Computer Methods in Mechanics: Lectures of the CMM 2009 (Advanced Structured Materials)

Advanced Structured Materials Series Editors: Andreas Öchsner (Editor-in-chief), Lucas Filipe Martins da Silva, and Hol...

Author: Mieczyslaw Kuczma | Krzysztof Wilmanski

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Advanced Structured Materials Series Editors: Andreas Öchsner (Editor-in-chief), Lucas Filipe Martins da Silva, and Holm Altenbach

Mieczyslaw Kuczma · Krzysztof Wilmanski

Computer Methods in Mechanics Lectures of the CMM 2009

With 292 Figures

ABC

Editors Prof. Mieczyslaw Kuczma University of Zielona Góra Inst. Building Engineering ul. Z. Szafrana 1 65-516 Zielona Góra Poland E-mail: [email protected]

Prof. Krzysztof Wilmanski Rue Diderot 4 13469 Berlin Germany E-mail: [email protected]

ISBN 978-3-642-05240-8

e-ISBN 978-3-642-05241-5

DOI 10.1007/978-3-642-05241-5 Advanced Structured Materials

ISSN 978-3-642-05241-5

Library of Congress Control Number: 2009940899 c 2010 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Cover design: WMX Design, Heidelberg Printed in acid-free paper 987654321 springer.com

Dedicated to the memory of Professor Olgierd Cecil Zienkiewicz: outstanding scholar, our dear teacher, colleague, and friend

Foreword

This book contains collected plenary and keynote papers presented during the 18th International Conference on Computer Methods in Mechanics - CMM 2009, which took place in Zielona Gra on May 18-21, 2009. The CMM 2009 was organized by the Polish Academy of Sciences (PAS), the Polish Association for Computational Mechanics and the University of Zielona Gra. The chairmen of the conference were Prof. M. Kuczma and Prof. K. Wilma´ nski. The Honorary Chairman of the CMM 2009 was Professor O.C. Zienkiewicz who passed away during the final preparation stage of the Conference. The conference was organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS) and the Central European Association for Computational Mechanics (CEACM). The CMM 2009 continued an over 35-year tradition of the Polish Conferences on Computer Methods in Mechanics. The 1st conference of this series was held in Pozna´ n in 1973, and ever since every two years in different academic centres of Poland the conferences on computational mechanics have been organized under the auspices of PAS and a selected local university of technology. In 2001, instead of the regular CMM conference, the European Conference on Computational Mechanics (ECCM 2001) took place in Cracow. The tradition of these conferences on computer methods in mechanics is one of the longest in Europe. Polish scientists and scientists of Polish descent have had great influence on the development of the field of computer methods in mechanics. One of the most distinguished scientist in this field was Prof. O.C. Zienkiewicz. He was a guest of numerous CMM conferences. Since 2003 he was Honorary Chairman of the CMM conferences. This book is dedicated to His memory. The CMM 2009 brought together 300 researchers, practitioners, professors, and students from European countries as well as scholars from overseas. Its aim was to promote the development of computational methods in civil, mechanical, material science and bioengineering, and their applications in engineering practice. In particular, it reflects the state-of-the-art of computational mechanics in science. The main focus was on applied mathematics

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Foreword

and computational methods, meshless and boundary element methods, solid mechanics, structural mechanics, coupled problems, mechanics of materials, multiscale modelling and nanomechanics, optimization and identification, geomechanics, heat transfer, biomechanics, contact problems, computational intelligence, experimental mechanics and industrial applications. The warm gratitude is given to the members of the Organizing Committee for their great effort and their enthusiasm, which contributed to a big success of this scientific meeting. The reviewing of all contributed papers by the members of the International Scientific Committee and the National Scientific Committee is also gratefully acknowledged. August 2009

Tadeusz Burczy´ nski

Professor Olgierd Cecil Zienkiewicz (1921 - 2009)

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Foreword

Professor Olgierd Cecil Zienkiewicz, Professor of applied mechanics, a civil engineer, one of the pioneers of the finite element method, one of the greatest mechanicians of the 2nd half of the 20th century, the recipient of many honours and medals died on January 2nd, 2009 in Swansea. Prof. O.C. Zienkiewicz was the highly respected Nestor of the world-wide computational mechanics and far beyond this field and the very influential great scientist. Moreover he had a humanistic view of the world, and he was a reliable personal friend. We loose an outstanding scientist and true friend whose memory will surely remain vivid in all our minds. Prof. O.C Zienkiewicz was born in 1921 in Caterham, Great Britain, as a son of Kazimierz Zienkiewicz and Edith Violet Penny. Since 1922 he lived in Katowice, Poland, where his father was a district judge. He finished secondary school in Katowice in 1939. Due to the outbreak of the World War Two he did not start studies at Warsaw University of Technology. He studied in Great Britain. In 1943 he graduated with first class honours from Imperial College and obtained his B.Sc. there. In 1945 he obtained his Ph.D. and DIC (Diploma Imperial College) as a member of Sir Richard Southwell’s research team. In 1946 he obtained his engineer’s degree at Polish University of Technology in London and in 1965 his D.Sc. (Eng) at the University of London. In the late 1940s Prof. Zienkiewicz worked as a Consulting Engineer, from 1949 till 1957 he was a lecturer at the University of Edinburgh. From 1957 till 1961 he was a professor of structural mechanics in Northwestern University, Evanstone, Illinois, USA. Since 1961 he was associated with the Department of Civil Engineering, the University of Wales, Swansea, UK. Prof. Zienkiewicz established the Institute for Numerical Methods in Engineering and was its Director till 1988, when he retired. He created a scientific school in Swansea, which soon became the world leading centre in the field of numerical methods. Here he developed the finite element method (FEM) – the main work of his life. After retirement from Swansea in 1987, Olek spent two months each year at the International Center for Numerical Methods in Engineering (CIMNE) at Universitat Politecnica de Catalunya (UPC) in Barcelona, Spain. In 1989 Olek was appointed to the UNESCO Chair of Numerical Methods in Engineering at UPS. This was the very first UNESCO Chair in the world. FEM has become an irreplaceable method in the analysis of complex problems of solid and fluid mechanics, in all fields of engineering, e.g. acoustics, heat conduction, biomedical engineering, electromagnetics, coupled and multiphysics problems. Professor’s activities aiming at the inner development of FEM are also worth mentioning. Among many problems developed in Swansea one can find the relations between FEM and other approximation methods (like finite difference method, the boundary element method, meshless methods), especially in the field of coupling these methods and taking into account an adaptive FE choice.

Foreword

XI

Prof. Zienkiewicz’s genius caused that his knowledge and intuition, his abilities to model and formulate algorithms opened not only new scientific horizons but also encompassed application possibilities of solving different engineering problems. Additionally, his capability to build research teams attracted outstanding students and young academics. It is worth emphasizing that Professor Zienkiewicz school soon achieved world standards and was distinguished by openness and wide international co-operation. Software created in Swansea was from the beginning in the public domain. It was continuously modernized and widened to fit the growing applications and abilities of better and better computers and computer networks. Professor Zienkiewicz published nearly 600 papers and wrote or edited 25 books. His first paper published in 1947 dealt with numerical stress analysis of dams. He wrote the first book on FEM (The Finite Element Method in Structural Mechanics, McGrow Hill, 1967). His next book was published in 1971 and then many times revised till its three-volume 6th edition in 2005. It was edited with Prof. R.L. Taylor from Berkeley University, California – a former student of Prof. Zienkiewicz – is regarded as a basic reference and the best textbook on FEM. Another field in which Professor Zienkiewicz was very active is the organization of science. In 1968 in Swansea he founded with Prof. R.H. Gallagher from USA International Journal for Numerical Methods in Engineering, which has became the major journal concerning FEM and other computational methods. Professor was the editor-in-chief of the journal till 1988, he was also a member of editorial boards of other leading journals dealing with numerical methods. Professor Zienkiewicz was the founder and the 1st President of the International Association of Computational Mechanics (IACM) 1986-1990. Nowadays, national associations of computational mechanics from all states that count on the world scientific map and regional associations are affiliated to IACM. He was also one of the founders of the European Community of Computational Mechanics and Applied Science (ECCOMAS). He was a member, the honorary chairman and the author of a huge number of general lectures on many regional and national computational mechanics conferences, as well as on conferences dealing with the applications of numerical methods FEM in science and technology. Professor Zienkiewicz great prestige is the result of not only his achievements and scientific activity but also of his personal features, among which it is worth to mention the communicativeness of his papers and lectures, the criticism concerning his own results, kindness to young academics, openness and will to share his achievements with others. He supervised over 70 Ph.D. students, not only in Swansea, but also in many other universities, where he was Visiting Professor or during short-term visits. Professor Zienkiewicz was also a scientific consultant of many companies and concerns (e.g. Rolls Royce, English Electric, Sir William Halerow & Partners).

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Foreword

He was a member of many academies of science, among others: Fellowship of the Royal Society, U.K., 1979; Fellowship of the Royal Acad. of Engng., U.K., 1979; US National Acad. of Engng., 1981; the Polish Academy of Science, 1985; Accademia Nazionale dei Lincei, Roma, 1999; Accademia di Science Lettere, Milano, 1999. He received Honorary degrees from many universities: Laboratorio Nacional de Engen. Civil, Lisboa, Portugal, 1972; Univ. of Ireland, 1975; Vrije Univ. Brussel, Belgium, 1982; Northwest Univ., USA, 1984; Techn. Univ. Trondheim, Norway, 1985; Chalmers Univ. of Technol., Goteborg, Sweden, 1987; Univ. of Dundee, Scotland, 1987; Dalion Inst. of Technology, Chine, 1987; Warsaw University of Technology, Poland, 1989; Cracow University of Technology, Poland, 1990; Techn. Univ. Budapest, Hungary, 1992; Univ. of Hong Kong, 1992; Compiegne Univ. of Technol., France, 1992; Univ of Wales, UK,1993; Brunel Univ., London, UK, 1993; Aristotle Univ. of Thessaloniki, Greece 1993; Imperial College of Sci., Technol and Medicine, London, UK, 1993; Ecole Normale Sup. de Cachan, Paris, France, 1997; Uniwersitat Politechnica de Madrid, Spain, 1998; Univ. Buenos Aires, Argentina 1998; Chinese Acad. Sci., 1998; Techn. Univ. Lisboa, Portugal, 2001; Silesian University of Technology, Gliwice, Poland, 2001; Politecnico di Milano, Italy, 2001; Czestochowa University of Technology, Poland, 2005. He received also many awards and distinctions. Professor Zienkiewicz was in touch with his family in Poland and his schoolmates in Katowice. In Swansea he received trainees and Ph.D. students from Poland. Since the mid 1960s he was in close contact with Prof. J. Szmelter from Lodz University of Technology, later Military University of Technology, Warsaw . the creator of the Polish school of FEM and Prof. I. Kisiel from Wroclaw University of Technology, who translated into Polish Professor’s book The Finite Element Method (Arkady, 1972). Prof. Zienkiewicz attended Polish Conferences on Computational Mechanics since 1981, was its honorary chairman since 2003, the invited honorary member of the Scientific Committee of the 2nd European Conference on Computational Mechanics ECCM2001, Cracow, 2001, promoted continuous cooperation with the Institute of Fundamental Technological Research of the Polish Academy of Science, and Warsaw, Cracow and Silesian Universities of Technology. Researchers and engineers on mechanics and computational methods regard Professor Zienkiewicz as one of the most outstanding scientists of the 20th century. He spoke beautiful and rich Polish and often emphasized his Polish roots. Professor’s undisturbed activity after he had retired from University of Wales, Swansea in 1989 is worth mentioning. For instance, he chaired the UNESCO Chair of Numerical Methods in Engineering at University of Technology of Catalunya, Barcelona, Spain. He was an honorary member of the Polish Association for Computational Mechanics (PACM). In recognition of his achievements PACM founded in 2007 Prof. O.C. Zienkiewicz Medal for outstanding Polish and foreign

Foreword

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scientists who contributed significantly to the development of computational methods. Gliwice, May 2009 Barcelona, May 2009

Tadeusz Burczy´ nski, PACM President Eugenio O˜ nate, IACM President

Preface

This book contains the written versions of some general and keynote lectures given at the 18th International Conference on C omputer Methods in Mechanics, CMM 2009, which took place from 18 to 21 May 2009 at the University of Zielona G´ ora, Poland. This prestigious event was overshadowed by the death of one of the pioneers of computer sciences, the honorary chairman of the Conference, prof. dr. Olgierd Cecil Zienkiewicz, Professor Emeritus of the Swansea University (United Kingdom) who passed away on 2nd January 2009. His vivid memories were perceptible during the whole meeting in Zielona G´ ora. From thirty nine outstanding scientists presenting the general and keynote lectures at the conference, twenty seven decided to contribute their presentations in the form of articles to this volume. Among them are students or coworkers of Professor Zienkiewicz, a highly respected researcher and author. We all have greatly benefited from his seminal works and activities. Therefore, we and the authors would like to pay tribute to the late Professor Zienkiewicz by dedicating the present book to his memory. The Conference was attended by 300 participants from 24 countries. It served both as a forum for the review and dissemination of recent scientific developments and technical information regarding all aspects of computer methods (250 presentations in 6 parallel sessions), and as a means to encourage cooperation and stimulate future research. The general lecturers and the keynote speakers are leading world-renowned scientists of this field of science, carefully selected by the International and National Scientific Committees. We have divided the contributions, rather arbitrarily, into 6 groups which appear as Parts of the book. Since most of the contributed articles are interdisciplinary, we do not claim this classification to be definite. However, we hope our grouping of papers will be helpful for readers who look for a particular issue in the field of computer science. Part I, Mathematical Methods, encompasses works where emphasis is laid upon mathematical analysis of the considered models supported with numerical experiments for finite dimensional approximations. These works are concerned with wave propagation and coupled multiphysics problems,

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meshless methods, non-smooth non-convex frictional contact problems, shape and topology sensitivity analysis, boundary integral equation on a sphere, hp-adaptivity and adaptive hierarchical modelling. The tools used include finite, spectral, and boundary elements, variational equations and inequalities, Clarke subdifferentials, hemivariational inequalities and evolution inclusions, as well as topological derivatives. A variety of types of materials were considered: elastic, poroelastic, viscoelastic, and piezoelectric, subjected to static or dynamic deformation processes. Part II, Soft Computing and Optimization, contains papers concerned with sensor network design and identification problems, and application of immune computing in computational bioengineering and in structural optimization. The techniques used include bio-inspired approaches like Artificial Neural Networks, Evolutionary Methods, Particle Swarm Optimization as well as classical Finite Element Method and optimization algorithms, which allow for different types of numbers (interval, fuzzy, random) and box constraints. In Part III, Multiscale Methods, we have collected the contributions which are strongly related to multilevel description. They combine molecular dynamics and an extended finite element method to simulate dynamic fracture, atomistic/continuum models to investigate crystal defects in semiconductor structures, or make use of the theory of Cosserat point to develop a 3-D brick element for finite elasticity, the particle finite element method to analyse coupled fluid/structure interaction problems, and finally of a meso-macro homogenization procedure to study electro-mechanically coupled material behaviour. Part IV, Geomechanics, is primarily related to the numerical analysis of multicomponent and porous media. The physical coupling of solids and fluids in those media requires special computational methods. These are a combination of percolation and forest fire algorithms for modelling cementitious materials at early age, mesh-free and strain enhancement procedures for material point and low-order element methods to solve problems involving incompressibility, and enhancements of a micro-polar hypoplastic constitutive model for granular materials in the framework of finite element study of shear localization. Computational problems presented in Part V, Biomechanics, are even more special. These deal with tissue – implant interactions accounting for bone adaptation to mechanical stresses, sequential stages of the tooth-implant life cycle design process with focus on mechanical behavior and optimization of dental implants using finite elements and a genetic optimization algorithm, and mechanobiology to model the response of tissues to changes in their mechanical environment by combining finite elements and a lattice with algorithms for cell activities. Part VI, Structural Mechanics, features the most common application field of computer mechanics in engineering sciences. This part is devoted to complex practical problems: unilateral frictional contact of beams, parameter indentification of material/structural responses for real-life structures and

