(u, p) = − u ∇u g(u, p)

(70)

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Structural design optimization considering uncertainties

The geometrical meaning of Equation (70) is that the gradient of g(u, p) = 0 is perpendicular to the vector of direction cosines of u∗ . The basic idea mentioned first in (Madsen and FriisHansen 1992) and elaborated in (Kuschel and Rackwitz 1997) now is to use these conditions as constraints in the cost optimization problem thus avoiding a bi-level optimization. It will turn out that this concept is crucial for further numerical analysis. For example, for the model in Equation (43) this leads to: Z(p) = B − C(p) − (C(p) + L)

subject to

λPf (p) γ

(71)

g(u, p) = 0 ui ∇u g(u, p) + ∇u g(u, p)i u = 0; i = 1, . . . , n − 1 hk (p) ≤ 0, k = 1, . . . , q

where hk (p) ≤ 0, k = 1, . . . , q are some constraints on the admissible parameter range. One may also add a constraint on the failure rate λPf (p). It is important to reduce the set of the gradient conditions in the Kuhn-Tucker conditions by one. Otherwise the system of Kuhn-Tucker conditions is overdetermined. It is also important that the remaining Kuhn-Tucker conditions are retained under all circumstances, for example, if one or more gradient Kuhn-Tucker conditions become co-linear with one or more of the other constraints. Otherwise the β-point conditions are not fulfilled. λP (p) must simply be replaced by If there are multiple failure modes (C(p) + L) γf  s λ (C(p) + L)(1 − P( j=1 F j (p))) (see Equation (44)). In this case γ ⎞⎤ ⎡ ⎛ s : λ⎣ Z(p) ≤ B − C(p) − (C(p) + L) 1 − P⎝ F j (p)⎠⎦ γ

(72)

j=1

subject to gk (uk , p) = 0; k = 1, . . . , s ui,k ∇u gk (uk , p) + ∇u gk (uk , p)i uk  = 0; i = 1, . . . , nk − 1; k = 1, . . . , s h (p) ≤ 0,  = 1, . . . , q where the Kuhn-Tucker conditions have to be fulfilled separately for each failure mode. Note that there are s distinct independent vectors uk . It may be noted that all failure mode equations are fulfilled simultaneously. For (locally) non-stationary problems, especially aging problems and for problems with non-Poissonian failures, it is possible to propose a numerical solution. More precisely, the Laplace transform is taken numerically and each value of the failure density is computed by FORM/SORM. The same scheme, however, applies to the full

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

361

Laplace transform of non-stationary problems as well. Z(p) ≈ B − C(p) − (C(p) + H) g(uj , p,tj ) = 0

f ∗ (γ, p) 1 − f ∗ (γ, p)

(73)

for j = 0, 1, . . . , m

ui,j ∇u g(uj , p,tj ) + ∇u g(uj , p,tj )i uj  = 0

i = 1, . . . , n − 1; j = 0, . . . , m

h (p) ≤ 0,  = 1, . . . , q where f ∗ (γ, p) ≈

m 

wj exp[−γtj ]fT (tj , p)

(74)

j=0

with m the number of time steps and wj the weights for numerical integration of Equation (74). In order to solve the optimization problem a suitable optimization algorithm is required. Unfortunately, off-shelf sequential quadratic programming methods turned out to have problems, possibly due to the many equality constraints. Based on sequential linear programming methods a new optimization algorithm JOINT 5 (Pshenichnyj 1994) has been developed from an earlier algorithm proposed by Enevoldsen and Sørensen (Enevoldsen and Sørensen 1992). This turned out necessary because the tasks in Equations (71), (72) and (73) require special precautions which are not necessarily available in most of the off-shelf algorithms. For example, the algorithm includes a reliable and robust slow down strategy to improve stability of the algorithm instead of an exact (or approximate) line search which too often is the reason for non-convergence. A special ‘extended’ equation system is solved in case of failure in the quadratic subalgorithm, e.g. due to linear dependence of the linearized constraints. In addition, the algorithm contains a careful active set strategy (for further details see (Streicher 2004)). As in the bi-level method a suitable scaling of the objective is advantageous. Gradient-based methods need first derivatives of the objective and all active constraints. In case of cost optimization under reliability constraints first order KuhnTucker optimality conditions for a design point are restrictions to the optimization problem. These equations are given in terms of the first derivatives of the limit state function. The gradients of these conditions involve second derivatives. Thus, the solution of the quadratic subproblem needs second derivatives, i.e. the complete Hessian of g(u, p). The determination of the Hessian in each iteration step is laborious and can be numerically inexact. In order to avoid this, an approximation by iteration is proposed. The Hessian is first preset with zeros. Note that linear limit state functions always have a zero Hessian matrix. This implies loss of efficiency, but the overall numerical effort needs not to rise, because calculation of the Hessian is no more necessary. In order to improve the results in case of strongly nonlinear limit state functions, it is possible to evaluate the Hessian after the first optimization run and restart the algorithm. For the improved solution the starting point is the solution of the previous run and the Hessian matrix is fixed for the whole run. This iterative improvement with subsequent restarts continues until the results differ only with respect to a given precision which is usually after very few steps. The results can be simultaneously improved by including

362

Structural design optimization considering uncertainties

second-order corrections during reiteration (see Kuschel and Rackwitz 2000). Any other more exact improvement can be taken into account in a similar manner. The techniques proposed enable the solution of quite general problems. They are based on a one-level optimization but rather strong requirements on differentiability of the objective, limit state functions and other restrictions must be made. Also, a possibly substantial increase of the problem dimension must be expected in extreme cases and, hence, much computing time will be necessary. In passing it is worthwhile to remark that for the bi-level approach a proof of convergence is not yet available whereas it is available for the one-level approach.

6 Illustrating example – Chloride attack in an existing building The following, slightly academic example shows several interesting features and is an appropriate test case. Chloride attack due to salting and subsequent corrosion, for example, in the entrance area of a parking house or in a concrete bridge is considered. A simplified, approximate model for chloride concentration in concrete is C(x, t) = Cs (1 − erf( 2√xDt )) where Cs = surface chloride content (measured ≈ 0.5 cm below surface and extrapolated), x = depth and D = diffusion parameter. A suitable criterion for the time to the start of chloride corrosion of the reinforcement is: 

 c Ccr − Cs 1 − erf √ ≤0 (75) 2 Dt where Ccr = critical chloride content and c = concrete cover. Inversion gives the initiation time

  Ccr −2 c2 −1 Ti = 1− erf (76) 4D Cs The stochastic model is Variable

[unit]

Distr. function

Parameters

C cr Cs c D

% % cm

Uniform Uniform Log-normal Uniform

0.4, 0.6 0.8, 1.2 mc , 0.8 0.5, 0.8

cm2 year

The planned concrete cover is mc = 5.0 cm. By drilling small holes and analyzing chemically the drill dust one has determined a chloride concentration of 0.4 at a depth of 3 cm with measurement error 0.05. Applying Equation (26) and truncation at t = 12 years gives an updated distribution function of the time to the start of corrosion as shown in Figure 13.1, where it is compared with the initiation time distribution. It is seen that chloride penetration occurred slightly more rapid than expected in renewed structures. During initiation time the structure can fail due to time-variant, stationary extreme loading. It is assumed that each year there is an independent extreme realization of the load. Load effects are normally distributed with mean 2.0 and coefficient of

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363

1.0

0.8

0.6

0.4

0.2 Regular distribution Updated distribution 24

48

72 96 Time [years]

120

144

Figure 13.1 Updated distribution for first failure time and subsequent failure time distributions.

variation of 40%. Structural resistance is also distributed normally with mean 3-times as large as the mean load effect and coefficient of variation 30% (p = 6.0). Once corrosion has started mean resistance deteriorates with rate δ(t) = 1 − 0.07t + 0.00002t 2 . The distribution and density functions of the time to first failure are computed using SORM in Equation (13) with the failure time distributions in the initiation phase and in the deterioration phase determined by Equation (7). The structural states in two arbitrary time steps have constant correlation coefficient of ρ = ρij = 0.973. First, the mean times of the various distributions in Equation (13) are determined. One finds E[Ti ] = 51 and E[Td ] = 9.4. The mean of Te does not exist. Using the distribution in Equation (13) one determines E[T] = 61. These mean times indicate that virtually no failures occur during initiation. The risk functions for both distributions assuming the repair probabilities in Figure 13.2 are first increasing but decrease slightly for larger t. Visual inspections, inspections by half-cell potential measurements and chemical analyses are performed at regular intervals a(1) . They are followed by renewals (repairs) with probability   ) *  mc,(1) ≤0 (77) PA (a(1) ) = P r(1 + 0.05UR ) − Cs,R 1 − erf  2 DR a(1) shown in Figure 13.2 if a chloride concentration of r at the reinforcement was observed. The term (1 + 0.2UR ) models the measurement error with UR a standard normal variable. Repair times are assumed negligibly short. Remember, the existing structure is already 12 years old and has suffered from chloride attack during the whole period. The first inspection is undertaken after 5 years. For all subsequent renewed structures the first inspection is after 8 years. Erection cost are C(mc , mr ) = C0 + C1 m2c + C2 mr , inspection cost are I = 0.02C0 , and we have C0 = 106 , C1 = C2 = 104 , L = 10C0 , γ = 0.03. For preventive repairs the cost are R(mc , mr ) = 0.6C(mc , mr ) · mr is the safety

364

Structural design optimization considering uncertainties

1.0

Probability of repair

0.8

0.6

0.4

0.2 Regular probability, r
0.42 Updated probability, r
0.43 16

32 48 Time [years]

64

80

Figure 13.2 Repair probabilities.

Expected maintenance cost [106 MU]

8 1 unit 2 units 5 units

7 6 5 4 3 2 1

6

12 Replacement age a [a]

18

24

Figure 13.3 Age replacement.

factor separating the means of load effect and resistance. The benefit is determined by using a decaying rate b(t) = b exp [−0.0001t 2 ], b = 0.15C0 , in the model in Equation (35). All cost are in appropriate currency units. It is noted that the physical and cost parameters are somewhat extreme but not yet unrealistic. When optimizing with respect to the inspection interval the Laplace transforms are taken numerically using Simpson’s integration formula. We first show the total cost (preventive and corrective) for the case of systematic age-dependent repairs and system sizes of s = 1, 2 and 5 (Figure 13.3). As expected, the total cost are higher for larger systems and the optimum replacement interval decreases.

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365

Expected maintenance cost in [106 MU]

8 1 unit, r
0.41 2 units, r
0.36 5 units, r
0.33

7 6 5 First inspection after 8 years 4 3 2 1

6

12 18 Inspection interval a [a]

24

Figure 13.4 Total cost for inspection and repair.

6 12 18 24 Inspection interval a [a]

18 12 6

4 4.4

1

Inspection interval a [a]

1

Inspection interval a [a]

1.9

*

1

n
5, r
0.35, r
0.32, D
2.30×106 24

3.6 3.2 2.8

12 6

6

2.4

6 12 18 24 Inspection interval a [a]

18

2.9 2.7

1.6

6

*

3.1

2

12

1

n
2, r
0.40, r
0.36, D
1.84 ×10 24

2.5 2.3 1 2.

1.8

2.2

18

6

1

*

Inspection interval a [a]

1

n
1, r
0.43, r
0.42, D
1.55×10 24

6 12 18 24 Inspection interval a [a]

Figure 13.5 Expected maintenance cost of an existing n-unit structure in [106 MU], with periodic inspections at an interval of a1 and a beginning after 5 and 8 years, respectively.

Figure 13.4 shows the results for the inspection/repair strategy. Here, we have also optimized the repair thresholds r. They become more stringent for larger systems. Also, the optimum inspection/replacement intervals are much smaller than in the simple agedependent case. The differences in cost between systematic age-dependent repairs and repairs after inspections are not large in this example. By parameter changes it is, however, easy to make them larger. The result of an optimization with respect to a1 and a is shown in Figure 13.5 for mc = 5 and mr = 6. One sees that the contour lines are spaced more narrow for a1 than for a. The optima with respect to a and a1 are rather flat. If, however, the repair probabilities are much smaller than given in (2) no optimum would be found. The inspection intervals depend strongly on the system size.

366

Structural design optimization considering uncertainties

7 Conclusions The theory developed earlier for optimal design and maintenance of aging structural components and systems is extended to optimal repair and retrofit of existing structures. It is assumed that structures are maintained (inspected and repaired with certain probability) at regular time intervals and systematically reconstructed after failure. Age-dependent and block repairs are studied assuming negligibly short repair times. Three models for the benefit are discussed. Due to updating by additional investigations, the time to first failure usually has different probabilistic characteristics than all other times. Appropriate objective functions for cost-benefit optimization are derived. It is pointed out that inspections and possible repair events and failure events must address the same realization of the damage process if preventive maintenance makes at all sense. Even if the risk function initially was increasing, maintenance operations will let the risk function drop. Perfect inspections and repairs will reduce the risk function down to zero. For imperfect inspections the risk function will drop down to finite values. This generally requires the numerical computation of the renewal intensity by differentiating the renewal function for which tight bounds can be given.

References Aitchison, J. & Dunsmore, I.R. 1975. Statistical Prediction Analysis. New York: Cambridge University Press. Ambartzumian, R., Der Kiureghian, A., Ohaniana, V. & Sukiasiana, H. 1998. Multinormal probability by sequential conditioned importance sampling: Theory and application. Probabilistic Engineering Mechanics 13(4):299–308. Au, S.-K. & Beck, J.L. 2001. First excursion probabilities for linear systems by very efficient importance sampling. Probabilistic Engineering Mechanics 16(3):193–207. Ayhan, H., Limón-Robles, J. & Wortman, M.A. 1999. An approach for computing tight numerical bounds on renewal functions. IEEE Transactions on Reliability 48(2): 182–188. Barlow, R.E. & Proschan, F. 1965. Mathematical Theory of Reliability. New York: John Wiley & Sons. Barlow, R.E. & Proschan, F. 1975. Statistical Theory of Reliability and Life Testing: Probabilistic Models. New York: Holt, Rinehart & Winston. Brown, M. & Proschan, F. 1983. Imperfect repair. Journal of Applied Probability 20: 851–859. Cox, D.R. 1962. Renewal Theory. Monographs on Applied Probability and Statistics. London: Chapman & Hall. Cox, D.R. & Isham, V. 1980. Point Processes. Monographs on Applied Probability and Statistics. London: Chapman & Hall. Cramér, H. & Leadbetter, M.R. 1967. Stationary and Related Stochastic Processes. New York: John Wiley & Sons. Dunnett, C.W. & Sobel, M. 1955. Approximations to the probability integral and certain percentage points of multivariate analogue of Student’s t-distribution. Biometrika 42: 258–260. Enevoldsen, I. & Sørensen, J. 1992. Optimization algorithms for calculation of the joint design point in parallel systems. Structural and Multidisciplinary Optimization 4(2):121–127. Fox, B. 1966. Age replacement with discounting. Operations Research 14(3):533–537.

C o s t-b e n e f i t o p t i m i z a t i o n f o r m a i n t a i n e d s t r u c t u r e s

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Genz, A. 1992. Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics 1:141–149. Gollwitzer, S. & Rackwitz, R. 1983. Equivalent components in first-order system reliability. Reliability Engineering 5:99–115. Gollwitzer, S. & Rackwitz, R. 1988. An efficient numerical solution to the multinormal integral. Probabilistic Engineering Mechanics 3(2):98–101. Hasofer, A. 1974. Design for infrequent overloads. Earthquake Engineering and Structural Dynamics 2(4). Hasofer, A.M. & Rackwitz, R. 2000. Time-dependent models for code optimization. In R.E. Melchers & M.G. Stewart (eds), Proceedings of the 8th International Conference on Applications of Statistics and Probability (ICASP8), Sydney, Australia, December 1999, Vol. 1, Rotterdam/Brookfield, pp. 151–158. CERRA: A.A. Balkema. Hohenbichler, M. & Rackwitz, R. 1981. Non-normal dependent vectors in structural safety. ASCE Journal of the Engineering Mechanics Division 107(6):1227–1249. Hohenbichler, M. & Rackwitz, R. 1983. First-order concepts in system reliability. Structural Safety 1(3):177–188. Joanni, A.E. & Rackwitz, R. 2006. Stochastic dependencies in inspection, repair and failure models. In C. Guedes Soares & E. Zio (eds), Proceedings of the European Safety and Reliability Conference, Estoril, Portugal, September 2006, London, pp. 531–537. Taylor & Francis. Kuschel, N. & Rackwitz, R. 1997. Two basic problems in reliability-based structural optimization. Mathematical Methods of Operations Research (ZOR) 46(3):309–333. Kuschel, N. & Rackwitz, R. 2000. Time-variant reliability-based structural optimization using SORM. Optimization 47(3/4):349–368. Madsen, H.O. & Friis-Hansen, P. 1992. A comparison of some algorithms for reliabilitybased structural optimization and sensitivity analysis. In R. Rackwitz & P. Thoft-Cristensen (eds), Proceedings of the 4th IFIP WG 7.5 Working Conference on Reliability and Optimization of Structural Systems, Munich, Germany, September 1991, Berlin, pp. 443–451. IFIP: Springer. Pandey, M.D. 1998. An effective approximation to evaluate multinormal integrals. Structural Safety 20(1):51–67. Powell, M.J.D. 1994. A direct search optimization method that models the objective and constraint functions by linear interpolation. In S. Gomez & J.-P. Hennart (eds), Proceedings of the 6th Workshop on Optimization and Numerical Analysis, Oaxaca, Mexico, January 1992, Dordrecht, pp. 51–67. Kluwer Academic Publishers. Pshenichnyj, B.N. 1994. The Linearization Method for Constrained Optimization, Volume 22 of Computational Mathematics. Berlin: Springer. Rackwitz, R. 2000. Optimization – the basis of code making and reliability verification. Structural Safety 22(1):27–60. Rackwitz, R. 2001. Reliability analysis – a review and some perspectives. Structural Safety 23(4):365–395. Rackwitz, R., Lentz, A. & Faber, M.H. 2005. Socio-economically sustainable civil engineering infrastructures by optimization. Structural Safety 27(3):187–229. Rosenblueth, E. 1976. Optimum Design for Infrequent Disturbances. ASCE Journal of the Structural Division 102(9):1807–1825. Rosenblueth, E. & Mendoza, E. 1971. Reliability optimization in isostatic structures. ASCE Journal of the Engineering Mechanics Division 97(6):1625–1642. Streicher, H. 2004. Zeitvariante zuverlässigkeitsorientierte Kosten-Nutzen-Optimierung für Strukturen unter Berücksichtigung neuer Ansätze für Erneuerungs- und Instandhaltungsmodelle. PhD dissertation, Technische Universität München, Munich, Germany. In German.

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Streicher, H., Joanni, A. & Rackwitz, R. 2006. Cost-benefit optimization and target reliability levels for existing, aging and maintained structures. Structural Safety. Accepted for publication. Streicher, H. & Rackwitz, R. 2004. Time-variant reliability-oriented structural optimization and a renewal model for life-cycle costing. Probabilistic Engineering Mechanics 19(1–2):171–183. Van Noortwijk, J.M. (2001). Cost-based criteria for obtaining optimal design decisions. In Proceedings of the 8th International Conference on Structural Safety and Reliability, Newport Beach, CA, U.S.A., June 2001. CD-ROM.

Chapter 14

A reliability-based maintenance optimization methodology Wu Y.-T. Applied Research Associates Inc., Raleigh, NC, USA

ABSTRACT: Many mechanical and structural systems, including aircraft, ship, car, oil and gas pipeline, utilize a structural integrity program to monitor and sustain structural integrity throughout the service life. Developing optimal maintenance plans under various uncertainties requires probabilistic analyses of damage accumulations, damage detections, and mitigation actions. Given the wide spectrum of the options and the complexities in modeling, the most practical way to conduct maintenance optimization is by random simulations, preferably efficient sampling methods. This chapter presents a reliability-based maintenance optimization (RBMO) methodology with a focus on computational strategies that involve physics-based models. In particular, a two-stage importance sampling (TIS) approach that drastically reduces computational time is described. Stage 1 computes failure probability and systematically generates failure samples, given no inspections. The failure samples are then repeatedly used in Stage 2 for inspection optimization. The RBMO methodology is demonstrated using analytical examples as well as applications related to aircraft and helicopter structural components.

1 Introduction For economical and reliability/safety reasons, many mechanical and structural systems apply maintenance practices to sustain structural integrity and reliability over the design life or extend the life beyond the original design for un-anticipated reasons. Since fatigue and fracture is one of the main failure modes for such systems, this chapter will focus on fracture failure analysis even though the methodology is applicable to more general damage accumulation models including corrosion. Most existing computational fracture mechanics methods and tools used in the design of structures apply safety margins to deterministic models. With the realization that many design parameters including defect or flaw characteristics, crack growth law, crack detection, loads, and usages are uncertain, various conservative assumptions are often employed to help ensure structural integrity. As an example, a comparison between deterministic and probabilistic damage tolerance analyses is shown in Table 14.1. The safety-factor based approach applies bounds, either explicitly or implicitly, to key design variables. The probabilistic approach, on the other hand, requires relatively more precise characterizations of the input uncertainties based on data and expert knowledge. In the more traditional safe-life design approach (Palmberg et al. 1987), the fatigue and facture life of a structure is assumed to be governed by crack initiation time, and

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Table 14.1 Example of deterministic vs. probabilistic damage tolerance.

Reliability principals Flaw/defect size Existence Inspection schedule Safety measure Other variables

Deterministic

Probabilistic

Bounds, safety factors A given crack size Probability = 1 Life/N Safety margin Bounds, safety factors

Probability & confidence Distribution of crack size 0 0.1 f (xNew ) Accept the new point with a probability of ρ = min f (x · Current )

(6)

Reject the candidate point and save the current point as the “next’’ point with a probability of 1 − ρ.

2

q(xCurrent |xNew ) ,1 q(xNew |xCurrent )

.

Step 4 ensures that candidate points in the safe domain will be rejected with a probability of one; consequently all the generated points will be in the failure domain. Steps 5 and 6 can be executed using a uniform random number generator. The effectiveness of the algorithm depends on the proposal distribution, which defines how to randomly move around in the failure region and the acceptance rate. A simple proposal distribution is a uniform distribution centered at the current point. In this case the proposal distribution is symmetrical and q(xCurrent |xNew )/q(xNew |xCurrent ) = 1; as a result, the ratio of the PDFs of the current and candidate points drives the random movement in a way that ensures that the frequency of visits at any point will be asymptotically proportional to the JPDF in the failure region. The selection of the range of the proposal distribution, which characterizes the step sizes of the random moves, can significantly affect the rejection rate and the convergence rate towards the target distribution, and therefore needs to be tuned. Additionally, a “burn-in’’ period (in which the samples would be thrown away) may be needed to improve the quality of the samples, especially if the number of samples used is relatively small. For higher dimension problems, the use of f (xNew )/f (xCurrent ) could decrease the acceptance rate drastically and slow down the converging process. To address the issue, a modified M-H algorithm has been proposed (Au and Beck, 2001) in which a one-dimensional symmetrical proposal distribution is used in combination with individual ratio f (xi_New )/f (xi_Current ) to allow for random movements in some of the variables and increase the acceptance rate. The M-H algorithm has been used in one of the RBMO example presented below. Note that the M-H algorithm itself does not provide an answer to the pf calculation, but the generated samples can be used with a pf computed from other methods such as importance sampling. 3.3.4 F a il u re-sa m p le b a s e d a n a ly s is f o r s in g l e i ns pe c t i o n Before applying the above methods to general inspection optimization applications, we first analyze a simple but practical case where only one inspection is feasible due to economical and other constraints. The objective of inspection optimization is to find the best inspection time that minimizes pf . Define the inspection time as t1 and the service time t2 . The probability of failure at t1 , prior to inspection, is pf (t1 ), which is the lower bound of pof (t2 ). The defect population includes the stronger parts that would not fail by t2 and the weaker parts, defined here as the “critical parts,’’ that would fail by t2 . The probability of the parts that would survive t1 is pof (t2 ) − pf (t1 ). Given the critical parts, which have a probability of pof (t2 ), there are three failure paths as shown in Figure 14.6, an event diagram. In summary, a defect of size a from the critical parts can (1) fail by t1 with a probability of pf (t1 )/pof , (2) survive t1 , escape inspection with PND(a), and fail by t2 with a probability of [pof − pf (t1 )] · E[PND(a(t1 ))]/pof , and (3) survive t1 , be detected with POD(a), be replaced by a part from the original population, and fail by t2 with a probability of

A r e l i a b i l i t y-b a s e d m a i n t e n a n c e o p t i m i z a t i o n m e t h o d o l o g y

a(t1) > a* (F)

383

- > Fail path 1

pf (t1) a(0) > a*(0)

Reach Insp. Time t1

- > Fail path 2

[pof (t2)

a(t2) > a* (F) Missed Detection

a(t1) < a* a*: Critical defect size a(t): Defect size of critical part at t F: Fail by t2 t1: Time of inspection t2: End of service life

(t2)

[pof pf (t1)]•E[PND(a)]

NDE at t1

Detected/Replaced

(t2 - t1) a (t2-t1) > a * (F )

- > Fail path 3

[pof  pf (t1)]•E[POD(a)]•pf (t2  t1)

Figure 14.6 Probabilities of failure events for one inspection.

[pof − pf (t1 )] · E[POD(a(t1 ))] · pf (t2 − t1 )/pof . Therefore, the total pf with inspection can be summarized in Equation 20. o pW f = pf (t1 ) + [pf − pf (t 1 )] · E[PND(a(t1 ))]

+ [pof

(20)

− pf (t 1 )] · E[POD(a(t1 ))] · pf (t 2 − t 1 )

and pof is the amount of risk reduction, pr , which is: The difference between pW f pr = [pof − pf (t1 )] · E[POD(a(t1 ))] · [1 − pf (t2 − t1 )]

(21)

In Equations 20 and 21, pof is computed from Stage 1 of TIS; the failure samples are used to compute pf (t 1 ) and pf (t 2 − t 1 ), and E[POD(a(t1 ))] can be computed for any t1 using the defect growth history data. Therefore, the Stage 1 failure samples, including defect growth histories, should be saved to calculate risk reduction for any inspection time without additional stress or life analysis. Since pf (t 2 − t 1 ) < pof , the last product term in Equation 21 is 1 − pf (t 2 − t 1 ) > 1 − pof , which is approximately 1 for small pof . This suggests that for small pof , a replacement using an original part can be approximated by assuming a “perfect repair,’’ meaning the part is “fail-proof,’’ and therefore pf (t 2 − t 1 ) = 0. In practice, this condition can be achieved if the flawed parts can be detected in time and can be either mitigated to eliminate the re-occurrence of failure (e.g., a corroded pipe section is wrapped with a corrosion-free composite sleeve) or replaced with new or better-grade parts that are guaranteed to survive the remaining service life. In the undesirable scenario where a bad repair is likely, the impact of the repair can be simulated using a worse-than-new distribution, and a fresh analysis is needed to compute pf (t 2 − t 1 ). If pf (t 2 − t 1 ) can be neglected, risk reduction becomes a product of two timedependent terms, where the first term, [pof − pf (t 1 )], is the risk-reduction potential, a monotonically decreasing function of time, and the second term, E[POD(a)], is a monotonically increasing function. This suggests that in practice, optimal inspection

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Structural design optimization considering uncertainties

time is neither at time zero (when the risk reduction potential is at the highest but E[POD(a)] is relatively small because of small initial defects), nor at the end of service life (when the risk-reduction opportunity is approaching zero). Equation 21 also implies that better POD(a) will create more risk reduction and the best inspection time is earlier with a better POD(a). An example of single-inspection analysis using an event-tree analysis is shown in Figure 14.7. In this example, the initiating event is the critical parts with pof = 0.01. There are four failure paths. The first two are the same as mentioned above. Given the critical parts, 30% of the parts fail by t1 . At inspection, 10% of the survived critical parts is missed and subsequent failures by t2 cannot be avoided. The last two are the results of two types of repairs, each with a 50% chance. The first is replacement by original part and the second is repair with worse-than-new part. Both the replacement/repair parts only need to survive a time of (t2 − t1 ). The result (see the right-hand side column in Figure 14.7 showing pf contribution) demonstrates that the risk contribution related to pf (t 2 − t 1 ) is insignificant (less than 0.5%) as expected. The use of Equation 21 will be demonstrated further in the RBDT examples described below. 3.3.5 TIS error a n a ly s is s in g le in s p e ct io n A typically small error is inherent in the TIS approach due to ignoring the samples that are originally in the safe region (Wu & Shin 2005). Such an error would arise if an originally safe part is unfortunately repaired to a worse condition. This scenario could be the result of poor workmanship and lack of a quality assurance process. The probability of failure due to ignoring the “safe’’ parts is a product of the probability of safe parts and the conditional probabilities of a sequence of events (detected, repaired if detected, bad repair that causes failure if repaired, and failure before service life) that lead to a failure: p∗f (t2 ) = P(Safe) · POD · P(Repair|Detected) · P(Bad Repair) · pf (t2 − t1 )

(22)

Using the relations P(Safe) ≤ 1(which is close to 1 for high reliability parts) and pf (t2 − t1 ) ≤ pof , and also assuming the worst case that POD = 1 (worse in the sense that more chances are created for bad repairs), the TIS error with respect to pof is dominated by two factors: TIS Error =

p∗f pof

≤ P(Repair|Detected) · P(Bad Repair)

(23)

The first factor, P(Repair|Detection), is expected to be small (at least for high reliability products) because it is unlikely that a large percentage of products will be repaired regardless of the detected defect sizes and knowing that repairs may produce negative (un-safe) results. The second term is also expected to be small assuming a good quality control procedure is in place. In the unlikely worst case scenario, the error is 100% if every defect is detected, every detection leads to a decision to repair, and every repair is a bad repair. In summary, the above error analysis suggests that the TIS error is small if the safe parts are ignored, which is the basis for the high efficiency of TIS compared with standard Monte Carlo sampling.

1%

a(t1) < a*

70%

Reach insp. time t1

30 %

10 %

PND

30%

Detected

Figure 14.7 An event tree analysis example for one inspection.

90%

POD

Inspection at t1

Missed

--> Fail path 1

a*: Critical defect size a(t): Defect size of critical part at t t1: Time of inspection t2: End of service life F: Fail by t2 Replacement: Original parts Repair: Better than before but worse than replacement

Critical parts

a (0) > a*(0)

pf = 1% if no inspection

a(t 1 ) > a* (F)

10 0%

50 %

0%

Repair

50%

Repair/Replacement

Replacement

a(t2) < a*

Reach sevice time t2

a (t2 ) > a* (F)

a( t2 t1 ) < a*

Reach t2

a( t2 t1) > a* (F)

a( t2 t1) < a*

Reach t2

a( t2 t1) > a* (F)

--> Fail path 2

99.75%

0.25%

99.85%

0.15%

7%

Total risk Risk reduced

0.079%

--> Fail path 4

0.047%

--> Fail path 3

0.00371 0.00629

7.88E-06

4.73E-06

100% 63%

0.2%

0.1%

18.9%

80.8%

0.003

0.0007

% Pf

Pf

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Structural design optimization considering uncertainties

3.3.6 G en era l re p a ir a n d m u lt ip le in s p e ct io ns The above analysis procedure for a single inspection can, in principle, be easily extended for multiple inspections, assuming replacement by original-part (i.e., same defect distribution) or ideal repair. However, when the effect of bad-repair needs to be carefully studied, a full set of MC samples is recommended. When there are multiple inspections, a brute-force MC for inspection optimization would require a set of MC runs for every selected option of inspection schedules. Clearly this is computationally very challenging. Recently, a recursive-probability-integration (RPI) procedure has been developed (Shiao 2006; Shiao & Wu 2004) to more rapidly calculate probability of failure for any number of inspections and types of distributions where the bookkeeping of the risk contributions from every inspection result becomes very tedious. In the RPI approach, the sum of the probabilities of failures from a potentially very large number of failure paths (created by multiple inspections and repairs) is formulated using a condensed formula that involves recursive calculations at every branch (with sub-branches and sub-sub branches, etc.). The formulation provides a systematic way to manage failure paths. Similar to the TIS approach, RPI also uses saved Monte Carlo crack growth histories to compute all the probabilities after each inspection and repair. RPI requires a baseline MC for the original defect distribution and an additional MC for each new repair distribution, the number of which is typically very small. The computational efficiency can be improved further by integrating RPI with the conditional expectation method (CEM), where the random variables are separated into two groups, X 1 and X 2 , and the failure probability is formulated as:

(24) pf = . . . P[g(X 1 , X 2 ) ≤ 0]fX1 fX2 dX 1 dX 2 = E[P[g c (X 2 |x1 ) ≤ 0]] To compute Equation 24, a set of realizations of X 1 is randomly generated and the corresponding P[g c (X2 |x1 ) < 0] values are computed using numerical integration or fast probability integrators. As demonstrated, with proper grouping of X , E[P[g c (X2 |X1 ) ≤ 0]] can be estimated using a relatively small number of samples (Shiao 2006). 3.3.7 S a mpli n g b a s e d r is k s e n s it iv it y a n a ly s i s As a by-product, the failure samples from Stage 1 and Stage 2 can be used to conduct risk sensitivities. The sensitivity of pf with respect to changes in the distribution parameters (mean or standard deviation) θi of a random variable Xi can be evaluated from: 

∂pf > ∂θi σi ∂fx σi ∂fx Sθi = = ··· fx dx = E (25) pf σi pf fx ∂θi pf fx ∂θi !

!

where Sθi are the sensitivity coefficients. Equation 25 leads to the following two non-dimensional sensitivities that can be computed easily using TIS samples. Sµi =

∂p/p = E[ui ]! ∂µui /σui

(26)

A r e l i a b i l i t y-b a s e d m a i n t e n a n c e o p t i m i z a t i o n m e t h o d o l o g y

Sσi =

∂p/p = E[u2i ]! − 1 ∂σui /σui

387

(27)

The expectations in Equations 26 and 27 are over the failure region !; µui is the mean of ui with a nominal value of zero and σui is the standard deviation of ui with a nominal value of one. These two sensitivities have been found to be useful for identifying and ranking important random variables (Karamchandani 1990; Enright & Wu 1999; Wu & Mohanty 2006).