Preface

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infrastructures based on a combination of computational and non-destructive measurements methods exemplified by analyses of steel pipelines, large concrete dams, and of marine propulsion system’s alignment of aged ships, as well as assessment of impact-induced damage in laminate composites, and pre- and postbuckling of composite shells. The list of contributors, which is also included in the book, follows the alphabetic order organized according to the names of the speakers who presented a particular paper. Next to them the names of their co-authors are included as well. We would like to thank all authors for the cooperation on this book. At the same time, we would also like to express our gratitude to the members of the International and National Scientific Committees of CMM 2009 for their kind help in the reviewing procedures, as well as to the members of the Organizing Committee for their commitment to the organization of the Conference. Finally, we would like to thank Springer Verlag and particularly Dr. Dieter Merkle and Dr. Christoph Baumann, who kindly agreed to publish this book and further helped us to bring it to its present fine printed form. We greatly appreciate the publishing of the book in the new series Advanced Structured Materials. Zielona G´ ora, Poland August 2009

Mieczyslaw Kuczma Krzysztof Wilma´ nski

Contents

Part I Mathematical Methods Explicit Discrete Dispersion Relations for the AcousticWave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark Ainsworth, Hafiz Abdul Wajid

3

hp-Adaptive Finite Elements for Coupled Multiphysics Wave Propagation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leszek Demkowicz, Jason Kurtz, Frederick Qiu

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Nonconvex Inequality Models for Contact Problems of Nonsmooth Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stanislaw Migórski, Anna Ochal

43

Quadrature for Meshless Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John E. Osborn

59

Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˙ Jan Sokołowski, Antoni Zochowski

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A Boundary Integral Equation on the Sphere for High-Precision Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ernst P. Stephan, Thanh Tran, and Adrian Costea

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Unresolved Problems of Adaptive Hierarchical Modelling and hp-Adaptive Analysis within Computational Solid Mechanics . . . . . . . . . . 111 Grzegorz Zboiński

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Contents

Part II Soft Computing and Optimization Granular Computing in Evolutionary Identification . . . . . . . . . . . . . . . . . 149 Witold Beluch, Tadeusz Burczyński, Adam Długosz, Piotr Orantek Immune Computing: Intelligent Methodology and Its Applications in Bioengineering and Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . 165 Tadeusz Burczyński, Michał Bereta, Arkadiusz Poteralski, Mirosław Szczepanik Bioinspired Algorithms in Multiscale Optimization . . . . . . . . . . . . . . . . . . 183 Wacław Ku´s, Tadeusz Burczyński Sensor Network Design for Spatio–Temporal Prediction of Distributed Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Dariusz Uciński Part III Multiscale Methods A Multiscale Molecular Dynamics / Extended Finite Element Method for Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Pascal Aubertin, Julien Réthoré, René de Borst Nonlinear Finite Element and Atomistic Modelling of Dislocations in Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Paweł Dłu˙zewski, Toby D. Young, George P. Dimitrakopulos, Joseph Kioseoglou, Philomela Komninou Accuracy and Robustness of a 3-D Brick Cosserat Point Element (CPE) for Finite Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 M. Jabareen, M.B. Rubin Possibilities of the Particle Finite Element Method in Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Eugenio Oñate, Sergio R. Idelsohn, Miguel Angel Celigueta, Riccardo Rossi, Salvador Latorre A Framework for the Two-Scale Homogenization of ElectroMechanically Coupled Boundary Value Problems . . . . . . . . . . . . . . . . . . . 311 Jörg Schröder, Marc-André Keip Part IV Geomechanics Modeling Concrete at Early Age Using Percolation . . . . . . . . . . . . . . . . . . 333 Lavinia Stefan, Farid Benboudjema, Jean Michel Torrenti, Benoˆıt Bissonette

Contents

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Simulation of Incompressible Problems in Geomechanics . . . . . . . . . . . . . 347 Dieter Stolle, Issam Jassim, Pieter Vermeer Effect of Boundary, Shear Rate and Grain Crushing on Shear Localization in Granular Materials within Micro-polar Hypoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Jacek Tejchman Part V Biomechanics Biomechanical Basis of Tissue–Implant Interactions . . . . . . . . . . . . . . . . . 379 Romuald Bedzinski, Krzysztof Scigala Tooth-Implant Life Cycle Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Tomasz Łodygowski, Marcin Wierszycki, Krzysztof Szajek, Wiesław He¸dzelek, Rafał Zagalak Predictive Modelling in Mechanobiology: Combining Algorithms for Cell Activities in Response to Physical Stimuli Using a Lattice-Modelling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Sara Checa, Damien P. Byrne, Patrick J. Prendergast Part VI Structural Mechanics The Beam-to-Beam Contact Smoothing with Beziers Curves and Hermites Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Przemysław Litewka Synergic Combinations of Computational Methods and Experiments for Structural Diagnoses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Giulio Maier, Gabriella Bolzon, Vladimir Buljak, Tomasz Garbowski, Bartosz Miller Optimization of Marine Propulsion System’s Alignment for Aged Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Lech Murawski, Wieslaw Ostachowicz Experimental-Numerical Assessment of Impact-Induced Damage in Cross-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Gigliola Salerno, Stefano Mariani, Alberto Corigliano, Francesco Caimmi, Luca Andena, Roberto Frassine Finite Element Modeling of Stringer-Stiffened Fiber Reinforced Polymer Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Werner Wagner, Claudio Balzani Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

List of Contributors

Mark Ainsworth University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland [email protected] Hafiz Abdul Wajid University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland [email protected] ´ Krzysztof Sciga la Division of Biomedical Engineering and Experimental Mechanics, Wroclaw University of Technology, ul. Lukasiewicza 7/9, 50-371 Wroclaw, Poland Romuald Bedzi´ nski Division of Biomedical Engineering and Experimental Mechanics, Wroclaw University of Technology, ul. Lukasiewicza 7/9, 50-371 Wroclaw,

Poland [email protected] Witold Beluch Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland [email protected] Tadeusz Burczy´ nski Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland, and Institute of Computer Modelling, Cracow University of Technology, Cracow, Poland [email protected] Adam Dlugosz Department of Strength of Materials and

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Computational Mechanics, Silesian University of Technology, Gliwice, Poland [email protected] Piotr Orantek Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland Sara Checa Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland Damien P. Byrne Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland Patrick J. Prendergast Trinity Centre for Bioengineering, School of Engineering, Trinity College, Dublin, Ireland [email protected] Alberto Corigliano Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Luca Andena Politecnico di Milano,


Dipartimento di Chimica, Materiali e Ingegneria Chimica ”Giulio Natta”, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Francesco Caimmi Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica ”Giulio Natta”, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Roberto Frassine Politecnico di Milano, Dipartimento di Chimica, Materiali e Ingegneria Chimica ”Giulio Natta”, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Stefano Mariani Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Gigliola Salerno Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza L. Da Vinci 32, 20133 Milano (Italy) [email protected] Ren´ e de Borst Eindhoven University of Technology, Department of Mechanical


Engineering, P.O. Box 513, 5600 MB Eindhoven, Netherlands [email protected] Pascal Aubertin Université de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, France [email protected] Julien R´ ethor´ e Université de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, France [email protected] Leszek Demkowicz Institute for Computational Engineering and Sciences, University of Texas at Austin, ACES 6.326, 201 E. 24th Street, Austin, TX 78712, USA [email protected] Jason Kurtz Institute for Computational Engineering and Sciences, University of Texas at Austin, ACES 6.326, 201 E. 24th Street, Austin, TX 78712, USA [email protected] Frederick Qiu Institute for Computational Engineering and Sciences, University of Texas at Austin, ACES 6.326, 201 E. 24th Street, Austin, TX 78712, USA [email protected]

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Pawel Dlu˙zewski Instytut Podstawowych Problemow Techniki PAN, ul. Swietokrzyska 21, 00-049 Warszawa, Poland [email protected] George P.Dimitrakopulos Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece Joseph Kioseoglou Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece Philomela Komninou Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece [email protected] Toby D. Young Instytut Podstawowych Problemow Techniki PAN, ul. Swietokrzyska 21, 00-049 Warszawa, Poland Waclaw Ku´ s Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, ul. Konarskiego 18a, 44-100 Gliwice, Poland Waclaw:[email protected] Tadeusz S. Burczy´ nski Department for Strength of Materials and Computational Mechanics,

XXVI

Silesian, University of Technology, ul. Konarskiego 18a, 44-100 Gliwice, Poland [email protected] and Institute of Computer Modelling, Cracow University of Technology, ul. Warszawska 24, 31-155 Krak´ ow, Poland Przemyslaw Litewka Institute of Structural Engineering, Poznan University of Technology, ul. Piotrowo 5, 60-965 Poznan, Poland przemyslaw.litewka@ put.poznan.pl Tomasz L odygowski Department of Structural Mechanics, Poznan, University of Technology, ul. Piotrowo 5, 65 246 Poznan, Poland [email protected] Wieslaw H¸ edzelek University of Medical Sciences, ul. Fredry 10, 61-701 Poznan, Poland [email protected] Krzysztof Szajek Department of Structural Mechanics, Poznan, University of Technology, ul. Piotrowo 5, 65 246 Poznan, Poland [email protected]


ul. Piotrowo 5, 65 246 Poznan, Poland [email protected] Rafal Zagalak Foundation of University of Medical Sciences, ul. Teczowa 3, 60-275 Poznan [email protected] Giulio Maier Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected] Gabriella Bolzon Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected] Vladimir Buljak Department of Structural Engineering, Politecnico (Technical University) di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected]

Tomasz Garbowski Department of Structural Engineering, Politecnico Marcin Wierszycki Department of Structural Mechanics, (Technical University) di Milano, piazza Leonardo da Poznan, University of Technology,


XXVII

Vinci 32, 20133 Milano, Italy [email protected]

Gdansk, Poland [email protected]

Bartosz Miller Department of Structural Mechanics, Rzeszów University of Technology, ul. Pozna´ nska 2, 35-959 Rzeszów, Poland [email protected]

Eugenio O˜ nate International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain [email protected]

Stanislaw Mig´ orski Jagiellonian University, Institute of Computer Science, Faculty of Mathematics and Computer Science, ul. Stanislawa L ojasiewicza 6, 30-348 Krakow, Poland {migorski,ochal}@ii.uj.edu.pl Anna Ochal Jagiellonian University, Institute of Computer Science, Faculty of Mathematics and Computer Science, ul. Stanislawa L ojasiewicza 6, 30-348 Krakow, Poland Lech Murawski Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952 Gdansk, Poland [email protected] Wieslaw Ostachowicz Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-952

M.A. Celigueta International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain S.R. Idelsohn ICREA Research Professor at CIMNE S. Latorre International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain R. Rossi International Center for Numerical Methods in Engineering (CIMNE) Universidad Politecnica de Cataluna, Campus Norte UPC, 08034 Barcelona, Spain

XXVIII

John E. Osborn Department of Mathematics, University of Maryland, College Park, MD 20742, USA [email protected] Miles B. Rubin Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel, [email protected] Mahmood Jabareen Faculty of Civil and Environmental Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel, [email protected] J¨ org Schr¨ oder Institute for Mechanics, Faculty of Engineering Sciences, Department of Civil Engineering, University of Duisburg-Essen, Universit¨ atsstraße 15, 45141 Essen, Germany [email protected] Marc-Andr´ e Keip Institute for Mechanics, Faculty of Engineering Sciences, Department of Civil Engineering, University of Duisburg-Essen, Universit¨ atsstraße 15, 45141 Essen, Germany [email protected] Ernst P. Stephan Institute for Applied Mathematics, Leibniz Universit¨ at Hannover, Welfengarten 1, Hannover,


Germany [email protected] Adrian Costea Institute for Applied Mathematics, Leibniz Universität Hannover, Welfengarten 1, Hannover, Germany [email protected] Thanh Tran School of Mathematics and Statistics, Sydney 2052 , Australia [email protected] Dieter Stolle Department of Civil Engineering, McMaster University, Hamilton, Ontario, Canada, L8S4L7 [email protected] Issam Jassim Institute of Geotechnical Engineering, Stuttgart University, Pfaffenwaldring 35D, 70569 Stuttgart, Germany issam.jassim@ igs.uni-stuttgart.de Pieter Vermeer Institute of Geotechnical Engineering, Stuttgart University, Pfaffenwaldring 35D, 70569 Stuttgart, Germany pieter.vermeer@ igs.uni-stuttgart.de Jacek Tejchman Faculty for Civil and Environmental Engineering, Gdansk University


of Technology, 80-952 Gda´ nsk-Wrzeszcz, Narutowicza 11/12, Poland [email protected] Jean Michel Torrenti LCPC, 58 boulevard Lefebvre, 75732 Paris cedex 15, France [email protected] Farid Benboudjema LMT, ENS Cachan, 61 Avenue du président Wilson, 94230 CACHAN, France Benoˆıt Bissonette Département de Génie Civil Pavillon Adrien-Pouliot local 2928B Université Laval Québec, Canada, G1V 0A6 Lavinia Stefan LMT, ENS Cachan, 61 Avenue du président Wilson, 94230 CACHAN, France, and Département de Génie Civil, Pavillon Adrien-Pouliot, local 2928B, Université Laval, Québec, Canada, G1V 0A6 [email protected] Dariusz Uci´ nski Institute of Control and Computation Engineering, University of Zielona G´ ora, ul. Licealna 9, 65–417 Zielona Góra,

XXIX

Poland [email protected] Werner Wagner KIT – Karlsruhe Institute of Technology, Institute for Structural Analysis, Kaiserstr. 12, D-76131 Karlsruhe, Germany werner.wagner@ bs.uni-karlsruhe.de Claudio Balzani KIT – Karlsruhe Institute of Technology, Institute for Structural Analysis, Kaiserstr. 12, D-76131 Karlsruhe, Germany claudio.balzani@ bs.uni-karlsruhe.de Grzegorz Zboi´ nski Polish Academy of Sciences, Institute of Fluid Flow Machinery, ul. Fiszera 14, 80-952 Gda´ nsk, Poland, and University of Warmia and Mazury, Faculty of Technical Sciences, ul. Oczapowskiego 11, 10-736 Olsztyn, Poland [email protected] ˙ Antoni Zochowski Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland [email protected] Jan Sokolowski Institut Elie Cartan,

XXX

UMR 7502 Nancy-Université-CNRS-INRIA, Laboratoire de Mathématiques, Université Henri


Poincaré Nancy 1, B.P. 239, 54506 Vandoeuvre lés Nancy Cedex, France [email protected]

Part I

Mathematical Methods

Chapter 1

Explicit Discrete Dispersion Relations for the Acoustic Wave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes Mark Ainsworth and Hafiz Abdul Wajid

Abstract. We study the dispersive properties of the acoustic wave equation for finite element, spectral element and optimally blended schemes using tensor product elements defined on rectangular grid in d-dimensions. We prove and give analytical expressions for the discrete dispersion relations for the above mentioned schemes. We find that for a rectangular grid (a) the analytical expressions for the discrete dispersion error in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blending parameter is p/(p + 1) for all p ∈ N and for any number of spatial dimensions; (c) the optimal scheme guarantees two additional orders of accuracy compared with both finite and spectral element schemes; and (d) the absolute accuracy of the optimally blended scheme is O(p−3 ) and O(p−2 ) times better than that of the pure finite and spectral element schemes respectively.