4 Examples 4.1

Rotor disk

Consider a rotor disk subject to fracture failure due to rare manufacturing anomalies such as the alpha defect (Leverant et al. 1997). The potential random variables in such an application include defect size and location, stress, material property, and the time and effectiveness of the inspections. The following numerical example was one of the several test examples developed for TIS methodology development. The example represents the analysis at a highly stressed zone assuming a circular imbedded crack with a specified probability of occurrence. For the entire disk, the risk can be integrated using a zone-based risk integration approach (Wu et al. 2002). In the example, all the units are MKS-based. The analyses were conducted using a computer program written in Matlab language. The fracture mechanics model is: da = C( K)m dN

(28)

√ where m = 3.0, C = 1.021E-11, a = crack radius, K = Yσ πa, σ = 414, and the crack geometry factor is Y = 0.636. Simplified stress and life random variables are used. Stress uncertainty is modeled as σ = X1 · σmodel where X1 is a random variable accounting for the errors in geometry and numerical (such as finite element) modeling. Similarly, a simplified stochastic life model is defined as N = X2 · Nmodel where Nmodel is the life model, and X2 is a life scatter random variable. Both X1 and X2 are modeled using log-normally distributed random variables with a median value of one and a specified coefficient of variation (COV). Assume the defect occurrence rate is 0.00348 per disk and the initial crack size follows a three-parameter Weibull distribution with a location parameter of 0.0028 m, a scale parameter of 0.00043 m, and a shape parameter of = 0.41. The critical crack size for fracture is:

 1 KC 2 (29) ac = π YS where KC = 60. The time of inspection is assumed to be normally distributed with a specified COV. The inspection has a POD(a) of:

 ln (a) − 2.996 POD(a) =
(30) 0.4724

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Structural design optimization considering uncertainties

1 0.9 0.8

PDF or POD

0.7 PDF(a)/1000 POD(a) POD(a)*PDF(a)/10 Crit. defect size

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.01

0.02

0.03 0.04 Defect size (m)

0.05

0.06

Figure 14.8 Initial defect PDF and POD, and their product.

The normalized initial-defect PDF and the POD(a) are plotted in Figure 14.8. Also plotted are the nominal critical defect size (diameter) and PDF(a)∗POD(a). The later curve, when integrated, is the percentage of defects that can be detected. Given all the disks with a defect, the percentage is 1.45. Assuming there are 1000 disks, the number of disks that have a defect is 3.48, and the number of defected disks that can be detected at time zero is near zero (3.48*0.0145 = 0.05). Thus, the best inspection time should be after the defect population has grown much bigger. After integration, Equation 28 becomes: 1−m/2

N = X2 ·

1−m/2

− ac ao (m/2 − 1) · C · Y m · (X1 S)m/2 · πm/2

(31)

For zero stress and life scatters, leaving defect as the only random variable, pof can be analyzed by using Equation 27 and setting N = NService . Figure 14.9 compares the pW ’s f using TIS and Monte Carlo and the analytical solution (solid curve). Given a defect, the conditional pof calculated analytically is 0.109. The unconditional pof , plotted in Figure 14.9, is 0.109∗0.00348 = 3.807e-04, which results in an average of 1.90e-08 per flight cycle. With inspection at 10 000 cycles, the unconditional pW is 1.29e-04, f o which is approximately one third of pf or 67% risk reduction. For this example with a relatively high pof (0.109), the TIS (500 samples) is ten times as efficient as Monte Carlo (5000 samples). In general, the efficiency of TIS will be higher with smaller pof . Figure 14.10 shows the increase of risk after adding uncertainties to inspection time, stress scatter, and life scatter. 500 TIS samples proved to be sufficient for the analysis.

A r e l i a b i l i t y-b a s e d m a i n t e n a n c e o p t i m i z a t i o n m e t h o d o l o g y

5

389

104

No inspection (Exact) With inspection (MC 500) With inspection (MC 5000) With inspection (IS 500)

4.5 4

Prob. of failure

3.5 3 2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8 1 1.2 Flight cycles

1.4

1.6

1.8

2 104

Figure 14.9 Risk with and without inspection (at 10 000 cycles) fixed stress and life scatter.

5

104 No insp. fixed stress and life With insp. random stress and life

4.5 4

Prob. of failure

3.5 3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8 1 1.2 Flight cycles

1.4

1.6

1.8

2 104

Figure 14.10 Random inspection, stress, and life scatter (All with COV = 0.1); 500 IS samples.

We will now demonstrate the use of failure samples for inspection optimization. Figure 14.11 shows the failure samples (i.e., for N < 20 000 cycles) in the threedimensional u-space of defect, stress scatter, and life scatter. These samples were created using FPA, a Fast Probability Analyzer software that integrates the Metropolis-Hasting

390

Structural design optimization considering uncertainties

u3 (Life scatter)

3

0

3

3 0

0

u1 (Defect size)

3

u2 (Life scatter)

Figure 14.11 Failure samples in the three-dimensional u-space.

1.2

104

1

Pf reduction

0.8 0.6 0.4 Optimal time ⴝ 13 300 cycles

0.2 0

0

0.2

0.4

0.6

0.8 1 1.2 Inspection time

1.4

1.6

1.8

2 104

Figure 14.12 Risk reduction plot for inspection optimization.

algorithm and importance sampling method. In the example, the selected proposal distribution was a uniform distribution with a range of 1. The samples were generated in the failure region that has a probability of 0.195 with approximately +/−20% error at 90% confidence. Using the failure samples and applying Equation 21, the risk-reduction versus inspection can be computed easily to create figure 14.12. The optimal time of 13 300 cycles is approximate due to the relatively small sample size. Figure 14.13 shows pf versus time for three inspection times: 10 000, 13 300, and 16 000 cycles. Clearly, inspection

A r e l i a b i l i t y-b a s e d m a i n t e n a n c e o p t i m i z a t i o n m e t h o d o l o g y

391

104

5

Insp. at N = 10 000 Insp. at N = 13 300 Insp. at N = 16 000

4.5 4

Prob. of failure

3.5 3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8 1 1.2 Flight cycles

1.4

1.6

1.8

2 104

Figure 14.13 Probability of failure curves for one inspection at 1000, 13 300, and 16 000 cycles.

1.2 1 0.8

S

0.6 0.4 0.2 0 0.2 0.4

1 Defect

2 Stress scatter

3 Life scatter

Figure 14.14 Probability sensitivities for three random variables.

at 10 000 is too early, at 16 000 is too late, and at 13 300 is significantly better. Using Equation 21, the failure samples were used to compute the mean sensitivities, as displayed in Figure 14.14, which shows that the initial defect size is the most influential random variable, followed by stress scatter and life scatter.

392

Structural design optimization considering uncertainties

6R

t P
140 KN

2R r

P 2R

4R

P

R

Reference R
0.25 m, Thickness
67 mm, Initial flaw size
0.4 mm

Figure 14.15 Spindle lug (Forth et al. 2002).

CC03 P/Wt

W

P S3
— Dt c

P

a

t

D c

Figure 14.16 NASGRO model for the lug example.

4.2 L u g ex a m p le A helicopter spindle lug model is shown in Figure 14.15 (Forth et al. 2002) with its fracture mechanics model (using NASGRO software) shown in Figure 14.16. Figure 14.17 shows a one-hour load spectra, FELIX/28 based on the main rotor blade of a military helicopter with four mission types and 140 flights (Everett et al. 2002). This study was conducted using an RBDT software that integrated a probabilistic function evaluation system (Wu and Shin 2005; Wu et al. 2006), a finite element software (ANSYS), and a fracture mechanics software, NASGRO. The random variables are listed in Table 14.2, where the load random variable represents the point load applied to the center of the pin, P, in Figure 14.16. The initial flaw size distribution, shown in Figure 14.18, is based on an equivalent initial flaw size distribution (Forth et al. 2002), EIFS, derived from stress-life experiments. We will compare the performance of three PODs shown in Figure 14.19 representing poor, fair, and good NDE devices.

A r e l i a b i l i t y-b a s e d m a i n t e n a n c e o p t i m i z a t i o n m e t h o d o l o g y

393

Percent of max. load in spectrum

400 300 200 100 0 Cycles : 2755 Flight hours ⴝ 3.26 hr

100 200

0

500

1000 1500 2000 Number of cycles

2500

3000

Figure 14.17 Felix/28 Helicopter load spectra (Everett et al. 2002). Table 14.2 Random variables for the Lug model.

Thickness, t (mm) Max. load (N) Initial flaw size (mm) Delta Kth Life scatter

Distribution

Mean

Std. Dev.

COV(%)

LN LN User-defined LN LN

28 145 000 0.074 48 1

0.14 10 000 0.0224 4 0.1

0.50 6.9 30.2 8.33 10.0

Equivalent initial flaw size 1 0.8

CDF

0.6 0.4 0.2 0

0

0.05

0.1 0.15 0.2 Flaw size (mm)

0.25

0.3

Figure 14.18 Equivalent initial flaw size distribution.

We will use conservative limit states to illustrate the TIS steps discussed in Section 3.3.3.2. Table 14.3 shows the probabilistic analysis results using three A values: 1, 1.07, and 1.33, which are associated with three target lives. Because the conditional probability of failure given a flaw is relatively high (about 7%), 1000 Monte Carlo samples are sufficient for illustration purposes.

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Three POD curves (Log-logistic function)

Probability of detection

1 POD I POD II POD III

0.8 0.6

0.4 0.2

0

2

4 6 Defect size (mm)

8

10

Figure 14.19 Three POD curves for the lug example.

Table 14.3 Lug example results.

A 1.00 1.07 1.33 MCS

N = Nf 750 800 1000 750

No. of samp. in samp. region

FORM and samp. Beta

Angle from FORM MPP

No. of failures in samp. region

Prob. in samp. region (Ps)

P f in samp. region (Pc)

Pf

n

Beta 1.495 1.709 1.682 1.587

Deg. 0.0 33.6 22.1 17.3

[N < 750] – 216 222 208

[N < Nf ] – 0.081 0.157 1.000

[N < 750] – 0.864 0.444 0.069

[N < 750] 0.0675 0.0699 0.0696 0.0693

– 250 500 3000

Based on A = 1.067, the probability in the IS (importance sampling) region is 0.081. Of the 250 samples generated in this region, there are 216 failure samples with a conditional pof of 0.864. Therefore pof is 0.0699, which is close to the FORM solution, 0.0675. The agreement, and the fact that the angles between the FORM and the sampling MPPs are reasonably small, suggest that the IS region covers the failure region. Now consider using A = 1.33. The probability in the IS region is 0.151, about twice larger than for A = 1.067. Of the 500 samples generated in this region, there are 222 failure samples with a conditional pof of 0.444, about half smaller than for A = 0.167. The resulting pof is still 0.0699, as before. This means that by doubling the IS region, no additional failure region has been found. This further suggests that the IS region is sufficient, provided that the MPP-based model is reasonably good (which was true as determined using an independent check). The above results are very close to the Monte Carlo result (0.0693) that took 60 hours of CPU time. Note that 500 IS samples are equivalent to 500/0.151 = 3311

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Reducible risk (Pr) given a flaw POD 1 POD II POD III

0.03 0.025

Pr

0.02 0.015 0.01 0.005 0 0

100

200

300 400 500 600 Inspection time (flight hours)

700

800

Figure 14.20 Lug inspection optimization.

Monte Carlo samples, and therefore, for this example, IS provides a slightly better accuracy. However, unlike MCS, IS could miss failure regions. In general, a sufficient number of Monte Carlo samples should be used to ensure that IS has not missed any significant failure region. For complex applications with possible multiple MPPs, more robust error-checking should be considered, including Markov Chain Monte Carlo methods. Using the 222 saved simulated crack growth histories and applying Equation 21, the risk reduction curves for three PODs are obtained as shown in Figure 14.20. The unsmooth curves are due to lack of samples, but are still reasonable for illustration purposes. The results show that POD I is superior and that there is an “effective inspection window,’’ roughly between 300 to 650 hours, with an optimal inspection time at about 550 flight hours. When POD II and III are used, the best inspection time is around 650 hours with a narrower inspection window. These results confirm the earlier suggestions that (1) the best inspection time is earlier for a better POD capability, and (2) a better POD capability will always produce a better optimal risk reduction. Figure 14.21 displays the simulation results of pf versus flight hours with and without inspection for POD I. Clearly the slope of pf changes more drastically at about 550 hours, coinciding with the best inspection time.

5 Conclusions A reliability-based maintenance optimization (RBMO) methodology was presented, with a focus on computational strategies. The RBMO methodology was demonstrated using examples related to aircraft and helicopter structures. The examples suggest that the RBDT methodology is well suited for inspection planning, and appears to be applicable to other structures such as ships, cars, and oil and gas pipelines, to more

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Pf given a flaw for various PODs 0.08

No inspection POD III at 650 hours POD II at 650 hours POD I at 550 hours

0.07 0.06

Pf

0.05 0.04 0.03 0.02 0.01 0 0

100

200

300

400 500 Flight hours

600

700

800

Figure 14.21 Probability of failure for one inspection at three different times using POD-I.

systematically design reliable and economical structures with associated maintenance programs to sustain structural integrity and reliability. RBMO involves time-dependent damage accumulation models, NDE detections, repairs, replacements and other risk control measures, and an optimal maintenance plan must consider a potentially large number of options including inspection schedules, mitigation options, and selection of NDE devices. In addition, the planning must factor in uncertainties. Given the wide spectrum of the options and the complexities in modeling, the best practical way to conduct RBMO is through the use of random simulations, preferably efficient sampling methods. The TIS approach has been developed to face the challenge. At its core, TIS is a type of Monte Carlo method that uses the power of random simulation. However, drastic efficiency improvement can be achieved by systematically generating samples in the failure domain. When mitigation effects can be reasonably modeled using ideal repairs or replacements with original parts, additional speed improvement can be realized by reusing crack growth histories for various maintenance options. Methods for generating failure-only samples were discussed, including one built on the MPP-based linear surface of a conservative limit state and another based on the Metropolis-Hasting algorithm. It is emphasized that the MPP methods, while widely known and used, are limited to well-behaved functions. For TIS, MPP offers an easy way to generate independent failure samples. M-H, on the other hand, can handle more difficult (non-smooth and nonlinear) functions, but the generated Markov-chain failure samples are correlated and therefore more samples are needed to reach the target distribution. Thus, both methods provide useful tools for RBMO with different strengths and limitations. The disk and lug examples represented the feasibility of the RBMO method to physics-based modeling applications. The software used for the lug example integrated a probabilistic analysis module, a finite element module, and a fracture mechanics module. As an illustration of the analysis CPU time needed for RBMO, in one lug analysis

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using a 2 GHz desktop PC, it took several hours, including time for a FE analysis, to carry out a thousand NSAGRO analyses with the rotorcraft load spectra, which took the most time, and a probabilistic analysis, which took the least time. The CPU time would increase if larger FE models were used or more failure samples were generated. This example suggests that, even with the efficient TIS method, for complex problems involving physics-based models, RBDT can still be very time-consuming unless further model approximations are made. Potential approximation methods for RBMO analysis include kriging (Sacks et al. 1989; Martin & Simpson 2005) and moving least squares (Krishnamurthy 2003) with error-checking procedures.

References Ang, A.H.-S. & Tang, W.H. 1984. Probability Concepts in Engineering Planning and Design, Volume II; Decision, Risk, and Reliability, New York, John Wiley & Sons. Au, S.K. & Beck, J.L. 2001. Estimation of small failure probabilities in high dimensions by subset simulation, Probabilistic Engineering Mechanics, Vol. 16, No. 4, pp. 263–277. Berens, A.P., Hovey, P.W. & Skinn, D.A. 1991. Risk Analysis for Aging Aircraft Fleets, Air Force Wright Lab Report, WL-TR-91-3066, Vol. 1. Bucher, C.G. 1988. Adaptive Sampling – An Iterative Fast Monte Carlo Procedure, Structural Safety, Vol. 5, pp. 119–26. Cunha, S.B., De Souza, A.P.F., Nicolleti, E.S.M. & Aguiar, L.D. 2006. A Risk-Based Inspection Methodology to Optimize Pipeline In-Line Inspection Programs, Journal of Pipeline Integrity, Q3. Ditlevsen, O., Bjerager, Olesen, R. & Hasofer, A.M. 1989. Directional Simulation in Gaussian Space, Probabilistic Engineering Mechanics, Vol. 3, No. 4, pp. 207–217. Ditlevsen, O. & Madsen, H.O. 1996, Structural Reliability Methods. J. Wiley & Sons, New York, 384 pp. Der Kiureghian, A. & Dakessian, T. 1998. Multiple Design Points in First and Second-order Reliability, Structural Safety, Vol. 20, pp. 37–50. Der Kiureghian, A. 2005. First- and Second-Order Reliability Methods, Chapter 14 in Engineering Design Reliability Handbook, E. Nikolaidis, D.M. Ghiocel & S. Singhal, (eds), CRC Press, Boca Raton, FL. Everett, Jr. R.A. 2002. Crack-Growth Characteristics of Fixed and Rotary Wing Aircraft, 6th Joint FAA/DoD/NASA Aging Aircraft Conference. Enright, M.P. & Wu, Y.-T. 1999. Probabilistic Fatigue Life Sensitivity Analysis of Titanium Rotors, Proceedings of the AIAA 41st SDM Conference, Atlanta, GA. Forth, S.C., Everett, Jr. R.A. & Newman, J.A. 2002. A Novel Approach to Rotorcraft Damage Tolerance, 6th Joint FAA/DoD/NASA Aging Aircraft Conference. Gamerman, D. 1997. Markov Chain Monte Carlo, Chapman & Hall. Gentle, J.E. 1998. Random Number Generation and Monte Carlo Methods, Springer-Verlag New York. Harbitz, A. 1986. An Efficient Sampling Method for Probability of Failure Calculation. Structural Safety, Vol. 3, pp. 109–115. Harkness, H.H., Fleming, M., Moran, B. & Belytschko, T. 1994. Fatigue Reliability With In-Service Inspections, FAA/NASA International Symposium on Advanced Structural Integrity Methods for Airframe Durability and Damage Tolerance. Hohenbichler, R. & Rackwitz, R. 1988. Improvement of Second-order Reliability Estimates by Importance Sampling. J. Eng. Mech. ASCE, Vol. 114, No. 12, pp. 2195–2199. Kale, A., Haftka, R.T. & Sankar, B.V. 2007. Efficient Reliability Based Design and Inspection of Stiffened Panels Against Fatigue. Journal of Aircraft.

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Karamchandani, A. 1990. New Methods in Systems Reliability, Ph.D. dissertation, Stanford University. Karamchandani, A. & Cornell, C.A. 1991. Adaptive Hybrid Conditional Expectation Approaches for Reliability Estimation, Structural Safety, Vol. 11, pp. 59–74. Krishnamurthy, T. 2003. Response Surface Approximation with Augmented and Compactly Supported Radial Basis Functions, Proceedings of the AIAA 44th SDM Conference. Leverant, G.R., Littlefield, D.L., McClung, R.C., Millwater, H.R. & Wu, Y.-T. 1997. A Probabilistic Approach to Aircraft Turbine Rotor Material Design, The International Gas Turbine & Aeroengine Congress & Exhibition, Paper No. 97-GT-22, Orlando, FL. Liu, P.-L. & Der Kiureghian, A. 1986. Multivariate Distribution Models with Prescribed Marginals and Covariances, Probabilistic Engineering Mechanics, Vol. 1, No. 2, pp. 105–112. Martin, J.D. & Simpson, T.W. 2005. Use of Kriging Models to Approximate Deterministic Computer Models, AIAA Journal, Vol. 43, No. 4. Madsen, H.O., Krenk, S. & Lind, N.C. 1986. Methods of Structural Safety, Englewood Cliffs, New Jersey; Prentice Hall. Madsen, H.O., Skjong, R.K., Talin, A.G. & Kirkemo, F. 1987. Probabilistic Fatigue Crack Growth Analysis of Offshore Structures, with Reliability Updating Through Inspection, SNAME, Arlington, VA. Melchers, R.E. 1987. Structural Reliability: Analysis and Prediction, Wiley. Millwater, H.R., Wu, Y.-T., Cardinal, J.W. & Chell, G.G. 1996. Application of Advanced Probabilistic Fracture Mechanics to Life Evaluation of Turbine Rotor Blade Attachments, Journal of Engineering for Gas Turbines and Power, Vol. 118, pp. 394–398. Millwater, H.R., Fitch, S., Riha, D.S., Enright, M.P., Leverant, G.R., McClung, R.C., Kuhlman, C.J., Chell, G.G. & Lee, Y.-D. 2000. A Probabilistically-Based Damage Tolerance Analysis Computer Program for Hard Alpha Anomalies In Titanium Rotors, Proceedings, 45th ASME International Gas Turbine & Aeroengine Technical Congress, Munich, Germany. Nikolaidis, E., Ghiocel, D.M. & Singhal, S. (eds). 2005. Engineering Design Reliability Handbook, CRC Press, Boca Raton, FL. Palmberg, B., Blom, A.F. & Eggwertz. 1987. Probabilistic Damage Tolerance Analysis of Aircraft Structures, In Probabilistic Fracture Mechanics and Reliability, J.W. Provan (ed.). Martinus Nijhoff Publishers. Rackwitz, R. 2001. Reliability Analysis – A Review and Some Perspectives, Structural Safety, Vol. 23, pp. 365–395. Robert, C.P. & Casella, G. 2004. Monte Carlo Statistical Methods. New York: Springer. Rosen Group, www.Roseninspection.net, 2004. Metal Loss Inspection Performance Specifications, Standard_CDP_POFspec_56_rev3.62.doc. Rosenblatt, M. 1952. Remarks on a Multivariate Transformation, The Annals of Mathematical Statistics 23(3), pp. 470–472. Sacks, J., Schiller, S.B. & Welch, W.J. 1989. Design for Computer Experiments, Technometrics, Vol. 31, No. 1. Schuëller, G.I. 1998. Structural Reliability – Recent Advances, Proc. 7th ICOSSAR’97, pp. 3–33. Shiao, M.C. & Wu, Y.-T. 2004. An Efficient Simulation-Based Method for Probabilistic Damage Tolerance Analysis With Maintenance Planning, Proceedings of the ASCE Specialty Conference on Probabilistic Mechanics and Reliability. Shiao, M.C. 2006. Risk-Based Maintenance Optimization, Proceedings of the International Conference on Structural Safety and Reliability. Thoft-Christensen, P. & Murotsu, Y. 1986. Application of Structural Systems Theory, Springer. Volker, A.W.F., Dijkstra, F.H., Heerings, J.H.A.M. & Terpstra, S. 2004. Modeling of NDE Reliability; Development of A POD-Generator, 16th WCNDT 2004 – World Conference on NDT.

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White, P., Barter, S. & Molent, L. 2002. Probabilistic Fracture Prediction Based On Aircraft Specific Fatigue Test Data, 6th Joint FAA/DoD/NASA Aging Aircraft Conference. Wu, Y.-T., Millwater, H.R. & Cruse, T.A. 1990. An Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions, AIAA Journal, Vol. 28, No. 9, pp. 1663–1669. Wu, Y-T., Enright, M.P. & Millwater, H.R. 2002. Probabilistic Methods for Design Assessment of Reliability With Inspection, AIAA Journal, Vol. 40, No. 5, pp. 937–946. Wu, Y.-T. & Shin, Y. 2004. Probabilistic Damage Tolerance Methodology For Reliability Design And Inspection Optimization, Proceedings of the AIAA 45th SDM Conference. Wu, Y.-T., Shiao, M., Shin, Y. & Stroud, W.J. 2005. Reliability-Based Damage Tolerance Methodology for Rotorcraft Structures, Transactions Journal of Materials and Manufacturing. Wu, Y.-T. & Shin, Y. 2005. Probabilistic Function Evaluation System for Maintenance Optimization, Proceedings of the AIAA 46th SDM Conference. Wu, Y.-T., Shin, Y., Sues, R. & Cesare, M. 2006. Probabilistic Function Evaluation System (ProFES) for Reliability-Based Design, Journal of Structural Safety, Vol. 28, Issues 1–2, pp. 164–195. Wu, Y.-T. & Mohanty, S. 2006. Variable Screening and Ranking Using Several Sampling Based Sensitivity Measures, Journal of Reliability Engineering and System Safety, Vol. 91, Issue 6, pp. 634–647.

Chapter 15

Overview of reliability analysis and design capabilities in DAKOTA with application to shape optimization of MEMS Michael S. Eldred Sandia National Laboratories, Albuquerque, NM, USA∗

Barron J. Bichon Vanderbilt University, Nashville, TN, USA

Brian M. Adams Sandia National Laboratories, Albuquerque, NM, USA

Sankaran Mahadevan Vanderbilt University, Nashville, TN, USA

ABSTRACT: Reliability methods are probabilistic algorithms for quantifying the effect of uncertainties in simulation input on response metrics of interest. In particular, they compute approximate response function distribution statistics (such as response mean, variance, and cumulative probability) based on specified probability distributions for input random variables. In this chapter, recent algorithm research in first and second-order local reliability methods is overviewed for both the forward reliability analysis of computing probabilities for specified response levels (the reliability index approach (RIA)) and the inverse reliability analysis of computing response levels for specified probabilities (the performance measure approach (PMA)). A number of algorithmic variations have been explored, and the effect of different limit state approximations, probability integrations, warm starting, most probable point search algorithms, and Hessian approximations is discussed. In addition, global reliability methods are presented for performing reliability analysis in the presence of nonsmooth, multimodal limit state functions. This set of reliability analysis capabilities is then used as the algorithmic foundation for reliability-based design optimization (RBDO) methods, and bi-level and sequential formulations are presented. These RBDO formulations may employ analytic sensitivities of reliability metrics with respect to design variables that either augment or define distribution parameters for the uncertain variables. Relative performance of these reliability analysis and design algorithms is presented for a number of benchmark test problems using the DAKOTA software, and algorithm recommendations are given. These recommended algorithms are subsequently applied to real-world applications in the probabilistic analysis and design of microelectromechanical systems (MEMS), and the calculation of robust and reliable MEMS designs is demonstrated.

∗ Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,

for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.

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1 Introduction Uncertainty quantification (UQ) is the process of determining the effect of input uncertainties on response metrics of interest. These input uncertainties may be characterized as either aleatory uncertainties, which are irreducible variabilities inherent in nature, or epistemic uncertainties, which are reducible uncertainties resulting from a lack of knowledge. Since sufficient data is generally available for aleatory uncertainties, probabilistic methods are commonly used for computing response distribution statistics based on input probability distribution specifications. Conversely, for epistemic uncertainties, data is generally sparse, making the use of probability theory questionable and leading to nonprobabilistic methods based on interval specifications. Reliability methods are probabilistic algorithms for quantifying the effect of aleatory input uncertainties on response metrics of interest. In particular, they perform UQ by computing approximate response function distribution statistics based on specified probability distributions for input random variables. These response statistics include response mean, response standard deviation, and cumulative or complementary cumulative distribution function (CDF/CCDF) response level and probability level pairings. These methods are often more efficient at computing statistics in the tails of the response distributions (events with low probability) than sampling-based approaches since the number of samples required to resolve a low probability can be prohibitive. Thus, these methods, as their name implies, are often used in a reliability context for assessing the probability of failure of a system when confronted with an uncertain environment. A number of classical reliability analysis methods are discussed in (Haldar and Mahadevan 2000), including Mean-Value First-Order Second-Moment (MVFOSM), First-Order Reliability Method (FORM), and Second-Order Reliability Method (SORM). More recent methods which seek to improve the efficiency of FORM analysis through limit state approximations include the use of local and multipoint approximations in Advanced Mean Value methods (AMV/AMV+ (Wu, Millwater, and Cruse 1990)) and Two-point Adaptive Nonlinearity Approximation-based methods (TANA (Wang and Grandhi 1994; Xu and Grandhi 1998)), respectively. Each of the FORM-based methods can be employed for “forward’’ or “inverse’’ reliability analysis through the reliability index approach (RIA) or performance measure approach (PMA), respectively, as described in (Tu, Choi, and Park 1999). The capability to assess reliability is broadly useful within a design optimization context, and reliability-based design optimization (RBDO) methods are popular approaches for designing systems while accounting for uncertainty. RBDO approaches may be broadly characterized as bi-level (in which the reliability analysis is nested within the optimization, e.g. (Allen and Maute 2004)), sequential (in which iteration occurs between optimization and reliability analysis, e.g. (Wu, Shin, Sues, and Cesare 2001; Du and Chen 2004)), or unilevel (in which the design and reliability searches are combined into a single optimization, e.g. (Agarwal, Renaud, Lee, and Watson 2004)). Bi-level RBDO methods are simple and general-purpose, but can be computationally demanding. Sequential and unilevel methods seek to reduce computational expense by breaking the nested relationship through the use of iterated or simultaneous approaches, respectively. In order to provide access to a variety of uncertainty quantification capabilities for analysis of large-scale engineering applications on high-performance parallel

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computers, the DAKOTA project (Eldred, Brown, Adams, Dunlavy, Gay, Swiler, Giunta, Hart, Watson, Eddy, Griffin, Hough, Kolda, Martinez-Canales, and Williams 2006) at Sandia National Laboratories has developed a suite of algorithmic capabilities known as DAKOTA/UQ (Wojtkiewicz, Jr., Eldred, Field, Jr., Urbina, and Red-Horse 2001). This package contains the reliability analysis capabilities described in this chapter and provides the foundation for the RBDO approaches. DAKOTA is freely available for download worldwide through an open source license. This chapter overviews recent algorithm research activities that have explored a variety of approaches for performing reliability analysis. In particular, forward and inverse local reliability analyses have been explored using multiple limit state approximation, probability integration, warm starting, Hessian approximation, and optimization algorithm selections. New global reliability analysis methods based on Gaussian process surrogate models have also been explored for handling response functions which may be nonsmooth or multimodal. Finally, these reliability analysis capabilities are used to provide a foundation for exploring bi-level and sequential RBDO formulations. Sections 2 and 3 describe these algorithmic components, Section 4 summarizes computational results for several analytic benchmark test problems, Section 5 presents deployment of these methodologies to the probabilistic analysis and design of MEMS, and Section 6 provides concluding remarks.

2 Reliability method formulations 2.1

Mean Value method

The Mean Value method (MV, also known as MVFOSM in (Haldar and Mahadevan 2000)) is the simplest, least-expensive reliability method because it estimates the response means, response standard deviations, and all CDF/CCDF responseprobability-reliability levels from a single evaluation of response functions and their gradients at the uncertain variable means. This approximation can have acceptable accuracy when the response functions are nearly linear and their distributions are approximately Gaussian, but can have poor accuracy in other situations. The expressions for approximate response mean µg , approximate response standard deviation σg , response target to approximate probability/reliability level mapping (z → p, β), and probability/reliability target to approximate response level mapping (p, β → z) are µg = g(µx ) σg =

 i

j

(1) Cov(i, j)

dg dg (µx ) (µx ) dxi dxj

(2)

βcdf =

µg − z σg

(3)

βccdf =

z − µg σg

(4)

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z = µg − σg βcdf

(5)

z = µg + σg βccdf

(6)

respectively, where x are the uncertain values in the space of the original uncertain variables (“x-space’’), g(x) is the limit state function (the response function for which probability-response level pairs are needed), and βcdf and βccdf are the CDF and CCDF reliability indices, respectively. With the introduction of second-order limit state information, MVSOSM calculates a second-order mean as µg = g(µx ) +

1  d2g Cov(i, j) (µx ) 2 dxi dxj i

(7)

j

This is commonly combined with a first-order variance (Eq. 2), since second-order variance involves higher order distribution moments (skewness, kurtosis) (Haldar and Mahadevan 2000) which are often unavailable. The first-order CDF probability p(g ≤ z), first-order CCDF probability p(g > z), βcdf , and βccdf are related to one another through p(g ≤ z) =
(−βcdf )

(8)

p(g > z) =
(−βccdf )

(9)

βcdf = −
−1 (p(g ≤ z))

(10)

βccdf = −
−1 (p(g > z))

(11)

βcdf = −βccdf p(g ≤ z) = 1 − p(g > z)

(12) (13)

where
() is the standard normal cumulative distribution function. A common convention in the literature is to define g in such a way that the CDF probability for a response level z of zero (i.e., p(g ≤ 0)) is the response metric of interest. The formulations in this chapter are not restricted to this convention and are designed to support CDF or CCDF mappings for general response, probability, and reliability level sequences. 2.2

L oc al MPP s e ar c h me t ho d s

Other local reliability methods solve a nonlinear optimization problem to compute a most probable point (MPP) and then integrate about this point to compute probabilities. Regardless of specified input probability distributions, the MPP search is performed in uncorrelated standard normal space (“u-space’’) since it simplifies the probability integration: the distance of the MPP from the origin has the meaning of the number of input standard deviations separating the median response from a particular response threshold. The transformation from correlated non-normal distributions (x-space) to uncorrelated standard normal distributions (u-space) is denoted as u = T(x) with the reverse transformation denoted as x = T −1 (u).

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These transformations are nonlinear in general, and possible approaches include the Rosenblatt (Rosenblatt 1952), Nataf (Der Kiureghian and Liu 1986), and Box-Cox (Box and Cox 1964) transformations. The nonlinear transformations may also be linearized, and common approaches for this include the RackwitzFiessler (Rackwitz and Fiessler 1978) two-parameter equivalent normal and the Chen-Lind (Chen and Lind 1983) and Wu-Wirsching (Wu and Wirsching 1987) threeparameter equivalent normals. The results in this chapter employ the Nataf nonlinear transformation which occurs in the following two steps. To transform between the original correlated x-space variables and correlated standard normals (“z-space’’), the CDF matching condition is used:
(zi ) = F(xi )

(14)

where F( ) is the cumulative distribution function of the original probability distribution. Then, to transform between correlated z-space variables and uncorrelated u-space variables, the Cholesky factor L of a modified correlation matrix is used: z = Lu

(15)

where the original correlation matrix for non-normals in x-space has been modified to represent the corresponding correlation in z-space (Der Kiureghian and Liu 1986). The forward reliability analysis algorithm of computing CDF/CCDF probability/reliability levels for specified response levels is called the reliability index approach (RIA), and the inverse reliability analysis algorithm of computing response levels for specified CDF/CCDF probability/reliability levels is called the performance measure approach (PMA) (Tu, Choi, and Park 1999). The differences between the RIA and PMA formulations appear in the objective function and equality constraint formulations used in the MPP searches. For RIA, the MPP search for achieving the specified response level z is formulated as minimize uT u subject to G(u) = z

(16)

and for PMA, the MPP search for achieving the specified reliability/probability level β, p is formulated as minimize subject to

±G(u) 2 uT u = β

(17)

where u is a vector centered at the origin in u-space and g(x) ≡ G(u) by definition. In the RIA case, the optimal MPP solution u∗ defines the reliability index from β = ±u∗ 2 , which in turn defines the CDF/CCDF probabilities (using Eqs. 8–9 in the case of first-order integration). The sign of β is defined by G(u∗ ) > G(0): βcdf < 0, βccdf > 0 G(u∗ ) < G(0): βcdf > 0, βccdf < 0

(18)

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where G(0) is the median limit state response computed at the origin in u-space (where βcdf = βccdf = 0 and first-order p(g ≤ z) = p(g > z) = 0.5). In the PMA case, the sign applied to G(u) (equivalent to minimizing or maximizing G(u)) is similarly defined by β βcdf < 0, βccdf > 0: maximize G(u) βcdf > 0, βccdf < 0: minimize G(u)

(19)

and the limit state at the MPP (G(u∗ )) defines the desired response level result. When performing PMA with specified p, one must compute β to include in Eq. 17. While this is a straightforward one-time calculation for first-order integrations (Eqs. 10–11), the use of second-order integrations complicates matters since the β corresponding to the prescribed p is a function of the Hessian of G (see Eq. 36), which in turn is a function of location in u-space. The β target must therefore be updated in Eq. 17 as the minimization progresses (e.g., using Newton’s method to solve Eq. 36 for β given p and κi ). This works best when β can be fixed during the course of an approximate optimization, such as for the AMV2 + and TANA methods described in Section 2.2.1. For second-order PMA without limit state approximation cycles (i.e., PMA SORM), the constraint must be continually updated and the constraint derivative should include ∇u β, which would require third-order information for the limit state to compute derivatives of the principal curvatures. This is impractical, so the PMA SORM constraint derivatives are only approximated analytically or estimated numerically. Potentially for this reason, PMA SORM has not been widely explored in the literature.