1.1 Introduction Partial differential equations model many physical processes and real world problems posed over complex domains encountered in different fields of life. Most of the time it is not possible to obtain closed form solutions to these problems and Mark Ainsworth University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland e-mail: [email protected] Hafiz Abdul Wajid University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland e-mail: [email protected] Support of MA by the Engineering and Physical Sciences Research Council under grant EP/E040993/1 and of HAW by COMSATS Institute of Information Technology, Pakistan through a research studentship is gratefully acknowledged.

M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 3–17. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

4

M. Ainsworth and H. Abdul Wajid

numerical methods are widely used. Amongst the most common numerical methods, finite element [5] and spectral element [6] methods have been extensively used. The latter one is highly attractive as it can be used to solve problems with complex geometrical domains and attains the desired accuracy with less computational cost, thanks to the fact that the mass matrix is diagonal [16, 17]. Both finite and spectral element methods have been widely used to study wave propagation [8, 13, 14, 15] but fail to propagate waves with the correct physical speed [1, 3] and numerical dispersion is introduced. The finite element scheme results in phase lead whereas with the spectral element scheme phase lag is observed [3]. Mulder [19] studied the dispersive properties of the acoustic wave equation in one dimension using both finite and spectral element methods and concluded that spectral element methods with Gauss-Lobatto quadrature rules perform better than both the spectral element method with Chebyshev quadrature points and standard finite element methods. In [8] explicit expressions for the dispersion error were obtained using eigenvalues and Taylor series for low order elements for the transient wave equation. Basabe and Sen [9] studied dispersion in 2D with both finite and spectral elements for both acoustics and elastic wave equations. They provided analytical expressions for the dispersion error and stability conditions for first order elements and showed dispersion curves numerically for higher-order elements. The first detailed study of the dispersive properties of the standard finite element and spectral element methods valid for elements of arbitrary order appeared in [1, 3] where an extension of these schemes to any number of dimensions using tensor product meshes is also given. Numerical approximations obtained with linear finite and spectral elements lead and lag respectively with an equal magnitude of the dispersion error [3, 20]. Superior phase accuracy is obtained with the spectral element method compared with the finite element method for higher order elements. Marfurt [18] suggested that the most cost-effective scheme for computational wave propagation would be obtained by forming an appropriate weighted averaging of the spectral and finite element schemes. This idea was taken up in [4] where the optimal blending of the methods was obtained and shown to give an additional two orders of accuracy in the dispersion error. The case of first and second order elements had been previously considered by Fried [10]. Moreover, in [4] an equivalence between the blended scheme and non-standard quadrature rules was established. In particular this meant that the optimally blended scheme could be efficiently implemented merely by replacing the standard Gaussian quadrature rules by non-standard quadrature rules. Following an entirely different line of reasoning, Challa [7] arrived at the same non-standard quadrature rules in his thesis in the particular cases of linear and quadratic finite elements. Subsequently, Guddati and Yue [11, 12] studied such schemes for linear finite elements and commented on the relation with blended spectral-finite element schemes in the case of first order elements. In the present work we adopt the same approach used in [2] where we showed that the discrete dispersion relation in higher dimensions may be expressed in terms of the approximation of the scalar Helmholtz equation in one dimension. We use tensor product meshes to study the dispersive properties of the d-dimensional acoustic

1 Explicit Discrete Dispersion Relations

5

wave equation and obtain explicit expressions for the dispersion error for finite, spectral and the optimal blending of both finite and spectral element schemes in terms of the results in the special case of d = 1 dimensional approximation. The remainder of the chapter is organised as follows. In Section 1.2, we develop discrete dispersion relations using tensor product meshes valid for d-dimensions. In Section 1.3, discrete dispersion relations are derived for finite, spectral and optimally blended schemes. In the final section the numerical results obtained with these schemes are shown.

1.2 Acoustic Wave Equation Consider the acoustic wave equation in d-dimensions

∂ 2u − u = 0 ∂ t2

in Rd .

(1.1)

We seek time-harmonic solutions of the form u(x,t) = eiω t U(x) to the above equation, so that U satisfies the Helmholtz equation

ω 2U + U = 0

in Rd

(1.2)

where ω ∈ R is the given angular frequency.

1.2.1 Continuous Dispersion Relation Observe that in one dimension, the function u(ω ; x) = eiω x satisfies (u , v ) = ω 2 (u, v) where

1 ∀v ∈ Hloc (R)

(1.3)

1 Hloc (Rd ) = {v : Rd → R, v ∈ H 1 (Ω ) for all Ω ⊂⊂ Rd }

and (u, v) =

R

uvdx

is the L2 -inner product on R. We note that (1.3) is the variational formulation of (1.2) in one dimension. Furthermore, it is trivial to verify that the function u(ω ; x) satisfies u(ω ; x + nh) = eiω hn u(ω ; x) ∀n ∈ Z, x, h ∈ R which is the so called Bloch wave property . To obtain the dispersion relation in d-dimensions for the acoustic wave equation (1.2), we start with the variational formulation of equation (1.2) which is given by

6


ω2

Rd

Uvdx −

Rd

gradU · gradvdx = 0

1 ∀v ∈ Hloc (Rd ).

(1.4)

Now choose U and v as the product of uni-variate functions given by d

d

=1

=1

U(x1 , x2 , · · · , xd ) = ∏ u(ω ; x ) and v(x1 , x2 , · · · , xd ) = ∏ v (x ) 1 (R) and u(ω ; ·) where {ω }d=1 ∈ R are constants to be determined, {v }d=1 ∈ Hloc is defined above. Substituting U and v into (1.4), we get d

d

=1

r=1

ω 2 ∏ (u(ω ; ·), v ) − ∑ (u (ωr ; ·), vr ) ∏(u(ω ; ·), v ) = 0.

(1.5)

=r

Now, exploiting the identity (1.3), and performing straightforward manipulations, the above equation simplifies to d

ω 2 − ∑ ωr2 r=1

d

∏(u(ω; ·), v ) = 0,

(1.6)

=1

from which we see that non-trivial solutions of (1.6) exist only when the parameters {ω }d=1 satisfy ω 2 = ω12 + ω22 + · · · + ωd2 . (1.7) Equation (1.7) is the well known continuous dispersion relation of the acoustic wave equation (1.1) which is usually derived by inserting u directly into the differential equation (1.1).

1.2.2 Framework for Discrete Dispersion Relation To obtain the dispersion relation for the discrete case, we partition the real line R into infinitely many subintervals of uniform length h > 0 with nodes located at 1 (R) is the corresponding space of continuous piecewise hZ. The space Vhp ⊂ Hloc polynomials of degree p relative to the grid. We seek an approximation uhp ∈ Vhp such that (uhp , v ) − ω 2 uhp , v = 0 ∀v ∈ Vhp where ·, · is an appropriate discrete L2 -inner product on Vhp. Examples of suitable choices for ·, · will be given later, but we shall require that for v, w ∈ Vhp , v , w = (v , w ). To obtain the corresponding bilinear form in d-dimensions we consider the tensor product grid where each side of the grid has length h > 0, = 1, . . . , d. Let d ⊂ H 1 (Rd ) denote the space of continuous piecewise polynomials of degree p Vhp loc in each variable relative to the grid in d-dimensions, then we seek an approximate d such that udhp ∈ Vhp


7

∇udhp , ∇v d − ω 2 udhp , v d = 0

d ∀v ∈ Vhp

(1.8)

where ·, · d is the tensor product bilinear form obtained from ·, · . Motivated by the arguments leading to the dispersion relation in the continuous case, we have the following theorem for the discrete dispersion relation. Theorem 1. Suppose there exists a non-trivial function uhp(ω ; ·) ∈ Vhp such that uhp (ω ; ·) has 1. the discrete Bloch wave property uhp (ω ; x + nh) = einhξ uhp (ω ; x),

∀n ∈ Z, x, h ∈ R

(1.9)

with discrete frequency ξ = ξ (ω ) and satisfies 2.

(uhp , v ) = ω 2 uhp , v ,

Let

∀v ∈ Vhp.

(1.10)

Ehp (ω ) = ξ 2 (ω ) − ω 2,

(1.11)

then the discrete dispersion relation for the acoustic wave equation in d-dimensions is given by d

ωh2 = ω 2 + ∑ Ehp (ω ).

(1.12)

=1

Proof. By analogy with the derivation of the dispersion relation in the continuous (d) case, we seek a non-trivial solution Uhp ∈ Vhp of the form: d

Uhp (x1 , x2 , · · · , xd ) = ∏ uhp (ω ; x )

(1.13)

=1

where {ω }d=1 ∈ R are again constants to be determined. The corresponding discrete variational formulation of (1.2) is given by (1.8). Now, substituting v of the form d

v = ∏ v (x )

(1.14)

=1

and Uhp from (1.13) into (1.8), we obtain

ω

2

d

d

=1

r=1

∏ uhp(ω ; ·), v − ∑

(uhp (ωr ; ·), vr )

∏ uhp(ω ; ·), v

=r

Now, exploiting the property (1.10), we get d 2 2 ω − ω1 + ω22 + · · · + ωd2 ∏ uhp (ω ; ·), v = 0 =1

which has a non-trivial solution only when

= 0.

(1.15)

8


ω 2 = ω12 + ω22 + · · · + ωd2 .

(1.16)

Now, consider d

Uhp (x1 + n1h1 , x2 + n2 h2 , · · · , xd + nd hd ) = ∏ uhp(ω ; x + nh ) =1

which, on using property (1.9), gives Uhp (x1 + n1 h1 , x2 + n2h2 , · · · , xd + nd hd ) = ei[h1 n1 ξ (ω1 )+h2 n2 ξ (ω2 )+···+hd nd ξ (ωd )]Uhp (x1 , x2 , · · · , xd ).

(1.17)

This is the discrete Bloch wave property for Uhp , and hence, the frequency ωh of the discrete solution satisfies

ωh2 = ξ (ω1 )2 + ξ (ω2 )2 + · · · + ξ (ωd )2 . Finally, upon using (1.11) together with (1.16) and after applying simple manipulations, the above equation gives (1.12) which is what the claimed result. Theorem 1 means that we can obtain the discrete dispersion relation for a scheme on a tensor product mesh in Rd using results for the discrete dispersion relation for the scheme in R1 . We use this result in the following section to analyse the finite element, spectral element and a novel, so-called optimally blended scheme that was introduced in [4].

1.3 Higher Order Discrete Dispersion Relations for Finite Element, Spectral Element and Optimally Blended Schemes in d-Dimensions In this section we will derive the explicit expressions of the discrete dispersion relations valid in d-dimensions for finite, spectral and optimally blended schemes of arbitrary order.

1.3.1 Standard Finite Element Scheme For finite elements we evaluate the stiffness and mass matrices using the Gaussian quadrature rule 1 −1

(p)

f (x)dx ≈ QG ( f ) =

p

∑ w f (ζ )

=0

(1.18)


9

p where {ζ }=0 are the zeros of L p+1 and L p+1 is the (p + 1)-th order Legendre p polynomial. Moreover, weights {w }=0 are given by

2

w =

∀ ∈ {0, 1, . . ., p}.

(1 − ζ2)[Lp+1 (ζ )]2

(1.19)

The Gaussian quadrature rule (1.18) is exact for all polynomials of degree at most 2p + 1, and as a consequence 1 −1

1

u v dx = QG (u v ) and (p)

−1

(p)

uvdx = QG (uv) (p)

for all u, v ∈ P p . Now a composite quadrature rule IG on R given by (p)

R

f (x)dx ≈ IG ( f ) =

h 2

p

∑ ∑ w f (ζj,h )

(1.20)

j∈Z =0

h 1 j,h is constructed using (1.18) where ζ = ( j + )h + ζ , ∀ j ∈ Z and = 0, 1, . . . , p. 2 2 The discrete L2 -inner product is taken to be (p)

u, v G = IG (uv) and will be exact for u, v ∈ Vhp. Theorem 2. Let ω ∈ R be given. There exists a non-trivial uhp ∈ Vhp which satisfies (uhp , v ) = ω 2 (uhp , v),

∀v ∈ Vhp

(1.21)

and the Bloch wave property (1.9) with frequency ξ (ω ), where

p! ξ (ω ) = ω − (2p)! 2

2

2

h2p ω 2p+2 + O(h)2p+2. 2p + 1

(1.22)

Consequently, the discrete dispersion relation for finite elements in Rd is given by

ωh2 = ω 2 −

p! (2p)!

2

d 1 h2p ω 2p+2 + O(h)2p+2 ∑ 2p + 1 =1

(1.23)

where ω12 + ω22 + · · · + ωd2 = ω 2 . Proof. The existence of uhp is proved in Theorem 3.1 of [1] where it is also shown that p! 2 h2p ω 2p+2 2 2 + O(h)2p+2. ξ (ω ) = ω − (2p)! 2p + 1 Hence, applying Thorem 1, we obtain (1.23) at once.

10


1.3.2 Spectral Element Scheme The only difference for spectral elements compared with finite elements is the replacement of the Gaussian quadrature rule (1.18) with the Gauss-Lobatto quadrature (p) rule QGL defined by 1 −1

p

(p)

f (x)dx ≈ QGL ( f ) =

∑ w f (ζ˜ )

(1.24)

=0

p are taken to be the zeros of Lp (x)(1 − x2 ) with weights w given by where {ζ˜ }=0

w =

2

∀ ∈ {0, 1, . . . , p}.

p(p + 1)[L p(ζ˜ )]2

(1.25)

The Gauss-Lobatto quadrature rule (1.24) is exact for all polynomials of degree at most 2p − 1. Hence the stiffness matrix is integrated exactly 1 −1

u v dx = QGL (u v ) (p)

whereas the mass matrix is underintegrated 1 −1

(p)

uvdx ≈ QGL (uv).

We use this quadrature rule to develop a composite quadrature rule on R, which (p) we denote by IGL (·), following the same construction used in the case of finite elements. The discrete L2 -inner product is taken to be (p)

u, v L = IGL (uv). The only difference now is that the mass matrix will be under-integrated. Theorem 3. Let ω ∈ R be given. There exists a non-trivial uhp ∈ Vhp which satisfies (uhp , v ) = ω 2 uhp , v L ,

∀v ∈ Vhp

(1.26)


ξ (ω )2 = ω 2 +

p! 2 h2p ω 2p+2 1 + O(h)2p+2. p (2p)! 2p + 1

(1.27)

Consequently, the discrete dispersion relation for spectral elements in Rd is given by

ωh2 = ω 2 +

d p! 2 1 1 2p+2 + O(h)2p+2 ∑ h2p ω p (2p)! 2p + 1 =1

where ω12 + ω22 + · · · + ωd2 = ω 2 .

(1.28)


11

Proof. The existence of uhp is established in Theorem 4.1 of [3] where it is also shown that p! 2 h2p ω 2p+2 1 + O(h)2p+2. ξ (ω )2 = ω 2 + p (2p)! 2p + 1 Equation (1.28) then follows at once from Theorem 1. Interestingly, from (1.28) it is clear that for higher orders the spectral element scheme provides p-times better phase accuracy as compared to the phase accuracy obtained with finite element scheme (1.23).

1.3.3 Optimally Blended Scheme We now apply Theorem 1 to a novel scheme introduced in [4] for the wave equation, whereby the finite and spectral element schemes are blended in such a way that the order of accuracy of the resulting discrete dispersion relation is optimised. If the blending parameter is denoted by τ ∈ [0, 1], then we base the blended scheme on the blended quadrature rule 1 −1 (p)

(p)

(p)

(p)

f (x)dx ≈ Qτ ( f ) = (1 − τ )QG ( f ) + τ QGL ( f ) (p)

where QG and QGL are the (p + 1)-point Gauss-Legendre and Gauss-LegendreLobatto quadrature rules defined in the previous sections and give us the standard finite and spectral element schemes for τ = 0 and τ = 1 respectively. Furthermore, (p) Qτ is the (p+1)-point non-standard quadrature rule given in [4] valid for elements p of arbitrary order with nodes {ζτ }=0 chosen as the zeros of L p+1 − τ L p−1 , where L p+1 and L p−1 are the Legendre polynomials of degrees p + 1 and p − 1 respectively, with weights given by w =

2[p(1 + τ ) + τ ] , ∀ = 0, 1, . . . , p. p(p + 1)L p(ζτ )[Lp+1 (ζτ ) − τ Lp−1 (ζτ )] (p)

Furthermore, Qτ

(1.29)

satisfies the following identity [4]

(p)

(p)

(p)

Qτ ( f ) = (1 − τ )QG ( f ) + τ QGL ( f )

∀ f ∈ P2p+1

(1.30)

and is exact for all polynomials of degrees at the most 2p − 1. We use this quadrature (p) rule to develop a composite quadrature rule on R, which we denote by Iτ (·), and follow the same construction used in the previous sections for finite and spectral element schemes. The discrete L2 -inner product is taken to be (p)

u, v τ = Iτ (uv). Once again the mass matrix is under-integrated.

12


Theorem 4. Let ω ∈ R be given. There exists a non-trivial uhp ∈ Vhp which satisfies (uhp , v ) = ω 2 uhp , v τ ,

∀v ∈ Vhp

(1.31)


p! 2 h2p ω 2p+2 1 −1 + O(h2p+2). ξ (ω ) = ω + τ 1 + p (2p)! 2p + 1 2

2

(1.32)

Consequently the discrete dispersion relation for optimally blended scheme in Rd is given by

d 1 p! 2 1 2p 2p+2 −1 ωh2 = ω 2 + τ 1 + h ω ∑ p (2p)! 2p + 1 =1

(1.33)

where ω12 + ω22 + · · · + ωd2 = ω 2 . Proof. The existence of uhp is proved in Theorem 3.1 of [4] where it is also shown that

p! 2 h2p ω 2p+2 1 −1 + O(h2p+2). ξ (ω )2 = ω 2 + τ 1 + (1.34) p (2p)! 2p + 1 Now applying Theorem 1, we obtain (1.33) at once. It is not difficult to check that the above expressions leads to expression (1.23) for τ = 0 and (1.28) for τ = 1 which are the discrete dispersion relations corresponding to finite and spectral element schemes respectively. More importantly, the first term in expression (1.33) vanishes if we choose blending parameter τ = p/(p + 1) which shows that the optimal blending parameter is independent of the number of spatial dimensions. Theorem 4 gives rise to the following corollary. Corollary 1. Let p ≥ 2. Then for the optimal choice of the blending parameter τ = p/(p + 1), the error in the discrete dispersion relation (1.33) is given by

ωh2 = ω 2 +

d 8 (p + 1)! 2 1 h2p+2ω2p+4 + O(h2p+4). ∑ (2p − 1) (2p + 2)! 2p + 3 =1

Proof. Substituting τ = p/(p + 1) in (1.33) and applying trivial manipulations gives us the required result. Whilst the cost of all of the schemes is virtually identical, remarkably the leading error term for the optimal scheme is two orders more accurate compared with the standard spectral and finite element schemes given in the previous sections. Moreover, the coefficient of the leading term in the error obtained with the blended scheme for the optimum value of τ is −2/(4p2 − 1)(2p + 3) and 2p/(4p2 − 1)(2p + 3) times


13

better compared with the leading terms in the error obtained with finite and spectral element schemes respectively. 2

1.5

Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

1.5


1

1 0.5

0

u

u

0.5

−0.5

0

−0.5

−1 −1 −1.5 −1.5

−2 0

0.5

1

1.5 x

2

2.5

3

1.3

1.35

1.4

1.45 x

(a)

1.5

1.55

1.6

(b)

1.5

3 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution


2.5

1

2 1.5

u

1

u

0.5

0

0.5 0 −0.5

−0.5

−1 −1.5

−1 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

−2 2

2.1

2.2

(c)

2.3

2.4

2.5 x

2.6

2.7

2.8

2.9

3

(d)

Fig. 1.1 Numerical approximations of the solution to equation (1.35) obtained for p = 1 with ω = 30 and ρ = 9. Furthermore 35 and 300 elements are used outside and inside the slab respectively.

1.4 Numerical Examples In order to study the behaviour of finite, spectral and optimally blended schemes in practical computations, we consider a simple one dimensional scattering problem on the interval Ω = (0, 3) with fixed ω ∈ R, and ωs ∈ R given by −u − ω 2 (x)u = f (x) where

ω (x) =

ω , for x ∈ / (1, 2), and f (x) = ωs , for x ∈ (1, 2)

(1.35)

0, for x ∈ / (1, 2), (ω 2 − ωs2 )eiω x , for x ∈ (1, 2)

14


with the following non-reflecting boundary conditions applied at both ends of the domain u (0) + iω u(0) = 0 and u (3) − iω u(3) = 0. Evidently, the model problem corresponds to scattering of an incoming plane wave by a slab of relative density ωs2 /ω 2 located on (1, 2). In Figure 1.1 (a), we approximate scattered wave using 35 and 300 linear elements outside and inside the slab respectively for spectral, finite and optimal schemes with given frequency ω = 30 and relative density ρ = 9. Scattered waves on the left and right side of the slab are shown in Figure 1.1 (c) − (d) to analyse better the numerical approximations obtained with all the schemes. The phase lead and lag of equal magnitudes are clearly visible and correspond to finite and spectral element schemes which is consistent with error expressions given in (1.23) and (1.28). The same observation was made in [3, 4, 20] in the case of linear elements. Furthermore the numerical approximation corresponding to the optimal scheme is noticeably better than that of finite and spectral element schemes which was also observed in [4]. Figure 1.1 (b), represents the scattered wave inside the slab and once again optimal scheme performs better than that of finite and spectral element schemes nonetheless phase lead and lag of equal magnitudes with linear elements are still prominent even inside the slab. In 2


1.5

1

1

0.5

0.5

0

0

u

u

1.5

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 0

0.5

1

1.5 x

(a)

2

2.5

3

−2 1


1.1

1.2

1.3

1.4

1.5 x

1.6

1.7

1.8

1.9

2

(b)

Fig. 1.2 Numerical approximations of the solution to equation (1.35) obtained for p = 2 with ω = 10 and ρ = 49. Furthermore 10 and 30 elements are used outside and inside the slab respectively.

Figures 1.2 and 1.3, we show numerical approximations obtained for all the schemes using quadratic and cubic elements. It is clear from Figures 1.2(a) and 1.3(a) that with piecewise quadratic and cubic elements both spectral and optimal schemes are performing much better than that of finite element scheme. This conjecture is consistent with analytical results (1.28) and (1.33) of dispersion error obtained for spectral and optimal schemes. The magnitude of the leading order error term for spectral and optimal schemes are O(p−1 ) and O(p−3 ) times better than that of the pure finite element scheme. Moreover, the numerical approximation obtained with finite element scheme is unresolved both for quadratic and cubic elements in each


15

2


1.5

1

1

0.5

0.5

0

u

u

1.5

−0.5

0

−0.5

−1

−1

−1.5

−1.5

−2 0


0.5

1

1.5 x

2

2.5

−2 1

3

1.1

1.2

1.3

1.4

(a)

1.5 x

1.6

1.7

1.8

1.9

2

(b)

Fig. 1.3 Numerical approximations of the solution to equation (1.35) obtained for p = 3 with ω = 10 and ρ = 115. Furthermore 10 and 30 elements are used outsides and inside the slab respectively.

region. The same conjecture is observed even inside the slab which is presented in Figures 1.2(b) and 1.3(b) for quadratic and cubic elements respectively. In Figure 2


1.5

1

1

0.5

0.5

0

u

u

1.5

−0.5

0

−0.5

−1

−1

−1.5

−1.5

−2 0


0.5

1

1.5 x

(a)

2

2.5

3

−2 0

0.5

1

1.5 x

2

2.5

3

(b)

Fig. 1.4 Numerical approximations of the solution to equation (1.35) obtained with ω = 30 and ρ = 9 for (a) 35 linear and 50 cubic elements (b) 5 quartic and 15 fifth order elements used outside and inside the slab respectively.

1.4, we show the effect of using polynomials of different orders in different regions. In Figure 1.4(a), we show numerical results approximated outside the slab with first order (p = 1) elements whereas cubic elements are used inside the slab. We use the same number of elements i.e. n1 = n3 = 35 outside the slab as we used in Figure 1.1 but inside the slab using n2 = 50 cubic elements instead of 300 linear elements gives us much better results but phase leads and lags of equal magnitude are visible outside the slab as we are using linear elements there. Now using n1 = n3 = 5 quartic elements outside the slab and n2 = 15 elements of fifth order provides very accurate

16

M. Ainsworth and H. Abdul Wajid 1.5

1

0.5

0.5

0

0

u

u

1

1.5 Exact Solution Finite Element Solution Spectral Element Solution Optimal Solution

−0.5

−0.5

−1

−1

−1.5 0

0.5

1

1.5 x

2

2.5

−1.5 0

3


0.5

1

(a)

1.5 x

2

2.5

3

(b)

Fig. 1.5 Numerical approximations of the solution to equation (1.35) obtained with ω = 10 and ρ = 9 using 10 linear elements outside the slab and using (a) ten cubic (b) ten fifth order elements inside the slab.

2.5 2


1.5


1.5 1 1 0.5

0

u

u

0.5

−0.5

0

−0.5

−1 −1 −1.5 −1.5

−2 −2.5 0

0.5

1

1.5 x

(a)

2

2.5

3

−2 0

0.5

1

1.5 x

2

2.5

3

(b)

Fig. 1.6 Numerical approximations of the solution to equation (1.35) obtained with ω = 10 and ρ = 10 for (a) 10 and 50 (b) 200 and 50 linear elements used outside and inside the slab respectively.

results as shown in Figure 1.4 (b). In Figure 1.5, we show that when the waves are fully resolved inside the slab than increasing the order or increasing the number of elements inside the slab does not help the waves outside the slab to converge. Hence when the waves are almost resolved inside the slab then waves outside the slab can be resolved either increasing the number of elements or using the higher order elements. Now consider the case where the wave is not resolved inside the slab then increasing the number of elements or using the higher order elements do not give us a completely resolved wave where this behaviour is shown in Figure 1.6.


17

References [1] Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004) (electronic) [2] Ainsworth, M.: Dispersive properties of high-order Nédélec/edge element approximation of the time-harmonic Maxwell equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362(1816), 471–491 (2004) [3] Ainsworth, M., Wajid, H.A.: Dispersive and dissipative behaviour of the spectral element method. SIAM J. Numer. Anal. (accepted, 2009) [4] Ainsworth, M., Wajid, H.A.: Optimally blended spectral-finite element scheme for wave propagation, and non-standard reduced integration. SIAM J. Numer. Anal. (submitted, 2009) [5] Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, 3rd edn., vol. 15. Springer, New York (2008) [6] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. In: Evolution to complex geometries and applications to fluid dynamics. Springer, Berlin (2007) [7] Challa, S.: High-order accurate spectral elements for wave propagation. Masters thesis in mechanical engineering, Clemson University (1998) [8] Cohen, G.C.: Higher-order numerical methods for transient wave equations. In: Scientific Computation. Springer, Heidelberg (2002) With a foreword by R. Glowinski [9] De Basabe, J., Sen, M.: Grid dispersion and stability criteria of some common finiteelement methods for acoustic and elastic wave equation. Geophysics 72(6), T81–T95 (2007) [10] Fried, I., Chavez, M.: Superaccurate finite element eigenvalue computation. Journal of sound and vibration 275, 415–422 (2004) [11] Guddati, M., Yue, B.: Modified integration rules for reducing dispersion error in finite element methods. Comput. Methods Appl. Mech. Engrg. 193, 275–287 (2004) [12] Guddati, M., Yue, B.: Dispersion-reducing finite elements for transient acoustics. J. Acoust. Soc. Am. 118(4), 2132–2141 (2005) [13] Ihlenburg, F.: Finite element analysis of acoustic scattering. In: Applied Mathematical Sciences, vol. 132. Springer, New York (1998) [14] Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995) [15] Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. II. The h-p version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997) [16] Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-I. Validation. Geophys. J. Int. 149(2), 390–412 (2002) [17] Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-II. 3-D models, oceans, rotation, and self-gravitation. Geophys. J. Int. 150(1), 303–318 (2002), doi:10.1046/j.1365-246X.2002.01716.x [18] Marfurt, K.J.: Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics 49(5), 533–549 (1984) [19] Mulder, W.A.: Spurious modes in finite-element discretizations of the wave equation not be all that bad. Appl. Numer. Math. 30(4), 425–445 (1999) [20] Thompson, L.L., Pinsky, P.M.: Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech. 13(4), 255–275 (1994)

Chapter 2

hp-Adaptive Finite Elements for Coupled Multiphysics Wave Propagation Problems Leszek Demkowicz, Jason Kurtz, and Frederick Qiu

Abstract. The paper describes a generalization of hp-adaptive finite elements technology to coupled multiphysics problems. Three representative examples are used: dual-mixed formulation with weakly imposed symmetry for linear elasticity, coupled acoustics/viscoelasticity and coupled acoustics/poroelasticity problem, to illustrate variational formulations and concept of weak couplings. We discuss then necessary changes in data structures, constrained approximation and the hp mesh optimization algorithm. Sample numerical results for the three problems illustrate the new methodology.