2.2.1 L i m it sta te a p p r o x im a t io n s There are a variety of algorithmic variations that can be explored within RIA/PMA reliability analysis. First, one may select among several different limit state approximations that can be used to reduce computational expense during the MPP searches. Local, multipoint, and global approximations of the limit state are possible. (Eldred, Agarwal, Perez, Wojtkiewicz, Jr., and Renaud 2007) investigated local first-order limit state approximations, and (Eldred and Bichon 2006) investigated local second-order and multipoint approximations. These techniques include: 1.

a single Taylor series per response/reliability/probability level in x-space centered at the uncertain variable means. The first-order approach is commonly known as the Advanced Mean Value (AMV) method: g(x) ∼ = g(µx ) + ∇x g(µx )T (x − µx )

(20)

and the second-order approach has been named AMV2 : g(x) ∼ = g(µx ) + ∇x g(µx )T (x − µx ) +

1 (x − µx )T ∇x2 g(µx )(x − µx ) 2

(21)

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same as AMV/AMV2 , except that the Taylor series is expanded in u-space. The first-order option has been termed the u-space AMV method: G(u) ∼ = G(µu ) + ∇u G(µu )T (u − µu )

(22)

where µu = T(µx ) and is nonzero in general, and the second-order option has been named the u-space AMV2 method: G(u) ∼ = G(µu ) + ∇u G(µu )T (u − µu ) + 3.

1 (u − µu )T ∇u2 G(µu )(u − µu ) 2

(23)

an initial Taylor series approximation in x-space at the uncertain variable means, with iterative expansion updates at each MPP estimate (x∗ ) until the MPP converges. The first-order option is commonly known as AMV+: g(x) ∼ = g(x∗ ) + ∇x g(x∗ )T (x − x∗ )

(24)

and the second-order option has been named AMV2 +: g(x) ∼ = g(x∗ ) + ∇x g(x∗ )T (x − x∗ ) + 4.

1 (x − x∗ )T ∇x2 g(x∗ )(x − x∗ ) 2

(25)

same as AMV+/AMV2 +, except that the expansions are performed in u-space. The first-order option has been termed the u-space AMV+ method. G(u) ∼ = G(u∗ ) + ∇u G(u∗ )T (u − u∗ )

(26)

and the second-order option has been named the u-space AMV2 + method: G(u) ∼ = G(u∗ ) + ∇u G(u∗ )T (u − u∗ ) + 5.

1 (u − u∗ )T ∇u2 G(u∗ )(u − u∗ ) 2

(27)

a multipoint approximation in x-space. This approach involves a Taylor series approximation in intermediate variables where the powers used for the intermediate variables are selected to match information at the current and previous expansion points. Based on the two-point exponential approximation concept (TPEA, (Fadel, Riley, and Barthelemy 1990)), the two-point adaptive nonlinearity approximation (TANA-3, (Xu and Grandhi 1998)) approximates the limit state as: g(x) ∼ = g(x2 ) +

1−p n n   xi,2 i pi ∂g 1 p p p (x2 ) (xi − xi,2i ) + (x) (xi i − xi,2i )2 ∂xi pi 2 i=1

i=1

(28)

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Structural design optimization considering uncertainties

where n is the number of uncertain variables and: ?   ∂g (x1 ) xi,1 ∂xi ln pi = 1 + ln ∂g xi,2 (x2 ) ∂x

(29)

i

H n pi pi 2 pi pi 2 (x − x ) i=1 i=1 (xi − xi,2 ) i i,1 +   1−p n  xi,2 i pi ∂g pi (x2 ) (xi,1 − xi,2 ) H = 2 g(x1 ) − g(x2 ) − ∂xi pi

(x) = n

(30)

(31)

i=1

6.

and x2 and x1 are the current and previous MPP estimates in x-space, respectively. Prior to the availability of two MPP estimates, x-space AMV+ is used. a multipoint approximation in u-space. The u-space TANA-3 approximates the limit state as: G(u) ∼ = G(u2 ) +

n  ∂G i=1

∂ui

1−pi

(u2 )

ui,2

pi

 p 1 p (u) (ui i − ui,2i )2 2 n

p

p

(ui i − ui,2i ) +

(32)

i=1

where:  ∂G pi = 1 + ln (u) = n

(u1 ) ∂ui ∂G (u2 ) ∂ui

pi i=1 (ui



?

 ln

H −

p ui,1i )2

+

H = 2 G(u1 ) − G(u2 ) −

ui,1 ui,2

n

7.

p

i=1

(34)

p

(ui i − ui,2i )2

n  ∂G i=1

(33)

∂ui



1−pi

(u2 )

ui,2

pi

p (ui,1i

−

p ui,2i )

(35)

and u2 and u1 are the current and previous MPP estimates in u-space, respectively. Prior to the availability of two MPP estimates, u-space AMV+ is used. the MPP search on the original response functions without the use of any approximations. Combining this option with first-order and second-order integration approaches results in the traditional first-order and second-order reliability methods (FORM and SORM).

The Hessian matrices in AMV2 and AMV2 + may be available analytically, estimated numerically, or approximated through quasi-Newton updates (see Section 2.2.3). The quasi-Newton variant of AMV2 + is conceptually similar to TANA in that both approximate curvature based on a sequence of gradient evaluations. TANA estimates curvature by matching values and gradients at two points and includes it through the use of exponential intermediate variables and a single-valued diagonal Hessian approximation. Quasi-Newton AMV2 + accumulates curvature over a sequence of points and then uses it directly in a second-order series expansion. Therefore, these methods may be expected to exhibit similar performance.

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The selection between x-space or u-space for performing approximations depends on where the approximation will be more accurate, since this will result in more accurate MPP estimates (AMV, AMV2 ) or faster convergence (AMV+, AMV2 +, TANA). Since this relative accuracy depends on the forms of the limit state g(x) and the transformation T(x) and is therefore application dependent in general, DAKOTA/UQ supports both options. A concern with approximation-based iterative search methods (i.e., AMV+, AMV2 + and TANA) is the robustness of their convergence to the MPP. It is possible for the MPP iterates to oscillate or even diverge. DAKOTA/UQ contains checks that monitor for this behavior; however, implementation of a robust model management approach (Giunta and Eldred 2000; Eldred and Dunlavy 2006) is an important area for future work. Another concern with TANA is numerical safeguarding. First, there is the possibility of raising negative xi or ui values to nonintegral pi exponents in Eqs. 30–32, and 34–35. This is particularly likely for u-space. Safeguarding techniques include the use of linear bounds scaling for each xi or ui , offseting negative xi or ui , or promotion of pi to integral values for negative xi or ui . In numerical experimentation, the offset approach has been the most effective in retaining the desired data matches without overly inflating the pi exponents. Second, there are a number of potential numerical difficulties with the logarithm ratios in Eqs. 29 and 33. In this case, a safeguarding strategy is to revert to either the linear (pi = 1) or reciprocal (pi = −1) ∂g ∂G (x1 ) or ∂u approximation based on which approximation has lower error in ∂x (u1 ). i i 2.2.2

Prob a b ilit y i ntegrati ons

The second algorithmic variation involves the integration approach for computing probabilities at the MPP, which can be selected to be first-order (Eqs. 8–9) or second-order integration. Second-order integration involves applying a curvature correction (Breitung 1984; Hohenbichler and Rackwitz 1988; Hong 1999). Breitung applies a correction based on asymptotic analysis (Breitung 1984): p =
(−βp )

n−1  i=1

1  1 + β p κi

(36)

where κi are the principal curvatures of the limit state function (the eigenvalues of an orthonormal transformation of ∇u2 G, taken positive for a convex limit state) and βp ≥ 0 (select CDF or CCDF probability correction to obtain correct sign for βp ). An alternate correction in (Hohenbichler and Rackwitz 1988) is consistent in the asymptotic regime (βp → ∞) but does not collapse to first-order integration for βp = 0: p =
(−βp )

n−1  i=1

1  1 + ψ(−βp )κi

(37)

φ() and φ() is the standard normal density function. (Hong 1999) applies where ψ() =
() further corrections to Eq. 37 based on point concentration methods. To invert a second-order integration and compute βp given p and κi (e.g., for second-order PMA as described in Section 2.2), Newton’s method can be applied as described in (Eldred and Bichon 2006). Additional probability integration approaches

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Structural design optimization considering uncertainties

can involve importance sampling in the vicinity of the MPP (Hohenbichler and Rackwitz 1988; Wu 1994), but are outside the scope of this chapter. While secondorder integrations could be performed anywhere a limit state Hessian has been computed, the additional computational effort is most warranted for fully converged MPPs from AMV+, AMV2 +, TANA, FORM, and SORM, and is of reduced value for MVFOSM, MVSOSM, AMV, or AMV2 . 2.2.3 Hessi a n a p p r o x im a t io n s To use a second-order Taylor series or a second-order integration when second-order information (∇x2 g, ∇u2 G, and/or κ) is not directly available, one can estimate the missing information using finite differences or approximate it through use of quasiNewton approximations. These procedures will often be needed to make second-order approaches practical for engineering applications. In the finite difference case, numerical Hessians are commonly computed using either first-order forward differences of gradients using ∇ 2 g(x) ∼ =

∇g(x + hei ) − ∇g(x) h

(38)

to estimate the ith Hessian column when gradients are analytically available, or secondorder differences of function values using ∇ 2 g(x) ∼ =

g(x + hei + hej ) − g(x + hei − hej ) − g(x − hei + hej ) + g(x − hei − hej ) 4h2 (39)

to estimate the ijth Hessian term when gradients are not directly available. This approach has the advantage of locally-accurate Hessians for each point of interest (which can lead to quadratic convergence rates in discrete Newton methods), but has the disadvantage that numerically estimating each of the matrix terms can be expensive. Quasi-Newton approximations, on the other hand, do not reevaluate all of the second-order information for every point of interest. Rather, they accumulate approximate curvature information over time using secant updates. Since they utilize the existing gradient evaluations, they do not require any additional function evaluations for evaluating the Hessian terms. The quasi-Newton approximations of interest include the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update Bk+1 = Bk −

Bk sk sTk Bk sTk Bk sk

+

yk ykT ykT sk

(40)

which yields a sequence of symmetric positive definite Hessian approximations, and the Symmetric Rank 1 (SR1) update Bk+1 = Bk +

(yk − Bk sk )(yk − Bk sk )T (yk − Bk sk )T sk

(41)

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which yields a sequence of symmetric, potentially indefinite, Hessian approximations. Bk is the kth approximation to the Hessian ∇ 2 g, sk = xk+1 − xk is the step and yk = ∇gk+1 − ∇gk is the corresponding yield in the gradients. The selection of BFGS versus SR1 involves the importance of retaining positive definiteness in the Hessian approximations; if the procedure does not require it, then the SR1 update can be more accurate if the true Hessian is not positive definite. Initial scalings for B0 and numerical safeguarding techniques (damped BFGS, update skipping) are described in (Eldred and Bichon 2006). 2.2.4 O ptimizat ion al gori thms The next algorithmic variation involves the optimization algorithm selection for solving Eqs. 16 and 17. The Hasofer-Lind Rackwitz-Fissler (HL-RF) algorithm (Haldar and Mahadevan 2000) is a classical approach that has been broadly applied. It is a Newton-based approach lacking line search/trust region globalization, and is generally regarded as computationally efficient but occasionally unreliable. DAKOTA/UQ takes the approach of employing robust, general-purpose optimization algorithms with provable convergence properties. This chapter employs the sequential quadratic programming (SQP) and nonlinear interior-point (NIP) optimization algorithms from the NPSOL (Gill, Murray, Saunders, and Wright 1998) and OPT++ (Meza 1994) libraries, respectively. 2.2.5 Wa rm starti ng of MPP searches The final algorithmic variation for local reliability methods involves the use of warm starting approaches for improving computational efficiency. (Eldred, Agarwal, Perez, Wojtkiewicz, Jr., and Renaud 2007) describes the acceleration of MPP searches through warm starting with approximate iteration increment, with z/p/β level increment, and with design variable increment. Warm started data includes the expansion point and associated response values and the MPP optimizer initial guess. Projections are used when an increment in z/p/β level or design variables occurs. Warm starts were consistently effective in (Eldred, Agarwal, Perez, Wojtkiewicz, Jr., and Renaud 2007), with greater effectiveness for smaller parameter changes, and are used for all computational experiments presented in this chapter.

2.3 G lobal reliability methods Local reliability methods, while computationally efficient, have well-known failure mechanisms. When confronted with a limit state function that is nonsmooth, local gradient-based optimizers may stall due to gradient inaccuracy and fail to converge to an MPP. Moreover, if the limit state is multimodal (multiple MPPs), then a gradientbased local method can, at best, locate only one local MPP solution. Finally, a linear (Eqs. 8–9) or parabolic (Eqs. 36–37) approximation to the limit state at this MPP may fail to adequately capture the contour of a highly nonlinear limit state. For these reasons, efficient global reliability analysis (EGRA) is investigated in (Bichon, Eldred, Swiler, Mahadevan and McFarland 2007).

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Structural design optimization considering uncertainties

In this approach, ideas from Efficient Global Optimization (EGO) (Jones, Shonlau, and Welch 1998) are adapted for use in reliability analysis. This approach employs a Gaussian process (GP) model to approximate the true response function ˆ G(u) = h(u)T β + Z(u)

(42)

where h( ) is the trend of the model, β is the vector of trend coefficients, and Z() is a stationary Gaussian process with zero mean and covariance defined from a squaredexponential correlation function that describes the departure of the model from its underlying trend. Gaussian process (GP) models are set apart from other surrogate models because they provide not just a predicted value at an unsampled point, but a full Gaussian distribution with an expected value and a predicted variance. This variance gives an indication of the uncertainty in the model, which results from the construction of the covariance function. This function is based on the idea that when input points are near one another, the correlation between their corresponding outputs will be high. As a result, the uncertainty associated with the model’s predictions will be small for input points which are near the points used to train the model, and will increase as one moves further from the training points. In EGO, the mean and variance estimates from the GP are used to form an expected improvement function (EIF), which calculates the expectation that any point in the search space will provide a better solution than the current best solution. An important feature of the EIF is that it provides a balance between exploiting areas of the design space where good solutions have been found, and exploring areas of the design space where the uncertainty is high. To adapt this concept to forward reliability analysis (z → p), an expected feasibility function (EFF) is used to provide an indication of how well the true value of the response is expected to satisfy the equality constraint G(u) = z by integrating over a region in the immediate vicinity of the threshold value z ± :
z+ ˆ ˆ EF(G(u)) = dG (43) [ − |z − G|] G(u) z−

where  is proportional to the standard deviation predicted by the GP at the point u. This integral can be evaluated analytically, as described in (Bichon, Eldred, Swiler, Mahadevan, and McFarland 2007), to create a simple GP-based function to maximize with a global optimization algorithm. Once a new point or points are computed which maximize the EFF, the GP is updated and the process is continued until the maximum EFF value falls below a tolerance. With a converged GP representation of the limit state, multimodal adaptive importance sampling is then applied to the GP to evaluate an approximation to the probabilities of interest.

3 Reliability-based design optimization Reliability-based design optimization (RBDO) methods are used to perform design optimization accounting for reliability metrics. The reliability analysis capabilities described in Section 2 provide a rich foundation for exploring a variety of RBDO formulations. (Eldred, Agarwal, Perez, Wojtkiewics, Jr., and Renaud 2007) investigated bi-level, fully-analytic bi-level, and first-order sequential RBDO approaches employing

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underlying first-order reliability assessments. (Eldred and Bichon 2006) investigated fully-analytic bi-level and second-order sequential RBDO approaches employing underlying second-order reliability assessments. These methods are overviewed in the following sections. 3.1

Bi-level RBDO

The simplest and most direct RBDO approach is the bi-level approach in which a full reliability analysis is performed for every optimization function evaluation. This involves a nesting of two distinct levels of optimization within each other, one at the design level and one at the MPP search level. Since an RBDO problem will typically specify both the z level and the p/β level, one can use either the RIA or the PMA formulation for the UQ portion and then constrain the result in the design optimization portion. In particular, RIA reliability analysis maps z to p/β, so RIA RBDO constrains p/β: minimize f subject to β ≥ β or p ≤ p

(44)

And PMA reliability analysis maps p/β to z, so PMA RBDO constrains z: minimize subject to

f z≥z

(45)

where z ≥ z is used as the RBDO constraint for a cumulative failure probability (failure defined as z ≤ z) but z ≤ z would be used as the RBDO constraint for a complementary cumulative failure probability (failure defined as z ≥ z). Note that many other objective and constraint formulations are possible (see (Eldred, Giunta, Wojtkiewicz Jr., and Trucano 2002) for general mappings which allow flexible use of statistics within multiple objectives, inequality constraints, and equality constraints); formulations with a deterministic objective and a single probabilistic inequality constraint are just convenient examples. An important performance enhancement for bi-level methods is the use of sensitivity analysis to analytically compute the gradients of probability, reliability, and response levels with respect to the design variables. When design variables are separate from the uncertain variables (i.e., they are not distribution parameters), then the following firstorder expressions may be used (Hohenbichler and Rackwitz 1986; Karamchandani and Cornell 1992; Allen and Maute 2004): ∇d z = ∇d g 1 ∇d g  ∇u G  = −φ( − βcdf )∇d βcdf

(46)

∇d βcdf =

(47)

∇d pcdf

(48)

where it is evident from Eqs. 12–13 that ∇d βccdf = −∇d βcdf and ∇d pccdf = −∇d pcdf . In the case of second-order integrations, Eq. 48 must be expanded to include the curvature

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Structural design optimization considering uncertainties

correction. For Breitung’s correction (Eq. 36), ⎡

⎛

⎢ ∇d pcdf = ⎢ ⎣
(−βp )

n−1 

⎜ −κi ⎜ ⎝ 2(1 + β κ ) 32 p i i=1

⎞ n−1  j=1 j =i

⎤ n−1 

⎟ ⎥ 1 1 ⎟ − φ(−βp ) ⎥ ∇d βcdf   ⎠ 1 + βp κj 1 + βp κi ⎦ i=1

(49) where ∇d κi has been neglected and βp ≥ 0 (see Section 2.2.2). Other approaches assume the curvature correction is nearly independent of the design variables (Rackwitz 2002), which is equivalent to neglecting the first term in Eq. 49. To capture second-order probability estimates within an RIA RBDO formulation using well-behaved β constraints, a generalized reliability index can be introduced where, similar to Eq. 10, ∗ βcdf = −
−1 (pcdf )

(50)

for second-order pcdf . This reliability index is no longer equivalent to the magnitude of u, but rather is a convenience metric for capturing the effect of more accurate probability estimates. Since reliability levels behave more linearly under design variable change than probability levels, replacing a second-order probability constraint with a generalized reliability constraint can improve optimization performance. The corresponding generalized reliability index sensitivity, similar to Eq. 48, is avoid numerical differencing across full reliability analyses, which can be costly or (worse) inaccurate. ∗ =− ∇d βcdf

1 ∇d pcdf ∗ φ(−βcdf )

(51)

where ∇d pcdf is defined from Eq. 49. Even when ∇d g is estimated numerically, Eqs. 46–51 can be used to avoid numerical differencing across full reliability analyses. When the design variables are distribution parameters of the uncertain variables, ∇d g is expanded with the chain rule and Eqs. 46 and 47 become ∇d z = ∇d x∇x g ∇d βcdf =

1 ∇d x∇x g  ∇u G 

(52) (53)

where the design Jacobian of the transformation (∇d x) may be obtained analytically for uncorrelated x or semi-analytically for correlated x (∇d L is evaluated numerically) by differentiating Eqs. 14 and 15 with respect to the distribution parameters. Eqs. 48–51 remain the same as before. For this design variable case, all required information for the sensitivities is available from the MPP search. Since Eqs. 46–53 are derived using the Karush-Kuhn-Tucker optimality conditions for a converged MPP, they are appropriate for RBDO using AMV+, AMV2 +, TANA, FORM, and SORM, but not for RBDO using MVFOSM, MVSOSM, AMV, or AMV2 .

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415

Sequential/Surrogate-bas ed RBDO

An alternative RBDO approach is the sequential approach, in which additional efficiency is sought through breaking the nested relationship of the MPP and design searches. The general concept is to iterate between optimization and uncertainty quantification, updating the optimization goals based on the most recent probabilistic assessment results. This update may be based on safety factors (Wu, Shin, Sues, and Cesare 2001) or other approximations (Du and Chen 2004). A particularly effective approach for updating the optimization goals is to use the p/β/z sensitivity analysis of Eqs. 46–53 in combination with local surrogate models (Zou, Mahadevan, and Rebba 2004). In (Eldred, Agarwal, Perez, Wojtkiewicz, Jr., and Renaud 2007) and (Eldred and Bichon 2006), first-order and second-order Taylor series approximations were employed within a trust-region model management framework (Giunta and Eldred 2000; Eldred and Dunlavy 2006) in order to adaptively manage the extent of the approximations and ensure convergence of the RBDO process. Surrogate models were used for both the objective function and the constraints, although the use of constraint surrogates alone is sufficient to remove the nesting. In particular, RIA trust-region surrogate-based RBDO employs surrogate models of f and p/β within a trust region k centered at dc . For first-order local surrogates: minimize f (dc ) + ∇d f (dc )T (d − dc ) subject to β(dc ) + ∇d β(dc )T (d − dc ) ≥ β or p(dc ) + ∇d p(dc )T (d − dc ) ≤ p  d − dc ∞ ≤ k

(54)

and for second-order local surrogates: minimize f (dc ) + ∇d f (dc )T (d − dc ) + 12 (d − dc )T ∇d2 f (dc )(d − dc ) subject to β(dc ) + ∇d β(dc )T (d − dc ) + 12 (d − dc )T ∇d2 β(dc )(d − dc ) ≥ β or p(dc ) + ∇d p(dc )T (d − dc ) + 12 (d − dc )T ∇d2 p(dc )(d − dc ) ≤ p  d − dc ∞ ≤ k

(55)

For PMA trust-region surrogate-based RBDO, surrogate models of f and z are employed within a trust region k centered at dc . For first-order surrogates: minimize f (dc ) + ∇d f (dc )T (d − dc ) subject to z(dc ) + ∇d z(dc )T (d − dc ) ≥ z  d − dc ∞ ≤ k

(56)

and for second-order surrogates: minimize f (dc ) + ∇d f (dc )T (d − dc ) + 12 (d − dc )T ∇d2 f (dc )(d − dc ) subject to z(dc ) + ∇d z(dc )T (d − dc ) + 12 (d − dc )T ∇d2 z(dc )(d − dc ) ≥ z  d − dc ∞ ≤ k

(57)

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Structural design optimization considering uncertainties

where the sense of the z constraint may vary as described previously. The second-order information in Eqs. 55 and 57 will typically be approximated with quasi-Newton updates. 3.3 Pro b l em f or mulat io n is s ues When performing RBDO in practice, a number of formulation issues arise. In particular, a flexible set of design parameterizations are needed for the input random variables and a rich set of output statistical metrics are needed for the optimization objectives and constraints. 3.3.1 In pu t p a ram e t e r iza t io n As described in Section 3.1, design variables in RBDO may be separate from the uncertain variables or they may define distribution parameters for the random variables. In the latter case, an implementation should first support design variable insertion into any of the native distribution parameters (e.g., mean, standard deviation, lower and upper bounds) for the supported probability distributions. While this supplies sufficient design authority for many distributions (e.g., normal, lognormal, extreme value distributions), other distributions (e.g., uniform, loguniform, triangular) do not directly support location and scale control within the native parameters. In these cases, location and scale are derived quantities and the native distribution parameters may be insufficient for design purposes, depending on the application. For example, the distribution parameters for a triangular distribution are lower bound, mode, and upper bound. Design control of any one of these three parameters independent of the other two is useful in some applications, but it will be insufficient to arbitrarily translate or scale the distribution in other applications. To provide additional design control in these cases, supporting the ability to design derived distribution parameters (from which the native parameters are updated) is an important extension. When gross distribution control (location, scale) and fine distribution control (native parameters) can both be provided, a broad range of design scenarios can be supported. 3.3.2 Ou tpu t m e t r ics Similar to the input parameterization, output metric characterization requires careful thought when developing optimization under uncertainty capabilities. In particular, a rich, expressive set of metrics is needed for arbitrary control of the shape of output distributions. Generally speaking, designing for robustness involves the control of moments; for example, minimizing an output variance statistic. On the other hand, design for reliability requires the control of tail statistics; for example, constraining a probability of failure statistic. Reliability methods are better suited for computing tail statistics, as MPP methods do not directly calculate moments. To control output variance, a PMAbased response interval, e.g. |zβ=3 − zβ = −3 |, may be substituted for the variance in order to achieve similar goals. To control both robustness and reliability, a multiobjective formulation can provide for general trade-off analysis. If, however, a particular reliability goal is known (e.g., β > 3), then formulations such as the two shown in Section 5 can be effective in reducing output variance while achieving prescribed reliability.

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Finally, model calibration under uncertainty studies typically involve the estimation of random variable distribution parameters which result in the best match in statistics between simulation and experiment. When using reliability methods, a convenient formulation is a nonlinear least squares objective which sums the discrepancies in CDF values (e.g., probabilities in an RIA RBDO formulation or response levels in a PMA RBDO formulation). In this case, all of the same analytic machinery applies (i.e., the sensitivity analysis of Section 3.1), only a broader set of distribution parameters may be of interest and a more complete set of CDF points may be required.

4 Benchmark problems (Eldred, Agarwal, Perez, Wojtkiewicz, Jr., and Renaud 2007) and (Eldred and Bichon 2006) have examined the performance of first and second-order local reliability analysis and design methods for four analytic benchmark test problems: lognormal ratio, short column, cantilever beam, and steel column. (Bichon, Eldred, Swiler, Mahadevan, and McFarland, 2007) has examined the performance of global reliability analysis methods for two additional analytic benchmark test problems that cause problems for local methods. 4.1

Local reliability analys is res ults

Within the reliability analysis algorithms, various limit state approximation (MVFOSM, MVSOSM, x-/u-space AMV, x-/u-space AMV2 , x-/u-space AMV+, x-/u-space AMV2 +, x-/u-space TANA, FORM, and SORM), probability integration (first-order or second-order), warm starting, Hessian approximation (finite difference, BFGS, or SR1), and MPP optimization algorithm (SQP or NIP) selections have been investigated. A sample comparison of reliability analysis performance, taken from the short column example, is shown in Tables 15.1 and 15.2 for RIA and PMA analysis, respectively, where “*’’ indicates that one or more levels failed to converge. Consistent with the employed probability integrations, the error norms are measured with respect Table 15.1 RIA results for short column problem. RIA Approach

SQP Function evaluations

NIP Function evaluations

CDF p Error norm

Target z Offset norm

MVFOSM MVSOSM x-space AMV u-space AMV x-space AMV2 u-space AMV2 x-space AMV+ u-space AMV+ x-space AMV2 + u-space AMV2 + x-space TANA u-space TANA FORM SORM

1 1 45 45 45 45 192 207 125 122 245 296* 626 669

1 1 45 45 45 45 192 207 131 130 246 278* 176 219

0.1548 0.1127 0.009275 0.006408 0.002063 0.001410 0.0 0.0 0.0 0.0 0.0 6.982e-5 0.0 0.0

0.0 0.0 18.28 18.81 2.482 2.031 0.0 0.0 0.0 0.0 0.0 0.08014 0.0 0.0

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Table 15.2 PMA results for short column problem. PMA Approach

SQP Function evaluations

NIP Function evaluations

CDF z Error norm

Target p Offset norm

MVFOSM MVSOSM x-space AMV u-space AMV x-space AMV2 u-space AMV2 x-space AMV+ u-space AMV+ x-space AMV2 + u-space AMV2 + x-space TANA u-space TANA FORM SORM

1 1 45 45 45 45 171 205 135 132 293* 325* 720 535

1 1 45 45 45 45 179 205 142 139 272 311* 192 191*

7.454 6.823 0.9420 0.5828 2.730 2.828 0.0 0.0 0.0 0.0 0.04259 2.208 0.0 2.410

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.598e-4 5.600e-4 0.0 6.522e-4

to fully-converged first-order results for MV, AMV, AMV2 , AMV+, and FORM methods, and with respect to fully-converged second-order results for AMV2 +, TANA, and SORM methods. Also, it is important to note that the simple metric of “function evaluations’’ is imperfect, and (Eldred and Bichon 2006) provides more detailed reporting of individual response value, gradient, and Hessian evaluations. Overall, reliability analysis results for the lognormal ratio, short column, and cantilever test problems indicate several trends. MVFOSM, MVSOSM, AMV, and AMV2 are significantly less expensive than the fully-converged MPP methods, but come with corresponding reductions in accuracy. In combination, these methods provide a useful spectrum of accuracy and expense that allow the computational effort to be balanced with the statistical precision required for particular applications. In addition, support for forward and inverse mappings (RIA and PMA) provide the flexibility to support different UQ analysis needs. Relative to FORM and SORM, AMV+ and AMV2 + has been shown to have equal accuracy and consistent computational savings. For second-order PMA analysis with prescribed probability levels, AMV2 + has additionally been shown to be more robust due to its ability to better manage β udpates. Analytic Hessians were highly effective in AMV2 +, but since they are often unavailable in practical applications, finite-difference numerical Hessians and quasi-Newton Hessian approximations were also demonstrated, with SR1 quasi-Newton updates being shown to be sufficiently accurate and competitive with analytic Hessian performance. Relative to first-order AMV+ performance, AMV2 + with analytic Hessians had consistently superior efficiency, and AMV2 + with quasi-Newton Hessians had improved performance in most cases (it was more expensive than first-order AMV+ only when a more challenging second-order p problem was being solved). In general, second-order reliability analyses appear to serve multiple synergistic needs. The same Hessian information that allows for more accurate probability integrations can also be applied to making MPP

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solutions more efficient and more robust. Conversely, limit state curvature information accumulated during an MPP search can be reused to improve the accuracy of probability estimates. For nonapproximated limit states (FORM and SORM), NIP optimizers have shown promise in being less susceptible to PMA u-space excursions and in being more efficient than SQP optimizers in most cases. Warm starting with projections has been shown to be consistently effective for reliability analyses, with typical savings on the order of 25%. The x-space and u-space approximations for AMV, AMV2 , AMV+, AMV2 +, and TANA were both effective, and the relative performance was strongly problemdependent (u-space was more efficient for lognormal ratio, x-space was more efficient for short column, and x-space and u-space were equivalent for cantilever). Among all combinations tested, AMV2 + (with analytic Hessians if available, or SR1 Hessians if not) is the recommended approach. An important question is how Taylor-series based limit state approximations (such as AMV+ and AMV2 +) can frequently outperform the best general-purpose optimizers (such as SQP and NIP) which may employ similar internal approximations. The answer likely lies in the exploitation of the structure of the RIA and PMA MPP problems. By approximating the limit state but retaining uT u explicitly in Eqs. 16 and 17, specific problem structure knowledge is utilized in formulating a mixed surrogate/direct approach. 4.2

G lobal reliability analys is res ults

Our test problem for demonstrating global reliability analysis is taken from (Bichon, Eldred, Swiler, Mahadevan, and McFarland 2007). It has a highly nonlinear response defined by: g(x) =

(x21 + 4)(x2 − 1) 5x1 − sin −2 20 2

(58)

The distribution of x1 is Normal(1.5, 1), x2 is Normal(2.5, 1), and the variables are uncorrelated. The response level of interest for this study is z = 0 with failure defined by g > z. Figure 15.1(a) shows a plot in u-space of the limit state throughout the ±5 standard deviation search space. This problem has several local optima to the forwardreliability MPP search problem (see Eq. 16). Figure 15.1(b) shows an example of an EGRA execution, with the total set of truth model evaluations performed from building the initial surrogate model and then repeatedly maximizing the expected feasibility function derived from the GP model. It is evident that the algorithm preferentially selects the data points needed to accurately resolve the limit state contour of interest. This multimodal problem was also solved using a number of local reliability methods for comparison purposes. Two approximation-based methods (AMV2 + and TANA) were investigated in x-space and u-space as well as the no approximation case (FORM/SORM). To produce results consistent with an implicit response function, numerical gradients and quasi-Newton Hessians from Symmetric Rank 1 updates were used. For each method, at the converged MPP, both first-order and second-order integration (using Eqs. 9 and 37) were used to calculate the probability.

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5 4 3 2 1 0 1 2 3 4 5 5 4 3 2 1

0

1

2

3

4

(a) Contour of the true limit state function

5

5 4 3 2 1 0 1 2 3 4 5 5 4 3 2 1 0 1 2 3 (b) Gaussian process approximation with data points generated by EGRA

4

5

Figure 15.1 Multimodal test problem.