2.1 Introduction The hp-adaptive Finite Elements have been successfully applied to a large number of wave propagation problems including time-harmonic acoustics and electromagnetics, see [3, 5] for numerous examples. The energy-, or goal-oriented automatic hp-adaptivity enables solution of challenging problems with large dynamic range and strong material contrasts resulting in point and edge (3D) singularities and boundary layers. In this paper, we report on an extension of the hp-technology to coupled, multiphysics wave propagation problems. Discretization of multiphysics problems necessitates frequently the simultaneous use of H 1-, H(curl)-, H(div)- and L2-conforming1 elements that enforce different continuity across interelement boundaries. In the classical H 1 -conforming discretizations FE solution is globally continuous. In contrast, the L2 -conformity implies no condition on continuity across interelement boundaries whatsoever. Contrary to H 1 - and L2 -conforming discretizations that are based on use Leszek Demkowicz, Jason Kurtz, and Frederick Qiu Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected], [email protected], [email protected] 1

Frequently referred to as continuous, edge, face and discontinuous elements.


20

L. Demkowicz, J. Kurtz, and F. Qiu

of scalar-valued shape functions, the H(curl)- and H(div)-conforming discretizations use vector-valued shape functions, and enforce a partial continuity across the interelement boundaries: tangential components of H(curl)- and normal components of H(div)-conforming vector fields are globally continuous. These continuity requirements are not merely a mathematical construction but they reflect the worst possible scenarios in the case of nonhomogeneous data, e.g. a jump in dielectric constant results in a discontinuity of normal component of electric field across the material interface. The properly constructed H 1 -, H(curl)-, H(div)- and L2 -conforming FE element spaces form the so-called exact sequence, both at a single element and the whole finite element mesh levels (including curvilinear elements). The exact sequence structure is absolutely crucial for constructing stable discretizations of multiphysics problems including examples discussed in this paper. Our new 2D hp code [4] uses a single hp mesh that supports the use of H 1 -, H(curl)-, H(div)- and L2 -conforming elements forming the exact sequence. The polynomial order for H 1 - conforming elements implies the corresponding order for the other involved elements. Additionally, the nature of coupled problems calls for the possibility of supporting different fields over different parts of the domain. In the discussed implementation we have restricted ourselves only to the so-called weak couplings where coupling of different physical models results in the presence of integrals over the interface boundaries. (Hybrid of mortar methods are examples of more general techniques.) From the point of view of data structures, this means that different fields may overlap at interfaces. The structure of the paper is as follows. We begin in Section 2.2 with a few examples of coupled multiphysics problems illustrating the points made in this introduction. Section 2.3 is devoted to a short discussion of our implementation including data structures, constrained approximation, and a generalization of hp algorithm to the coupled multiphysics problems. In Section 2.4, we present a few illustrative numerical results for sample problems from Section 2.2. A short discussion in Section 2.5 concludes the presentation.

2.2 Examples of Coupled Multiphysics Problems We start with the classicial example of linear elasticity.

2.2.1 Dual Mixed Formulation for Elasticity with Weakly Imposed Symmetry We begin with the time-harmonic linear elasticity problem. Let Ω ⊂ R I n , n = 2, 3, denote a bounded domain occupied by the elastic body with the boundary Γ = ∂ Ω split into two disjoint subsets Γ1 and Γ2 .

2 hp-Adaptive for Multiphysics Wave Propagation

21

We seek: • displacement vector ui (x), x ∈ Ω , • linearized strain tensor εi j (x), x ∈ Ω , • stress tensor σi j (x), x ∈ Ω that satisfy the following system of equations and boundary conditions. • Cauchy’s geometrical relation between the displacement and strain,

εi j = 12 (ui, j + u j,i ),

x∈Ω

• Equations of motion resulting from the principle of linear momentum, −σi j, j − ρ (x)ω 2 ui = fi (x),

x∈Ω

• Symmetry of the stress tensor being a consequence of the principle of angular momentum, σi j = σ ji , x ∈ Ω , • The constitutive equations of linear elasticity,

σi j = Ei jkl (x)εkl ,

x∈Ω

• kinematic boundary conditions, ui = u0i ,

x ∈ Γ1

• Traction boundary conditions,

σ ji n j = gi (x),

x ∈ Γ2

Here: • • • • • •

ρ is the density of the body, fi are volume forces prescribed within the body, gi are tractions prescribed on Γ2 part of the boundary, u0i are displacements prescribed on Γ1 part of the boundary, n j is the unit outward normal vector for boundary Γ Ei jkl = μ (δik δ jl + δil δ jk ) + λ δi j δkl is the tensor of elasticities for an isotropic solid, where μ , λ are the Lamé constants.

The constitutive equation can be inverted to represent strains in terms of stresses,

εkl = Ckli j σi j where Ckli j = Ei−1 jkl is the compliance tensor. Of particular interest is the case of nearly incompressible material corresponding to λ → ∞. Notice that the norm of the elasticities blows up then to infinity but the norm of the compliance tensor remains uniformly bounded. This suggests that

22


formulations based on the compliance relation have a chance to remain uniformly stable for nearly incompressible materials.

Classical, displacement-based formulation The discussed equations can be treated in a strong, pointwise sense, or can be interpreted in the sense of distributions. The equations understood in the strong sense imply then the possibility of eliminating some of the unknowns. Different choices lead to different ultimate variational formulations, and different energy spaces. In the classical formulation corresponding to the Lamé equations, the momentum equations are treated in a distributional sense. In other words, we multiply the momentum equations with a test function vi (sum up with respect to i), integrate over domain Ω , and integrate the first term by parts, to obtain

Ω

σi j vi, j −

Γ

σi j n j vi − ω 2

Ω

ρ u i vi =

Ω

f i vi

The symmetry of the stress tensor implies that the first term can be rewritten in the form, σi j vi, j = σi j 12 (vi, j + v j,i ) + σi j 12 (vi, j − v j,i ) = σi j εi j (v) where by εi j (v) we understand the strain tensor (symmetric part of the gradient) corresponding to test function vi . All the remaining equations are treated in the strong sense. We use the strain-displacement relations to eliminate the strain tensor, and the constitutive equation to eliminate the stress tensor in the domain integral. We obtain,

Ω

Ei jkl εi j (u)εkl (v) −

Γ

σi j n j vi − ω 2

Ω

ρ u i vi =

Ω

f i vi

This leads to the H 1 energy setting for displacement ui . The displacement boundary conditions can then be understood in the sense of the Trace Theorem. The same energy setting is used for the test functions. We assume that the test functions vanish (in the sense of traces) on Γ1 , and reduce the boundary term that has resulted from integration by parts, to Γ2 part only. Finally, we “build in” the traction boundary condition into the formulation assuming that

Γ2

σi j n j vi =

Γ2

g i vi ,

for all eligible test functions. The traction boundary condition is thus satisfied also in a weak sense only. The ultimate formulation looks as follows.


23

⎧ ⎪ ui ∈ H 1 (Ω ), ui = u0i on Γ1 ⎪ ⎪ ⎪ ⎪ ⎨ Ei jkl εi j (u)εkl (v) − ω 2 ρ u i vi = f i vi + g i vi ⎪ Ω Ω Ω Γ2 ⎪ ⎪ ⎪ ⎪ ∀vi ∈ H 1 (Ω ) : vi = 0 on Γ1 ⎩ In the static case ω = 0, the variational problem reduces to the classical Principle of Virtual Work, and it is equivalent to the problem of minimizing the total potential energy (Lagrange’s Theorem). The H 1 energy setting leads to the use of classical continuous (H 1 -conforming) finite element discretization. The classical dual formulation for the static problem in terms of stresses leading to the Castigliano’s Principle (maximization of complementary energy) is expressed in terms of stresses σi j that satisfy a-priori the equilibrium equations in the strong sense. The principle does not lead to practical discretization schemes as the assumption cannot be satisfied easily for piece-wise polynomial functions.

Dual formulation We start by eliminating the strain tensor from the constitutive equation in the compliance form, Ci jkl σkl = 12 (ui, j + u j,i ) In this alternative, dual approach, we solve the momentum equations in the strong sense, but treat the combined constitutive/geometry relations above in a distributional way. We multiply the equation with a symmetric tensor-valued test function τi j , integrate over Ω , and integrate by parts to obtain,

Ω

Ci jkl σkl τi j = −

Ω

ui τi j, j +

Γ

ui τi j n j

We treat the momentum equations in a strong form and solve them for the displacements, 1 ui = [−σi j, j − fi ] ρω 2 Substituting the expression into the weak statement, we get,

Ω


Ω

1 [−σi j, j − fi ] τi j, j + ρω 2

Γ

ui τi j n j

We restrict ourselves now to stress fields that satisfy the traction boundary conditions in a strong sense and assume the corresponding homogeneous boundary conditions for test functions, τi j n j = 0 on Γ2 Finally, substituting the kinematic boundary condition into the boundary term, we arrive at the alternative variational formulation,

24


⎧ σi j n j = gi on Γ2 ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Ω

Ci jkl σkl τi j −

Ω

1 σi j, j τi j, j = ρω 2

Ω

1 fi τi j, j + u0i τi j n j ρω 2 Γ1 ∀τi j : τi j n j = 0 on Γ2

The formulation leads to the energy setting:

σ , τ ∈ H(div, Ω , S) where H(div, Ω , S) denotes the space of square integrable symmetric tensor-valued functions, whose divergence is also in L2 . The dual formulation degenerates as ω → 0 and it does not cover the static case.

Dual-mixed formulation The idea is based on satisfying the momentum equations in the strong sense, but without eliminating the displacements. We multiply the momentum equations with a test function vi and integrate over Ω to obtain2, −

Ω

σi j, j vi − ω 2

Ω

ρ u i vi =

Ω

f i vi

The final formulation reads as follows. ⎧ σ ∈ H(div, Ω , S) : σi j n j = gi on Γ2 , u ∈ L2 (Ω , V) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ci jkl σkl τi j + ui τi j, j = u0i τi j n j ∀τ ∈ H(div, Ω , S) : τi j n j = 0 on Γ2 Ω Ω Γ1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ρ u i vi = f i vi ∀v ∈ L2 (Ω , V) ⎩ − σi j, j vi −ω Ω

Ω

Ω

Here L2 (Ω , V) denotes the square integrable vector-valued functions. The traction conditions in both dual and dual-mixed formulations are satisfied in the sense of traces for functions from H(div, Ω ) (they live in H −1/2(Γ )). The corresponding static case can be derived formally by considering the socalled Hellinger-Reissner functional, and the variational formulation is frequently identified as the Hellinger-Reissner variational principle.

Dual-mixed formulation with weakly imposed symmetry The symmetry condition is difficult to enforce on the discrete level. This has led to the idea of relaxing the symmetric function and satisfying it in a weaker, integral 2

Notice that we do not integrate by parts.


25

form. This is obtained by introducing tensor-valued test functions q with values in the space of antisymmetric tensors K := {qi j : qi j = −q ji }, and replacing the symmetry condition with an integral condition,

Ω

σi j qi j = 0,

∀q ∈ L2 (Ω , K)

On the continuous level, the integral condition implies the pointwise condition (understood in the L2 sense), but on the discrete level, with an appropriate choice of spaces, the integral condition does not necessary imply the symmetry condition pointwise. The extra condition leads to an extra unknown. The derivation of the weak form of the constitutive equation has to be revisited. We start by introducing the tensor of infinitesimal rotations, pi j = 12 (ui, j − u j,i ) Upon eliminating the strain tensor, the constitutive equation in the compliance form is now rewritten as, Ci jkl σkl = 12 (ui, j + u j,i) = ui, j − pi j Multiplication with a test function τ (now, not necessarily symmetric), integration over Ω , and integration by parts, leads to a new relaxed version of the equation,

Ω


Ω

ui τi j, j +

Γ

ui τi j n j −

Ω

pi j τi j

After similar considerations as before, we obtain a new variational formulation in the form: ⎧ σ ∈ H(div, Ω , M) : σi j n j = gi on Γ2 , u ∈ L2 (Ω , V), p ∈ L2 (Ω , K) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ci jkl σkl τi j + ui τi j, j + pi j τi j = u0i τi j n j ⎪ ⎪ ⎪ Ω Ω Ω Γ ⎪ 1 ⎨ ∀τ ∈ H(div, Ω , M) : τi j n j = 0 on Γ2 ⎪ ⎪ ⎪ ⎪ ⎪ − σi j, j vi −ω 2 ρ u i vi = f i vi ∀v ∈ L2 (Ω , V) ⎪ ⎪ ⎪ Ω Ω Ω ⎪ ⎪ ⎪ ⎪ ⎩ σi j qi j =0 ∀q ∈ L2 (Ω , K) Ω

In the above, H(div, Ω , M) denotes the space of (general) tensor-valued square integrable fields with square integrable divergence. The formulation can be derived formally by looking for a stationary point of the so-called Generalized HellingerReissner functional, and it is frequently identified as the Generalized HellingerReissner Variational Principle. Summarizing, in two space dimensions, the dual-mixed variational formulation with the weakly imposed symmetry involves the use of two H(div)-conforming

26


fields representing rows of the stress tensor σi, j , j = 1, 2, i = 1, 2, and three L2 conforming fields representing two components of the displacement vector ui , and scalar variable p representing the antisymmetric tensor of infinitesimal rotations

0p pi j = −p 0 All fields are supported over the whole domain Ω . Notice that the problem is symmetric.

2.2.2 Acoustics Coupled with (visco-)Elasticity In this example, we present a coupled acoustics/visco-elasticity problem encountered in modeling of streamers. Streamers are several kilometer long hollow plastic cylinders strengthened with a couple of ropes (strength elements) and filled with a (very) soft viscoelastic “filler”. Placed in the filler is a periodic array of spacers (made of hard plastic) housing hydrophones. Periodically spaced buoyancy system (the “birds”) maintains the streamer floating below the surface of a sea. An array of six to ten streamers, deployed on the sea, and pulled by a tugboat is used then to record acoustical waves initiated near the surface and reflected from layers of the ocean floor. The recorded signals are ultimately used to solve an inverse problem to reconstruct the geological formation and produce survey maps necessary for oil exploration. A simple, axisymmetric mechanical model of the streamer is presented in Fig. 2.1. The strength elements have been lumped into a single rope placed in the middle of the streamer. The z axis coincides with the axis of symmetry and the picture has been magnified by a factor of 10 in the radial direction to enable visualization.