Table 15.3 Results for the multimodal test problem. Reliability method

Function evaluations

First-order pf (% Error)

Second-order pf (% Error)

Sampling pf (% Error, Avg. Error)

No Approximation x-space AMV2 + u-space AMV2 + x-space TANA u-space TANA x-space EGRA u-space EGRA True LHS solution

66 26 26 506 131 50.4 49.4 1M

0.11798 (276.3%) 0.11798 (276.3%) 0.11798 (276.3%) 0.08642 (175.7%) 0.11798 (276.3%) — — —

0.02516 (−19.7%) 0.02516 (−19.7%) 0.02516 (−19.7%) 0.08716 (178.0%) 0.02516 (−19.7%) — — —

— — — — — 0.03127 (0.233%, 0.929%) 0.03136 (0.033%, 0.787%) 0.03135 (0.000%, 0.328%)

Table 15.3 gives a summary of the results from all methods. To establish an accurate estimate of the true solution, 20 independent studies were performed using 106 Latin hypercube samples per study. The average probability from these studies is reported as the “true’’ solution. Because the EGRA method is stochastic, it was also run 20 times and the average probability is reported. To measure the accuracy of the methods, two errors are reported for the EGRA results: the error in the average probability, and the average of the absolute errors from the 20 studies. For comparison, the same errors are given for the 20 LHS studies. Most of the MPP search methods converge to the same MPP (in the vicinity of (0.5, 1) in u-space) and thus report the same probability. These probabilities are more accurate when second-order integration is used, but still have significant errors. However, x-space TANA converges to a secondary MPP, which lies in a relatively flat region of the limit state (in the vicinity of (2, 1) in u-space). This local lack of curvature means that first-order and second-order integration produce approximately the same probability. In isolation, this second-order result could be viewed as a verification of the first-order probability and thus provide a misguided confidence in the local reliability

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Table 15.4 Analytic bi-level RBDO results, short column test problem. RBDO Approach

Function evaluations

Objective function

Constraint violation

RIA z → p x-space AMV+ RIA z → p x-space AMV2 + RIA z → p FORM RIA z → p SORM RIA z → β x-space AMV+ RIA z → β x-space AMV2 + RIA z → β FORM RIA z → β SORM PMA p, β → z x-space AMV+ PMA p → z x-space AMV2 + PMA β → z x-space AMV2 + PMA p, β → z FORM PMA p → z SORM PMA β → z SORM

149 129 911 1204 72 67 612 601 100 98 98 285 306 329

217.1 217.1 217.1 217.1 216.7 216.7 216.7 216.7 216.8 216.8 216.8 216.8 217.2 216.8

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

analysis. For this problem, the new EGRA method is more expensive than AMV2 +, but cheaper than all the other methods, and provides much more accurate results. Thus, global reliability analysis can provide accuracy similar to that of exhaustive sampling with expense comparable to local reliability. It handles both multimodal and nonsmooth limit states and does not require any derivative information from the response function. The primary limitation of the technique is dimensionality. For larger scale uncertainty quantification problems, the expense of building a global approximation grows quickly with dimension, although this can be mitigated to some extent by requiring accuracy only along a single contour and only in the highest probability regions. 4.3 RBDO results These reliability analysis capabilities provide a substantial foundation for RBDO formulations, and bi-level and sequential RBDO approaches based on local reliability analyses have been investigated. Both approaches have utilized analytic gradients for z, β, and p with respect to augmented and inserted design variables, and sequential RBDO has additionally utilized a trust-region surrogate-based approach to manage the extent of the Taylor-series approximations. A sample comparison of RBDO performance, taken again from the short column example, is shown in Tables 15.4 and 15.5 for bi-level and sequential surogate-based RBDO, respectively. Overall, RBDO results for the short column, cantilever, and steel column test problems build on the reliability analysis trends. Basic first-order bi-level RBDO has been evaluated with up to 18 variants (RIA/PMA with different p/β/z mappings for MV, x-/u-space AMV, x-/u-space AMV+, and FORM), and fully-analytic bi-level and sequential RBDO have each been evaluated with up to 21 variants (RIA/PMA with different p/β/z mappings for x-/u-space AMV+, x-/u-space AMV2 +, FORM, and SORM). Bi-level RBDO with MV and AMV are inexpensive but give only approximate

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Table 15.5 Surrogate-based RBDO results, short column test problem. RBDO Approach

Function evaluations

Objective function

Constraint violation

RIA z → p x-space AMV+ RIA z → p x-space AMV2 + RIA z → p FORM RIA z → p SORM RIA z → β x-space AMV+ RIA z → β x-space AMV2 + RIA z → β FORM RIA z → β SORM PMA p, β → z x-space AMV+ PMA p → z x-space AMV2 + PMA β → z x-space AMV2 + PMA p, β → z FORM PMA p → z SORM PMA β → z SORM

75 86 577 718 65 51 561 560 76 58 79 228 128 171

216.9 218.7 216.9 216.5 216.7 216.7 216.7 216.7 216.7 216.8 216.8 216.7 217.2 216.8

0.0 0.0 0.0 1.110e-4 0.0 0.0 0.0 0.0 2.1e-4 0.0 0.0 2.1e-4 0.0 0.0

optima. These approaches may be useful for preliminary design or for warm-starting other RBDO methods. Bi-level RBDO with AMV+ was shown to have equal accuracy and robustness to bi-level FORM-based approaches and be significantly less expensive on average. In addition, usage of β in RIA RBDO constraints was preferred due to it being more well-behaved and more well-scaled than constraints on p. Warm starts in RBDO were most effective when the design changes were small, with the most benefit for basic bi-level RBDO (with numerical differencing at the design level), decreasing to marginal effectiveness for fully-analytic bi-level RBDO and to relative ineffectiveness for sequential RBDO. However, large design changes were desirable for overall RBDO efficiency and, compared to basic bi-level RBDO, fully-analytic RBDO and sequential RBDO were clearly superior. In second-order bi-level and sequential RBDO, the AMV2 + approaches were consistently more efficient than the SORM-based approaches. In general, sequential RBDO approaches demonstrated consistent computational savings over the corresponding bi-level RBDO approaches, and the combination of sequential RBDO using AMV2 + was the most effective of all of the approaches. With initial trust region size tuning, sequential RBDO computational expense for these test problems was shown to be as low as approximately 40 function evaluations per limit state (35 for a single limit state in short column, 75 for two limit states in cantilever, and 45 for a single limit state in steel column). At this level of expense, probabilistic design studies can become tractable for expensive engineering applications.

5 Application to MEMS In this section, we consider the application of DAKOTA’s reliability algorithms to the design of micro-electro-mechanical systems (MEMS). In particular, we summarize results for the MEMS application described in (Adams, Eldred, and Wittwer 2006).

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Force

Actuation force

Switch contact Fmax

Anchors

Shuttle E2

E3

E1 vernier Fmin

(a) Scanning electron micrograph of a MEMS bistable mechanism in its second stable position. The attached vernier provides position measurements.

Displacement

(b) Schematic of force–displacement curve for bistable MEMS mechanism. The arrows indicate stability of equilibria E1 and E3 and instability of E2.

Figure 15.2 Bi-stable MEMS mechanism.

These types of application studies provide essential feedback on the performance of algorithms for real-world design applications, which may contain computational challenges not well-represented in analytically defined test problems. The reliability analysis and design results in (Adams, Eldred, and Wittwer 2006) are extended to include parameter-adaptive solution verification through the use of finite element a posteriori error estimation in (Adams, Bichon, Eldred, Carnes, Copps, Neckels, Hopkins, Notz, Subia, and Wittwer 2006; Eldred, Adams, Copps, Carnes, Notz, Hopkins, and Wittwer 2007). Pre-fabrication design optimization of microelectromechanical systems (MEMS) is an important emerging application of uncertainty quantification and reliability-based design optimization. Typically crafted of silicon, polymers, metals, or a combination thereof, MEMS serve as micro-scale sensors, actuators, switches, and machines with applications including robotics, biology and medicine, automobiles, RF electronics, and optical displays (Allen 2005). Design optimization of these devices is crucial due to high cost and long fabrication timelines. Uncertainty in the micromachining and etching processes used to manufacture MEMS can lead to large uncertainty in the behavior of the finished products, resulting in low part yield and poor durability. RBDO, coupled with computational mechanics models of MEMS, offers a means to quantify this uncertainty and determine a priori the most reliable and robust designs that meet performance criteria. Of particular interest is the design of MEMS bistable mechanisms which toggle between two stable positions, making them useful as micro switches, relays, and nonvolatile memory. We focus on shape optimization of compliant bistable mechanisms, where instead of mechanical joints, material elasticity enables the bistability of the mechanism (Kemeny, Howell, and Magleby 2002; Ananthasuresh, Kota, and Gianchandani 1994; Jensen, Parkinson, Kurabayashi, Howell, and Baker 2001). Figure 15.2(a) contains an electron micrograph of a MEMS compliant bistable mechanism in its second stable position. The first stable position is the as-fabricated position. One achieves transfer between stable states by applying force to the center shuttle via a thermal actuator, electrostatic actuator, or other means to move the shuttle past an unstable equilibrium.

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Tapered beam

Anchor Shuttle

Actuation force

(a) Schematic of a tapered beam bistable mechanism in as fabricated position (not to scale).

(b) Scale rendering of tapered beam leg for bistable mechanism.

Figure 15.3 Tapered beams for bistable MEMS mechanism.

Bistable switch actuation characteristics depend on the relationship between actuation force and shuttle displacement for the manufactured switch. Figure 15.2(b) contains a schematic of a typical force–displacement curve for a bistable mechanism. The switch characterized by this curve has three equilibria: E1 and E3 are stable equilibria whereas E2 is an unstable equilibrium (arrows indicate stability). A device with such a force–displacement curve could be used as a switch or actuator by setting the shuttle to position E3 as shown in Figure 15.2(a) (requiring large actuator force Fmax ) and then actuating by applying the comparably small force Fmin in the opposite direction to transfer back through E2 toward the equilibrium E1 . One could utilize this force profile to complete a circuit by placing a switch contact near the displaced position corresponding to maximum (closure) force as illustrated. Repeated actuation of the switch relies on being able to reset it with actuation force Fmax . The device design considered in this chapter is similar to that in the electron micrograph in Figure 15.2(a), for which design optimization has been previously considered (Jensen, Parkinson, Kurabayashi, Howell, and Baker 2001), as has robust design under uncertainty with mean value methods (Wittwer, Baker, and Howell 2006). The primary structural difference in the present design is the tapering of the legs, shown schematically in Figure 15.3(a). Figure 15.3(b) shows a scale drawing of one tapered beam leg (one quarter of the full switch system). A single leg of the device is approximately 100 µm wide and 5–10 µm tall. This topology is a cross between the fully compliant bistable mechanism reported in (Jensen, Parkinson, Kurabayashi, Howell, and Baker 2001) and the thickness-modulated curved beam in (Qiu and Slocum 2004). As described in the optimization problem below, this tapered geometry offers many degrees of freedom for design. The tapered beam legs of the bistable MEMS mechanism are parameterized by the 13 design variables shown in Figure 15.4, including widths and lengths of beam segments as well as angles between segments. For simulation, a symmetry boundary condition allowing only displacement in the negative y direction is applied to the right surface (x = 0) and a fixed displacement condition is applied to the left surface. With appropriate scaling, this allows the quarter model to reasonably represent the full four-leg switch system.

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1 0.5

W4 W3

0 0.5

W2

1 y (m)


−5.0). Reliability index βccdf ≥ 2 is required. The RBDO problem utilizes the RIA z → β approach (Eq. 44) with z = −5.0: max subject to

2 ≤ 50 ≤

A @ E Fmin (d, x) @ βccdf (d) A E F @ max (d, x)A ≤ 150 E@ E2 (d, x) A ≤ 8 E Smax (d, x) ≤ 3000

(60)

although the PMA β → z approach (Eq. 45) could also be used. The use of the Fmin metric in both the objective function and the reliability constraint results in a powerful problem formulation, because in addition to yielding a design with specified reliability, it also produces a robust design. By forcing the expected value of Fmin toward the −5.0 target while requiring two input standard deviations of surety, the optimization problem favors designs with less variability in Fmin . This renders the design performance less sensitive to uncertainties. The response PDF control is depicted in Figure 15.5(a), where the mean is maximized subject to a reliability constraint on the right tail. Alternatively, the response PDF control depicted in Figure 15.5(b) could be employed by maximizing the PMA z level corresponding to β = −2. This has the advantage of controlling both sides of the response PDF, but it is more computationally expensive since it requires the solution of two MPP optimization problems per design cycle instead of one. For this reason, the RIA RBDO formulation in Eq. 60 is used for all results in this section. Results using the MVFOSM, AMV2 +, and FORM methods are presented in Table 15.7 and the optimal force–displacement curves are shown in Figure 15.6. Optimization with MVFOSM reliability analysis offers substantial improvement over the initial design, yielding a design with actuation force Fmin nearer the −5.0 target and

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Table 15.7 Reliability formulation RBDO: design variable bounds and optimal designs from MVFOSM, AMV2 +, and FORM methods for MEMS bistable mechanism. Variable/metric l.b.

Name

AMV 2 + Optimal

FORM Optimal

−26.29 5.376 68.69 4.010 470 3.771 3.771

−5.896 2.000 50.01 5.804 1563 1.804 1.707

−6.188 1.998 57.67 5.990 1333 – 1.784

−6.292 1.999 57.33 6.008 1329 – –

150 8 1200

70

2

60

2.5

50

3

30

AMV2 Target force

4

4.5

20

5

10

5.5

0

6

10

MVFOSM

3.5

40

Force (µN)

Force (µN)

MVFOSM Optimal

u.b.

E[F min ] (µN) β E[F max ] (µN) E[E2 ] (µm) E[Smax ] (MPa) AMV2 + verified β FORM verified β

2 50

Initial

0

2

4 6 8 Displacement (µm)

10

6.5

6

6.5

7 7.5 Displacement (µm)

8

Figure 15.6 Optimal force–displacement curves resulting from RBDO of MEMS bistable mechanism with mean value and AMV2 + methods. The right plot shows the area near the minimum force. Two input standard deviations (as measured by the method used during optimization) separate F min from the target Fmin = −5.0.

tight reliability constraint β = 2. However, since mean value analyses estimate reliability based solely on evaluations at the means of the uncertain variables, they can yield inaccurate reliability metrics in cases of nonlinearity or nonnormality. In this example, the actual reliability (verified with MPP-based methods) of the optimal MVFOSMbased design is only 1.804 (AMV2 +) or 1.707 (FORM); both less than the prescribed reliability β ≥ 2. In this example, the additional computational expense incurred when using MPP-based reliability methods appears to be justified. Reliability-based design optimization with either the AMV2 + or FORM methods for reliability analysis yield constraint-respecting optimal beam designs with significantly different geometries than MVFOSM. The MPP-based methods yield a more conservative value of Fmin due to the improved estimation of β. Each of the three methods

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Fmin(∆W, Sr)

1.5 2

2.74

2.5

Residual stress Sr (MPa)

3 6.87

3.5 4

11

4.5 5

15.13

5.5 6

19.26

6.5 0.36

0.28

0.2

0.12

0.04

Width bias ∆W (µm)

Figure 15.7 Contour plot of F min (d, x) as a function of uncertain variables x (design variables d fixed at MVFOSM optimum). Dashed line: limit state g(x) = F min (x) = −5.0; plus sign: MPP from AMV2 +; circle: MPP from FORM; triangle indicates contour corresponding to F min = −6.2 (optimal expected value from MPP-based RBDO runs).

yields an improved design that respects the reliability constraint. The variability in Fmin has been reduced from approximately 5.6 (initial) to 0.52 (MVFOSM design), 0.67 (AMV2 + .design), or /0.65 (FORM design) µN per (FORM verified) input standard deviation E[Fminβ]−Fmin , resulting in designs that are less sensitive to input uncertainties. For the MVFOSM optimal design, the verified values of β calculated by AMV2 + and FORM differ by 6%, illustrating a typical challenge engineering design problems pose to reliability analysis methods. Figure 15.7 displays the results of a parameter study for the metric Fmin (d, x) as a function of the uncertain variables x for design variables fixed at the optimum from MVFOSM RBDO. Since the uncertain variables are both normal, the transformation to u-space used by AMV2 + and FORM is linear, so the contour plot is scaled to a ±3 standard deviation range in the native x-space. The relevant limit state for MPP searches, g(x) = Fmin (x) = −5.0, is indicated by the dashed line. For some design variable sets d (not depicted), the limit state is relatively well-behaved in the range of interest and first-order probability integrations would be sufficiently accurate. For the design variable set used to generate Figure 15.7, the limit state has significant nonlinearity, and thus demands more sophisticated probability integrations. The most probable points converged to by the AMV2 + and FORM methods are denoted in Figure 15.7 by the plus sign and circle, respectively. While the distance from each point

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to the origin differs slightly (see verified β values in Table 15.7), there clearly exist multiple candidates for the most probable point u satisfying Eq. 16. This appears to be a common occurrence when using RBDO methods: the tendency of the optimizer to push the design into a corner where the mean response is encircled by the failure domain. Unfortunately, even second-order probability integration does a poor job in these situations due to exception handling requirements for negative principal curvatures in Eqs. 36–37. This motivates future use of global reliability methods within RBDO to properly estimate the probabilities in these situations. Another computational difficulty observed during design optimization of an earlier bistable mechanism design is simulation failure resulting from model evaluation at extreme values of physical and/or geometric parameters. For example, during an MPP search, edge bias W might grow in magnitude into its left tail causing the effective width of the beam to shrink, possibly resulting in too flimsy a structure to simulate. In summary, highly nonlinear limit states, nonsmooth and multimodal limit states, and simulation failures caused by, e.g., evaluations in the tails of input distributions, pose challenges for RBDO in engineering applications, and must be mitigated through development of algorithms hardened against these challenges, careful attention to problem formulation, and ongoing simulation refinement.

6 Conclusions This chapter has overviewed recent algorithm research in first and second-order local reliability methods. A number of algorithmic variations have been presented, and the effect of different limit state approximations, probability integrations, warm starting, most probable point search algorithms, and Hessian approximations has been discussed. These local reliability analysis capabilities have provided the foundation for reliability-based design optimization (RBDO) methods, and bi-level and sequential formulations have been presented. The RBDO formulations employ analytic sensitivities of reliability metrics with respect to design variables that either augment or define distribution parameters for the uncertain variables. An emerging algorithmic capability is global reliability analysis methodologies which address the common limitations of local methods. In particular, nonsmooth limit states can cause convergence problems with gradient-based optimizers and probability integrations for highly nonlinear or multimodal limit states cannot be performed accurately using a low order polynomial representation from a single MPP solution. Efficient global reliability analysis (EGRA), on the other hand, can handle highly nonlinear and multimodal limit states and is insensitive to nonsmoothness since it does not require any derivative information from the response function. Relative performance of these reliability analysis and design algorithms has been measured for a number of benchmark test problems using the DAKOTA software. The most effective local techniques in these computational experiments have been AMV2 + for reliability analysis and second-order sequential/surrogate-based approaches for RBDO. In a low-dimensional multimodal example problem, global reliability analysis has been shown to provide accuracy similar to that of exhaustive sampling with expense comparable to local reliability. Continuing efforts in algorithm research will build on these successful methods through investigation of trust-region model management for approximation-based local reliability analysis, sequential RBDO with mixed surrogate

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and direct models (for probabilistic and deterministic components, respectively) and RBDO formulations based on global reliability assessment. These reliability analysis and design algorithms have been applied to real-world applications in the shape optimization of micro-electro-mechanical systems, and experiences with this deployment have been presented. Issues identified in deploying reliability methods to complex engineering applications include highly nonlinear, nonsmooth/noisy, and multimodal limit states, and potential simulation failures when evaluating parameter sets in the tails of input distributions. In addition, RBDO methods tend to exacerbate the reliability analysis challenges by exhibiting the tendency to push the design into a corner where the mean response is encircled by the failure domain. To mitigate these challenges, continuing development of new algorithms that have been hardened for engineering design applications, careful attention to design under uncertainty problem formulations, and refinements to modeling and simulation capabilities are recommended.

Acknowledgments The authors would like to express their thanks to the Sandia Computer Science Research Institute (CSRI) for support of this collaborative work between Sandia National Laboratories and Vanderbilt University.

References Adams, B.M., Bichon, B.J., Eldred, M.S., Carnes, B., Copps, K.D., Neckels, D.C., Hopkins, M.M., Notz, P.K., Subia, S.R. & Wittwer, J.W. 2006. Solution-verified reliability analysis and design of bistable mems using error estimation and adaptivity. Technical Report SAND2006-6286, Sandia National Laboratories, 2006, October, Albuquerque, NM. Adams, B.M., Eldred, M.S. & Wittwer J.W. 2006. Reliability-based design optimization for shape design of compliant micro-electro-mechanical systems. In Proceedings of the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Number AIAA2006-7000, 2006, September 6–8, Portsmouth, VA. Agarwal, H., Renaud, J.E., Lee, J.C. & Watson, L.T. 2004. A unilevel method for reliabilitybased design optimization. In Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Number AIAA-2004-2029, 2004, April 19–22, Palm Springs, CA. Allen, J.J. 2005. Micro Electro Mechanical System Design. Boca Raton: Taylor and Francis. Allen, M. & Maute, K. 2004. Reliability-based design optimization of aeroelastic structures. Struct. Multidiscip. Optim. 27:228–242. Ananthasuresh, G.K., Kota, S. & Gianchandani, Y. 1994. A methodical approach to the design of compliant micromechanisms. In Proc. IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, pp. 189–192. Bichon, B.J., Eldred, M.S., Swiler, L.P., Mahadevan, S. & McFarland, J.M. 2007. Multimodal reliability assessment for complex engineering applications using efficient global optimization. In Proceedings of the 9th AIAA Non-Deterministic Approaches Conference, Number AIAA2007-1946, 2007, April 23–26, Honolulu, HI. Box, G.E.P. & Cox, D.R. 1964. An analysis of transformations. J. Royal Stat. Soc. 26: 211–252. Breitung, K. 1984. Asymptotic approximation for multinormal integrals. J. Eng. Mech., ASCE 110(3):357–366.

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Chen, X. & Lind, N.C. 1983. Fast probability integration by three-parameter normal tail approximation. Struct. Saf. 1:269–276. Der Kiureghian, A. & Liu, P.L. 1986. Structural reliability under incomplete probability information. J. Eng. Mech. ASCE 112(1):85–104. Du, X. & Chen, W. 2004. Sequential optimization and reliability assessment method for efficient probabilistic design. J. Mech. Design 126:225–233. Eldred, M.S. & Bichon, B.J. 2006. Second-order reliability formulations in DAKOTA/UQ. In Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Number AIAA-2006-1828, 2006, May 1–4, Newport, RI. Eldred, M.S. & Dunlavy, D.M. 2006. Formulations for surrogate-based optimization with data fit, multifidelity, and reduced-order models. In Proceedings of the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Number AIAA-2006-7117, 2006, September 6–8, Portsmouth, VA. Eldred, M.S., Adams, B.M., Copps, K.D., Carnes, B., Notz, P.K., Hopkins, M.M. & Wittwer, J.W. 2007. Solution-verified reliability analysis and design of compliant microelectro-mechanical systems. In Proceedings of the 9th AIAA Non-Deterministic Approaches Conference, Number AIAA-2007-1934, 2007, April 23–26, Honolulu, HI. Eldred, M.S., Agarwal, H., Perez, V.M., Wojtkiewicz, S.F. Jr. & Renaud, J.E. 2007. Investigation of reliability method formulations in DAKOTA/UQ. Structure & Infrastructure Engineering: Maintenance, Management, Life-Cycle Design & Performance, Vol. 3, No. 3, Sept. 2007, pp. 199–213. Eldred, M.S., Brown, S.L., Adams, B.M., Dunlavy, D.M., Gay, D.M., Swiler, L.P., Giunta, A.A., Hart, W.E., Watson, J.-P., Eddy, J.P., Griffin, J.D., Hough, P.D., Kolda, T.G., MartinezCanales, M.L. & Williams, P.J. 2006. DAKOTA, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: Version 4.0 users manual. Technical Report SAND2006-6337, Sandia National Laboratories, Albuquerque, NM. See http://www.cs.sandia.gov/DAKOTA/software. html. Accessed October, 2006. Eldred, M.S., Giunta, A.A., Wojtkiewicz, S.F. Jr. & Trucano, T.G. 2002. Formulations for surrogate-based optimization under uncertainty. In Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Number AIAA-2002-5585, 2002, September 4–6, Atlanta, GA. Fadel, G.M., Riley, M.F. & Barthelemy, J.-F.M. 1990. Two point exponential approximation method for structural optimization. Structural Optimization 2(2):117–124. Gill, P.E., Murray, E.W., Saunders, M.A. & Wright, M.H. 1998. User’s guide for npsol 5.0: A fortran package for nonlinear programming. Technical Report SOL 86-1, System Optimization Laboratory, Stanford University, Stanford, CA. Giunta, A.A. & Eldred, M.S. 2000. Implementation of a trust region model management strategy in the DAKOTA optimization toolkit. In Proceedings of the 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Number AIAA-2000-4935, 2000, September 6–8, Long Beach, CA. Haldar, A. & Mahadevan, S. 2000. Probability, Reliability, and Statistical Methods in Engineering Design. New York: Wiley. Hohenbichler, M. & Rackwitz, R. 1986. Sensitivity and importance measures in structural reliability. Civil Eng. Syst. 3:203–209. Hohenbichler, M. & Rackwitz, R. 1988. Improvement of second-order reliability estimates by importance sampling. J. Eng. Mech., ASCE 114(12):2195–2199. Hong, H.P. 1999. Simple approximations for improving second-order reliability estimates. J. Eng. Mech. ASCE 125(5):592–595. Jensen, B.D., Parkinson, M.B., Kurabayashi, K., Howell, L.L. & Baker, M.S. 2001. Design optimization of a fully-compliant bistable micro-mechanism. In Proc. 2001 ASME Intl. Mech. Eng. Congress and Exposition, 2001, November 11–16, New York, NY.

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Jones, D.R., Shonlau, M. & Welch, W. 1998. Efficient global optimization of expensive blackbox functions. INFORMS J. Comp. 12:272–283. Karamchandani, A. & Cornell, C.A. 1992. Sensitivity estimation within first and second order reliability methods. Struct. Saf. 11:95–107. Kemeny, D.C., Howell, L.L. & Magleby, S.P. 2002. Using compliant mechanisms to improve manufacturability in MEMS. In Proc. 2002 ASME DETC, Number DETC2002/DFM-34178. Meza, J.C. 1994. OPT++: An object-oriented class library for nonlinear optimization. Technical Report SAND94-8225, Sandia National Laboratories, 1994, March, Albuquerque, NM. Qiu, J. & Slocum, A.H. 2004. A curved-beam bistable mechanism. J. Microelectromech. Syst. 13(2):137–146. Rackwitz, R. 2002. Optimization and risk acceptability based on the Life Quality Index. Struct. Saf. 24:297–331. Rackwitz, R. & Fiessler, B. 1978. Structural reliability under combined random load sequences. Comput. Struct. 9:489–494. Rosenblatt, M. 1952. Remarks on a multivariate transformation. Ann. Math. Stat. 23(3): 470–472. Tu, J., Choi, K.K. & Park, Y.H. 1999. A new study on reliability-based design optimization. J. Mech. Design 121:557–564. Wang, L. & Grandhi, R.V. 1994. Efficient safety index calculation for structural reliability analysis. Comput. Struct. 52(1):103–111. Wittwer, J.W., Baker, M.S. & Howell, L.L. 2006. Robust design and model validation of nonlinear compliant micromechanisms. J. Microelectromechanical Sys. 15(1). To appear. Wojtkiewicz, Jr. S.F., Eldred, M.S., Field, Jr. R.V., Urbina, A. & Red-Horse, J.R. 2001. A toolkit for uncertainty quantification in large computational engineering models. In Proceedings of the 42rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Number AIAA-2001-1455, 2001, April 16–19, Seattle, WA. Wu, Y.-T. 1994. Computational methods for efficient structural reliability and reliability sensitivity analysis. AIAA J. 32(8):1717–1723. Wu, Y.-T. & Wirsching, P.H. 1987. A new algorithm for structural reliability estimation. J. Eng. Mech. ASCE 113:1319–1336. Wu, Y.-T., Millwater, H.R., & Cruse, T.A. 1990. Advanced probabilistic structural analysis method for implicit performance functions. AIAA J. 28(9):1663–1669. Wu, Y.-T., Shin, Y., Sues, R. & Cesare, M. 2001. Safety-factor based approach for probabilitybased design optimization. In Proceedings of the 42rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Number AIAA-2001-1522, 2001, April 16–19, Seattle, WA. Xu, S. & Grandhi, R.V. 1998. Effective two-point function approximation for design optimization. AIAA J. 36(12):2269–2275. Zou, T., Mahadevan, S. & Rebba, R. 2004. Computational efficiency in reliability-based optimization. In Proceedings of the 9th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability, 2004, July 26–28, Albuquerque, NM.

Part 2

Robust Design Optimization (RDO)

Chapter 16

Structural robustness and its relationship to reliability Jorge E. Hurtado National University of Colombia, Manizales, Colombia

ABSTRACT: Two main ways of incorporating structural uncertainties in design optimization under a probabilistic point of view have been proposed in the international literature: robust and reliability-based design options. While the former is oriented to a reduction of the spread of critical responses, the latter aims to control the probabilities of failure. However, since the reduction of response spread does not preclude a regard to extreme cases, both methods can be considered as complementary. This makes desirable to have at a disposal methods yielding a design satisfying reliability and robustness criteria. In this chapter, methods allowing a simultaneous calculation of the leading probabilistic quantities used in these approaches are examined. It is shown that the the combination of the saddlepoint expansion of probability density together with the method of point estimates for approximating the response statistical moments yields good results. The chapter also deals with a rigorous definition of robustness, which is somewhat loose in the literature. It is shown that the entropy concept is highly appealing to such a purpose, as it shows similarities with that of controllability and stability concepts in dynamic systems theory. On this basis the concept of Robustness Assurance in structural design is introduced, paralleling that of Quality Assurance in the construction phase. A practical method for robust optimal design interpreted as entropy minimization is presented for the common case of linear structures.

1 Introduction 1.1 Theoretic al and practical approache s to unc e rtai nty In an essay that deserves attention (Sexsmith 1999), R. G. Sexsmith remarks that, despite the rapid development of structural reliability theories and methods, their inclusion into the design practice by structural engineers has been little. This rejection is attributed by the author mainly to educational problems. In fact, modern natural science, which arose from mechanical sciences developed in the sixteenth century on the basis of a mathematical interpretation of nature, showed in its beginnings a trend to interpret the random results of experiments as a deficiency of mathematical models rather than as a property of nature itself. In another interesting essay, G. F. Klir quotes the explicit objection common in the XIX century uncertainty jargon (Klir 1997). In those times, uncertainty was rejected as a natural phenomenon because the illusion of a science providing exact answers was still alive and enthusiastic. However, the introduction of the mathematical models for probability and randomness became absolutely

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necessary to explain phenomena in thermodynamics and quantum mechanics. From that time on, the old paradigm of a exact science was abandoned in those areas where the evidence and the magnitude of randomness could no longer be ignored. Nowadays, as a consequence of the need of considering complex systems, we assist to the development of proposals intended to enhance or overcome the modeling of randomness and uncertainty offered by probability theory, such as possibility theory (Dubois and Prade 1988), fuzzy set theory (Kosko 1992, e.g.), interval analysis (Kharitonov 1997; Hansen and Walster 2004), clustering analysis (Ben-Haim 1985; Ben-Haim 1996; Elishakoff 1999), ellipsoidal modeling (Chernousko 1999), etc. Structural and mechanical engineering continue the tradition of classical mechanics as developed by Galileo and Newton. This fact may be invoked to explain the above mentioned reluctance to include uncertainty models in structural design, at a difference to the well established central consideration of randomness in quantum mechanics and other, later branches of physics. In the author’s opinion, the explanation of this fact cannot entirely be attributed to this historical reason. In addition to this, there is the smaller randomness present in most structural and mechanical situations (with the exemption of earthquake loads and others), as compared to that present in quantum mechanics. But more important is the nature of the challenge posed to the structural engineer, namely, the design of an object. In Kant’s terms, modern science is oriented by the two-way approach of analysis (a priori mathematical principles, which are exact) and synthesis (a posteriori empirical facts, to which mathematical models must accommodate), and it offers a knowledge that remains valid for some time until it is falsified, according to Popper’s theory of science. Engineering, on the contrary, aims to offer not a knowledge but a product, which is defended not by arguments but by its quality, whose cost must be a minimum and whose design must be produced in most cases with resort to simplifying rules. This implies taking decisions, which is a challenge not hanging over knowledge discovering. All this may perhaps explain the somewhat paradoxical fact that structural engineers, on the one hand, do not include probabilities into their calculations, but, on the other, have for long recognized the importance of uncertainties in the design practice, as expressed in the use of safety factors of several kinds and of statistical analysis of experiments for fixing their code values. From the practical design viewpoint it is not realistic to expect that in actual structural designs failure probabilities will ever be calculated as a part of conventional design process of most structures. Besides the educational and computational problems involved, there is the lack of sufficient probabilistic information on load and material parameters, the difficulties for interpreting such probabilities, the high sensitivity of these values to probabilistic models, the randomness of the results conveyed by the most universal method (Monte Carlo simulation), etc. But the main reason is and will be the pragmatism of design. Thus, it can be said that randomness is in fact considered in structural design, but in a manner that results quite unsatisfactory from the analytical, argumentative, mathematically-oriented point of view. In fact, the close examination of the relationship between safety factors and failure probabilities conducted recently by I. Elishakoff (Elishakoff 2005) shows that a link between them strongly depends on the probabilistic structure of the random variables x in hand1 . But this is just the information from 1 In this chapter an underlined letter indicates a random variable or a vector, if it is written boldface.

Non-underlined letters are used to denote either their realizations, their deterministic counterparts or deterministic variables in general.

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whose need safety factors intend to dispense with. The contradiction lies in the fact that reliability requires a scientifically-oriented calculation, whereas safety factors are a mere practical tool for producing a qualified product. The requirement of practical, sometimes simplifying approaches can be considered as the implicit but dominating rule in engineering design. This has fostered the development of the concept of robustness, meaning a product that exhibits strength with respect to variations or fluctuations of parameters, sometimes random, sometimes uncontrollable and sometimes unknown. A robust product assures the engineer that it can absorb such fluctuation without compromising its quality, which is its main feature. Such, in fact, is the rationale behind the concept of BIBO (bounded input, bounded output) stability of dynamic systems (Szidarovszky and Bahill 1992, e.g.). In that field, the certainty that under an action of a bounded input the system response will be also bounded is considered sufficient by the engineer to be relieved from the need of tracing the exact trajectory of the response in particular situations. In general, robust design orientation aims at overcoming the need of considering particular uncertain situations and to assure the designer the imperturbability of the system under the presence of unknown, unpredictable or random parameters. Notice, however, that the question for a quantitative measure of the uncertainty in extreme situations is not solved by the robustness approach. 1.2

Optimal structural des ign under un c e rtai nty

Arising from the deterministic orientation of modern engineering mentioned above, structural optimization is normally performed without regard to random fluctuations of the parameters. It consists in minimizing a cost function C(y) subject to deterministic constraints posed upon responses (displacements, stresses, etc) and geometrical quantities. Formally, this problem is expressed as (Haftka et al. 1990; Kirsch 1993, e.g.) Problem Deterministic optimization : find : y minimizing : C(y) subject to : fi (y) < Fi , i = 1, 2, . . . y− ≤ y ≤ y+

(1)

In this equation y is the vector of design variables, fi (y) are system responses depending on them and Fi are their limiting values. These constitute the so-called behavioral constraints. On the other hand y− and y+ are bounds imposed to the design variables, normally constituting geometric constraints. It is evident that the uncertainties present in loads, materials and elements are not explicitly taken into account. Thus the result may be a fragile structure with respect to random changes in the design parameters. Notice, however, that in the definition of the upper bounds in behavioral and geometric constraints there is an implicit recognition of the risk associated to values excessively low. The decision on these bounds is normally taken on the basis of safety factors, which simply express a caution with respect to randomness and uncertainty. Anyhow, the robustness and the reliability of a structure designed using safety factors without regard to the probabilistic definition of the random variables present in it remain rather uncertain.

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Structural design optimization considering uncertainties

The explicit consideration of uncertainties in structural design optimization is a challenging task, as it demands the minimization of cost functions in a noisy environment generated by the presence of certain random variables. However, it can be considered an analysis of maximum importance, because it yields the solution to the ideal of producing a structural model that is both economical and safe. Two main families of methods have been proposed to this end: 1) Robust Design Optimization (RDO), which is oriented to minimizing the spread of the structural responses, as measured by low-order statistical moments. 2) Reliability-Based Design Optimization (RBDO), which minimizes the cost function with probabilistic constraints (Rosenblueth and Mendoza 1971; Gasser and Schuëller 1997; Frangopol 1995; Royset et al. 2001; Royset and Polak 2004). A common formulation of RBDO can be formally presented as follows: Problem Reliability-based optimization : find : y minimizing : C(y) (2) subject to : P[fi (x, y) > Fi ] ≤ Pi , i = 1, 2, . . . y− ≤ y ≤ y+ where x is a set of random variables, P[A] the probability of the random event A and Pi its limiting value. Function gi (x, y) = Fi − fi (x, y) is known in structural reliability as the limit state function. Other formulations than Eq. (2) are, however, possible. The following are some criticisms that have been addressed to the application of structural reliability for performing a structural optimization under uncertainty: • • •

•

The lack of information about actual probability models for materials and loads and the concern about the applicability of the published ones in every case. The sensitivity of the tails of the input probability density models to their parameters (Elishakoff 1991; Ben-Haim 1996; Elishakoff 1999). This undoubtedly affects the value of the failure probability. The difficulty of interpretation among the engineering community of the meaning of the failure probability. Despite structural safety researchers stress that it is a nominal failure indicator, there is a natural tendency to interpret it in the frequentist sense. On the other hand the Bayesian interpretation as a belief measure has gained favor but in the limited field of health monitoring and other tasks associated to existing structures. It is difficult to accommodate such an interpretation for new projects. These and other phenomena explain the little use of structural reliability concepts in design practice (Sexsmith 1999). The limitations of some popular methods of calculating such probabilities. For instance, FORM presents problems of accuracy for non-linear limit state functions and convergence problems (Schuëller and Stix 1987); the Response Surface Method exhibits problems of instability (Guan and Melchers 2001); Monte Carlo simulation requires high computational efforts, etc. There is a continuous effort among researchers for improving these techniques and developing new ones (Au and Beck 2001, e.g.), but there is no agreement about a method that satisfies both the requirements of generality, accuracy and low computational cost. An updated, general benchmark study is presently lacking.

Structural robustness and its relationship to reliability

•

439

The need of calculating one or more failure probabilities, which in some cases is a time-consuming task, for each trial model, increasing enormously the computational effort with respect to conventional, deterministic optimization and to reliability analysis as well. If Monte Carlo simulation is used this problem can be greatly alleviated if use is made of convenient to apply solver surrogate techniques such as neural networks (Papadrakakis et al. 1996; Hurtado and Alvarez 2001; Hurtado 2001) or support vector machines (Hurtado 2004a; Hurtado 2004b; Hurtado 2007). For performing the optimization, these methods can be combined with optimization techniques with biological optimization such as genetic algorithms, evolutionary strategies (Papadrakakis et al. 1998; Lagaros et al. 2002; Lagaros and Papadrakakis 2003), particle swarm optimization (Hurtado 2006), etc.

Some structural designers tend to favor the concept of robustness, understood as safety against unpredictable variations of the design parameters, over the concept of failure probability, which is normally a very low value lacking significant meaning in practice. This may be explained by the production-oriented approach of design discussed above. Robustness can be defined in several forms, depending on whether use is made of the clustering (Ben-Haim 1985; Ben-Haim 1996) or conventional, frequentist interpretation of uncertainty. In this chapter the second interpretation is adopted. The following formulation of robust design optimization corresponds to the proposal in (Doltsinis and Kang 2004; Doltsinis et al. 2005): Problem Robust optimization : find : y minimizing : subject to :

 C(y) = (1 − α)E[f (y)]/µ∗ + α Var[f (y)]/σ ∗ (3)  E[gi (y)] + βi Var[gi (y)] ≤ 0, i = 1, 2, . . .  Var[hj (y)] ≤ σj+ , j = 1, 2, . . . y− ≤ y ≤ y+

where f (y) is a performance function, 0 < α < 1 is a factor weighting the minimization of its mean and standard deviation, βi > 0 is a factor defining the control of the response gi (y) in the tail of its distribution, σj+ an upper bound to the standard deviation of response hj (y) and µ∗ , σ ∗ are normalizing factors. Many other formulations are, however, possible. Globally speaking, the essential of robustness optimization is the control of low order statistical moments of the response. The nature of these two alternative methods can be explained with the help of Fig. 16.1, which shows three alternative probability density functions of a structural response. While RDO aims to reduce the spread, RBDO is intended to bound the probability of surpassing the critical threshold. Notice that in applying RDO the effect pursued by RBDO is indirectly obtained, because the reduction of the spread implies a reduction of the failure probability. The reliability (or its complement, the failure probability) refers to the occurrence of extreme events, whereas the robustness refers to the low spread of the structural responses under large variation of the input parameters.

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Structural design optimization considering uncertainties

pz(z)

z

Figure 16.1 Robust and reliability-based design options.While the first aims at reducing the spread of the response function, the second attempts to control the probability of surpassing a critical threshold (dashed line). However, low failure probabilities may correspond to large spreads (dotted line).

This is assumed to assure a narrow response density function, which in turn assures a low failure probability, if it is unimodal, as is common case. However, this is not necessarily true: to a significant spread of the structural response may correspond a low failure probability because the definition of the limit state can be such that the possibility of surpassing it is very rare, as the situation it describes is rather extreme (See Fig. 16.1). In applying the RDO other possibilities exist, such as moving the probability density function away from a critical threshold or a combination of both approaches. 1.3 Ai m s and s c o pe The above discussion means that a comparison between RDO and RBDO on the basis of some examples cannot be conclusive, because, as Fig. 16.1 shows, it all depends on the critical thresholds selected for reliability estimations. Besides, the relationship between statistical moments and probabilities is severely nonlinear. For these reasons, to a good consideration of the uncertainties in structural design both approaches are valuable and complementary. This justifies the search of techniques that allow establishing a link between them, which is the purpose of the research reported herein. In fact, since both kinds of designs correspond to a different way of incorporating the uncertainties and to different goals, a method allowing a joint monitoring both the moments (mean and variance), on the one hand, and the failure probabilities, on the other, at a low computational cost, would be of avail. A simple link between RDO and RBDO is given by inequalities involving low order moments and the probability of exceeding a certain threshold. Two of them are the following (Abramowitz and Stegun 1972): •

Bienaymé – Markov inequality: P[x > ω] ≤

E(x) , ω

ω>0

(4)

Structural robustness and its relationship to reliability

•

441

Chebyshev inequality: P[|x − E(x)| ≥ tσx ] ≤

1 , t2

t>0

(5)

where σx is the standard deviation of the random variable x. These bounds, however, are reputed to be not tight when the probabilities are very low, as is common case in structural safety. Bounds such as those expressed by the above inequalities are employed when the probabilistic information is not sufficient to calculate exceedance probabilities. In probability theory the concept of entropy associates information and uncertainty in a clear, positive manner. For this reason, a second aim of the chapter is to discuss the concept of robustness from this point of view. From the production-oriented, boundassuring approach of engineering design discussed above, robustness with respect to uncontrollable external actions can be controlled in a similar fashion as the randomness of the material properties can be subjected to quality control. For this reason the concept of Robustness Assurance is introduced, referring to the control of the response spread under the influence of the uncertainty of external actions, in a similar manner as Quality Assurance in construction industry subjects to control the randomness of material properties and structural member dimensions. It is shown that robustness assurance defined in this manner can easily be incorporated into conventional deterministic optimization. The chapter is organized as follows. First, the methods for estimating the failure probability upon this information are discussed: They can be grouped into (a) global expansions and (b) local expansions. It is shown that the later offers significant advantages for the purpose in hand. However, one of the global expansion techniques, namely the maximum entropy method, is useful for linking robustness and reliability and therefore it is discussed in some detail. Next the methods allowing estimation of high order moments of the response are briefly presented, with an emphasis on the point estimates technique. The application of this method to robust design is then discussed. An example illustrates the accuracy and the low computational cost of the joint computation of moments and probability estimates by the proposed procedures. Then, the definition of robustness in terms of entropy and the ensuing derivations for optimization of linear structures is introduced. It is shown that Robustness Assurance can easily be incorporated into conventional deterministic optimization. The practical application of this concept is developed and illustrated for the case of linear structures. The chapter ends with some conclusions. Since the information on concepts and methods involved in the exposition is disperse in journals and books published along five decades, the chapter is as self-contained as possible.

2 Probability estimation based on moments In this section the estimation of probability density function based on the information provided by statistical moments is reviewed, as it offers a general link between robust and reliability-based design methods. This problem can be approached by means of Pearson, Johnson or other families of distributions (Johnson et al. 1994). However,

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Structural design optimization considering uncertainties

the discussion herein limits to polynomial and maximum entropy families of methods. A digression into the latter is instructive as it sheds light on the nature of robustness discussed at the end of the chapter. The above mentioned proposals for density estimation are global, i.e. they are valid for all values of the random variable. For estimating the reliability it would be more interesting to expand the density about a critical threshold. This is the purpose of the saddlepoint expansion, explained in the last paragraph of the present section.

2.1 Pol yno m i al expans io ns Classical probability theory provides two expansions of the probability density function based on moments. They are the Gram-Charlier and the Edgeworth expansions given respectively by (Muscolino 1993, e.g.):

pz (z) =

1 κ3 1 κ4 10 κ32 H (z) + H (z) + H6 (z) 3 4 3! σz3 4! σz4 6! σz6  280 κ33 35 κ3 κ4 H7 (z) + H9 (z) φ(z) + 7! σz7 9! σz9 1+

(6)

and

pz (z) =

1+ +

1 κ3 1 κ4 10 κ32 H (z) + H (z) + H6 (z) 3 4 3! σz3 4! σz4 6! σz6

 35 κ3 κ4 280 κ33 1 κ52 H (z) + H (z) + H (z) φ(z) 5 7 9 5! σz5 7! σz7 9! σz9

(7)

where φ(z) is the standard Gaussian density 

1 1 φ(z) = √ exp − z2 2 2π

(8)

κi and Hi (z) are respectively the cumulant and the Hermite polynomial of order i. As is well known the following relationships hold for the first cumulants and moments µj = E[zj ] (Kolassa 1997): κ1 = µ1 κ2 = µ2 − µ21 κ3 = µ3 − 3µ1 µ2 + 2µ31 κ4 = µ4 − 4µ1 µ3 − 3µ22 + 12µ2 µ21 − 6µ41

(9)

Structural robustness and its relationship to reliability

443

The coefficients are those of the Pascal triangle. The first Hermite polynomials are given by (Abramowitz and Stegun 1972, e.g.) H1 (z) = z H2 (z) = z2 − 1 H3 (z) = z3 − 3z H4 (z) = z4 − 6z2 + 3 H5 (z) = z5 + 10z3 + 15z H6 (z) = z6 − 15z4 + 45z2 − 15

(10)

Despite the similarities between Gram-Charlier and Edgeworth expansions it is worth noticing that they emerge from rather different approaches: The first from an orthogonal expansion of the probability density function and the second from the Fourier transform of a non-Normal characteristic function. In practice the use of Edgeworth expansion is more often recommended. However, notice first that they use polynomials, whose behavior is more oscillating as their order increase. Hermite polynomials, in particular, are negative for intervals that are the larger, the higher their order (Abramowitz and Stegun 1972, e.g.). As a consequence, the probability density estimate may not be strictly positive in certain intervals and, in compensation, may exhibit undesirable multimodality in other ones. A discussion on the use of polynomial approximations to the density function based on moments can be found in (Kennedy and Lennox 2000; Kennedy and Lennox 2001), where the authors propose a method based on non-classical orthogonal polynomials. As far as the examples presented in the mentioned chapters may be conclusive, the method seems to overcome the deficiencies of the classical approaches. It consists in an approximation of the form pz (z) = w(z)

r 

ai Qi (z)

(11)

i=0

where w(z) is a weighting function selected upon judgement of the moments in hand, Qi (z) are orthogonal polynomials and ai coefficients to be determined. Notice, however, that the possibility of having a negative value of the density remains. This problem is corrected in Er’s method (Er 1998) with an approximation of the form pz (z) = C exp(Q(z))

(12)

due to the strict nonnegativity of the exponential function. Here C is a normalizing constant and Q(z) =

r  i=1

ai zi

(13)

444

Structural design optimization considering uncertainties

The coefficients are obtained by solving the following algebraic problem: ⎞⎛ ⎞ ⎛ ⎞ a1 0 1 2µ1 . . . rµr−1 ⎟ ⎜ µ1 2µ2 . . . rµr ⎟ ⎜ a2 ⎟ ⎜ −1 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟ = ⎜ ⎟ ⎜ .. .. .. . . .. ⎝ ⎝ ⎠ ⎠ ⎠ ⎝ . . . . . . −(r − 1)µr−2 µr−1 2µr . . . rµ2(r−1) ar ⎛

(14)

Notice that for calculating r coefficients the number of moments needed equals 2(r − 1).

2.2

Ma x i m u m e nt r o py me t ho d

In probability theory, entropy is a simultaneous measure of the information and uncertainty given by the present samples (Shannon 1948; Jaynes 1957). In fact, a deterministic event offers no information at all, while a purely random event (having uniform distribution) offers the maximum. Therefore, entropy establishes a connection between information and uncertainty. The random samples of an event A can be expressed by means of many possible partitions U, i.e. collections of mutually exclusive subsets Ai , i = 1, 2, . . . of A in which the random occurrences are allocated. The entropy of the partition is defined by Shannon (Shannon 1948) as H(U) = −



pi ln pi

(15)

i

where pi is the probability associated to subset Ai . Empirically, if there are N samples of the event and Ni are located in subset Ai , then pi ≈ Ni /N. A continuous expression of a partition is a probability density function, in terms of which entropy is defined as
Hx = −

px (x) ln px (x)dx

(16)

There is an important difference between entropy definitions for discrete and continuous cases: It is an absolute measure of uncertainty in the former case, while a relative one in the latter, as it changes with the coordinate system (Shannon 1948). (See Eq. (76)). This remark is important for the development of a robustness assurance method proposed in the final section of present chapter. The principle of maximum entropy states that the most unbiased estimate of the probability density function of a random event is that maximizing Eq. (16). The principle determines a method for estimating the density function upon the availability of knowledge about the random event, such as e.g. statistical moments. If, for instance, such a knowledge consists of ordinary moments µk , k = 1, 2, . . . , the

Structural robustness and its relationship to reliability

445

method of maximum entropy (MEM) consists in solving the following optimization problem: Problem Maximum entropy : find : px (x)
maximizing : H = − px (x) ln px (x)dx
subject to : gk (x)px (x)dx = θk , k = 1, 2, . . .

(17)

where gk (x), θk are known functions and their expected values, respectively. When these correspond to ordinary moments, i.e. gk (x) = xk , θk = µk the result is px (x) = exp(−λ0 − λ1 x − λ2 x2 − λ3 x3 · · · ) where λk are Lagrange multipliers, with λ0 acting as a normalizing constant: 
λ0 = ln exp(−λ1 x − λ2 x2 − λ3 x3 − · · · )

(18)

(19)

Upon replacing these equations into the definition (16) the following important result is obtained: Hx = λ0 + λ1 µ1 + λ2 µ2 + λ3 µ3 + · · ·

(20)

It is worth noticing that the maximum entropy method is not limited to moment information but it applies to expected values of function in general. In (Shore and Johnson 1980) it is proved that this is the uniquely correct method that satisfies all consistency axioms. Two families of methods have been proposed to find the Lagrange multipliers. The first consists in solving the set of nonlinear equations by means of Newton methods (Mead and Papanicolau 1984; Sobczyk and Tr¸ebicki 1990; Tr¸ebicki and Sobczyk 1996; Hurtado and Barbat 1998; Ching and Hsieh 2007). The other consists in the unconstrained minimization of the concave functional (Agmon et al. 1979; Pandey and Ariaratman 1996) F(ζ0 , ζ1 , . . . ) = ζ0 + ζ1 µ1 + ζ2 µ2 + ζ3 µ3 + · · ·

(21)

because its minimum is the entropy given by Eq. (20). According to the author’s experience, the second approach is much faster and numerically more stable. Notice that Er’s method mentioned above (Er 1998) is based on the same functional form of the density as that resulting from applying the MEM to the case when the information is given by ordinary moments. This is shown by a simple comparison of Eqs. (12) with (13), on the one hand, and (18) with (19), on the other, indicating that λ0 is equivalent to − ln C. However, Er’s method requires a larger number of moments as said in the preceding and, therefore, the results are not coincident.

446

Structural design optimization considering uncertainties

2.3 Sad d l epoint e xpans io n The saddlepoint approximation to an ordinate of a density or distribution function was originally proposed by Daniels (Daniels 1954). In contrast to the classical Gram-Charlier or Edgeworth expansions, it has the advantage of producing good approximations far into the tails, because it is a local rather than a global approximation method. In other words, it is aimed at estimating the functions at a single point only. For a detailed exposition see (Barndorff-Nielsen and Cox 1979; Reid 1988; Cheah et al. 1993; Kolassa 1997). The saddlepoint approximation is based on the idea of embedding the target function within a family of parameterized functions and to select one member of the family for the approximation. Let us first approximate the density function pg (g) by the family rg (g, η) = exp(ηg − Kf (η))pg (g)

(22)

where Kf (η) is the cumulant generating function of the density pg (g), Kf (η) = ln Mf (η) = ln E[ exp(ηg)]
= ln exp(ηg)pg (g)dg

(23)

and η is a parameter. In the preceding equation Mf (η) is the moment generating function of pg (g). Notice that function rg (g, η) satisfies the normalization condition for a density since
exp(ηg − Kf (η))pg (g)dg = exp(−Kf (η))Mf (η) = 1 (24) The parameter η is selected such that the mean of the family of functions equals the ordinate at which the density is to be estimated, g, ¯ which in structural reliability is normally zero. The mean of the family is
g exp(ηx − Kf (η))pg (g)dg


d (exp(ηg))pg (g)dg dη
d = exp(−Kf (η)) (exp(ηg))pg (g)dg dη = exp(−Kf (η))

=

1 d Mf (η) Mf (η) dη

=

d ( ln Mf (η)) dη

= Kf
(η)

(25)

Structural robustness and its relationship to reliability

447

It can also be easily shown that the variance of the saddlepoint density Kf

(η). Hence the parameter is the solution of Kf
(¯η) = g¯

(26)

A convenient choice for the family of approximating functions is the standard Normal (8). Using ¯  this density implies standardizing variable g in the form q = (g − g)/σ, where σ = Kf

(¯η) is the standard deviation. The density of the standardized variable is σ pg (σq + g, ¯ η¯ ) according to the probability transformation rules. Hence we have 

. 1 2 / . / Kf

(¯η) exp η¯ Kf

(¯η)q + g¯ − Kf (¯η) pg Kf

(¯η)q + g¯ = φ(q)

(27)

Setting q ≡ 0 and solving for pg ( · ) yields 1 pg (g) ¯ = ¯ exp(Kf (¯η) − η¯ g) 2πKf

(¯η)

(28)

which is the sought-after saddlepoint approximation for the density at g. ¯ The computation of a probability Q = P[G ≥ g] ¯ eventually requires the calculation of the integral
∞ Q=

 g¯

1 2πKf

(¯η(u))

exp(Kf (¯η(u)) − η¯ u)du

(29)

where η¯ (u) is the solution of Kf
(¯η(u)) = u. Since this should be solved at each integration point, the computational demands and the accumulation of errors in approximating this integral can be large. As an alternative, direct formulas for computing Q have been proposed (Robinson 1982; Lugannani and Rice 1980). In this chapter use will be made of the proposal in (Lugannani and Rice 1980), since in the comparisons made in (Kolassa 1997) it yields the best performance. It is given by

1 1 − Q = 1 −
(ω) ¯ + φ(ω) ¯ ν¯ ω¯

 (30)

  where ν¯ = η¯ Kf

(¯η) and ω¯ = 2(¯ηg¯ − Kf (¯η)). The saddlepoint approximation is commonly applied in Statistics for estimating the density or the distribution at a given ordinate for sums of variables with widely different properties for which the Central Limit Theorem does not give good results (Lange 1999). This implies the solution of Eq. (26) using the derivative of the actual cumulant generating function by means of Newton methods. In our case such a function is not

448

Structural design optimization considering uncertainties

known and resort must be made to an approximation in terms of the cumulants using the series Kf (η) =

∞  κj ηj j=1

(31)

j!

Upon deriving Eq. (31) with respect to η and equating to the threshold, according to Eq. (26), one obtains a polynomial whose lowest real positive root yields the value η¯ . The probability of failure Pf can then be readily estimated with Eq. (30). To conclude the present exposition of the saddlepoint expansion, mention should be done to the use of Monte Carlo simulation for approximating the integral (29). To this end, random numbers are generated from the saddlepoint density and the probability is estimated as the average of the values of the indicator function located on the threshold g, ¯ as is usual in Monte Carlo integration. A method for doing this, using the Metropolis-Hastings simulation method has been proposed (Robert and Casella 1999). The method uses the alternative formulation of the integral, given by 1


2 Q= η¯



Kf

(ϑ) 2π

exp(Kf (ϑ) − ϑKf
(ϑ))dϑ

(32)

which can be obtained from (29) with the change of variable u = Kf
(ϑ). In the numerical experiments reported in the quoted reference the method gives quite similar results to the exact integration. However, notice that the value of the integral hinges upon the lower limit, which in the method proposed herein is known only approximately via the point estimate technique. Hence very small differences can be expected from this simulation approach in comparison to the simple application of the LugannaniRice formula. In addition, the randomness of the failure probability, common to all simulation-based methods, appears.

3 Structural response moment estimation 3.1 Perturb a ti o n appr o ac h Perturbation methods in structural analysis (Hisada and Nakagiri 1981; Liu et al. 1995; Kleiber and Hien 1992) are based on a basic result of the probability theory concerning the approximation of the mean vector and covariance matrix of a function h(x) of a set of r basic variables x = {x1 , xj , . . . , xr }. Function h( · ) can be expanded in Taylor series about the mean vector µx as r r r  ∂h 1   ∂2 h h(x)=h(µ ˙ (µ )[x −µxk ]+ (µ )(x −µxk )(xl −µxl ) x )+ ∂xk x k 2 ∂xk xl x k k=1

k=1 l=1

(33)

Structural robustness and its relationship to reliability

449

Applying the expectation operator to this equations we obtain 1   ∂2 h (µ )Ckl (x) 2 ∂xk xl x r

E[h(x)]=h(µ ˙ x) +

r

(34)

k=1 l=1

where it has been taken into account that E[xk − µxk ] = 0. Here Ckl (x) denotes the (k, l) element of the covariance matrix of the vector x. This equation is known as the second order approximation of the mean of function h( · ). Let us now derive a first order approximation to the covariance of two functions hi (x) and hj (x). To this end multiply the Taylor expansion of the two functions up to the first order derivative terms, i.e. r B C  ∂hi . (µx )[xk − µxk ] hi (x)hj (x) = hi (µx ) + ∂xk k=1

r B C  ∂hj hj (µx ) + (µx )[xl − µxl ] ∂xl

(35)

l=1

Arranging terms yields  ∂hi . hi (x)hj (x) = hi (µx )hj (µx ) + hj (µx ) (µ )[x − µxk ] ∂xk x k r

k=1

+ hi (µx )

r  l=1

+

r 

∂hj (µ )[x − µxl ] ∂xl x l

 ∂hj ∂hi (µx )[xk − µxk ] (µ )[x − µxl ] ∂xk ∂xl x l r

k=1

(36)

l=1

Moving the product hi (µx )hj (µx ) to the left-hand side and taking expectations at both sides of this equation leads to the final result: ∂hj .   ∂hi cov(hi (x)hj (x)) = (µx ) (µ )Ckl (x) ∂xk ∂xl x r

r

(37)

k=1 l=1

The variance of either of the two functions is but a particular case of this equation: ∂hi .   ∂hi (µ ) (µ )Ckl (x), var(hi (x)) = ∂xk x ∂xl x r

r

i = 1, 2

(38)

k=1 l=1

At least three objections can be addressed to perturbation methods for the purpose of present chapter. First, they are reputed to be accurate for low coefficient of variation of the basic variables x (Elishakoff and Ren 2003). Second, they require special computational codes to their application (Kleiber and Hien 1992). Third, as is evident, they do not yield equations for estimating moments of order higher than two, which

450

Structural design optimization considering uncertainties

are needed for applying local or global expansions of probability distributions in order to estimate the probability of failure. The method of point estimates summarized next overcomes these deficiencies.

3.2 Poi n t esti m at e me t ho d The method of Point Estimates (Rosenblueth 1975; Ordaz 1988; Christian and Baecher 1998; Harr 1989; Hong 1998) is a valuable tool for estimating the low order statistical moments of a system response with good accuracy. The reason explaining this property is that the method imposes the annihilation of some order terms in the Taylor expansion of the response and the concentration of their information in some weights located around the mean vector of the basic variables. This is the main difference with perturbation approaches based in the Taylor expansion, which are built over the assumption that the high order terms of the expansion are negligible. Besides, the method of point estimates has the additional advantage over perturbation schemes that it can be easily applied to the estimation of moments of order higher than the second. Last but not least, the method does not need special finite element codes for its application, as required by the perturbation approach. As a consequence, it can be used in connection to practically any structural problem using available numerical tools. In the basic formulation of the method, the total number of finite element solver calls is only twice the number of independent random variables, which in a problem determined by a few basic variables implies a low computational effort. These features make the method an accurate and practical technique for the stochastic performance analysis of mechanical systems. In the following lines the proposal in (Hong 1998) is summarized, because in the experience reported in (Hong et al. 1998) using actual structural models it offers by far the best approximation over the other point estimate alternatives cited above. In addition, the applicability of the method for higher order moment evaluation is discussed. Let us consider a structural function g(x) that is a function of a single variable x. The Taylor expansion of a power function g j (x) about the mean value of x is g j (x) = b(x) = b(µx ) +

∞  1 (l) b (µx )(x − µx )l l!

(39)

l=1

Taking expectations on both sides of the above equation one obtains ∞  1 ∂b (µ )E[(x − µx )l ] E[g (x)] = b(µx ) + l! ∂xl x j

(40)

l=1

which can be put in the form E[g j (x)] = b(µx ) +

∞  1 ∂b (µ )γ σ l l! ∂xl x x,l x l=1

(41)

Structural robustness and its relationship to reliability

451

where σx is the standard deviation of x and γx,l is a normalized central moment defined as
1 ∞ γx,l = l (x − µx )l px (x)dx (42) σx −∞ Multiplying successively equation (39) by two weights wi , l = 1, 2 assigned to the concentration points xl and summing up the result yields w1 b(x1 ) + w2 b(x2 ) = b(µx )(w1 + w2 ) +

∞  1 ∂b (µ )(w1 ξ1l + w2 ξ2l )σxl l! ∂xl x

(43)

l=1

where ξi , i = 1, 2 is the standardized random variable ξi =

xi − µx

(44)

σx

Solving equation (43) for b(µx ), imposing the condition w1 + w2 ≡ 1

(45)

and substituting back the result into equation (41) yields E[g j (x)] = w1 b(x1 ) + w2 b(x2 ) +

∞  1 ∂b (µ )[γ − (w1 ξ1l + w2 ξ2l )]σxl l! ∂xl x x,l

(46)

l=1

This equation suggests the approximation . E[g j (x)] = w1 b(x1 ) + w2 b(x2 ) = w1 g(x1 )j + w2 g(x2 )j

(47)

in which function g( · ) is evaluated at points xi = µx + ξi σx , i = 1, 2. Implicit in the above approximation is the condition that the concentration parameters must also satisfy the following constraint: w1 ξ1i + w2 ξ2i = γx,i

(48)

for an adequate number of normalized moments γx,i allowing the determination of ξl and wl . For determining two weights and concentration points three moment equations of the type (48) are necessary. These, appended to Eq. (45), yield the values of the four parameters. In this case the system has the following closed-form solution: ξi =

γx,3 2

 + ( − 1)

wi = (−1)i

ξ3−i ζ

3−i

1+

γx,3

2

2 (49)

452

Structural design optimization considering uncertainties

 2 /4. For the more general case of a function g(x) of n mutually where ζ = 2 1 + γx,3 uncorrelated random variables xk , k = 1, . . . , n, collected in vector x, it is possible to apply the same strategy as above by setting all variables at their means and applying the Taylor expansion about the mean of each xk in turn. The derivation of the equations for the weights and concentration points can be performed in the same way as for the one dimensional case. As a result, the approximation of the ordinary moment of order j of g(x) with m points per variable is given by .  E[g j (x)] = wk,i g(µ1 , . . . , xk,i , µk+1 , . . . )j m

n

(50)

k=1 i=1

The total number of solver calls will then be S = mn. In (Hong 1998) several alternatives for calculating the weights and point locations are offered, according to the number of concentration points. These are the following: •

S = 2n scheme: ξk,i =

γxk ,3 2

 + ( − 1)

3−i

n+

γ

xk ,3

2

2

ξk,3−i 1 (−1)i (51) n ζk  2 /4. Notice that, in this case, the approximation (50) is = 2 n + γx,3

wk,i = with ζk

•

accurate to the third order of the Taylor expansion, as determined by the number of normalized central moments used in the calculation of the weights and concentration points. S = 2n + 1 scheme: 

γ 2 γxk ,3 xk ,3 3−i + ( − 1) γxk ,4 − 3 ξk,i = 2 2 wk,i = ( − 1)3−i

1 ξk,i (ξk,1 − ξk,2 )

(52)

for i = 1, 2 and ξk,3 = 0 wk,3 =

1 − wk,1 − wk,2 n

(53)

Note that the repetition of the point ξk,3 = 0 makes this three-point scheme equivalent to a 2n + 1-point scheme  . E[g (x)] = w0 g(µ1 , . . . , µk , µk+1 , . . . )j + wk,i g(µ1 , . . . , xk,i , µk+1 , . . . )j (54) n

2

j

k=1 i=1

Structural robustness and its relationship to reliability

•

453

S = 3n scheme: ξk,j ξk,l , (ξk,j ξk,i )(ξk,l ξk,i )

wk,i =

i, j, l = 1, 2, 3;

i = j = l = i

(55)

The locations ξk,i , i = 1, 2, 3 are the roots of the polynomial ω0 + ω1 q1 + ω2 q22 + ω3 q33 = 0

(56)

in which . / ω0 = γxk ,5 − γxk ,3 2γxk ,4 − γx2 ,3

ω3 =

•



k

γx ,4  − γxk ,3 + γxk ,4 1 − k n n

 γ 1 x ,5 − k ω2 = γxk ,3 γxk ,4 + n n ω1 = γxk ,3

γ

xk ,5

γxk ,4 − (1 + γx2 ,3 ) k

(57)

n

The approximation obtained with these points is accurate to the fifth order because it supposes the cancelling of 2m − 1 = 5 terms of the Taylor expansion. S > 3n scheme: In the general case the size of the nonlinear system for determining the weights and location points of becomes 2m. This implies the solution of the following system of nonlinear equations for each variable k: m 

wk,i =

i=1 m 

1 n

j

wk,i ξk,i = γxk ,j

(58)

i=1

A system like this can be solved by an algorithm described in (Hamming 1973; Miller and Rice 1983). Let us expand the system of equations (58) + w2

w1 ξ1

+ w2 ξ2

+ · · · + wm + · · · + wm ξm

w1 ξ12 .. .

+ w2 ξ22 .. .

+ · · · + wm ξm2 .. .. . .

w1

= b0 = n1 = b1 = γ1 = b2 = γ2 .. .

w1 ξ12m−1 + w2 ξ22m−1 + · · · + wm ξm2m−1 = b2m−1 = γ2m−1

(59)

454

Structural design optimization considering uncertainties

where the subindex denoting the random variable has been dropped for clarity. Define a polynomial p(ξ) =

m 

ωl ξ l

(60)

l=0

whose roots are the desired values ξ1 , ξ2 , . . . , ξm , i.e p(ξ) = (ξ − ξ1 )(ξ − ξ2 ) · · · (ξ − ξm )

(61)

From this equation follows that ωm = 0 and that p(ξi ) = 0 for all i. Now, take the first m equations from system (59) and multiply the first by ω0 , the second by ω1 , etc., and add them to obtain: m 

ws p(ξs ) =

s=0

m 

ωl bl

(62)

l=0

Then take the groups made up by the r to the m + r −1 equations, for r = 1, 2, . . . and apply the same multiplications and sums. The result is the following linear set b0 ω0 + b1 ω1 + · · · + bm−1 ωm−1 = −γm b1 ω0 + b2 ω1 + · · · + bm ωm−1 = −γm+1 .. .. .. .. .. . . . . . bm−1 ω0 + bm ω1 + · · · + b2m−2 ωm−1 = −γ2m−1

(63)

The solution of this system gives the values of the coefficients ωl , l = 0, 1, . . . , m. Substituting them into the definition of the polynomial p(ξ) (Eqs. 60 and 61) yields the value of the roots ξi . Finally, the weights wi can be computed from Eq. (59) which now becomes a linear system. The treatment of correlated variables in this method can be consulted in (Hong 1998). It consists in rotating the basic variable space to a new one in which no correlation exists, using well-known spectral techniques. Let us now examine the limitations of the point estimate method using some simple examples. A first limitation concerns the simple 2n scheme. In fact, as noted in (Christian and Baecher 1998), when the number of input variables is very large the locations of the concentration points may be very far from the mean value thus making the concentration points meaningless from engineering viewpoint. With respect to the 2n + 1 scheme, it has been observed (Hong 1998) that, eventually, the following condition for applying Eq. (52) is not satisfied by some density functions:

γ

γxk ,4 − 3

xk ,3

2

2 >0

(64)

Further, in the 3n plot some roots of the polynomial (56) may be complex, rendering impossible the application of the method. Finally, in the S > 3n scheme, a solution may not exist.

Structural robustness and its relationship to reliability

455

As an example of this latter case, let us examine the application of the above numerical procedure for obtaining the location and weights of m = 4 concentration points for Normal variables. For a single variable x ∼ N(0, σx2 ) the moments are  µx,j =

1 · 3 · · · (n − 1)σxj , j even 0, j odd

(65)

Consequently, the right hand vector in Eq. (63) is [−3 0

− 15 0]T

and the solution of the system is α = [3 0

− 6 0]T

Hence, the locations of the weights are the roots of the polynomial 3 − 6ξ 2 + ξ 4 = 0 which are −2.334, −0.742, 0.742 and 2.334. The weights are calculated by Eq. (59), yielding 0.459, 0.454, 0.454, 0.459. Let us now turn to the case n = 2 for which b0 = 0.5 in Eq. (59). In this case the problem has a solution ξ = [ − 2.715, −1.275, 1.275, 2.715]. However, for n = 3 the matrix of coefficients in Eq. (63) becomes ill-conditioned, so there is no stable solution. And for n = 4 the linear system has a solution but two roots of the polynomial are complex.