Fig. 2.1 Axisymmetric model of a streamer


27

Sample material properties of the streamer and geometrical dimensions are recorded in Table 2.1. Under a large drag force resulting from pulling the streamers through water, the strength elements become very stiff. The spacers and streamer’s skin are only moderately stiff but the filler is very soft. Additionally, the viscoelastic properties of the filler depend upon the temperature. Table 2.1 Material properties of streamer components component E(GPa)

ρ (kg.m−3 ) ν

length(m)

height(m)

Spacer

1.8

1200

0.30

0.075

0.021843

Rope Skin

41.0 0.02

1400 1200

0.30 0.45

75 75

0.005657 0.0032

Filler

Egel (ω , T ) 1040

0.45

0.165

0.021843

E f iller (ω , T ) = Er (ω , T ) + iEr (ω , T ) . At T = 10◦C, we have Er (ω , T ) = 2.9 × 100.4125 log10 (ω )+3.0871 [Pa] and Ei (ω , T ) = 2.9 × 100.3977 log10 (ω )+2.9707 [Pa]. In the real, three-dimensional scenario, the filler extends into cavities in the spacers housing the microphones. The critical quantity of interest in simulating vibrations of such structures is thus the distribution of the pressure in the soft part of the structure - the gel. The streamer project is focusing on studying unwanted vibrations of the streamers propagating from the tugboat and polluting the recorded signals. The problem is formulated as a coupled acoustics-(visco)elasticity problem. The domain occupied by the streamer is denoted by Ωe and the rest of the domain, occupied by the acoustical fluid, is denoted by Ωa . In the acoustic domain, we wish to determine pressure p and velocity wi satisfying: • conservation of mass,

c−2 iω p + ρ f w j, j = 0

and • conservation of linear momentum,

ρ f iω w j + p, j = 0 where ρ f is the density of the fluid, c is the sound speed, ω is the angular frequency and i denotes the imaginary unit. In the (visco)elastic domain, we wish to determine stresses σi j and displacements ui satisfying the equations discussed in our first example: • conservation of linear momentum equations, −ρs ω 2 ui − σi j, j = 0

28


and • constitutive equations combined with geometrical relations,

σi j = μ (ui, j + u j,i) + λ uk,k δi j The system of equations is accompanied by the interface boundary conditions representing the continuity of normal velocity and tractions, iω ui ni = wi ni

and σi j n j = −pni

on interface ΓI

As the acoustic domain representing the ocean is infinite, the system must be accompanied by a radiation condition eliminating waves coming from infinity. Finally, we will assume that the system is driven by a kinematic boundary condition imposed on part of the streamer boundary representing the connector with the tugboat. ui = ui0

on Γ1e

Variational formulation and the weak coupling We proceed in the following steps. Step 1. Formulate conservation of mass (acoustics) and balance of momentum (elasticity) in a weak form: −

ω 2 Ωa

c Ωe

pq + iωρ f vi q,i −

ΓI

−ω 2 ρ ui vi + σi j vi, j −

iωρ f vi ni q = 0 ∀q ΓI

σi j n j vi = 0 ∀vi : vi = 0 on Γe1

Step 2. Use the remaining equations in the strong form to eliminate fluid velocity and elastic stresses: ω 2 − pq + p,iq,i − iωρ f vi ni q = 0 ∀q Ωa c ΓI Ωe

−ω 2 ρ ui vi + μ (ui, j + u j,i)vi, j + λ uk,k vk,k −

ΓI

σi j n j vi = 0 ∀vi : vi = 0 on Γe1

Step 3. Use the interface conditions to couple the two variational formulations: − Ωe

ω 2 Ωa

c

pq + p,iq,i + ω 2

ΓI

−ω 2 ρ ui vi + μ (ui, j + u j,i )vi, j + λ uk,k vk,k +

ρ f ui ni q = 0 ∀q ΓI

pvi ni = 0 ∀vi : vi = 0 on Γe1

We end up with the following variational formulation.


29

⎧ u ∈ u˜ 0 + V e , p ∈ Va ⎪ ⎪ ⎪ ⎪ ⎨ bee (u, v) + bae(p, v) = le (v) ∀v ∈ V e ⎪ ⎪ ⎪ ⎪ ⎩ bea (u, q) + baa(p, q) = la (q) ∀q ∈ Va where

V e = {v ∈ H 1 (Ωe ) : v = 0 on Γe1 } Va = H 1 (Ωa ) bee (u, v) =

Ωe

bae (p, v) =

ΓI

bea (u, q) = − baa (p, q) =

Ei jkl uk,l vi, j − ρs ω 2 ui vi dx

pvn dS un q dS

ΓI

1

ω 2ρ

f

Ωa

∇p∇q − k2 pq dx

le (v) = 0 la (q) = 0 Above, u˜ 0 represents a lift of the kinematic boundary data. The problem is driven by the kinematic boundary condition only. The radiation condition is handled by a Perfectly Matched Layer resulting in complex material data but preserving the symmetry of the formulation, see [5] for details. The problem is thus complex-symmetric (not Hermitian !). Discretization of a two-dimensional axisymmetric version of the problem requires the use of three H 1 -conforming components: two components of the elastic displacement ui , and one component for the pressure p. The elastic displacement components are supported in the elastic domain Ωe , and the pressure lives over Ωa . All three components must be supported over the elastic/acoustic interface ΓI .

2.2.3 Coupled Acoustics and Poroelasticity In this section we present an example of acoustics coupled with poroelasticity, a problem encountered in modeling of sonic tools used in a borehole environment. The hp technology was used simulate the experiments performed in [7]. A porous sandstone cylinder with a borehole is placed in a bath of saturating liquid. A transmitter excites waves within the borehole, and measurements of the velocity and attenuation of Stoneley waves along the borehole wall are made. The axisymmetric geometry is shown in Figure 2.2.

30


z

r

Fig. 2.2 Axisymmetric geometry for sandstone cylinder in liquid bath.

Assuming the time dependence eiω t , with angular frequency ω , the liquid subdomain Ωa is modeled by the linear acoustics equations, iω p = −c2 ρ ∇ · v

(2.1)

iωρ v = −∇p

(2.2)

involving the pressure p, velocity v, density ρ and sound speed c. The variational formulation is obtained by multiplying (2.1) by a test function δ p, integrating over Ωa , integrating by parts, and using equation (2.1) in the strong form to eliminate the velocity:

Ωa

1 ∇p · ∇δ p − k2 pδ p dx + iω ρ

Γi

vn δ p dS = −iω

Γs

vsn δ p dS

(2.3)

In fact, (2.3) is modified to incorporate a perfectly matched layer (PML) using the procedure described in [5]. The surface integrals in (2.3) are over the source Γs at the bottom of the borehole (where the normal velocity vsn is prescribed), and the interface Γi with the poroelastic cylinder. The poroelastic subdomain Ω p is modeled by the two-velocity approach [2]. The system involves the solid velocity u, liquid velocity v, pressure p and partial stress h (the total stress is σ = −h − pI)


31

ρs ∇p − ∇ · h − ρl2 χ (u − v) ρ ρl iωρl v = − ∇p + ρl2 χ (u − v) ρ iω p = ρ a3 ∇ · u − ρ a4∇ · v

ρs ρl K − λ ∇ · u + K ∇ · v I − 2 με (u) iω h = ρ ρ

iωρs u = −

(2.4) (2.5) (2.6) (2.7)

where ε (u) = (∇u + ∇uT )/2 is the linear strain operator. The total density ρ is the sum of the partial densities of liquid and solid, ρl and ρs , respectively. The parameter controlling dissipation, η χ= , kρρl is given in terms of the viscosity η and permeability k. The Lamé parameters, λ and μ , and a third parameter α3 , are determined by the three wave speeds for the medium. Finally, K = λ + 2 μ /3 ρl a3 = K 2 − α3 ρρs ρ ρl a4 = K 2 + α3 ρρl ρ The bottom of the cylinder Γb is fixed so that u = 0, vn = 0 and the variational formulation is Ωp

−

ρl a4 (∇ · v)(∇ · δ v) − ω 2 ρl v · δ v dx

Ωp

ρl a3 (∇ · u)(∇ · δ v) dx + iω

Γi

ρl p (δ v · n) dS = 0 ρ

(2.8)

K 2με (u) : ε (δ u) + λ − ρs + a3 (∇ · u)(∇ · δ u) − ω 2 ρs (u · δ u) dx ρ Ωp

ρs h + pI δ u · n dS = 0 (2.9) − ρl a3 (∇ · v)(∇ · δ u) dx + iω ρ Ωp Γi for all test functions δ u = 0, δ vn = 0 on Γb . The two formulations are weakly coupled by introducing a surface porosity d and the interface conditions van = dvn + (1 − d)un

ρl p = d pa ρ ρs hn + pn = (1 − d)pan ρ

(2.10) (2.11) (2.12)

32


where we have introduced the superscript “a” to denote quantities from the acoustic subdomain. The material parameters were set to model the Berea sandstone experiments in [7].

2.2.4 Exact Sequence and the de Rham Diagram The energy spaces form the classical grad-curl-div exact sequence: ∇ ∇× ∇◦ H 1 → H(curl) → H(div) → L2 that can be reproduced on the discrete level using hp finite element spaces for elements of all shapes: tetrahedra, hexahedra, prisms and pyramids, including curvilinear elements. The continuous and discrete spaces are connected through special Projection-Based Interpolation (PB) operators that make the following diagram commute. ∇ ∇× ∇◦ H 1 → H(curl) → H(div) → L2 ⏐ ⏐ ⏐ ⏐ ⏐ grad ⏐ curl ⏐ div ⏐ Π Π Π P ∇ Whp →

Qhp

∇× ∇◦ → V hp → Yhp

In the two-dimensional case, the 3D sequence reduces to two 2D sequences: ∇ ∇× R I −→ H 1 −→ H(curl) −→ L2 −→ 0 ∇× ∇◦ R I −→ H 1 −→ H(div) −→ L2 −→ 0 The discussed code supports hybrid meshes based on Nédélec triangles of the second type and Nédélec quads of the first type, see [5] for detailed explanations.

2.3 hp Technology In this section we discuss shortly the main implementation changes required to update the hp technology presented in [3, 5] to coupled, multiphysics problems.


33

2.3.1 Handling Multiphysics The code supports an arbitrary number of H 1 -, H(curl)-, H(div)- and L2 -conforming components. User specifies involved fields along with the dicretization type and number of components. For instance, some characteristics of the dual-mixed formulation for elasticity with weakly imposed symmetry are gathered in Tab. 2.2. Table 2.2 Characteristics of the dual-mixed formulation #

name

discretization type

number of components

1

stress

H(div)

2

displacement

L2

2

rotation

L2

1

2 3

Recall that the main idea of the hp data structure presented in [3, 5] is based on the concept of a node object, an abstraction for element vertices, edges, faces (3D) and interiors. Mesh refinements involve breaking of the nodes and growing nodal trees, in contrast to h-adaptive methods that “grow” only trees for elements. The nodal trees provide a sufficient information for a number of efficient algorithms supporting looping through elements in the current mesh (natural order of elements), determination of element neighbors for edges and faces (3D) and, most importantly, element to nodes connectivities. The nodal connectivities include information on constrained (hanging) nodes. Only 1-irregular meshes are supported and the minimum rule is enforced through the whole adaptivity process. Operations on nodes are identical for all types of discretization. The difference between different discretizations has to be accounted for only at the level of nodal degrees-of-freedom (d.o.f.). This two-level logic is crucial for the simultaneous support of the exact sequences and reduction of complexity of the code. The definition of a node has been modified to include a partial list of variables to be supported and pointers to arrays storing the corresponding d.o.f. (dynamically allocated arrays). For instance, for the coupled acoustics/elasticity problem, a node in the interior of acoustics subdomain supports only the pressure, a node inside of the elastic domain supports only the elastic displacement, but a node on elastic/acoustic interface supports both variables. The dimension of the arrays depends upon the (polynomial) order of the node and the actual allocation takes place during the initial mesh generation or mesh modifications. A C preprocessing is used to produce a real-valued or complex-valued version of the code. Our Geometrical Modeling Package (GMP) provides the Mesh Based Geometry (MBG) description [3] without any changes. The logic of GMP blocks is adopted to specify on input different fields to be supported in each block. The initial mesh generator automatically identifies then interfaces and the fields to be supported at interface vertex and edge nodes.

34


Postprocessing: Expanded Mode In order to simplify mathematical and graphical postprocessing, all solution d.o.f. are output elementwise in the so-called expanded mode that includes all fields specified by the user. Thus, e.g., for an element in the acoustical subdomain, we return also the d.o.f for the elastic displacement. For an element in the interior of the acoustical domain, all returned elastic d.o.f. are simply zero. However, for an element adjacent to the acoustic/elastic interface, the returned d.o.f. will represent a FE lift of the values on the interface.

2.3.2 Constrained Approximation The constrained approximation package described in [3, 5] is a result of many years of improvements and we have been very hesitant to modify it. The package utilizes the element to nodes connectivity information (including constraints), and generates the corresponding generalized d.o.f. connectivity information enabling two tasks: • Assembly of modified element matrices • Computation of local element geometry and solution d.o.f. Two versions of those routines were available: one for the H 1 -, and another one for H(curl)-conforming discretization. As in 2D, the H(div)-conforming elements are obtained from H(curl)-conforming elements by a 90 degrees rotation, the corresponding constrained approximation coefficients are identical. Finally, the L2 conforming elements involve only interior d.o.f. and do not involve any constraints. This has led us to the idea of computing the multiphysics element stiffness matrices and load vectors in the block form. Thus, for example, solution of the elasticity problem with the dual-mixed formulation with weakly imposed symmetry, involves computation of nine different stiffness matrices involving H(div) and L2 shape functions. The original, single physics, constrained approximation arrays are then used to transform every block into the corresponding block for the modified element. For example, for the coupled acoustics/elasticity problem, for an element in acoustic domain but adjacent to acoustic/elastic interface, we generate all four multiphysics element matrices even though the two field interact through the boundary. In order to eliminate the zero entries and, at the same time, account for Dirichlet boundary conditions, we perform one more logical step on element matrices, changing the ordering of the modified element d.o.f. from the multiphysics ordering to the geometrical ordering: • interior node H 1 , H(curl),H(div) and L2 d.o.f.; • edge nodes H 1 ,H(curl) and H(div) d.o.f.; • vertex nodes H 1 d.o.f.


35

The process of reordering of d.o.f. is accompanied by elimination of “empty nodes” (e.g. elastic interior node for an element in acoustic domain or nodes whose d.o.f. are determined by Dirichlet boundary conditions.) With the use of frontal solvers, the new ordering is equivalent to the static condensation of interior d.o.f., a well known trick to improve the conditioning.

2.3.3 Automatic hp Adaptivity The 3D package supporting automatic hp-adaptivity described in [5] has been modified and expanded to adopt the multiphysics setting. The hp-algorithm is a black box based on a two-grid paradigm and it is problem independent. Given a coarse mesh, we perform a global hp-refinement (each element is broken into four elementsons, and the order is raised uniformly by one) and solve the problem on the fine mesh. The logical information on the coarse grid and the corresponding fine grid solution is then entered into the black box which returns the optimal hp-refinement. The execution of the optimal refinement is accompanied with a “closing operation” enforcing 1-irregularity and minimum order mesh regularity rules. The principle of the energy-based hp algorithm is as follows. Given the fine grid solution uh/2,p+1, corresponding to the current coarse grid hp, we determine the next coarse grid hpopt by solving the following maximization problem: uh/2,p+1 − Πhpuh/2,p+12 − uh/2,p+1 − Πhpopt uh/2,p+12 → max Nhpopt − Nhp Here Πhp , Πhpopt denote the Projection Based Interpolation (PBI) operators mentioned in the Section 2.2, corresponding to the coarse grid, and the new, optimal coarse grid to be determined, and Nhp , Nhpopt denote the total number of d.o.f for the two meshes. The principle is widely accepted - we maximize the rate of our investment. The actual algorithm follows the logic of PB Interpolation. Step 0: We subtract from the H 1 components of the fine grid solution, the coarse grid vertex interpolants. Similarly, we subtract from the H(curl) and H(div) components of the fine grid solution, the coarse grid edge averages of tangential or normal components, respectively. Step 1: We determine optimal refinements for coarse grid element edges. This involves making three decisions on: • how to refine an edge, h or p ? • which edges to refine ? • and, in the case of an h-refined edge, what order for edge sons should be used, i.e. how many d.o.f. to invest ?