4 Robust analysis with point estimates Before describing the linkage between RDO and RBDO approaches, it is useful to make some remarks about the use of point estimates for robust optimization. Notice that the method gives estimates of the ordinary moments. Since robust optimization is oriented towards spread control, it is necessary to compute the variance of the response, given by ! "2 Var[g(x)] = E[g 2 (x)] − E[g(x)]

(66)

Evidently, the minimum of the variance corresponds to the maximum of the mean and the minimum of the mean square. However, since a requirement of the robust optimization is also a minimization of the mean (see Eq. (3)), it results that in using point estimates we eventually have to minimize a weighted cost function of the form C = ω1 E[g(x)] − ω2 E[g(x)] + ω3

!

" E[g 2 (x)]

(67)

456

Structural design optimization considering uncertainties

P

Q 10

12 9

2

14 11

4 1

16 13

6 3

15 [email protected] m

8 5

7

4@1 m

Figure 16.2 Finite element mesh for the numerical example.

where the dependence on the design variables y has been removed for clarity of notation. However, since ω1 and ω3 are both related to spread, they can be made equal, ω1 = ω3 ≡ ω, with the result ! " (68) C = (1 − ω)E[g(x)] + ω E[g 2 (x)] in which the basic requirement ω1 + ω2 + ω3 = 1 has been taken into account.

5 Uniting RDO and RBDO The proposed approach for performing simultaneously a robust and reliability-based design consists in the following steps: (a) To estimate the moments of the response by means of the point estimate method, as this requires a minimal number of solver calls and no other solver than that used for deterministic computations. (b) To estimate the failure probabilities using either the saddlepoint expansion at the critical point. Notice that several responses defining an equal number of limit states can be calculated simultaneously. 5.1 Ex am pl e In this example the method of point estimates will be applied to the estimation of the failure probabilities corresponding to surpassing a threshold by the von Mises yield stress τm in all the finite elements forming the elastic beam shown in Fig. 16.2: gi (x) = τ¯m,i − τm,i (x) = 0, i = 1, 2, . . . , 16

(69)

For a plane problem, the von Mises stress is  τm =

τ12 − τ1 τ2 + τ22 3

where τi , i = 1, 2 are the principal stresses.

(70)

Structural robustness and its relationship to reliability

457

Table 16.1 Random variable definition. Variable

Type

Mean

Standard deviation

P Q E

Lognormal Lognormal Normal

500 50 20,000,000

75 5 3,000,000

Table 16.2 Samples for the 2N scheme. Sample

P

Q

E

1 2 3 4 5 6

648.01 385.99 500.00 500.00 500.00 500.00

50.00 50.00 59.44 42.06 50.00 50.00

2.0e07 2.0e07 2.0e07 2.0e07 3.1374e07 1.7606e07

The elements are constant strain triangles. The beam is subject to two random loads. The elasticity modulus is also random and the Poisson modulus is fixed at 0.2. The stochastic properties of these independent random variables are shown in Table 16.1. A different von Mises stress threshold τ¯m was assigned to each element in order to assure a probability of failure around 10−3 for all elements. In order to have an idea of the estimation errors, 50,000 Monte Carlo simulations were calculated. Er’s method for density estimation mentioned above was also computed for comparison. Notice that the relationship between the limit state function and the input random variables is highly nonlinear. The attempts for calculating three and four concentration points per variable failed in that no real solutions were found for the roots. For this reason the 2n point estimate strategy was applied. Table 16.2 shows the coordinates of the point estimate samples. Only four cumulants were used for the estimation of the failure probabilities. In spite of the reduced number of concentration points and cumulants, the results given by both methods are reasonably good as shown by Table 16.3. The second column of the Table informs the threshold values for each element. Notice that in general the saddlepoint method exhibits better accuracy than Er’s technique. This is especially so for element No. 10, in which case the moment structure of the stress in the element implied that there were three real roots for Eq. (26), at a difference with the rest of elements, for which one real and two complex roots were found. Notice that the saddlepoint method somewhat underestimates the failure probability with respect to Monte Carlo simulation. Figure 16.3 compares a histogram density obtained with a subset of 10,000 Monte Carlo samples and Er’s estimation for element No. 1. Figure 16.4 depicts the standard deviations of von Mises element stresses and the failure probabilities multiplied by 104 for each finite element, as given by Monte Carlo simulation with 50,000 samples. It can be noticed that there is no clear-cut relationship between the two uncertainty

458

Structural design optimization considering uncertainties Table 16.3 Estimates of the failure probabilities. τ¯m,i

Finite element i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

600 550 750 300 325 700 625 425 425 350 750 950 225 1100 275 450

8

Pf,i Monte Carlo

Pˆ f,i 2N scheme + saddlepoint

Pˆ f,i 2N scheme + Er’s method

(50,000 samples)

(6 samples)

(6 samples)

0.0031 0.0042 0.0049 0.0041 0.0041 0.0063 0.0041 0.0048 0.0053 0.0029 0.0054 0.0033 0.0144 0.0005 0.0121 0.0054

0.0024 0.0034 0.0043 0.0033 0.0034 0.0057 0.0034 0.0041 0.0047 0.0019 0.0047 0.0027 0.0142 0.0004 0.0116 0.0048

0.0056 0.0072 0.0078 0.0069 0.0069 0.0094 0.0069 0.0076 0.0084 0.0107 0.0083 0.0060 0.0180 0.0021 0.0153 0.0084

 103 Er method (4 moments) Monte Carlo (10,000 samples)

7

Probability density

6 5 4 3 2 1 0 200

300

400 500 600 Von Mises stress in element 1

700

800

Figure 16.3 Comparison of Er’s method of density estimation and Monte Carlo histogram.

measures. In fact, in some cases to two similar deviations there correspond rather different probabilities; also, the correlation coefficient between the two measures is rather poor (0.542). Finally, notice that the highest probability there corresponds the lowest standard deviation (element No. 13), which contradicts the naive intuition

Structural robustness and its relationship to reliability

459

s, 104 Pf 180 160

s

140

104 Pf

120 100 80 60 40 20 0

Element No. 0

2

4

6

8

10

12

14

Figure 16.4 Comparison of standard deviation and amplified failure probability for each element.

expressed by Figure 16.1. Similar conclusions arise from the comparison of the failure probability and the coefficient of variation of the von Mises stress. In this case, the correlation is even poorer: −0.109. All this means that optimizing with respect to the statistical moments may yield rather different results than when optimizing with respect to the failure probability and that it is important to consider both kinds of approaches in designing safe structures in the noisy environment of random loads and structural material parameters.

6 Robustness as entropy minimization The main goal of robust design is to control the spread of the structural response. This can be regarded as a minimization of the entropy of the response. A simple illustration of this is given by the fact that the entropy of a Normal density function 

1 (x − µ)2 φx (x) = √ (71) exp − 2σ 2 2πσ increases along with the standard deviation: . √ / Hx = ln σ 2πe

(72)

In order to more rigorously define robustness as entropy minimization, let us consider a set of random variables x = [x1 , x2 , x3 , . . . , xn ] which is transformed to a set z = [z1 , z2 , z3 , . . . , zn ] by functions of the form zj = gj (x)

(73)

460

Structural design optimization considering uncertainties

According to the laws of probability transformation, the density function of vector z is given by (Papoulis 1991) pz (z) =

1 px (x) |J(x)|

(74)

where J(x) is the Jacobian of the transformation:    ∂g1 . . . ∂g1   ∂x1 ∂xn    J(x) =  ... ... ...   ∂gn ∂gn    ∂x . . . ∂x 1

(75)

n

Upon substituting this result in Hz = −

(

pz (z) ln pz (z)dz one obtains (Papoulis 1991)

@ A Hz ≤ Hx + E ln|J(x, y)|

(76)

In this result the dependence of the Jacobian on the vector of design variables y has been made explicit in order to emphasize the relevance of the design in entropy transformation. If the set of equations (73) has a unique inverse, as is the case for linear structures, then equality holds. Equation (76) is useful for understanding why entropy is relevant for defining structural robustness. In fact, assuming that x is the set of input random variables of the structural system and z that of observed random responses, Eq. (76) states that the system is an entropy dissipating system, i.e. it can reduce the scatter of the input variables if @ A E ln|J(x, y)| ≤ 0

(77)

because, in that case, @ A Hz − Hx ≤ E ln|J(x, y)| ≤ 0

(78)

implying Hz ≤ Hx

(79)

Recalling the remark made above about the relativity of the entropy measure of uncertainty in the case of continuous @ Adistributions, our focus will not be the terms Hx and Hz but on the term E ln|J(x, y)| . Let us delve into its expression for the common case of linear structures. 6.1 Rob ustn ess o f linear s t r uc t ur e s Consider a linear structure modeled with the finite element method, so that it is described with the classical equation Ku = q

(80)

Structural robustness and its relationship to reliability

461

where u is the displacement vector, K is the stiffness matrix and q the external force vector, respectively. Assume that the external loads are random. The structure may have some random properties of the materials, but since their dispersion can be controlled with Quality Assurance (QA), we are interested only in reducing the sensitivity of the response to random changes in external loads. By analogy with QA applied in the construction phase, we may call Robustness Assurance (RA) the control of randomness applied in the design phase. It may seem surprising that the RA analysis just proposed ignores the randomness of the material properties and of other structural variables such as geometrical dimensions, etc. The underlying reason for leaving it to the Quality Assurance in the construction phase is that the robustness approach stems from the product-oriented nature of engineering design and construction, that is not interested in establishing the actual risk of the structure, as in the knowledge-oriented reliability approach, but only in assuring a product capable of dissipating the randomness imposed by external actions as much as possible. Thus, the proposed approach to robustness separates design from the determination of the actual reliability, considering this latter as a specialized task whose need arises in certain situations. In addition, notice that the robustness defined in terms of entropy incorporates the available information on external actions in a positive manner, as indicated by the maximum entropy principle (see Eq. 17) and the quotation of Jaynes’ classical chapter above. Thus, beyond the second order analysis on which some proposals for robust design are based, which may invoke the lack of full probabilistic information to proceed in that way, the entropy approach to robustness incorporates such a deficiency as an element of design calculations. In fact, the maximum entropy principle establishes the distribution that accords with several situations of available information. If, for instance, both the mean and variance are prescribed and the variable can be either negative or positive, the principle indicates that the distribution is Gaussian. However, if it is known that the variable is strictly positive and the mean alone is prescribed, the solution is the exponential distribution. And so on. (See (Kapur 1989) for a detailed exposition). Formulating the problem as u = K −1 q,

(81)

a typical element of vector u is ui =



(K−1 )ij qj

(82)

The sensitivity of the i-th response with respect to the j-th random variable is, therefore, ∂ui = (K−1 )ij ∂qj

(83)

which does not depend on any element of vector x. Other responses, such as end forces and stresses can be expressed as other linear combinations of the displacements and,

462

Structural design optimization considering uncertainties

Hz

r r

2

r

3





r

r

2

1

1

y

Figure 16.5 Variation of response entropy of structures as a function of a structural dimension (y) and the coefficient of variation of an external load (ρ).

therefore, of the external loads. For this reason the ensuing development will be done in terms of the displacements. The expectation in Eq. (76) is     @ A (84) E ln|J(x, y)| = ln detK−1 (y) = −ln detK(y) Since K is a positive definite matrix, |detK(y)| = detK(y) > 0 yielding @ A E ln|J(x, y)| = −ln detK(y)

(85)

The term R(y) = ln detK(y)

(86)

will be simply called robustness. Figure 16.5 illustrates the behavior of Eq. (76) for a linear structure as a function of the single cross section dimension subject to design and the coefficient of variation of a single random load. According to this exposition a RA-design of linear structures with random external loads can be involved in a Deterministic Optimization program (see Eq. (1)) as follows: Problem Robustness and Cost Optimization : find :

y

minimizing : Z(y) = −αR(y)/R∗ + (1 − α)C(y)/C ∗ subject to :

fi (y) < Fi , i = 1, 2, . . . y− ≤ y ≤ y+

(87)

Here R∗ and C ∗ are normalizing factors. The meaning of this equation is that cost C(y) is minimized while robustness R(y) is maximized. Thus, the solution will be a saddlepoint instead of the global minimum of the cost. Notice that the robustness term prevents from a one-sided search of the minimum cost in a global manner not

Structural robustness and its relationship to reliability

463

Table 16.4 Comparison of Shannon (entropy) and Lyapounov (stability) functionals. Shannon

Lyapounov

H(U ) is a continuous function of pi If all pi , i = 1, . . . , n are equal, H(U ) is an increasing function of n H(U ) has a unique global maximum

V(x(t)) is a continuous function of t V(x(t)) is a non-increasing function of t V(x(t)) has a unique global minimum

provided by usual behavioral or geometric constraints, some of which could eventually be removed from optimization programs. Factor α, 0 ≤ α ≤ 1, should weight the relative importance of cost and robustness with regard to external loads and, therefore, it must be selected judiciously. The relative weight of robustness should consider the departure of the external loads from determinism, in such a way that the larger their spread, the higher α.

6.2 Analogy to s ys tem dynamics It is interesting to make some remarks on the analogy of entropy dissipation and the theory of dynamic systems and control (See, e.g., (Szidarovszky and Bahill 1992, e.g.)). In fact, one of the basic concerns in dynamic systems is that of controllability, meaning that given an initial state there exists an input capable of leading the system to another state. In dynamic system theory the control force is the result of a trade-off between its cost and the reduction of the responses. Similarly, in robust design under uncertainties the engineer is interested in obtaining a product such that, given an uncertain input, the uncertainty of the response can be controlled to a given value without excessive cost. Another relevant analogy is that with stability. While in the designing dynamical systems the engineer is not interested in particular trajectories of the system but only in assuring its overall stability, in robust design the designer is interested in assuring that the system will not be seriously perturbed by random fluctuations of the input parameters, without detailed stochastic characterizations of the paths of randomness inside the structural model. This is an important, practical reason motivating the development of robust alternatives to the more theoretical, argumentative approach of establishing failure probabilities, according to the exposition made in the Introduction. The theory of stable systems makes use of a Lyapounov functional V(x(t)) of the dynamic system state x(t), whose characteristics have close resemblance to those of the entropy function, as illustrated by Table 16.4. While the Lyapounov functional has its global minimum at the equilibrium state of the system, the entropy functional has its maximum at a density function which “may be asserted for the positive reason that it is uniquely determined as the one which is maximally noncommittal with regard to missing information, instead of the negative one that there was no reason to think otherwise’’ as stated by Jaynes in his classical chapter (Jaynes 1957).

464

Structural design optimization considering uncertainties

6.3 Ex am pl e 1: A s imple b ar in t e ns io n Let us consider the simplest structural model, i.e. an elastic bar of cross section A, length l, elasticity modulus E subject to a tension force P, as shown in Fig. 16.6. Let as assume that the set of random variables is E D x = P, E

(88)

of which it is known that both are positive. On the other hand, the set of responses is D E z = u, τ

(89)

where τ is the random tension in the bar. The single design variable is y=A

(90)

For the transformation Pl EA P τ = A

u =

(91)

the Jacobian is J(x, y) =

Pl A2 E2

(92)

Since all quantities are positive, ln|J(x, y)| = lnJ(x, y) and @ A E ln|J(x, y)| = E[lnP − 2lnE] + lnl − 2lnA

(93)

Thus, the bar is entropy dissipating if 2lnA > E[lnP − 2lnE] + lnl. Now, if A is constrained to lie in the range A1 ≤ A ≤ A2 and no reference is made to cost, the robust design consists simply in assigning A = A2

(94)

since this value minimizes the expectation in Eq. (93). It is evident that no probabilistic information is necessary to arrive to this result. However, without such an information it is not possible to ascertain whether the structure is entropy dissipating or not. 6.4

Ex am pl e 2: A t r us s

Consider the three-bar structure shown in Fig. 16.6. The random variables are x = (P, E), the design variables are y = (A1 , A2 ) and the the observed responses are

Structural robustness and its relationship to reliability

465

E, A

P

l

Figure 16.6 Simple bar in tension.

z = (u, τ) which are the horizontal component of the displacement of load point and the tension in the left bar, respectively. They are given by u = τ =

Pl EA1 A2 +

√

2A1 P √ 2A1 A2 + 2A21

(95)

The Jacobian of the transformation is √ Pl A2 + 2A1 J(x, y) = 2 √ E 2A21 A2 + 2A31

(96)

which has the separable form J(x, y) = Q(x)R(y). Therefore, without reference to cost, a robust design can be obtained with no probabilistic information of P and E by simply finding the values of (A1 , A2 ) that maximize ln

A2 + 2A21 A2

√

2A1 √ + 2A31

(97)

within the specific bounds assigned to each cross section area. If cost is considered, the solution must be a trade-off between cost minimization and robustness maximization. 6.5

Example 3: A clamped beam

Consider finally the clamped beam of variable shape shown in Fig. 16.8. The only random variable is the external load P. The cross section is a square tube of external dimension yi , i = 1, 2 and thickness t, so that the moment of inertia is Ii =

2 3 ty 3 i

(98)

The vertical displacement of the end point is u=

7Pl 3 Pl 3 + 3EI1 3EI2

(99)

466

Structural design optimization considering uncertainties

P

45°

E, A2

E, A1

l

45°

E, A1

Figure 16.7 A simple truss. P E, I1

E, I2

l

l

Figure 16.8 A clamped beam with variable section.

so that l3 J= 2tE

1 7 + 3 3 y1 y2

 (100)

Let us use E = 2100 t/cm2 , t = 1 cm and l = 200 cm. A beam with y1 = 51.2 cm, y2 = 31.5 cm minimizes the cost subject to the constraint that the end displacement is less than or equal to l/250 (Hernández 1990). This solution is entropy dissipating since ln |J| < 0. On the contrary, for y1 = 30 cm, y2 = 15 cm the structure increases entropy.

7 Conclusions The following conclusions stem from the research reported in this chapter: •

The concept of entropy is useful for clarifying the meaning of robustness in structural systems. In fact, the entropy of the structural responses decreases as the stiffness increases. Accordingly, in parallel to Quality Assurance operating on the spread control of structural material properties in the construction phase, one may define Robustness Assurance as the control of the entropy of response variables such as displacements and stresses due to the uncertainty in random external loads in the design phase. Such Robustness Assurance can easily be incorporated

Structural robustness and its relationship to reliability

•

•

•

•

467

into conventional Deterministic Optimization programs. A proposal in this regard has been exposed. It has also been shown that structural robustness defined in these terms exhibits similarities to the theory of controllability and stability studied with the assistance of Lyapounov functions in the context of dynamic systems. Both subjects are instances of the production-oriented approach that is characteristic of engineering design process at a difference to the knowledge-oriented process of scientific discovery. For optimizing a structure under uncertainty both robust and reliability-based approaches are valuable and therefore complementary. The first aims at a control of response spread whereas the second to reducing the probability of extreme undesirable situations. For the above reason, methods allowing simultaneous monitoring of the basic statistical quantities implied by both approaches (namely statistical moments and failure probabilities) are of importance. In this chapter, this has been sought by means of the method of saddlepoint local expansion of the density function about the critical threshold and the method of maximum entropy. While the former exhibits better accuracy for estimating the failure probability, the latter is highly useful for the assessment of the degree of robustness of the structural system as commented above. The method of point estimates allows a fast and simple estimation of response moments using the finite element solver employed for deterministic calculations. For these reasons it is highly for both robust and reliability-based design optimization.

Further research is needed to develop the entropy approach to structural robustness as well as on computational methods for the complex task of structural optimization granting the accomplishment of both reliability and robustness requirements.

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Chapter 17

Maximum robustness design of trusses via semidefinite programming Yoshihiro Kanno University of Tokyo, Tokyo, Japan

Izuru Takewaki Kyoto University, Kyoto, Japan

ABSTRACT: This chapter discusses evaluation and maximization of the robustness function of trusses, which is regarded as one of measures of structural robustness under the uncertainties of member stiffnesses and external forces. By using quadratic embedding of the uncertainty and the S-procedure, we formulate a quasiconvex optimization problem which provides a lower bound of the robustness function. We next formulate the maximization problem of the robustness function as a robust structural optimization scheme. An algorithm based on the semidefinite program is proposed to obtain the optimal truss design. Numerical examples are shown to demonstrate the validity of the algorithms presented.

1 Introduction Recently, the info-gap decision theory has been proposed as a non-probabilistic decision theory under uncertainties (Ben-Haim 2006), and has been applied to wide fields including neural networks (Pierce et al. 2006), biological conservation (Moilanen & Wintle 2006), financial economics (Ben-Haim 2005), etc. In the info-gap decision theory, the robustness function plays a key role as a measure of robustness of systems having uncertainties (Ben-Haim 2006). The robustness function is regarded to represent the immunity against failure, and is defined as the greatest level of uncertainty at which any failure cannot occur. In structural engineering, the robustness function represents the greatest level of uncertainty, caused by manufacture errors, limitation of knowledge of input disturbance, observation errors, etc., at which any constraint on mechanical performance cannot be violated. The constraints on mechanical performance can be violated only at great level of uncertainty in a structure with a large robustness function, while they can be violated at small level of uncertainty in a structure with a small robustness function. Thus, we can compare robustness of structures quantitatively in terms of robustness functions. Takewaki & Ben-Haim (2005) computed the robustness function of structures in a particular case where the worst case can be obtained analytically. Unfortunately it is difficult to compute exactly the robustness function of structures exactly, and no efficient method has ever been proposed to the authors’ knowledge. The first contribution of the work in this chapter is to propose a numerically tractable optimization problem to obtain a lower bound of the robustness function of trusses considering various constraints and circumstances of uncertainties. The solution of the problem presented can

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be obtained efficiently by solving some semidefinite programming (SDP) (Wolkowicz et al. 2000) problems. Note that a lower bound is regarded as a conservative estimate of the robustness function, i.e. a level of uncertainty at which the satisfaction of the constraints on mechanical performance is guaranteed. Hence, finding a lower bound, not an upper bound, is meaningful when it is difficult to find the exact value of the robustness function. Secondly, we consider a structural optimization problem in which we seek for the truss design maximizing the robustness function. Based on the stochastic uncertainty model of mechanical parameters, various methods were proposed for reliability-based optimization (see the other chapters and the references therein). One of the motivations for the info-gap theory is an awareness of the limitations of the probabilistic approaches as discussed by Ben-Haim (2004, Section 7). As a non-probabilistic uncertainty model, Ben-Haim & Elishakoff (1990) developed the so-called convex model, where the uncertainty of a system is expressed in terms of unknown-but-bounded parameters. Pantelides & Ganzerli (1998) proposed a robust truss optimization method based on the convex model. The mathematical programming problems including uncertain data have also been investigated extensively. For various classes of convex optimization problems, a unified methodology of robust counterpart was developed by Ben-Tal & Nemirovski (2002), where the data in optimization problems are assumed to be unknown but bounded. Calafiore & El Ghaoui (2004) proposed a method for finding the ellipsoidal bounds of the solution set of uncertain linear equations by using SDP. In this chapter, we deal with the robustness function of trusses that consist of members with uncertain stiffness and/or are subjected to uncertain external forces. The non-probabilistic uncertain parameters are assumed not to be known precisely but to be bounded. The details of background of decision strategies based on the info-gap theory may be consulted to the basic textbook (Ben-Haim 2006). To overcome the difficulty of computing the robustness function, we utilize the framework of SDP. For mathematical backgrounds and algorithms for SDP, the readers may refer to the review chapters (Helmberg 2002; Vandenberghe & Boyd 1996) and the handbook (Wolkowicz et al. 2000).

2 Preliminary results Some useful technical results used in this chapter are listed in Appendix A. Throughout this chapter, all vectors are assumed to be column vectors. However, for vectors u ∈ Rn and v ∈ Rm , we often simplify the notation (uT , v T )T as (u, v). The standard Euclidean norm (pT p)1/2 of a vector p ∈ Rn is denoted by p2 . The l∞ -norm of p, denoted by p∞ , is defined as p∞ = maxi∈{1,...,n} |pi |. Define Rn+ ⊂ Rn by Rn+ = {p ∈ Rn | p ≥ 0} For p = (pi ) ∈ Rn and q = (qi ) ∈ Rn , we write p ≥ 0 and p ≥ q, respectively, if p ∈ Rn+ and pi ≥ qi (i = 1, . . . , n). Let S n ⊂ Rn×n denote the set of all n × n real symmetric matrices. We write A  O if A ∈ S n is positive semidefinite, i.e. if all the eigenvalues of the matrix A are nonnegative.

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For A ∈ S n and B ∈ S n , we write A  B if the matrix A − B is positive semidefinite. The Moore–Penrose pseudo-inverse of C ∈ Rm×n is denoted by C† ∈ Rn×m . 2.1

Semidefinite program

The semidefinite program (SDP) is classified as a convex and nonlinear mathematical program. The SDP problem refers to the optimization problem having the form of (Wolkowica et al. 2000)   m  T max b y: C − A i yi  O (1) i=1

Here y ∈ Rm is a variable vector, b ∈ Rm is a constant vector, and Ai ∈ S n (i = 1, . . . , m) and C ∈ S n are constant symmetric matrices. Recently, SDP has received increasing attention for its wide fields of application (Ohsaki et al. 1999; Ben-Tal & Nemirovski 2001). It is well known that the linear program and the second-order cone program are included in SDP as particular cases. The primal-dual interior-point method, which has been first developed for LP, has been naturally extended to SDP. It is theoretically guaranteed that the primal-dual interiorpoint method converges to the global optimal solution of the SDP problem (1) within the number of arithmetic operations bounded by a polynomial of m and n (Ben-Tal & Nemirovski 2001; Wolkowicz et al. 2000). 2.2

Quasic onvex optimization problem

The α-sublevel set of a function f : Rn → R is defined as Lf (α) = {x ∈ Rn | f (x) ≤ α} A function f is called quasiconvex if its domain and all its sublevel sets Lf (α) for α ∈ R are convex. Let f0 : Rn → R be quasiconvex, and let f1 , . . . , fm : Rn → R be convex. The quasiconvex optimization problem refers to the optimization problem having the form of (Boyd & Vandenberghe 2004, Section 4.2.5) min{f0 (x): fi (x) ≤ 0 (i = 1, . . . , m), Ax = b}

(2)

where A ∈ Rm×n and b ∈ Rm . The difference between convex and quasiconvex optimization problems is that a quasiconvex optimization problem can have locally optimal solutions that are not globally optimal. It is known that the global optimal solution of a quasiconvex optimization problem can be obtained by using the bisection method in which some convex optimization problems are solved (Boyd & Vandenberghe 2004).

3 Modeling uncertain trusses and mechanical constraints Consider a linear elastic truss in the two- or three-dimensional space. Small rotations and small strains are assumed. Letting nd denote the number of degrees of freedom

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Structural design optimization considering uncertainties d

d

of displacements, u ∈ Rn and f ∈ Rn denote the vectors of nodal displacements and external forces, respectively. The system of equilibrium equations can be written as Ku = f

(3) d

where K ∈ S n denotes the stiffness matrix of the truss. In (3) we explicitly deal with the model- and data-uncertainties of K and f , that shall be rigorously defined below. m Let a = (ai ) ∈ Rn denote the vector of member cross-sectional areas, where nm denotes the number of members. For trusses, the stiffness matrix K is a function of a, and can be decomposed as m

K(a) =

n 

m

n 

ai K i =

i=1

ai bi bTi

(4)

i=1

d

d

where K i ∈ S n and bi = (bij ) ∈ Rn (i = 1, . . . , nm ) are constant matrices and constant vectors, respectively. 3.1

U n c erta i n t y mo d el

Assume that the uncertainty of K is caused only by the uncertainties of stiffness of members, while the locations of nodes are assumed to be certain. We represent the uncertainties of stiffness of members through the uncertainties of member cross-sectional areas a. m d f = (F fj ) ∈ Rn denote the nominal values (or the best estimates) Let F a = (F ai ) ∈ Rn and F m d of a and f , respectively. Let ζ a = (ζai ) ∈ Rn and ζ f = (ζfj ) ∈ Rn denote the parameter vectors that are considered to be unknown but bounded. We describe the uncertainties of a and f by using ζ a and ζ f , respectively. Suppose that a and f depend on ζ a and ζ f affinely, i.e. ai + a0i ζai , ai = F

i = 1, . . . , nm

(5)

fj + f 0 ζfj , fj = F

j = 1, . . . , nd

(6)

m

Here, a0 = (a0i ) ∈ Rn+ and f 0 ∈ R+ are constant coefficients satisfying F ai > a0i 0 m 0 (i = 1, . . . , n ). Note that ai and f represent the relative magnitude of uncertainties of ai and f , respectively. Moreover, a0 and f 0 make ζ a and ζ f have no dimensions. d For p = 1, . . . , nt , let mp ∈ {1, . . . , nd } and let T p ∈ Rmp ×n be a constant matrix. For m d a fixed α ∈ R+ , define two sets Za (α) ⊂ Rn and Zf (α) ⊂ Rn by m

Za (α) = {ζ a ∈ Rn | α ≥ ζ a ∞ } nd

(7)

Zf (α) = {ζ f ∈ R | α ≥ T p ζ f 2 (p = 1, . . . , n )} t

(8)

Here, we choose T 1 , . . . , T nt so that Zf (α) becomes bounded for any α ∈ R+ . It is obvious that Za (α) is bounded. Since a truss is an assemblage of nodes connected by some independent members, the perturbation of stiffness of a member from its nominal value does not affect those

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of the other members. Hence, in (7) we choose the l∞ -norm which represents the independent uncertainties of scalars ζa1 , . . . , ζanm . On the other hand, the definition (8) permits us to suppose that there exist correlation among some components of ζ f by choosing T p appropriately. Moreover, these matrices allow to represent the difference of magnitudes of uncertainties among some components of ζ f . For examples of T p , see Example 3.1 and Section 6.3. The uncertain parameters ζ a and ζ f in (5) and (6), respectively, are assumed to be running through the uncertain sets Za (α) and Zf (α) defined by (7) and (8), i.e. ζ a ∈ Za (α),

ζ f ∈ Zf (α)

(9)

For simplicity, we often write ζ = (ζ Ta , ζ Tf )T ,

Z(α) = Za (α) × Zf (α)

so that (9) is simplified as ζ ∈ Z(α) Roughly speaking, ζ a and ζ f perturb around the origin with the “width’’ of α. Then a and f , respectively, vary around the center-points F a and F f . The greater the value of α, the greater the range of possible variations of a and f , and hence α is called the uncertainty parameter (Ben-Haim 2004). Note that the value of α is usually unknown in structures actually built. Throughout the following robustness analysis based on the info-gap theory, we do not use any knowledge of the actual range of uncertainty of a truss, that is regarded as one of advantages of using the robustness function. It is easy to check that the uncertainty model of a and f defined by (5)–(8) obey a, and F f , let the info-gap model (Ben-Haim 2006) of uncertainty. For given α ∈ R+ , F m d T A(α,F a, F f ) ⊆ Rn × Rn be the set of all vectors (aT , f )T satisfying (5)–(8). Then A(α) satisfies the two basic axioms of the info-gap model: (i) (ii)

Nesting: 0 ≤ α1 < α2 implies A(α1 ,F a, F f ) ⊂ A(α2 ,F a, F f ); F Contraction: the info-gap model A(0,F a, f ) coincides with a singleton set T containing its center point, i.e. A(0,F a, F f ) = {(F a T,F f )T }.

From the nesting axiom we see that the uncertainty set A(α,F a, F f ) becomes more inclusive as α becomes larger. The contraction axiom guarantees that the estimates F a and F f are correct at α = 0. Example 3.1 (interval uncertainty of external load). The interval uncertainty model of the external load f is conventionally used in the so-called interval analysis of uncertain structures; see, e.g., Chen et al. (2002). We show in this example that the uncertainty model of f defined by (6), (8), and (9) includes the interval uncertainty

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Structural design optimization considering uncertainties d

model as a particular case. For each p = 1, . . . , nd , let ep ∈ Rn denote the pth row d vector of the identity matrix I ∈ S n , and let δp be a positive constant. Then, by putting Tp =

1 T e , δp p

p = 1, . . . , nd

with m1 = · · · = mnt = 1 and nt = nd , Zf (α) defined by (8) is reduced to       ζfj  nd  d   Zf (α) = ζ f ∈ R  α ≥   , j = 1, . . . , n δj

(10)

Consequently, the uncertainty of f obeying (6), (9), and (10) can be alternatively written as fj ∈ [F fj − αf 0 δj , F fj + αf 0 δj ],

j = 1, . . . , nd

which coincides with the conventional interval uncertainty model. 3.2 C o nstrai n t s o n mec hanic al per fo r m a n ce Consider the mechanical performance of trusses that can be expressed by the cond d straints in terms of displacements. Let Ql ∈ S n , ql ∈ Rn , and γl ∈ R. Suppose that the constraints on mechanical performance can be written in the following quadratic inequalities in terms of u: uT Ql u + 2qTl u + γl ≤ 0,

l = 1, . . . , nc

(11)

where nc denotes the number of constraints. Suppose that Ql , ql , and γl are functions r of r c ∈ Rn . Here, r c is regarded as the vector of parameters representing the level of r d+1 performance, and nr denotes the number of these parameters. Define H l : Rn → S n by ) * c c Q (r ) q (r ) l l H l (r c ) = − ql (r c )T γl (r c ) r

d

For a given vector r c ∈ Rn , define a set F ⊆ Rn as    

  T u c nd  u c c F(r ) = u ∈ R  H l (r ) ≥ 0 (l = 1, . . . , n ) 1  1

(12)

Then the constraint (11) is equivalently rewritten as u ∈ F(r c )

(13)

Note that we have restricted ourselves to cases in which the constraints on the truss can be represented by a finite number of quadratic inequalities. However, there exist various constraints that can be described via (12) and (13) from a practical point of

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view, because it is known that any single polynomial inequality can be converted into a system of (a finite number of) quadratic inequalities (Kojima & Tunçel 2000). Example 3.2 (stress constraints). We show the explicit reformulation of the stress constraints into (13). Let σi (u) denote the stress of the ith member compatible with u, m and let σ c = (σic ) ∈ Rn+ . Then the stress constraints may be written in the form of |σi (u)| ≤ σic ,

i = 1, . . . , nm

(14)

Here, we assume for simplicity that the lower and the upper bounds of stress of each member have the common absolute value σic . Let E denote the elastic modulus of truss members; let i denote the initial unstressed length of the ith member. From (4) we see  E T σi (u) = b u (15) i i From (15) it follows that (14) is equivalently rewritten as u ∈ F(σ c ) with        T u −(E/i )bi bTi 0 c nd  u m F(σ ) = u ∈ R  ≥ 0 (i = 1, . . . , n ) 1 0T −(σic )2  1 Thus, the stress constraints (14) can be embedded into the form of (13) with nc = nm . The parameters σ c determine the level of performance required. Hence, we have r c = σ c with nr = nm in (13).

4 Definition of robustness function for truss In this section, we show that the robustness function (Ben-Haim 2006) of trusses is obtained as the optimal objective value of a mathematical programming problem with infinitely many constraint conditions. d F = K(F For simplicity, we often write K a). By introducing auxiliary variables η ∈ Rn and from (5) and (6), the system (3) of uncertain equilibrium equations is reduced to m

F + Ku

n 

a0i ζai K i u = η,

ζ a ∈ Za (α)

(16)

i=1

F f + f 0 ζ f = η,

ζ f ∈ Zf (α)

(17) d

a) ⊂ Rn denote the set of all possible solutions to (16) and For a given α ∈ R+ , let U(α,F (17), that is defined by B C d (18) U(α,F a) = u ∈ Rn  (16), (17) Recall that the (nominal) constraint has been introduced in (13). We next consider the robust counterpart of (13). Let α ∈ R+ be fixed. Since the equilibrium equations (16) and (17) include the unknown parameters ζ = (ζ a , ζ t ), the nodal displacement u is regarded as a function of ζ, namely, we may write u(ζ) for ζ ∈ Z(α). To define the robustness function, we require that the constraint (13) should be satisfied by

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all possible realization of u(ζ) when ζ takes any vector satisfying ζ ∈ Z(α). This requirement can be written as u(ζ) ∈ F(r c ),

∀ζ ∈ Z(α)

(19)

By using the set U introduced in (18), the condition (19) is equivalently rewritten as u ∈ F(r c ),

∀u ∈ U(α,F a)

(20)

m

r

For a given F a ∈ Rn and r c ∈ Rn , the robustness function G α(F a, r c ) represents the largest α with which the robust constraint (20) is satisfied. Rigorously, the robustness m r function G α: Rn × Rn → (−∞, +∞] associated with the constraints (11) is defined by (Ben-Haim 2006, Chapter 3)  ∗ α , if Problem (22) is feasible G α(F a, r c ) = (21) 0, if Problem (22) is infeasible where α∗ = max{α: u ∈ F(r c ), ∀u ∈ U(α,F a)}

(22)

Problem (22) is classified to the semi-infinite programming. By semi-infinite we mean an optimization problem having a finite number of scalar variables and infinitely many inequality constraints. Note that α∗ defined by (22) depends on the level r c of constraints on mechanical performance as well as the nominal cross-sectional areas F a. Throughout the chapter, we assume U(0,F a) ⊆ F(r c ) for simplicity, and hence α or G α(F a). Problem (22) is feasible. In what follows, G α(F a, r c ) is often abbreviated by G m m a2 ∈ Rn , we say For the two different vectors of design variables F a1 ∈ Rn and F a2 if G α(F a1 , r c ) >G α(F a2 , r c ). Let ζ 1 ∈ Z(G α). If there exists an that F a1 is more robust than F c l ∈ {1, . . . , n } such that (11) becomes active at a given ζ 1 , then we say that ζ 1 is the worst case. Note that there exists typically more than a single worst case. Especially, optimum truss designs maximizing the robustness function or for specified robustness function often have many worst cases, as will be illustrated in Section 8.2. a), and G α with Figure 17.1 illustrates the schematic relations among F(r c ), U(α,F various values of α. Here, Figure 17.1(a) and 17.1(b), respectively, correspond to α and αb =G α, where we see that the constraint u ∈ F(uc ) is satisfied for all possible αa G

5 Illustrative example of robustness analysis As an illustrative example, consider a two-bar truss shown in Figure 17.2. The nodes (b) and (c) are pin-supported at (x, y) = (0, 100.0) and (0, 0) in cm, respectively, while the node (a) is free, i.e.√nd = nm = 2. The lengths of members (1) and (2), respectively, are 100.0 cm and 100 2 cm. The elastic modulus of each member is 200 GPa. Let f = (f1 , f2 )T denote the external force vector applied at the node (a). The nominal value F f of f is given as F f = (1000.0, 0)T kN The vector of nominal cross-sectional areas is denoted by F a = (F a1 ,F a2 ), and is given by F a = (20.0, 30.0)T cm2 Consider the uncertainty model introduced in section 3.1. In accordance with (5) and (6), define the uncertainties of a and f as ai = F ai + a0i ζai ,

i = 1, 2;

ζ a ∈ Za (α)

(23)

fj + f 0 ζfj , fj = F

j = 1, 2;

ζ f ∈ Zf (α)

(24)

where the coefficients of uncertainty are a0i = 5.0 cm2 ,

i = 1, 2;

f 0 = 200.0 kN

(25)

For a given α, the uncertain sets Za (α) and Zf (α) are defined as Za (α) = {ζ a ∈ R2 | α ≥ |ζai |, i = 1, 2}

(26)

Zf (α) = {ζ f ∈ R2 | α ≥ ζ f 2 }

(27)

y

u2, f2 (1) (a)

(b)

u1, f1

(2)

0 (c)

Figure 17.2 2-bar truss.

x

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Structural design optimization considering uncertainties

Here, we have put nt = 1, T 1 = I, and m1 = 2 in (8). For simplicity, we often write ζ ∈ Z(α) if ζ a ∈ Za (α) and ζ f ∈ Zf (α). Let σ1 and σ2 denote the stresses of members (1) and (2), respectively. Consider the stress constraints of all members defined by (14) with σ1c = σ2c = 1.0 GPa, i.e. the conditions |σi (u)| ≤ σic ,

i = 1, 2

(28)

should be satisfied for any ζ ∈ Z(α). As an example, putting α = 1.0, we randomly generate a number of ζ satisfying ζ ∈ Z(α) with (23) and (24). The corresponding generated a and f defined by (23) and (24) are shown in Figures 17.3 and 17.4, respectively.

36 34

a2 (cm2)

32 30 28 26 24

14

16

18

20 22 a1 (cm2)

24

26

Figure 17.3 The cross-sectional areas a for randomly generated ζ a ∈ Za (α) with α = 1.0.

200

f2 (kN)

100

0

100 200 700

800

900

1000

1100

1200

1300

f1 (kN)

Figure 17.4 The external forces f for randomly generated ζ f ∈ Zf (α) with α = 1.0.

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481

The axial forces q1 and q2 of the members (1) and (2), respectively, are written as q1 = f1 − f2 ,

q2 =

√ 2f2

(29)

Note that q1 and q2 are independent of a, because the truss is statitiscally determinate. From (24), (27), and (29), the maximum value of q1 under the uncertain external force f is obtained as max{q1 (ζ): ζ ∈ Z(α)} = F f1 +

√

2f 0 α

(30)

The minimum value of q1 and the maximum and minimum values of q2 are obtained similarly. Figure 17.5 depicts the variations of (q1 , q2 ) for randomly generated ζ ∈ Z(α) with α = 1.0, and Figure 17.6 shows the corresponding variation of (u1 , u2 ). Figure 17.7 shows the stress states (σ1 , σ2 ) computed from randomly generated ζ ∈ Z(α) with α = 1.0. By using (23), (26), and (30), the maximum value σ1max of σ1 among possible realization of uncertain parameters ζ can be computed analytically as σ1max (α):

√ f1 + 2f 0 α max{q1 (ζ): ζ ∈ Z(α)} F = = max{σ1 (ζ): ζ ∈ Z(α)} = min{a1 (ζ): ζ ∈ Z(α)} F a1 − a01 α

(31)

Similarly, we obtain σ1min (α):

√ F f1 − 2f 0 α = min{σ1 (ζ): ζ ∈ Z(α)} = F a1 + a01 α

(32)

300 200

q2 (kN)

100 0 100 200 300

700

800

900

1000 1100 q1 (kN)

1200

1300

Figure 17.5 The axial forces q for randomly generated (ζ a , ζ f ) ∈ Za (α) × Zf (α) with α = 1.0.

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Structural design optimization considering uncertainties

0 0.1

u2 (cm)

0.2 0.3 0.4 0.5 0

0.1

0.2

0.3 u1 (cm)

0.4

0.5

0.6

Figure 17.6 The nodal displacements u for randomly generated (ζ a ,ζ f ) ∈ Za (α) × Zf (α) with α = 1.0.

0.2 0.15

s2 (GPa)

0.1 0.05 0

0.05 0.1 0.15 0.2 0.2

0.3

0.4

0.5

0.6 0.7 s1 (GPa)

0.8

0.9

1

Figure 17.7 Stress states σ of the 2-bar truss with F a =F a1 for randomly generated (ζ a ,ζ f ) ∈ Za (α) × Zf (α) with α = 1.0.

√

2f 0 α F a2 − a02 α

(33)

σ2min (α): = min{σ2 (ζ): ζ ∈ Z(α)} = −σ2max (α)

(34)

σ2max (α):

= max{σ2 (ζ): ζ ∈ Z(α)} =

Substitution α = 1.0 and F a =F a1 into (31)–(34) results in σ1max = 855.2 MPa,

σ1min = 286.9 MPa,

σ2max = −σ2min = 113.1 MPa

(35)

Maximum robustness design of trusses via semidefinite programming

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0.2 0.15

s2 (GPa)

0.1 0.05 0

0.05 0.1 0.15 0.2 0.2

0.3

0.4

0.5 0.6 0.7 s1 (GPa)

0.8

0.9

1

Figure 17.8 Stress states σ of the 2-bar truss with F a =F a1 for randomly generated (ζ a ,ζ f ) ∈ 1 Za (α) × Zf (α) with α =G α(F a ) = 1.277.

It is verified by Figure 17.7 and (35) that the stress constraints (28) are always inactive for any ζ ∈ Z(α) with α = 1.0. This implies that the robustness function G α(F a1 , σ c ) is greater than 1.0. Observe that the definition (21) (with (22)) of the robustness function is alternatively rewritten as G α(F a1 , σ c ) = max{α: σimax (α) ≤ σic , σimin (α) ≥ −σic (i = 1, 2)}

(36)

By substituting F a =F a1 into (31)–(34), we see that σ1max (α) > σ2max (α) and σ1max (α) > min α(F a1 , σ c ) satisfies the condition |σ1 (α)| hold for any α ≥ 0. Hence, (36) implies that G α) = σ1c σ1max (G from which we obtain G α(F a1 , σ c ) =

σ1cF f1 a1 − F √ c 0 σ1 a1 + 2f 0

= 1.277

α(F a1 , σ c ) The stress states (σ1 , σ2 ) computed from randomly generated ζ ∈ Z(α) with α =G is shown in Figure 17.8. It is observed from Figure 17.8 that the stress constraints 17.8 are always satisfied for the generated ζ, and that the worst case corresponds to the case in which σ1 (ζ) = σ1c holds. The stress constraints on the member (2) are always inactive. We next consider the nominal cross-sectional areas F a2 = (31.7, 21.7)T cm2

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Structural design optimization considering uncertainties

as an alternative truss design. Note that F a1 and F a2 share the same structural volume, 2 and at F a =F a the condition σ1max (α) = σ2max (α)

(37)

is satisfied. Thus, the robustness function G α(F a2 , σ c ) now satisfies the condition σ1max (G α) = σ1c ,

σ2max (G α) = σ2c

from which we obtain G α(F a2 , σ c ) = 2.774 For the truss defined by F a =F a2 , Figure 17.9 depicts the stress states (σ1 , σ2 ) computed from randomly generated ζ ∈ Z(α) with α =G α(F a2 , σ c ). From Figure 17.9 it is seen that c c c the constraints σ1 ≤ σ1 , σ2 ≤ σ2 , and σ2 ≥ −σ2 become active in the worst cases, i.e. the constraints on both members can happen to be active. It is of interest to note that the robustness function of the truss design F a2 is larger 1 1 2 a and F a have the same structural than twice of that of F a in spite of the fact that F volume. This implies that the truss defined by F a2 violates the constraints only at larger ambient uncertainty compared with F a1 . Thus, we may naturally conclude that the truss 2 1 a . design F a is more robust than F Unfortunately, if a truss has moderately many degrees of freedom and/or the uncertainty set has a complicated structure, it is difficult to find the worst case parameters and the corresponding active constraint conditions. This is the crucial difficulty in evaluating the robustness function. This motivates us to propose a numerically tractable formulation for finding a lower bound of the robustness function in the following section.

1 0.8 0.6

s2 (GPa)

0.4 0.2 0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6 s1 (GPa)

0.8

1

Figure 17.9 Stress states σ of the 2-bar truss with F a =F a2 for randomly generated (ζ a ,ζ f ) ∈ 2 Za (α) × Zf (α) with α =G α(F a ) = 2.774.

Maximum robustness design of trusses via semidefinite programming

485

6 Computation of robustness function In this section, we propose an approximation algorithm for Problem (22), which provides a lower bound on the robustness function G α(F a, r c ). We also show that the exact value of the robustness function can be obtained by solving an SDP problem if a is certain. 6.1

Lower bounds of robus tnes s functio n

We start with embedding (16) and (17) into a finite number of quadratic inequalities. d m Define the matrix ∈ Rn ×n by

= (b1 , . . . , bnm ) where bi has been introduced in (4). In what follows, we assume nd < nm , which is usually satisfied for moderately large trusses. Define nn by nn = nm − rank

(38)

where rank denotes the row rank of . Then we see nn > 0. m d m n Let † ∈ Rn ×n denote the pseudo-inverse of . We denote by ⊥ ∈ Rn ×n a basis nm for the nullspace of , where the nullspace of is the set of all vectors β ∈ R satisfying n n d n d

β = 0. Letting ν ∈ Rn , define ξ ∈ Rn +2n +1 and H
l (r c ) ∈ S n +2n +1 (l = 1, . . . , nc ) by ξ = (ν, η, u, 1)

 O O H
l (r c ) = O H l (r c ) so that H
l ξ

 u = Hl 1

holds, where H l (r c ) has been introduced in (12). Let †i,· and ⊥ i,· denote the ith row of

the matrices † and ⊥ , respectively. Note that †i,· and ⊥ i,· are row vectors. Define n +2nd +1

n

d

(i = 1, . . . , nm ) and p (α2 ) ∈ S n +2n +1 (p = 1, . . . , nt ) as ⎞ ⎛ T −( ⊥ ⎛ ⎞ i,· ) 0 ⎟ ⎜ ⎜ ( † )T ⎟ ⎟ ⎜ ⎟ ⎜ i,· † † F 2 2⎜ 0 ⎟ T 0 i (α ) = α ⎝ 0 ⎠ (0T 0T ai bi 0) − ⎜ ⎟ (− ⊥ i,· i,· − i,· K 0) ai bi ⎜−( † K) F T⎟ ⎝ ⎠ i,· 0 0

i (α2 ) ∈ S n

⎛ ⎞ 0 ⎜ ⎟ 2 2⎜0⎟ p (α ) = α ⎝ ⎠ (0T 0 f0

⎞ O T ⎜ Tp ⎟ ⎟ f 0) − ⎜ ⎝ O ⎠ (O T −F f TT ⎛

0T

0T

p

Tp

O

f) −T pF

486

Structural design optimization considering uncertainties

Proposition 6.1.

The conditions (16) and (17) hold if and only if ξ satisfies

ξ T i (α2 )ξ ≥ 0,

i = 1, . . . , nm

(39)

ξ T p (α2 )ξ

p = 1, . . . , n

(40)

Proof.

≥ 0,

t

m

By introducing w = (wi ) ∈ Rn , we see that (16) is equivalently rewritten as

F − η,

w = Ku

(41)

wi = ζai (−a0i bTi u),

α ≥ |ζai |,

i = 1, . . . , nm

(42)

⊥

From the definition of and , we see that any solution to (41) can be written as †

F − η) + ⊥ ν w = † (Ku

(43)

nn

with ν ∈ R . On the other hand, the condition (42) is equivalent to wi2 ≤ (a0i α)2 (bTi u)2 ,

i = 1, . . . , nm

(44)

Consequently, by using (43) and (44), we see that (41) and (42) are equivalent to F − η) + ⊥ ν]2 ≥ 0, (a0i α)2 (bTi u)2 − [ †i,· (Ku i,·

i = 1, . . . , nm

Thus, the condition (16) is equivalent to (39). From the definition (8) of Zf it follows that ζ f ∈ Zf if and only if ζ f satisfies α2 ≥ ζ Tf T Tp T p ζ f ,

p = 1, . . . , nt

Hence, the condition (17) can be equivalently embedded into the following quadratic inequalities in terms of η: f )T T Tp T p (η − F f ), (f 0 α)2 ≥ (η − F

p = 1, . . . , nt

which can be rewritten as (17). Consequently, (17) is equivalent to (40). t c

Let ρ ∈ Rn n and τ ∈ Rn

m nc

be

ρ = (ρ11 , . . . , ρnt 1 , . . . , ρ1nc , . . . , ρnt nc )T τ = (τ11 , . . . , τnm 1 , . . . , τ1nc , . . . , τnm nc )T The following proposition, which plays a key role in constructing an approximation of Problem (22), shows a relaxation of infinitely many constraints by using a finite number of constraints: Proposition 6.2. u ∈ U(α,F a)

The implication =⇒

u ∈ F(r c )

(45)

holds if there exist ρ and τ satisfying t

H
l (rc )

−

n  p=1

m

ρpl p (α ) − 2

n  i=1

τil i (α2 )  O,

l = 1, . . . , nc

(46)

Maximum robustness design of trusses via semidefinite programming

ρ ≥ 0,

τ≥0

487

(47)

Proof. From Proposition 6.1 it follows that u ∈ U(α,F a) if and only if (39) and (40) are satisfied. Observe that the constraint (13) is reduced to ξ T H
l ξ ≥ 0,

l = 1, . . . , nc

Consequently, the implication (45) holds if and only if the implication ξ T p ξ ≥ 0,

p = 1, . . . , nt

ξ T i ξ ≥ 0 i = 1, . . . , nm

=⇒

ξ T H
l ξ ≥ 0

(48)

holds for each l = 1, . . . , nc . The assertion of this proposition is obtained by applying Lemmas A.1 and A.2 (ii) to (48). Proposition 6.2 implies that the set of a finite number of constraints (46) and (47) in terms of a finite number of variables corresponds to a sufficient condition for the infinitely many constraints of Problem (22). A lower bound of Problem (22) is then naturally constructed as follows: t c

Consider the following problem in variables (t, ρ, τ) ∈ R × Rn n × Rn ⎧ nt nm ⎨  
c ∗ t : = max ρpl p (t) − τil i (t)  O (l = 1, . . . , nc ), t : Hl (r ) − t,ρ,τ ⎩

Lemma 6.3.

p=1



m nc

:

i=1

ρ ≥ 0, τ ≥ 0

(49)

Then G α(F a, r c )2 ≥ t ∗ Proof. Recall that the robustness function G α is defined by (21) with Problem (22). It follows from Proposition 6.2 that the constraints of Problem (22) are satisfied if the constraints of Problem (49) are satisfied. This completes the proof. 6.2 Algorithm for computing lower bounds Lemma 6.4.

Problem (49) is a quasiconvex programming problem.

For a given t, define a set T by ⎧   nt nm ⎨   t m c  T (−t) = (ρ, τ) ∈ R(n +n )n  H
l − ρpl p (t) − τil i (t)  O (l = 1, . . . , nc ), ⎩  p=1 i=1 

Proof.

ρ ≥ 0, τ ≥ 0

488

Structural design optimization considering uncertainties

By regarding t ∈ R as an auxiliary variable, Problem (49) is equivalently rewritten as min{t : (ρ, τ) ∈ T (t)}

(50)

t,ρ,τ

Observe that T (t) is defined by nc linear matrix inequalities and (nt + nm )nc linear inequalities. Hence, T (t) is convex for any given t ∈ R. This implies that Problem (50) is a quasiconvex optimization problem. Let I denote the identity matrix with an appropriate size. For a fixed t, consider the t c m c following problem in the variables (s, ρ, τ) ∈ R × Rn n × Rn n :

∗

s : = min s,ρ,τ

⎧ ⎨

t

⎩

s : H
l (r c )

−

n 

m

ρpl p (t) −

p=1

n 

τil i (t) + sI  O (l = 1, . . . , nc ) ,

i=1

 ρ ≥ 0, τ ≥ 0

(51)

Problem (51) corresponds to a convex feasibility problem of Problem (49) at the given level t. Lemma 6.4 guarantees that the following bisection method solves Problem (49): Algorithm 6.5 (bisection method for Problem (49)). Step 0: Step 1: Step 2: Step 3: Step 4:

0

0

Choose t 0 and t satisfying 0 ≤ t 0 ≤ t ∗ ≤ t , and the small tolerance  > 0. Set k = 0. k k If t − t k ≤ , then stop. Otherwise, set t = (t k + t )/2. ∗ ∗ ∗ Find an optimal solution (s , ρ , τ ) to the SDP problem (51). k+1 k k+1 If s∗ ≤ 0, then set t k+1 = t and t = t . Otherwise, set t = t and t k+1 = t k . Set k ← k + 1, and go to Step 1.

Algorithm 6.5 finds a global optimal value t ∗ of Problem (49) by solving some SDP 0 problems, where exactly #log2 ((t − t 0 )/)$ iterations are required before the algorithm terminates. Here, we denote by #γ$ the minimum integer that is not smaller than γ ∈ R. From Lemma 6.3 it follows that (t ∗ )1/2 corresponds to a lower bound of the robustness function G α(F a, r c ). At Step 0, we may simply choose t 0 = 0, and a 0 sufficiently large t . At Step 2 of each iteration, we solve Problem (51), which can be embedded into the standard form of SDP problem (1) with m = nc (nt + nm + 1) + 1 and n = nc (nn + 2nd + nm + nt + 1). It should be emphasized that a global optimal solution to an SDP problem (51) can be obtained by using the primal-dual interior-point method, where the number of arithmetic operations is bounded by a polynomial of m and n (Wolkowicz et al. 2000).

Maximum robustness design of trusses via semidefinite programming

6.3

489

Spec ial cas e

The remainder of this section is devoted to investigating the case in which a is certain. The following result shows that, under some assumptions on the uncertainty set, the robustness function G α can be obtained by solving an SDP problem: By putting nt = 1 in (8), let Zf be

Proposition 6.6.

d

Zf (α) = {ζ ∈ Rn |α ≥ T1 ζ2 }

(52)

and let a0i = 0,

i = 1, . . . , nm

(53)

a) ∈ S n in (5). Assume that Hl  O (l = 1, . . . , nc ). Define 0 (α2 ,F ) *

 (T1F K)T 0 ! T 0" 2 2 0 f − (T1F K −F f ). 0 (α ,F a) = α T f0 −F f

d +1

by

Then the robustness function G α(F a, r c ) is obtained by solving the following SDP problem nc in the variables (t, µ) ∈ R × R with µ = (µl ) ∈ nc : D E G α(F a, r c )2 = max t : µl Hl (r c ) − 0 (t,F a)  O (l = 1, . . . , nc ), µ ≥ 0 (54) t,µ

Proof. From (52) and (53), the uncertain equilibrium equations (16) and (17) are reduced to F =F Ku f + f 0 ζ f , α ≥ T 1 ζ f 2 Hence, u ∈ U(α,F a) if and only if 1 2T 1 2 F −F F −F T 1 (Ku T 1 (Ku f) f ) ≤ α2 from which we obtain u ∈ U(α,F a)

⇐⇒

T

 u u 0 (α2 ,F a) ≥0 1 1

(55)

By using (55) and Lemmas A.2 (i) and A.1, we see that the implication (45) holds if and only if ∃ρl ≥ 0

subject to

H l  ρl 0 (α2 ,F a),

l = 1, . . . , nc

(56)

Note that H l  O implies that ρl = 0 does not satisfy (56). Hence, by putting µl = 1/ρl , l = 1, . . . , nc , the implication (56) is reduced to ∃µl ≥ 0

subject to µl H l − 0 (α2 ,F a)  O,

l = 1, . . . , nc

Consequently, Problem (22) is reduced to D E a)  O (l = 1, . . . , nc ), µ ≥ 0 G α(F a, r c ) = max α : µl H l (r c ) − 0 (α2 ,F α,µ

(57)

We see in Problem (57) that maximizing α is equivalent to maximizing α2 , which concludes the proof.

490

Structural design optimization considering uncertainties

7 Maximization of robustness function Throughout this section, we assume that the assumptions in Proposition 6.6 hold, i.e. only f possesses the uncertainty defined by (6) and (52) and a =F a is always satisfied. In Section 5, we have observed through an analytical example that the truss with the larger robustness function is considered to be more robust. We attempt in this section to find F a which maximizes the robustness function G α(F a, r c ). We call this structural optimization problem the maximization problem of robustness function. Consider the conventional constraints on F a which are dealt with in the usual structural optimization problems, e.g. the upper and lower-bound constraints of F a and m g the upper-bound constraint of structural volume. Letting g : Rn → Rn be a smooth function, we assume that these constraints can be written in the form of g(F a) ≥ 0

(58)

Note that g(F a) involves neither u nor f . For the given r c and g, the maximization problem of robustness function is formulated as max{G α(F a, r c ) : g(F a) ≥ 0} F a

(59)

In what follows, the argument r c is often omitted for brevity. m α(F a, r c ) > 0 and g(F a) ≥ 0. Then the objecAssume that there exists F a ∈ Rn satisfying G tive function of Problem (59) can be replaced by G α(F a, r c )2 without changing the optimal solution. From this observation and Proposition 6.6 it follows that Problem (59) is equivalent to the following problem: max{t : µl H l − 0 (t,F a)  O (l = 1, . . . , nc ), µ ≥ 0, g(F a) ≥ 0} t,µ,F a

(60)

Problem (60) is sometimes referred to as nonlinear semidefinite programming problem (Kanzow et al. 2005). To solve Problem (60), we next propose a sequential SDP method, which is an extension of the successive linearization method for standard nonlinear programming problems. Let DG(x
) denote the derivative of the smooth mapping G : Rm → S n at x
= (x
i ) ∈ Rm defined such that DG(x
)h is a linear function of h = (hi ) ∈ Rm given by DGl (x
)h =

 m  ∂Gl (x)  hi ∂xi x=x
i=1

The following is the sequential SDP method solving Problem (60) based on the successive linearization method: Algorithm 7.1 (Sequential SDP method for Problem (60)). Step 0: Step 1:

Choose F a0 satisfying g(F a0 ) ≥ 0 and G α(F a0 , r c ) > 0; choose cmax ≥ cmin > 0, 0 c ∈ [cmin , cmax ], and the small tolerance  > 0. Set k = 0. Find an optimal solution (t k , µk ) of Problem (54) by setting F a =F ak .

Maximum robustness design of trusses via semidefinite programming

Step 2:

c

491

m

Find the (unique) optimal solution ( t k , µk , F ak ) ∈ R × Rn × Rn of the SDP problem ⎫ 1 k ⎪ ⎪ c ( t, µ, F a)22 ⎪ ⎪

t, µ, F a 2 ⎬ k c subject to F l ( t, µ, F a)  O, l = 1, . . . , n , ⎪ ⎪

µ + µk ≥ 0, ⎪ ⎪ ⎭ k T k ∇g(F a ) F a + g(F a )≥0 max

t −

(61)

where a) = ( µl + µkl )H l − D 0 (t k ,F ak )( t, F a T )T − 0 (t k ,F ak ). F kl ( t, µ, F Step 3: Step 4:

If ( t k , µk , F ak )2 ≤ , then stop. k+1 k =F a + F ak . Set F a k+1 ∈ [cmin , cmax ]. Set k ← k + 1, and go to Step 1. Choose c

Essentially, Algorithm 7.1 solves the nonlinear SDP problem (60) by successively approximating it as the SDP problems. In Steps 1 and 2, we solve the SDP problems (54) and (61) by using the primal-dual interior-point method (Wolkowicz et al. 2000). The following proposition shows the global convergence property of Algorithm 7.1: Proposition 7.2. (Kanno & Takewaki 2006b). SupposeF f = 0 and that Problem (61) is strictly feasible at each iteration. Let {(t k , µk ,F ak )} be a sequence generated by Algorithm ak )} is a stationary point of Problem (60). 7.1. Then any accumulation point of {(t k , µk ,F

8 Numerical examples The lower bounds on the robustness functions are computed for various trusses by using Algorithm 6.5. Moreover, the optimal designs with the maximal robustness functions are computed for various trusses by using Algorithm 7.1 in the case where only the external forces possess uncertainties. In these algorithms, the SDP problems are solved by using SeDuMi Ver. 1.05 (Sturm 1999), which implements the primaldual interior-point method for the linear programming problems over symmetric cones. Computation has been carried out with MATLAB Ver. 6.5.1 (The MathWorks, Inc. 2002). 8.1

20-bar trus s

Consider a plane truss illustrated in Figure 17.10, where nd = 16 and nm = 20. Nodes (a) and (b) are pin-supported. The lengths of members in the x- and y-directions, respectively, are 100 cm and 50 cm. The elastic modulus of each member is 200 GPa. We assume that the cross-sectional areas of members (1)–(5) have uncertainty, whereas those of members (6)–(20) are certain. The external loads applied at nodes (e)–(j) have uncertainty, whereas those applied at nodes (c) and (d) are certain. No external loads are applied at nodes (c) and (d). The nominal cross-sectional areas are

492

Structural design optimization considering uncertainties

y

(11)

(19)

(16)

(c)

(18)

(3) (4)

0

(12)

(d)

(1)

(2) (a)

(13)

(f)

(6) (15)

(14)

(h)

(7)

(e) (9)

(20)

(17)

(g) (10)

(j)

(8)

(i)

(5) (b)

兵

兵

Uncertain loads & certain stiffness

Uncertain stiffness & certain loads

x

Figure 17.10 20-bar truss.

F ai = 20.0 cm2 (i = 1, . . . , 20). As the nominal external loads, we consider the following two cases: (Case 1): (Case 2):

(200.0, 0) kN, (500.0, 0) kN, (700.0, −400.0) kN, and (0, −400.0) kN are applied at the nodes (e), (g), (i), and (j), respectively; (200.0, 0) kN, (500.0, 0) kN, (700.0, −700.0) kN, and (0, −700.0) kN are applied at the nodes (e), (g), (i), and (j), respectively.

The coefficients of uncertainty in (5) and (6) are a0i = 2.5 cm2 (i = 1, . . . , 5) and (j) (j) f 0 = 50.0 kN. The uncertainty set for ζ f is given by (52) with T 1 = I. Let u(j) = (ux , uy )T denote the nodal displacement vector of the node (j). As the constraint (13) we consider the following conditions: |u(j) x | ≤ 5.0 cm,

(62)

|u(j) y |

(63)

≤ 2.0 cm.

The lower bound of the robustness function G α(F a, uc ) is computed by using Algorithm 0 0 6.5 for each case. We set t = 0, t = 10.0, and  = 10−4 . The lower bounds (t ∗ )1/2 are obtained as 2.672 and 2.412 for (Case 1) and (Case 2), respectively, after 17 SDP problems are solved. Thus, the robustness functions depend on the nominal external loads.

Maximum robustness design of trusses via semidefinite programming

493

0.8 1

u(j)y

1.2 1.4 1.6 1.8 2 2

2.5

3

3.5 u(j) x

4

4.5

5

Figure 17.11 Nodal displacements of the node ( j ) in (Case 1) for randomly generated ζ ∈ Z(α) with α = 2.6717. 0.8 1 1.2

u(j) y

1.4 1.6 1.8 2 2

2.5

3

3.5 u(j) x

4

4.5

5

Figure 17.12 Nodal displacements of the node ( j ) in (Case 2) for randomly generated ζ ∈ Z(α) with α = 2.4124.

We next randomly generate a number of ζ a and ζ f satisfying (7) and (8), respectively, by putting α = (t ∗ )1/2 , and compute the corresponding nodal displacements. Figures 17.11 and 17.12 depict the obtained displacement of the node (j) for (Case 1) and (Case 2), respectively. It is observed from Figures 17.11 and 17.12 that the

494

Structural design optimization considering uncertainties

constraints (62) and (63) are satisfied for all generated (ζ a , ζ f ), which verifies that α. In (Case 1), from Figthe obtained values (t ∗ )1/2 are certainly the lower bounds of G ure 17.11 we may conjecture that the worst case corresponds to the case in which the constraint (62) becomes active. On the other hand, in (Case 2), Figure 17.12 shows that the worst case corresponds to the case in which the constraint (63) becomes active. In (j) (j) both cases, at least one of |ux | and |uy | possibly becomes very close to its bound. This implies that Algorithm 6.5 provides sufficiently tight lower bounds, i.e. the obtained α in each case. value (t ∗ )1/2 is very close to the exact value of G 8.2

2 9 - b ar trus s

Consider a truss illustrated in Figure 17.13. The nodes (a) and (b) are pin-supported, where nd = 20 and nm = 29. The lengths of members both in x- and y-directions are 50.0 cm. The elastic modulus of each member is 200 GPa. Suppose that the force of (0, −10.0) kN is applied at the nodes (c) and (d) as the nominal external load F f . The uncertainty set for ζ f is given by (52) with T 1 = I. We put f 0 = 1.0 kN in (6). The member cross-sectional areas a are assumed to be certain. Hence, we can compute the exact value of the robustness function by using Proposition 6.6. Consider the stress constraints (14) with σic = 500 MPa for each member. The maximization problem (60) of the robustness function is solved by using Algorithm 7.1. As the constraints (58) in Problem (59), we consider the conventional constraint on structural volume as well as nonnegative constraints of F a, namely, g is defined as

g(F a) =

 F a V − V(F a)

y (a)

(9)

(10)

(18) (1)

(15)

0

(b)

(14)

(22) (4)

(23) (6)

(28)

(8)

(29) (17)

(16)

(c)

(d) ~ ƒ

Figure 17.13 29-bar truss.

(7)

(26)

(13)

(21) (27)

(5)

(25)

(12)

(2)

(20)

(19) (3)

(24)

(11)

x ~ ƒ

Maximum robustness design of trusses via semidefinite programming

495

Here, V(F a) denotes the total structural volume of a truss, which is a linear function a0i = 20.0 cm2 (i = 1, . . . , nm ). We of a, and ng = nm + 1. The initial solution is given as F first compute the robustness function at the initial solution F a =F a0 by using Proposition 6.6. Since only the external load possesses the uncertainty, the robustness function is obtained as G α(F a0 ) = 0.7261 by solving only one SDP problem. In Algorithm 7.1 we set  = 0.1, cmax = cmin = 10−5 , and V = 3.3971 × 104 cm3 so aopt found by that the volume constraint becomes active at F a =F a0 . The optimal design F Algorithm 7.1 after 53 iterations is shown in Figure 17.14, where the width of each member is proportional to its cross-sectional area. The corresponding robustness function is G α(F aopt ) = 11.0710. We compute the optimal designs for various V. Figure 17.15 depicts the relation between V and the robustness function at the optimal design. For comparison, we compute the robustness function for the cross-sectional areas that

Figure 17.14 Optimal design of the 29-bar truss. 12

Robustness function ^ a

10 8 6 4 2 0 0

0.5

1

1.5 2 Volume V (cm3)

2.5

3

3.5 104

Figure 17.15 Relation between V and G α of the optimal trusses (×: initial solution; •: optimal solution; ∗: solution obtained by scaling aopt ).