36


The only difference between the single- and multi-physics versions is that we have to interpolate now multiple fields, with a number and type of those fields varying from edge to edge. We conclude the edge optimization step by subtracting from the fine grid solution the corresponding edge interpolants corresponding to the optimal edge refinements. Step 2: We determine optimal refinements for the coarse grid element interiors. This involves now not only the choice between h and p-refinements but also a choice between different h-refinements (anisotropic and isotropic refinements). The established already optimal refinements for edges set the stage for the second step, and significantly limit the number of cases to be considered. For instance, if an edge of an element is to be h-refined, this eliminates the possibility of a p-refinement for the element, see [5] for details. In conclusion, we have been able to keep the logic of our hp algorithm practically unchanged.

2.4 Numerical Examples In this section, we present a few representative numerical results for sample coupled multiphysics problems discussed in Section 2.2.

2.4.1 Mixed-Dual Elasticity The hp discretization of this problem extends the theory developed by Arnold et al. [1] for uniform p, to the case of variable order of approximation. A stability analysis presented in [6] extends the results from [1] to the variable p but, at present, a convergence analysis for p- and hp-methods is not available. This does not stop us from numerical experimentation. The classical Airy stress function approach combined with separation of variables has been used to construct an analytical solution with finite energy and strongest singularity for an infinite L-shape sector of a plane (see [6]) for details). The solution is then used as a manufactured solution on the standard (bounded) L-shape domain with the corresponding traction boundary conditions on truncating boundary driving the problem. Although manufactured, the solution displays the same singular behavior at the origin as actual elasticity boundary-value problems set up in the same domain. Fig. 2.3 presents an optimal coarse hp mesh corresponding to relative energy error below 0.6 percent. Fig. 2.5 presents the corresponding convergence history for the hp algorithm and, for illustration, Fig. 2.4 displays the distribution of σ11 component of the stress tensor on the optimal mesh. Notice the use of algebraic scale on the convergence plot with which the straight line corresponds to exponential convergence.


31 22

33

42

34

90

44 45

97

100

99

106 109

37

108

55

117

23

56

20

153 146 156155162 165 164 173 118 209 211 218 220 212 221 229 147202 174 144 265 267 274 276 258 268 277 285 203 230 200 228 172 259 266 275 286 256 257 283 284 292 294 295 301 238 182 304 239 293 302 303 183 248 237 246 247 192 127 181 190 191 136

91 88

54 116 126 66 67 125

134

135 78 65 76

y z x

77

Fig. 2.3 Mixed-dual elasticity: L-shape domain problem. Optimal hp mesh corresponding to 0.6 percent error.

31 22

33

42

34

90

44 45

97

100

99

106 109

108

55

117

23

56

20 91 88

153 146 156155162 165 164 173 118 209 211 218 220 212 221 229 147202 174 144 265 267 274 276 258 268 277 285 203 230 200 228 172 259 266 275 286 256 257 283 284 292 294 295 301 238 182 304 239 293 302 303 183 248 237 246 247 192 127 181 190 191 136

54 116 126 66 67 125

134

135 78 65 76

y z x

77

Fig. 2.4 Mixed-dual elasticity: L-shape domain problem. Distribution of stress σ11 on the optimal mesh.

15.51

error SCALES: nrdof^0.33, log(error)

8.88 5.09

adaptive hp

2.91 1.67 0.96 0.55 0.31 0.18 0.10 0.06 243

594

1184

2075

3332

5018

7197

9933

nrdof 13292

Fig. 2.5 Mixed-dual elasticity: L-shape domain problem. Convergence history.

38


2.4.2 The Streamer Problem Fig. 2.6 presents an optimal hp coarse mesh corresponding to (estimated) relative H 1 error of 3.6 percent. Each junction of three or more different materials results in a singularity. Additional singularities occurred at reentrant corners at elastic/acoustic interface. The use of PML results in development of strong boundary layers whose resolution with hp-adaptivity is also critical for an high accuracy solution. Fig. 2.7 presents the pressure in both the structure and water in terms of decibels.

Fig. 2.6 Streamer problem: Optimal hp mesh corresponding to 3.6 percent error.

Fig. 2.7 Streamer problem: Pressure distribution on the optimal mesh. Range: -20 - 133.6 dB.


39

The next two figures illustrate the need for resolving the singularities for problems with large material contrasts. Fig. 2.8 presents the pressure distribution along a horizontal section through the soft filler, just next to the stiff strength element (rope). One should emphasize that, in order to avoid locking, we have started with horizontal order p = 5 and vertical order q = 2. In other words, singularities but not locking presents the main difficulty with getting an accurate solution. Fig. 2.9 presenting the same distribution but on the optimal mesh, is dramatically different, although the small oscillations indicate that the accuracy is still insufficient (the problem was solved on a laptop with 1GB memory only...).

Fig. 2.8 Streamer problem: Pressure profile for the initial mesh.

Fig. 2.9 Streamer problem: Pressure profile for the optimal mesh.

40


2.4.3 Modeling of Sonic Tools The following figures present illustrative examples for the Berea’s experiment. The history of hp-refinements is shown in Fig. 2.10. Final hp fine grid with the corresponding distribution of acoustic pressure is shown in Fig. 2.11. Note that only the acoustic part of the domain is meaningful (recall the discussion on the expanded mode). Figures 2.12 and 2.13 present the corresponding distribution of elastic displacements and fluid velocity (in pores) in the poroelastic region. Clearly, the entire action happens along the interface.

2

Percent relative error in energy norm

10

1

10

0

10

6162

7269

7775 8098 8226 8295 8700 91449420

10139

10843 11440 11439 1147111911

12715

Number of dof in algebraic scale N1/3

Fig. 2.10 Simulating Berea’s experiment: Sequence of hp coarse grid errors for the sandstone cylinder.

The results display a clear exponential convergence for which again, at present, we do not have any supporting stability analysis.


41

Fig. 2.11 Simulating Berea’s experiment: Final hp fine grid and acoustic pressure p for the sandstone cylinder.

Fig. 2.12 Simulating Berea’s experiment: r and z components of solid velocity u for the sandstone cylinder.

2.5 Conclusions The paper reports on our current effort to extend the technology of hp-adaptive finite elements to a class of coupled, multiphysics problems. Three representative examples presented in the paper reflect well the complexity and technical problems encountered in the course of the project. In order to account for the multiphysics and coupled problems setting, we had to develop a new code including a new data structure, initial mesh generator, constrained approximation, mesh modification package, postprocessing etc. However, we have been able to preserve the overall logical structure

42


Fig. 2.13 Simulating Berea’s experiment: r and z components of fluid velocity v for the sandstone cylinder (rescaled to obtain a nice picture: observed max |vr | ≈ 0.065, max |vz | ≈ 0.0074).

of or hp implementations and, through compromises discussed in the text, recycle the most difficult part of the constrained approximation package. The hp-algorithm based on the two-grid paradigm and the family of Projection-Based Interpolation operators seems to extend well to the class of discussed problems. Presented numerical examples document exponential convergence and illustrate the necessity of using adaptive methods for problems with high material contrasts. A detailed manual for our new 2D code is under development [4] and we hope to report on new numerical and theoretical results in forthcoming papers.

References [1] Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Mathematics of Computation 76, 1699–1723 (2007) [2] Blokhin, A., Dorovsky, V.: Mathematical Modelling in the Theory of Multivelocity Continuum. Nova Science Publishers, Inc., Bombay (1995) [3] Demkowicz, L.: Computing with hp Finite Elements. I. One- and Two-Dimensional Elliptic and Maxwell Problems. Chapman & Hall/CRC Press, Taylor and Francis (2006) [4] Demkowicz, L., Kurtz, J.: An hp framework for coupled multiphysics problems. Technical report, ICES (2009) (in preparation) [5] Demkowicz, L., Kurtz, J., Pardo, D., Paszyński, M., Rachowicz, W., Zdunek, A.: Computing with hp Finite Elements. II. Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications, October 2007. Chapman & Hall/CRC (2007) [6] Qiu, W., Demkowicz, L.: Mixed h p-finite element method for linear elasticity with weakly imposed symmetry. Comput. Methods Appl. Mech. Engrg (2009); in print, see also ICES Report 2008/30 [7] Winkler, K.W., Liu, H.L., Johnson, D.L.: Permeability and borehole Stoneley waves: Comparison between experiment and theory. Geophysics 54(1), 66–75 (1989)

Chapter 3

Nonconvex Inequality Models for Contact Problems of Nonsmooth Mechanics Stanislaw Migórski and Anna Ochal

Abstract. This review paper deals with selected nonsmooth and nonconvex inequality problems for dynamic frictional contact between a viscoelastic body and a foundation. The process is modeled by general nonmonotone possibly multivalued multidimensional Clarke subdifferential contact boundary conditions. The problems of frictional contact with both short and long memory, thermoviscoelastic frictional contact, bilateral frictional contact and bilateral contact for piezoelectric materials with adhesion are considered. The formulations and results on existence, and uniqueness of solutions are presented.

3.1 Introduction In this paper we present an up-to-date review of a part of our current results on dynamic contact problems described by nonmonotone subdifferential boundary conditions. The related results on static and quasistatic contact problems with and without additional effects are today quite well covered by still growing literature [13, 39, 40, 41], however, their nonconvex counterparts as well as dynamic models need further investigation. There is a fundamental need for existence theory of contact problems. Even when the constitutive laws are linear, the problems are nonlinear because of contact conditions. The aim of our study is to contribute towards this goal and present in a unified way, results on analysis that deal with models for contact processes involving multivalued contact laws. It is clear that such contact problems do not have classical solutions. We mention that the existence of a solution to the dynamic contact problem between an elastic body and a rigid obstacle, described by a unilateral contact Stanislaw Migórski · Anna Ochal Jagiellonian University, Institute of Computer Science, Faculty of Mathematics and Computer Science, ul. Stanisława Łojasiewicza 6, 30-348 Krakow, Poland e-mail: {migorski,ochal}@ii.uj.edu.pl


44

S. Migórski and A. Ochal

condition of the Signorini type, is still an open problem. Today the literature on various mathematical and numerical aspects of dynamic contact with or without friction is growing rapidly. We restrict ourselves to the description of viscoelastic contact problems with the Kelvin-Voigt constitutive law (Section 3.2), frictional contact with both short and long memory (Section 3.3), thermoviscoelastic frictional contact (Section 3.4), bilateral frictional contact (Section 3.5) and bilateral contact for viscoelastic piezoelectric materials with adhesion (Section 3.6). The frictional contact problem in Section 3.2 is presented with more details and for other models we give their classical and weak formulations and refer to the references for a precise statement of existence and uniqueness results and further explanations. We treat the problems within the infinitesimal strain theory. For all of these problems with nonconvex superpotentials, effective numerical methods are needed and reliable general three-dimensional codes should be delivered. Such studies allow for better understanding and more accurate, and realistic prediction of the long time behavior of the considered system. We present nonsmooth and nonconvex inequality problems for dynamic frictional contact between a viscoelastic body and a foundation. We establish the existence of weak solutions to different models by employing a unified approach based on evolution hemivariational inequalities. The latter can, in turn, be formulated as the Cauchy problem for the following evolution inclusion of second order t

u (t) + A(t, u(t)) + B(t, u(t)) +

C(t − s) u(s) ds + F(t, u(t), u (t)) f (t).

0

Here A, B and C correspond to viscosity, elasticity and relaxation operators in the mechanical problems and the multifunction F contains different subdifferential boundary contact conditions. The notion of hemivariational inequality was introduced and studied by P.D. Panagiotopoulos ([37, 38]) in the early 1980s as variational formulation for certain classes of mechanical problems with the nonconvex, nondifferentiable and nonsmooth energy functionals. Such inequality results from the d’Alembert principle for a dynamic mechanical system. During years a hemivariational inequality proved to be an effective tool to give positive answers to unsolved or partially unsolved problems, cf. [38, 35]. In this respect our approach can be considered as a continuation of works of Panagiotopoulos. In this paper the main interest lies in general nonmonotone and multivalued Clarke subdifferential boundary conditions. More precisely, it is supposed that on the boundary of the body under consideration, the subdifferential relations hold, for instance, between the normal component of the velocity and the normal component of the stress, between the tangential components of these quantities and between temperature and the heat flux vector. These subdifferential boundary conditions are the natural generalizations of the normal damped response condition, the accociated friction law and the well known Fourier law of heat conduction, respectively. They allow to incorporate in our models several types of boundary conditions considered earlier e.g. in [13, 38, 39, 40, 41].

3 Nonconvex Models in Contact Mechanics

45

3.2 Physical Setting of Dynamic Viscoelastic Problem In this section we present a dynamic viscoelastic frictional contact problem and describe a method which can be used to obtain results on its solvability and unique solvability. This method uses various concepts of nonlinear analysis including evolution inclusions and hemivariational inequalities. The main novelty of the approach arises in the fact that we deal with a large class of multivalued contact and friction boundary conditions for nonconvex superpotentials. In fact, it is the nonconvexity which makes the problem more difficult than those already studied in the literature. After the description of the mechanical problem, we will show that it leads to a hemivariational inequality for the displacement field. We begin with the description of the mechanical problem. In its reference configuration, a viscoelastic body occupies a subset Ω in R d , d = 2, 3 in applications. We suppose that Ω has a Lipschitz boundary Γ , thus the unit outward normal vector ν exists a.e. on Γ and the boundary is divided into three mutually disjoint parts ΓD , ΓN and ΓC such that m(ΓD ) > 0. The body is acted upon by volume forces and surface tractions and, as a result, its state is evolving. We are interested in dynamic evolution process of the mechanical state of the body on the time interval [0, T ] with T > 0. We assume that the body is clamped on ΓD , so the displacement field vanishes there. Volume forces of density f1 act in Ω and surface tractions of density f2 are applied on ΓN . The body may come in contact with an obstacle over the potential contact surface ΓC . We denote by Sd the linear space of second order symmetric tensors on Rd and we use the notation Q = Ω × (0, T ). For simplicity we skip the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ . Then, the frictional contact problem under consideration can be stated as follows: find the displacement field u : Q → Rd and the stress tensor σ : Q → Sd such that u (t) − div σ (t) = f1 (t) in Q

(3.1)

σ (t) = A (t, ε (u (t))) + B ε (u(t)) in Q

(3.2)

u(t) = 0 on ΓD × (0, T )

(3.3)

σ (t)ν = f2 (t) on ΓN × (0, T )

(3.4)

−σν (t) ∈ ∂ jν (t, uν (t)),

(3.5)

−στ (t) ∈ ∂ jτ (t, uτ (t)) on ΓC × (0, T )

u(0) = u0 , u (0) = v0 in Ω .