496

Structural design optimization considering uncertainties

are obtained by scaling F aopt . It is observed from Figure 17.15 that the optimal design cannot be obtained only by scaling F aopt . It is of interest to note that, from the definition of the robustness function, all truss designs are plotted in (or on the boundary of) the domain D in Figure 17.15. Thus, engineers may be able to make decisions incorporating the trade-off between the robustness and the structural volume.

9 Conclu sions Based on the info-gap theory (Ben-Haim 2006), the robustness function of trusses has been investigated extensively as a measure of robustness of a truss under load and structural uncertainties. We have proposed an approximation algorithm for computing the robustness functions of trusses under the load and structural uncertainties. A global convergent algorithm has been proposed for the maximization problem of the robustness function. We have introduced an uncertainty model of trusses, where the external forces as well as the member stiffness include uncertainties. We assume that the constraints on mechanical performance can be expressed by using some quadratic inequalities in terms of displacements. In fact, we can deal with the polynomial inequality constraints in terms of displacements by converting them into a finite number of quadratic inequalities. Then we have formulated a quasiconvex optimization problem, which provides a lower bound, i.e. a conservative estimation, of the robustness function. In order to obtain a global optimal solution to the present quasiconvex optimization problem, a bisection method has been proposed, where a finite number of SDP problems are successively solved by the primal-dual interior-point method. In order to solve the maximization problem of the robustness function for variable member cross-sectional areas, a sequential SDP approach has been presented, where the SDP problems are successively solved by the primal-dual interior-point method to obtain the optimal truss designs. The method has been shown to be globally convergent under certain assumptions.

Technical lemmas Lemma A.1 (homogenization). two conditions are equivalent: (a) (b)

 

T x x Q p ≥ 0, 1 1 pT r 

Q p O pT r

Let Q ∈ S n , p ∈ Rn , and r ∈ R. Then the following

∀ x ∈ Rn

Proof. The implication from (b) to (a) is trivial. We show that (a) implies (b) by the contradiction. Suppose that there exist x
∈ Rn and η ∈ R satisfying


T 
 x Q p x θall ) the probability that the drift θ10/50 for the 10/50 hazard level exceeds the allowable drift θall , that is bound by an upper allowable probability equal to pall . For rigid frames with W-shape cross sections as in this study, the design constraints were taken from the design requirements specified by Eurocode 3 (2003) and Eurocode 8 (2004). During the solution of the optimization problems a number of MCS runs are carried out for each different set of design variables. In order to replace the time consuming FE analyses by predicted results obtained with a trained NN, a training procedure is performed based on the data collected from a number of previously performed FE

584

Structural design optimization considering uncertainties

analyses. After the training phase is concluded the NN predictions replace all conventional FE analyses, for the current design. For the selection of the suitable training pairs, the sample space for each random variable is divided into equally spaced distances. The central points within the intervals are used as inputs for the FE analyses. The random variables considered are the cross-sectional dimensions of the structural members, the material properties (E, σy ) and the loading conditions. Under the proposed approach the FE analyses required during the MCS are replaced by NN predictions of the structural response. For every design a NN is trained utilizing available information generated from selected conventional FE analyses. The trained NN is then used to predict the structural response for different sets of random variables depending on the type of problem examined. 6.2.1 Ea rth q u a ke lo a d in g o f s t e e l f r a m e s In Eurocodes earthquake loading is taken as a random action, therefore it must be considered for the structural design with the following loading combination: Sd =



G “+’’ Ed “+’’



kj

ψ2i Qki

(8)

where “+’’ implies “to be combined with’’,  implies “the combined effect of’’, Gkj denotes the characteristic value of the permanent action j, Ed is the design value of the seismic action, and Qki refers to the characteristic value of the variable action i, while ψ2i is the combination coefficient for quasi permanent value of the variable action i, here taken as 0.30. Design code checks are implemented in the optimization algorithm as constraints. Each structural member should be checked for actions that correspond to the most severe load combination obtained from Eq. (8) and the permanent load combination: Sd = 1.35



Gkj “+’’ 1.50



Qki

(9)

It should be pointed out that the seismic action is obtained from the elastic spectrum reduced by the behaviour factor q. This is done because the structure is expected to absorb energy by deforming inelastically. Maximum values for the q-factor are suggested by design codes and vary according to the material and the type of the structural system. For the framed steel structures considered in this study q = 4.0. The most common approach for the definition of the seismic input is the use of design code response spectra, a general approach that is easy to implement. However, if higher precision is required, the use of spectra derived form natural earthquake records is more appropriate. Since significant dispersion on the structural response due to the use of different natural records has been observed, these spectra must be scaled to the same desired earthquake intensity. The most commonly applied scaling procedure is based on the peak ground acceleration (PGA). Dynamic analysis of simple frames is most frequently performed using the multi-modal response spectrum analysis, which is based on the mode superposition approach and is briefly described in the next section.

M e t a m o d e l-b a s e d c o m p u t a t i o n a l t e c h n i q u e s i n p r o b a b i l i s t i c o p t i m i z a t i o n

6.2.2

585

D yn a mic anal y si s usi ng Mul ti-modal Re s p o n s e Sp e ct ru m

In general, the equations of equilibrium for a finite element system in motion can be written in the usual form Mü(t) + C u(t) ˙ + Ku(t) = R(t)

(10)

where M, C, and K are the mass, damping and stiffness matrices; R(t) is the external load vector, while u(t), u(t) ˙ and u(t) ¨ are the displacement, velocity, and acceleration vectors of the finite element assemblage, respectively. A design approach based on the Multi-modal Response Spectrum (MmRS) analysis, which, in turn, is based on the mode superposition approach, has been used in the present study. The MmRS method is based on a simplification of the mode superposition approach with the aim to avoid time history analyses which are required by both the direct integration and mode superposition approaches. In the case of the multi-modal response spectrum analysis Eq. (10) is modified according to the modal superposition approach to a system of independent equations Mi ÿi (t) + C i y˙ i (t) + Ki yi (t) = Ri (t)

(11)

where Mi =
Ti M
i ,

C i =
Ti C
i ,

Ki =
Ti K
i

and R(t) =
Ti R(t)

(12)

are the generalized values of the corresponding matrices and the loading vector, while
i is the i-th eigenmode shape matrix. According to the modal superposition approach the system of N differential equations, which are coupled with the off-diagonal terms in the mass, damping and stiffness matrices, is transformed to a set of N independent normal-coordinate equations. The dynamic response can therefore be obtained by solving separately for the response of each normal (modal) coordinate and by superposing the response in the original coordinates. In the MmRS analysis a number of different formulas have been proposed to obtain reasonable estimates of the maximum response based on the spectral values without performing time history analyses for a considerable number of transformed dynamic equations. The simplest and most popular one is the Square Root of Sum of Squares (SRSS) of the modal responses. According to this estimate the maximum total displacement is approximated by

umax =

N  i=1

1/2 u2i,max

(13)

ui,max =
i yi,max where ui,max corresponds to the maximum displacement vector corresponding to the i-th eigenmode.

Victoria Mexico (06.09.80) Kobe (16.01.95) Imperial Valley (19.05.40) Duzce (12.11.99) San Fernando (09.02.1971) Gazli (17.05.1976) Friuli (06.05.1976) Aigion (17.05.90) Central California (25.04.54) Alkyonides (24.02.81) Northridge (17.01.94) Athens (07. 09.99) Cape Mendocino (25.04.92) Erzihan,Turkey (13.03.92) Kalamata (13.09.86) Iran (16.09.78) Loma Prieta1 (18.10.89) Loma Prieta2 (18.10.89) Mammoth Lakes (27.05.80) Irpinia, Italy (23.11.80)

*Ms : Surface moment magnitude.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Earthquake name (Date)

Table 21.3 List of natural accelerograms.

Cerro Prieto\Alluvium Kobe\Rock El Centro Array\CWB: D, USGS: C Bolu\CWB: D, USGS: C Pacoima dam\Rock Karakyr, CWB:A Bercis\CWB: B OTE building\Stiff soil Hollister City Hall\CWB: D, USGS: C Korinthos OTE building\Soft soil Jensen filter Plant\CWB: D, USGS: C Sepolia (Metro Station)\Unknown Petrolia\CWB: D, USGS: C Erzikan East-East Comp\CWB: D, USGS: S Kalamata, Prefecture\Stiff soil Tabas\CWB: S Hollister Diff Array\CWB: D Coyote Lke dam\CWB: D McGee Creek\CWB: D Sturno\Unknown

Site\Soil Conditions 45 0 180 90 164 90 0 90 271 90 292 0 90 270 0 0 255 285 0 270

Orientation 6.4 6.95 7.2 7.3 6.61 7.3 6.5 4.64 – 6.69 6.7 5.6 7.1 6.9 5.75 7.4 7.1 7.1 5 6.5

MS 0.62 0.82 0.31 0.82 1.22 0.72 0.03 0.20 0.05 0.31 0.59 0.24 0.66 0.49 0.21 0.85 0.28 0.48 0.33 0.36

PGA (g)

31.57 81.30 29.80 62.10 112.49 71.56 1.33 9.76 3.90 22.70 99.10 17.89 89.72 64.28 32.90 121.40 35.60 39.70 8.55 52.70

PGV (cm/sec)

19.30 9.91 10.32 12.99 10.69 9.83 21.17 20.00 12.77 13.34 5.86 13.32 7.24 7.56 6.41 6.89 7.69 11.95 37.29 6.66

a/v (sec)

M e t a m o d e l-b a s e d c o m p u t a t i o n a l t e c h n i q u e s i n p r o b a b i l i s t i c o p t i m i z a t i o n

5

5

5 2

1

4

4

2

1

2 3

3 2

1

5 2

2 

4

1 4

2

1 3

3 2

587

2

1

Figure 21.4 RBDO Test example – Geometry, member grouping.

6.2.3 Three-storey plane frame under earthquake loads RBDO example One test example has been considered in the present study in order to illustrate the efficiency of the proposed methodology for reliability-based sizing optimization problems under earthquake loading. This test example is a four-bay, three-storey moment resisting plane frame shown in Figure 21.4. The frame has been previously studied by Gupta and Krawinkler (2000), where a detailed description of the structure is given. The frame consists of rigid moment connections and fixed supports. Each bay has a span of 9.15 m (30 ft), while each storey is 3.96 m (13ft) high. The permanent action considered is equal to 5 kN/m2 while the variable action is equal to 2 kN/m2 , both distributed along the beams. The frame is considered to be part of a 3D structure where each frame is 4.5 m (15ft) apart. The median spectrum used for the determination of the base shear corresponds to a peak ground acceleration of 0.32 g. Structural members are divided into five groups, as shown in Figure 21.4, corresponding to the five design variables of a discrete structural optimization problem. The cross-sections are W-shape beam and column sections available from manuals of the American Institute of Steel Construction (AISC). The objective function is the weight of the structure, to be minimized. In this study a suite of twenty natural accelerograms, shown in Table 21.3, is used. It can be seen that each record corresponds to different earthquake magnitudes and soil conditions. The records of this suite comprise a wide range of PGA and peak acceleration over peak displacement ratio (a/v) values. The latter parameter is considered to describe the damage potential of the earthquake more reliably than PGA. The records are scaled to the same PGA and their response spectra that are subsequently derived are shown in Figure 21.5. It has been observed that the response spectra follow the lognormal distribution. Therefore the median spectrumx, ˆ also shown in Figure 21.5, and the standard deviation δ are calculated from the above suite of spectra using the following expressions:  n xˆ = exp

i=1

ln (Rdi (T)) n

(14)

588

Structural design optimization considering uncertainties

1.6

Spectral acceleration (g)

1.4 1.2 1 Median spectra 0.8 0.6 0.4 0.2 0

0.5

0

1

1.5

2 Period T (sec)

2.5

3

3.5

4

Figure 21.5 Natural record response spectra and their median.

Table 21.4 Characteristics of the random variables. Random variable

Probability density function

Mean value

Standard deviation

E σy Seismic load

N N Log-N

2.1 106 MPa 235 MPa Median Spectrum (Eq. 14)

0.10E 0.10σy δ (Eq. 15)

 n ! δ=

i=1

"2 1/ 2 ln (Rdi (T)) − ln (x) ˆ n−1

(15)

where Rdi (T) is the response spectrum value for period equal to T of the i-th record (i = 1, . . . , n, where n = 20 in this study). For a given period value, the acceleration Rd is obtained as a random variable following the log-normal distribution with its mean value equal to xˆ and the standard deviation equal to δ. The deterministic constraints are related to stress and displacement constraints for steel frames according to Eurocodes. The probabilistic constraint is imposed on the probability of structural collapse which is set equal to pall = 0.001. The probability of failure caused by uncertainties related to seismic loads and material properties of the structure is estimated using MCS with the LHS technique. The earthquake ground motion parameter, as described in Eq. (14), the yield stress and the elastic modulus are considered to be random variables. The type of probability density functions, mean values, and variances of the random parameters are shown in Table 21.4. The seismic action follows a log-normal probability density function, while the rest of

M e t a m o d e l-b a s e d c o m p u t a t i o n a l t e c h n i q u e s i n p r o b a b i l i s t i c o p t i m i z a t i o n

589

Table 21.5 Performance of the methods. Optimization procedure

ES cycles

pf

Time sequential (h)

Time parallel (p = 5) (h)

DBO RBDO-MCS (5,000 siml.) RBDO-LHS (1,000 siml.) RBDO-NN (100,000 siml.)

157 65 72 68

0.0932 0.0008 0.001 0.0009

0.3 557.3 149.1 42.1

0.08 140.1 40.6 16.2

the random variables follow a normal probability density function. For more details on probabilistic formulations of uncertainties the reader is referred to JCSS (2001) guidelines. For this test case the (µ + λ)-ES approach is used with µ = λ = 5 (equal to the number of design variables), while a sample size of 5000 simulations is taken. Table 21.5 depicts the performance of the optimization procedure for this test case. As it can be seen, the probability of failure corresponding to the optimum computed by the deterministic optimization procedure is much larger than the specified value of 10−3 , thus unacceptable. On the other hand, the increase in safety results also in a significant increase on optimum weight. When probabilistic constraints are considered the weight increase is approximately 26% compared to the deterministic one, from 125.3 to 167.4 tn. Furthermore, the computation times are also enlarged in the case of RBDO, however, the use of NN as well as parallel computation reduces drastically the excessive computational cost of the process. As far as the NN implementation is concerned, it was performed in a similar manner as the ES-NN1 algorithm that was described in the previous section. The NN configuration used has the typical architecture shown in Figure 21.1. It consists of three layers: one input, one hidden, and one output layer with varying number of nodes per layer. After an initial investigation on the optimum number of hidden layers and their nodes, one hidden layer was used having 10 nodes. The input data of the NN are the eleven random variables (two for each of the five element groups plus the seismic coefficient), while the output is one, i.e. the maximum interstorey drift value, which defines the limit-state violation. Thus, the NN configuration that was used was the following: 16-20-1. The training-testing set of the NN consisted of two hundred input/output pairs, twenty of which were used for testing the generalization capabilities of the trained NN. The application of NN reduces the computing time in a fraction of the time required for the conventional FE analyses. In addition, it does not affect the accuracy of the MCS method, in fact it can increase it since the fast NN approximations allow the use of much greater sampling size.

6.3

Hybrid RRDO 3D trus s tes t exampl e

For the purposes of this study a 3D steel truss structure has also been considered. For this test example, two objective functions have been taken into account, the initial construction cost and the standard deviation of a characteristic node displacement

590

Structural design optimization considering uncertainties

representing the response of the structure. Two sets of constraints are enforced, deterministic constraints on stresses, element buckling and displacements imposed by the European design codes and probabilistic ones. Furthermore, due to manufacturing limitations the design variables are not continuous but discrete since cross-sections belong to a certain predefined set provided by the manufacturers. The discrete design variables are treated in the same way as in single optimum design problems using the discrete evolution strategies. The design variables considered are the dimensions of the members of the structure taken from the Circular Hollow Section (CHS) table of the Eurocode. The random variables related to the cross-sectional dimensions, for both test examples, are two per design variable: the external diameter D and the thickness t of the circular hollow section. Apart from the cross-sectional dimensions of the structural members, the material properties (modulus of elasticity E and yield stress σy ) and the lateral loads have also been considered as random variables. The robustness of the constraints is also considered using the overall probability of maximum violation of the behavioural constraints, as a result of the variation of the uncertain structural parameters. The test example considered is the 3D truss tower shown in Figures 6(a) to 6(c). The height of the truss tower is 128 m, while its basis is a rectangle of side 17.07 m. The FE model consists of 324 nodes and 1254 elements which are divided into 12 groups that play the role of the design variables. The applied loading consists of: (i) self weight (dead load), (ii) live loads and (iii) wind actions according to the (Eurocode 1 2003). The type of probability density function, the mean value, and the variance of the random variables are shown in Table 21.6. In the present implementation an investigation is performed on the ability of the NN to predict the required data for the evolution of the RRDO process. The inputs of the NN correspond to the random variables, while the outputs are the characteristic node displacement and the maximum displacement, stress and compression force required for the calculation of the probability of violation. The appropriate selection of I/O training data constitutes the most important parts in the NN training. The number of training patterns may not be the only concern, as the distribution of samples is of great importance also. Having chosen the NN architecture and trained the neural network, the probability of violation and the standard deviation of the response can be obtained in orders of magnitude less computing time. The modulus of elasticity, yield stress, diameter D and thickness t of the circular hollow cross-section as well as the loading have been considered as random variables for the structures examined. The inputs of the NN correspond to the random variables, while the outputs are the characteristic node displacement and the maximum displacement, stress and compression force required for the calculation of the probability of violation. The previously described multi-criteria optimization (CEATm) algorithm employed is denoted as CEATm(µ + λ)nruns,csteps where µ, λ are the number of the parent and offspring vectors used in the ES optimization strategy, nruns is the number of independent CEA runs and csteps is the number of cascade stages employed. The basic steps inside an independent run of the multi-objective algorithm when the NN is embedded in the optimization process, as adopted in this test case, are described in Flowchart 21.5. For the solution of the multi-objective optimization problem in question the non-dominant CEATm(µ + λ)nrun,csteps optimization scheme was employed, where µ = λ = 5, nrun = 10 and csteps = 3. The resultant Pareto front curves for the RDO

M e t a m o d e l-b a s e d c o m p u t a t i o n a l t e c h n i q u e s i n p r o b a b i l i s t i c o p t i m i z a t i o n

591

Independent run do i = 1, nrun CEATm LOOP 1. Initial generation: do while sk not feasible k = 1, µ Generate sk (k=1,… , µ) vectors Analysis step Evaluation of the Tchebycheff metric Deterministic constraints check: if satisfied continue else regenerate sk design Monte Carlo Simulation step: Selection of the NN training set NN training for the limit load NN Monte Carlo Simulations Probabilistic constraint check end do 2. Global non-dominant search: Check if global generation is accomplished. If yes, then non-dominant search is performed, else wait until global generation is accomplished. 3. New generation: do while s not feasible  = 1, λ Generate s (k = 1, . . . , µ) vectors Analysis step Evaluation of the Tchebycheff metric Deterministic constraints check: if satisfied continue else regenerate sk design Monte Carlo Simulation step: Selection of the NN training set NN training for the limit load NN Monte Carlo Simulations Probabilistic constraint check end do 4. Selection step: selection of the next generation parents according to (µ + λ) or (µ, λ) scheme 5. Global non-dominant search: Check if global generation is accomplished. If yes, then non-dominant search is performed, else wait until global generation is accomplished. 6. Convergence check: If satisfied stop, else go to step 3 END OF CEATm LOOP End do of Independent run Flowchart 21.5 The CEATm algorithm combined with NN. Table 21.6 3D truss tower example: Characteristics of the random variables. Random variable

Description

Probability Mean density value (µ) function

E (kN/m2 ) σy (kN/m2 ) F (kN) D t

Young’s Modulus Allowable stress Horizontal loading CHS Diameter CHS Thickness

Normal Normal Normal Normal Normal

2.10E + 08 355000 Fµ d ∗i t ∗i

Standard σ/µ deviation (σ)

95% of Values interval

1.50E + 07 35500 0.4 Fµ 0.02 d i 0.02 t i

(1.81E + 08, 2.39E + 08) (2.85E + 05, 4.25E + 05) (2.16 Fµ , 17.84 Fµ ) (0.9618 d i , 1.039 d i ) (0.9618 t i , 1.039 t i )

7.14% 10.00% 40.00% 2% 2%

* Taken from the Circular Hollow Section (CHS) table of the Eurocode, for every design.

592

Structural design optimization considering uncertainties

29.26

4.27

40.23

x y

58.52

128.01

Characteristic node

17.07 19.21 (a)

(b)

(c)

Figure 21.6 3D truss tower example: (a) 3D view, (b) Side view, (c) Top view.

formulations are depicted in Figure 21.7, with the structural weight on the horizontal axis and the standard deviation of the characteristic node displacement on the vertical axis. The displacement in the x-direction of the top node is selected as the characteristic one (Figure 21.6c). As can be seen in Figure 21.7 the trend on the influence of the probabilistic constraint is similar to that of the first example, where the Pareto front curves coincide in different parts. Four different formulations of the RDO problem have been considered in this study: (i) the standard RDO formulation, (ii) RRDO with allowable probability equal to 2% denoted as RRDO_2%, (iii) RRDO with allowable probability equal to 0.1% denoted as RRDO_0.1% and (iv) RRDO with allowable probability equal to 0.01% denoted as RRDO_0.01%. As can be seen in Figure 21.7 the presence of the probabilistic constraint influences the Pareto curves near the DBO area (designs Ai, i = 1, . . . , 4) of the Pareto front, where the weight of the structure is the dominant criterion. On the contrary, the four Pareto front curves almost coincide at the areas

M e t a m o d e l-b a s e d c o m p u t a t i o n a l t e c h n i q u e s i n p r o b a b i l i s t i c o p t i m i z a t i o n

593

5.5E-02

Standard deviation (m)

5.0E-02

A4 (4497.3, 5.1E-02) A2 (4231.2, 5.0E-02) A1 (3874.7, 4.9E-02) A3 (4314.6, 4.9E-02)

RDO RRDO 2% RRDO 0.1% RRDO 0.01%

4.5E-02

4.0E-02

3.5E-02 B1 (4997.3, 3.0E-02) B3 (4982.8, 2.9E-02) B2 (4964.7, 3.0E-02) B4 (4983.7, 2.9E-02)

3.0E-02

2.5E-02 3750.0

4000.0

4250.0

4500.0 Weight (kN)

4750.0

5000.0

5250.0

Figure 21.7 3D truss tower example: Comparison of the Pareto front curves.

where the importance of the second criterion (standard deviation of the response) increases. The performance of the NN prediction is depicted in Figures 21.8a to 21.8d, where the prediction of the characteristic displacement, the maximum displacement, the maximum compressive force and the maximum tensile force are shown, respectively. Three different training sets, of size 100, 200 and 500, respectively have been examined, randomly generated using LHS, while 50 patterns have been used for testing. As can be seen in Figure 21.8, 100 samples are enough for efficiently training the NN. The MCS sample sizes used in this test example are 10,000, 100,000 and 500,000. In the RDO and RRDO_2% formulations a sample size of 10,000 simulations has been used, while in RRDO_0.1% and RRDO_0.01% a sample size of 100,000 and 500,000 simulations has been employed. The different formulations and consequently the different sample sizes lead to a significantly different computing cost. In order to reduce the increased computing cost, especially of the last two formulations, a neural network formulation has been applied. The NN configuration implemented in this example has one hidden layer with 50 nodes resulting in a 27-50-4 NN architecture (see Figure 21.1), which is used for all runs. The computing cost is depicted in Table 21.7, where the conventional and the corresponding NN computing times are reported. It has to be mentioned that the denoted basic computing costs for the RRDO_0.1% and RRDO_0.01% formulations are estimations due to the excessive computing cost required for these two cases. It can be seen that the NN-based methodology requires up to four orders of magnitude less computing time compared to the conventional one.

594

Structural design optimization considering uncertainties Table 21.7 3D truss tower example: Computing times. Formulation

No of simulations

RDO RRDO 2% RRDO 0.01% RRDO 0.001%

Time (hours)

10,000 10,000 100,000 500,000

Basic

NN

5.33E+01 5.42E+01 5.59E+02* 2.79E+03*

5.87E−01 5.96E−01 6.15E−01 6.14E−01

* Estimated.

6.0E-01

6.0E-01

100 200 500

4.0E-01

5.0E-01

100 200 500

Real

Real

5.0E-01

4.0E-01 3.0E-01 2.0E-01 2.0E-01

3.0E-01

4.0E-01 Predicted (a)

5.0E-01

3.0E-01 3.0E-01

6.0E-01

100 200 500

50.0 40.0

Real

180.0

60.0 Real

6.0E-01

200.0

70.0

30.0 30.0

4.0E-01 5.0E-01 Predicted (b)

100 200 500

160.0 140.0 120.0

40.0

50.0 Predicted (c)

60.0

70.0

100.0 100.0

120.0

140.0 160.0 Predicted (d)

180.0

200.0

Figure 21.8 3D truss tower: Performance of NN with respect to the number of the training patterns (a) characteristic displacement, (b) maximum displacement, (c) maximum compressive force, and (d) maximum tensile force.

7 Conclusions In most cases the optimum design of structures is based on nominal values of the design parameters and is focused on the satisfaction of the deterministically defined design code provisions. The deterministic optimum is not always a “safe’’ design, since there are many random factors that affect the design, i.e. manufacturing and performance of a structure during its lifetime. In order to find a “real’’ optimum the designer has to take into account all necessary random variables. In order to alleviate this deficiency, two types of formulations have been proposed in the past: RBDO and RDO. In the present work, apart from presenting successful RBDO applications, the combined RRDO is

M e t a m o d e l-b a s e d c o m p u t a t i o n a l t e c h n i q u e s i n p r o b a b i l i s t i c o p t i m i z a t i o n

595

also proposed, where probabilistic constraints are incorporated into the robust design optimization formulation. In the examined RBDO formulations, under static or dynamic loads, the reliability analysis of the structure has to be performed in order to determine its optimum design taking into account a desired level of probability of structural failure. Only after forming and solving this RBDO problem, even with additional cost in weight and computing time, can a “global’’ and realistic optimum structural design be found. The aim of the proposed RBDO procedure is to increase the safety margins of the optimized structures under various uncertainties, while at the same time minimizing its weight, and reducing substantially the required computational effort. The solution of realistic RBDO problems in structural mechanics is an extremely computationally intensive task. As it can be observed from the numerical results, the computational cost for the solution of realistic RBDO problems is orders of magnitude larger than the corresponding cost for a deterministic optimization problem. Due to the size and complexity of RBDO problems, a non-conventional, stochastic evolutionary optimization method – such as ES – appears to be a suitable choice. In a similar manner, in order to implement the hybrid RRDO formulation, structural reliability analyses for every candidate design have to be performed for the evaluation of the probability of violation. Depending on the value of the allowable probability of violation, different sample sizes are employed in order to calculate with sufficient accuracy the statistical quantities under consideration i.e. the standard deviation of the response and the probability of violation of the constraints. The Pareto front curves obtained for the presented RRDO formulation and the RDO formulation appear to be different when the weight objective function is predominant, while they approach each other in the areas of the Pareto fronts where the significance of the standard deviation of the response criterion increases. In other words, for the same standard deviation value, the optimum weight achieved by the RRDO formulation are larger than the corresponding weight achieved by the conventional RDO approach. Furthermore, it was observed that the presence of the standard deviation as an objective function forces the RDO formulation to produce results very close to those obtained by the RRDO formulation close to the right end of the Pareto front curve. Concluding, the aim of this work was twofold: to examine the influence of the probabilistic parameters and constraints in structural optimization, and to deal with computationally demanding tasks in probabilistic mechanics. The computational effort involved in the conventional MCS becomes excessive in large-scale problems, especially when earthquake loading is considered, due to the enormous sample size and the computing time required for each Monte Carlo run. Although the LHS technique has been implemented for improving the computational efficiency of the MCS method, the computational cost remains excessive, making the solution of large-scale probabilistic optimization problems computationally unsolvable. Thus, a neural network assisted methodology has been proposed in order to obtain the structural response results required during the Monte Carlo simulations inexpensively. The achieved reduction in computational time was several orders of magnitude compared to the conventional procedure making tractable the optimization of real world structures under probabilistic constraints. The use of NN can practically eliminate any limitation on the scale of the problem and the sample size used for MCS without deteriorating the accuracy of the results.

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Author index

Adams, B.M. 401 Agarwal, H. 281 Allen, M. 135 Aoues, Y. 217

Huh, J.S. 57 Hurtado, J.E. 435

Beck, J.L. 155 Ben-Haim, Y. 531 Bichon, B.J. 401

Kanno, Y. 471 Kharmanda, G. 189 Kokkolaras, M. 115 Kwak, B.M. 57

Chateauneuf, A. 3, 217 De Palma, P. 549 Doltsinis, I. 499 Eldred, M.S. 401 Fragiadakis, M. 567 Frangopol, D.M. 135 Ganzerli, S. 549

Patel, N.M. 281 Plevris, V. 567 Polak, E. 307

Joanni, A.E. 335

Lagaros, N.D. 567 Lee, S.H. 57 Liang, J. 87 Mahadevan, S. 401 Maute, K. 135 Mourelatos, Z.P. 87, 247 Nikolaidis, E. 87 Papadrakakis, M. 567 Papalambros, P.Y. 115

Renaud, J.E. 281 Royset, J.O. 307 Rackwitz, R. 335 Sørensen, J.D. 31 Taflanidis, A.A. 155 Takewaki, I. 471, 531 Tillotson, D. 281 Tovar, A. 281 Tsompanakis, Y. 567 Weickum, G. 135 Wu, Y.-T. 369 Zhou, J. 247

Subject index

Advanced Mean Value (AMV) 402, 406 Age-dependent Repairs 350 AMV+ 402, 406–407 AMV2 + 402, 406–407 Analytical Target Cascading 117, 121 Bi-level RBDO 413 Block repairs 353 Cascade Evolutionary Algorithm (CEA) 573, 590 Cellular automata 290 Chloride attack 362 Combined approximation 142 Common Random Numbers 168, 171, 180 Convergence 221, 223 Convex model 472, 549, 551, 554, 560 Cost function 5, 22 Cost-benefit optimization 34, 44, 343 Critical excitation 532, 538 DAKOTA 403 Decision theory 32 Decomposition-based Design 115 Design of Experiments 59 Desirability function 500, 509 Deterioration failure models 336–337 Deterministic Design Optimization 189, 199, 233 Earthquake input energy 534, 536, 546 Efficient global reliability analysis (EGRA) 411, 419 Elasticity 512 Entropy 444, 459 Evidence theory 248, 261 Evidence-based Design Optimization 261 Evolution Strategies (ES) 572

Failure-Region Sampling 378, 381 First-order reliability method (FORM) 146, 402, 408 Genetic Algorithms 554 Global reliability methods 411, 419 Hybrid Cellular Automaton method 282, 290–291 Importance Sampling 174, 180 Info-gap theory 471, 475, 532, 539, 546 Inspection and repair 349, 354 Inspection optimization 371, 386 Interval Analysis 127 Large displacements 514, 526 Latin Hypercube Sampling (LHS) 575 Life-cycle Cost 156, 175 Limit state 574 Maintenance Optimization 369 Maintenance strategies 348 Markov Chain Monte Carlo 381 Mean-Value First-Order Second-Moment (MVFOSM) 403 Microelectromechanical systems (MEMS) 422 Model Prediction 157, 159, 170 Moment estimation 448, 450 Monte Carlo Simulation (MCS) 307, 314, 574 Most Probable Failure Point 8, 15, 219, 222, 226 Multi-objective optimization 573 Nataf transformation 405 Neural networks (NN) 576 Nonlinear dynamics 516

634

Subject index

Optimal inspection 36 Optimization 31, 36 Optimization methods 359 Optimization under uncertainty 116 Optimum Safety Factor 191, 199, 209 Pareto optima 509, 526, 573 Passively controlled structure 532, 542, 547 Pearson System 64 Performance measure approach (PMA) 402, 405 Performance-based Earthquake Engineering (PBEE) 569 Plasticity 517 Poissonian disturbances 347 Polynomial Chaos expansion 147 Possibility theory 248, 253 Possibility-based Design Optimization 249, 258 Probabilistic Damage Tolerance 370 Probabilistic Design 124 Probabilistic Ground Motion Model 176 Probabilistic transformation 221, 223, 226 Probability estimation 441 Probability of detection 371, 387 Quasiconvex optimization 473, 487 Reduced order model 137–138 Reliability 89–90 Reliability analysis 191–192 Reliability index 6, 18, 22 Reliability index approach (RIA) 402, 405 Reliability methods 403 Reliability versus robustness 509 Reliability-Based Design Optimization (RBDO) 34, 76, 149, 189, 199, 233, 283, 309, 412, 570 Reliability-Based Robust Design Optimization (RRDO) 571, 589 Reliability-Based Topology Optimization 282, 296 Renewal model 340, 344 Response Surface Method 68 Risk Sensitivity Analysis 386 Robust analysis 455, 459

Robust Design Optimization (RDO) 439, 507, 509, 571 Robustness function 471, 477, 485, 490, 533, 541 Safety factor 3, 10, 26, 226, 228 Sample average approximations 314 Sample-adjustment rules 308, 315–316 Second-order reliability method (SORM) 402, 408 Semidefinite programming 471, 473, 490 Sensitivity analysis 42, 139, 413 Sequential Optimization and Reliability Assessment (SORA) 92 Sequential PBDO 265 Sequential RBDO 415 Series systems 339, 357 Simultaneous-perturbation Stochastic Approximation 171, 180 Single-Loop RBDO approach 95 Statistical Moments 58–59, 76 Stochastic constraints 504, 509 Stochastic Design 157, 171 Stochastic Optimization 499 Stochastic Simulation 157, 173 Stochastic Subset Optimization 159, 178 Structural Dynamics 138 System Reliability 9, 21, 156, 159, 230–231 System Reliability-based Design Optimization 89 Target reliability 221, 229 Taylor series approximation 499, 503 Topology Optimization 286 Two-point adaptive nonlinearity approximation (TANA) 402, 406–407 Two-Stage Importance Sampling (TIS) 377 Uncertainty analysis 144 Uncertainty Propagation 122 Unified RDO-RRDO 456 Updating 342 Wind turbines 48