(3.6)

Equation (3.1) is the equation of motion and div denotes the divergence operator for tensor valued functions. Relation (3.2) represents the Kelvin–Voigt viscoelastic constitutive law, where A is a nonlinear operator describing the purely viscous properties of the material and B is the linear elasticity operators. Note that the operator A may depend explicitly on the time variable and this is the case when the viscosity properties of the material depend on the temperature field. Conditions (3.3) and

46


(3.4) are the displacement and the traction boundary conditions, respectively. Conditions (3.5) represent the frictional contact condition in which jν and jτ are given contact and frictional potentials and the subscripts ν and τ for σ and u indicate normal and tangential components of tensors and vectors. The symbol ∂ j denotes the Clarke subdifferential of j with respect to the last variable. Note also that the explicit dependence of the functions jν and jτ with respect to the time variable allows to model situations when the frictional contact conditions depend on the temperature, which plays the role of a parameter, and which evolution in time is prescribed. Finally, conditions (3.6) represent the initial conditions where u0 and v0 denote the initial displacement and the initial velocity, respectively. We recall two notions introduced by Clarke (cf. [3, 9]) which are the basic tools in our studies. Let ϕ : X → R be a locally Lipschitz function defined on a Banach space X . Then the generalized directional derivative of ϕ at x ∈ X in the direction v ∈ X, denoted by ϕ 0 (x; v), is defined by

ϕ 0 (x; v) = lim sup

y→x, λ ↓0

ϕ (y + λ v) − ϕ (y) λ

and the generalized gradient of ϕ at x, denoted by ∂ ϕ (x), is a subset of a dual space X ∗ given by ∂ ϕ (x) = ζ ∈ X ∗ | ϕ 0 (x; v) ≥ ζ , v X ∗ ×X for all v ∈ X . A locally Lipschitz function ϕ is called regular (in the sense of Clarke) at x ∈ X if for all v ∈ X the one-sided directional derivative ϕ (x; v) exists and satisfies ϕ 0 (x; v) = ϕ (x; v) for all v ∈ X. Weak formulation of the problem. Throughout the paper, indices i, j, k, l, etc. run from 1 to d. In order to provide the weak form of the problem under consideration, we consider the following spaces H = L2 (Ω ; R d ),

H = L2 (Ω ; Sd )

and let V be the closed subspace of H 1 (Ω ; Rd ) given by V = v ∈ H 1 (Ω ; Rd ) | v = 0 on ΓD . The deformation and the divergence operators are given by 1 ε (u) = {εi j (u)}, εi j (u) = (∂ j ui + ∂i u j ), div σ = {∂ j σi j }. 2 We denote by V ∗ the dual space of V and by ·, · the duality pairing of V and V ∗ . Let V = L2 (0, T ;V ) and W = {w ∈ V | w ∈ V ∗ }, where the time derivative is understood in the sense of vector valued distributions. We obtain W ⊂ V ⊂ L2 (0, T ; H) ⊂ V ∗ , where all the embeddings are continuous (cf. [10]).


47

Moreover, let Z = H 1/2 (Ω ; Rd ) and let γ denote the norm of the trace operator γ : Z → L2 (Γ ; Rd ) which has adjoint γ ∗ : L2 (Γ ; R d ) → Z ∗ . In the study of problem (3.1)–(3.6) we need the following assumptions on the data. H(A ) :

A : Q × Sd → Sd is such that

(i) A (·, ·, ε ) is measurable on Q, for all ε ∈ Sd ; (ii) A (x,t, ·) is continuous on Sd for a.e. (x,t) ∈ Q; (iii) A (x,t, ε ) ≤ a(x,t) + c1 ε for all ε ∈ Sd , a.e. (x,t) ∈ Q with a ∈ L2 (Q), a ≥ 0 and c1 > 0; (iv) (A (x,t, ε1 ) − A (x,t, ε2 )) : (ε1 − ε2 ) ≥ m1 ε1 − ε2 2 for all ε1 , ε2 ∈ Sd , a.e. (x,t) ∈ Q with m1 > 0; (v) A (x,t, ε ) : ε ≥ c2 ε 2 for all ε ∈ Sd , a.e. (x,t) ∈ Q with c2 > 0. H(B) :

B : Ω × Sd → Sd is such that B(x, ε ) = b(x)ε and

(i) b(x) = {bi jkl (x)} with bi jkl = b jikl = blki j ∈ L∞ (Ω ); (ii) bi jkl (x)εi j εkl ≥ 0 for all ε = {εi j } ∈ Sd , a.e. x ∈ Ω . H( jν ) :

jν : ΓC × (0, T ) × R → R satisfies

(i) jν (·, ·, r) is measurable for all r ∈ R and jν (·, ·, 0) ∈ L1 (ΓC × (0, T )); (ii) jν (x,t, ·) is locally Lipschitz for a.e. (x,t) ∈ ΓC × (0, T ); (iii) |∂ jν (x,t, r)| ≤ cν (1 + |r|) for a.e. (x,t) ∈ ΓC × (0, T ), all r ∈ R with cν > 0; (iv) jν0 (x,t, r; −r) ≤ dν (1 + |r|) for all r ∈ R, a.e. (x,t) ∈ ΓC × (0, T ) with dν ≥ 0; (v) (η1 − η2 )(r1 − r2 ) ≥ −mν |r1 − r2 |2 for all ηi ∈ ∂ jν (x,t, ri ), ri ∈ R, i = 1, 2, a.e. (x,t) ∈ ΓC × (0, T ) with mν ≥ 0. H( jτ ) : jτ : ΓC × (0, T ) × R d → R satisfies (i) jτ (·, ·, ξ ) is measurable for all ξ ∈ R d and jτ (·, ·, 0) ∈ L1 (ΓC × (0, T )); (ii) jτ (x,t, ·) is locally Lipschitz for a.e. (x,t) ∈ ΓC × (0, T ); (iii) ∂ jτ (x,t, ξ ) ≤ cτ (1 + ξ ) a.e. (x,t) ∈ ΓC × (0, T ), all ξ ∈ Rd with cτ > 0; (iv) jτ0 (x,t, ξ ; −ξ ) ≤ dτ (1 + ξ ) a.e. (x,t) ∈ ΓC × (0, T ), all ξ ∈ Rd with dτ ≥ 0; (v) (η1 − η2 ) · (ξ1 − ξ2 ) ≥ −mτ ξ1 − ξ2 2 a.e. (x,t) ∈ ΓC × (0, T ), all ηi ∈ ∂ jτ (x,t, ξi ), ξi ∈ Rd , i = 1, 2 with mτ ≥ 0. H(0) :

f1 ∈ L2 (0, T ; H), f2 ∈ L2 (0, T ; L2 (ΓN ; R d )), u0 ∈ V , v0 ∈ H.

Now, we introduce the operators A : (0, T ) ×V → V ∗ and B : V → V ∗ defined by A(t, u), v = A (t, ε (u)), ε (v) H Bu, v = B ε (u), ε (v) H

for u, v ∈ V and t ∈ (0, T ),

for u, v ∈ V.

We also consider the function f : (0, T ) → V ∗ given by

(3.7) (3.8)

48


f (t), v = f1 (t), v H + ( f2 (t), v)L2 (ΓN ;Rd ) for v ∈ V, a.e. t ∈ (0, T ). The weak formulation of the contact problem (3.1)–(3.6) is obtain by the standard method. Assume that (u, σ ) is a couple of regular functions which solve (3.1)–(3.6), v ∈ V and t ∈ (0, T ). Using the equation of motion (3.1) and the Green formula, one finds u (t), v + σ (t), ε (v) H = f1 (t), v H + σ (t) ν · v dΓ . Γ

We take into account the boundary conditions (3.3) and (3.4) to get

Γ

σ (t) ν · v dΓ =

ΓN

f2 (t) · v d Γ +

ΓC

(σν (t)vν + στ (t) · vτ ) d Γ

and from the definition of the Clarke subdifferential and (3.5), we have −σν (t)vν ≤ jν0 (x,t, uν (x,t); vν ), −στ (t) · vτ ≤ jτ0 (x,t, uτ (x,t); vτ ) on ΓC × (0, T ), which imply that − ≤

ΓC

ΓC

(σν (t)vν + στ (t) · vτ ) d Γ ≤

jν0 (x,t, uν (x,t); vν (x)) + jτ0 (x,t, uτ (x,t); vτ (x)) dΓ .

Combining the above, we obtain u (t), v + σ (t), ε (v) H + 0 jν (x,t, uν (x,t); vν (x)) + jτ0 (x,t, uτ (x,t); vτ (x)) dΓ ≥ + ΓC

≥ f (t), v for all v ∈ V and a.e. t ∈ (0, T ). (3.9) We use (3.9), the constitutive law (3.2), (3.7)–(3.8) and the initial conditions (3.6) to obtain the following weak formulation of the mechanical problem (3.1)–(3.6): ⎧ ⎪ find u : (0, T ) → V such that u ∈ V , u ∈ W and ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u (t) + A(t, u(t)) + Bu(t), v + ⎪ ⎪ ⎨ 0 (3.10) jν (x,t, uν (x,t); vν (x)) + jτ0 (x,t, uτ (x,t); vτ (x)) dΓ ≥ + ⎪ ⎪ Γ ⎪ C ⎪ ⎪ ⎪ ≥ f (t), v for all v ∈ V and a.e. t ∈ (0, T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩u(0) = u , u (0) = v . 0 0 Unique weak solvability of the problem. The following result concerns the existence and uniqueness of solutions to problem (3.10). Theorem 1. Assume that H(A ), H(B), H( jν ), H( jτ ), H(0) hold and


49

m1 > (mν + mτ )γ 2 .

(3.11)

Then problem (3.10) admits at least one solution. If, in addition, ⎧ ⎨either jν (x,t, ·) or − jν (x,t, ·) is regular and ⎩either j (x,t, ·) or − j (x,t, ·) τ τ

is regular,

then problem (3.10) has a unique solution. We comment on the proof of Theorem 1. The method of the proof is based on the existence and uniqueness results obtained recently in [20, 26] for evolution inclusions of the form ⎧ ⎪ find u ∈ V with u ∈ W such that ⎪ ⎪ ⎨ (3.12) u (t) + A(t, u(t)) + Bu(t) + γ ∗ ∂ J(t, γ u (t)) f (t) a.e. t ∈ (0, T ), ⎪ ⎪ ⎪ ⎩u(0) = u , u (0) = v . 0 0 Theorem 10 of [20] provides hypotheses on the data under which the inclusion (3.12) is solvable while Proposition 15 ibid. gives conditions which guarantee the uniqueness of solution. Let us mention that the application of these results to the hemivariational inequality (3.10) requires the introduction of the integral functional J : (0, T ) × L2 (ΓC ; R d ) → R defined by J(t, v) =

ΓC

j(x,t, v(x)) d Γ a.e. t ∈ (0, T ) and v ∈ L2 (ΓC ; R d ),

where j : ΓC × (0, T ) × Rd → R is given by j(x,t, ξ ) = jν (x,t, ξν ) + jτ (x,t, ξτ )

a.e. (x,t) ∈ ΓC × (0, T ), all ξ ∈ Rd .

Furthermore, we can associate with the hemivariational inequality (3.10) the abstract inequality of the form ⎧ ⎪ find u ∈ V with u ∈ W such that ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ j0 (x,t, u (t); v) d Γ ≥ f (t), v

u (t) + A(t, u(t)) + Bu(t), v + Γ C ⎪ ⎪ ⎪ for all v ∈ V and a.e. t ∈ (0, T ), ⎪ ⎪ ⎪ ⎪ ⎩ u(0) = u0 , u (0) = v0 . We refer to [20, 26] to see that existence of solutions to (3.12) implies the existence of solution to the hemivariational inequality (3.10) and that both these problems are equivalent under the condition that either J(t, ·) or −J(t, ·) is regular. On the other hand, the solvability of the inclusion (3.12) is based on the known surjectivity result

50


for a class of pseudomonotone operators in a reflexive Banach space (cf. Theorem 6.3.73 of [10]). The existence result of Theorem 1 solves an open problem formulated by Panagiotopoulos in Chapter 7.2 of [38]. We also mention that the strong monotonicity of the operator A(t, ·), cf. H(A )(iv), in the existence part of Theorem 1 can be relaxed, cf. [20]. Finally, we underline that if in addition jν (x,t, ·) and jτ (x,t, ·) are convex potentials, then the hemivariational inequality (3.10) reduces to hyperbolic variational inequality and the conditions H( jν )(v) and H( jτ )(v) hold with mν = mτ = 0. This is due to a fact that the subgradient of a convex function is a maximal monotone operator (cf. Theorem 1.3.19 of [10]). In this case the condition (3.11) is trivially satisfied.

3.3 Integrodifferential Hemivariational Inequalities and Viscoelastic Frictional Contact The physical setting of the problem considered in this section is analogous to the one of (3.1)–(3.6) with one exception, namely the constitutive relation (3.2) is now replaced by the following law t

σ (t) = A (t, ε (u (t))) + B ε (u(t)) +

C (t − s) ε (u(s)) ds in Q.

0

The assumptions also remain the same as in Section 3.2 and, in addition, we assume that the relaxation operator C is linear and continuous (cf. [32] for details). One of the features of this viscoelastic model consists in gathering short and long memory effects in the same constitutive law. This leads to a new class of hemivariational inequalities which involve a Volterra integral term t

u (t) + A(t, u(t)) + Bu(t) +

C(t − s) u(s) ds + γ ∗ ∂ J(t, γ u (t)) f (t).

0

For this kind of hemivariational inequalities, the existence and uniqueness results have been delivered in ([32]), while the dependence of the solution on the memory term and a convergence result can be found in [15]. It can be shown that a sequence of solutions corresponding to a long memory material converges to a solution of the problem with short memory as the relaxation coefficient tends to zero.

3.4 Thermoviscoelastic Frictional Contact Problem In this section we give the mathematical formulation of a frictional contact problem of linear thermoviscoelasticity. The physical setting and notation remain the same as in Section 3.2. The volume Ω is occupied by a viscoelastic body and we assume


51

that the temperature changes accompanying the deformations are small and they do not produce any changes in the material parameters which are regarded temperature independent. We assume with no loss of generality that the material density and the specific heat at constant deformation are constants, both set equal to one. The system of equations of motion assuming small displacements and the law of conservation of energy takes the form ui − ∂ j σi j = f1i

in Q,

θ + ∂i qi = −ci j ∂ j ui + g in Q. Further the heat flux vector q ∈ Rd satisfies the Fourier law of heat conduction qi = −ki j ∂ j θ in Q. The behavior of the body is governed by the thermoviscoelastic constitutive equation of the Kelvin-Voigt type

σi j = ai jkl ∂l uk + bi jkl ∂l uk − ci j θ in Q, where {ai jkl } and {bi jkl } are the viscosity and elasticity tensors, respectively, and {ci j } are the coefficients of thermal expansion. We study the evolution of the system state in a time interval [0, T ]: find a displacement field u : Q → R d and a temperature θ : Q → R such that ui − ∂ j ai jkl ∂l uk − ∂ j bi jkl ∂l uk + ∂ j (ci j θ ) = f1i in Q

θ − ∂i (ki j ∂ j θ ) + ci j ∂ j ui = g in Q u(t) = 0 on ΓD × (0, T )

σ (t)ν = f2 (t) on ΓN × (0, T ) −σν (t) ∈ ∂ jν (t, uν (t)),

−στ (t) ∈ ∂ jτ (t, uτ (t)) on ΓC × (0, T )

−ki j ∂ j θ νi ∈ ∂ j(t, θ (t)) on ΓC × (0, T )

θ (t) = 0 on (ΓD ∪ ΓN ) × (0, T ) u(0) = u0 , u (0) = v0 , θ (0) = θ0 in Ω . The weak form of the problem consists of a system of coupled evolution inclusions: ⎧ ∗ ⎪ ⎪u (t) + A u (t) + B u(t) + C1 θ (t) + γ ∂ J1 (t, γ u (t)) f (t) a.e. t ⎪ ⎨ (3.13) θ (t) + C2 θ (t) + C3 u (t) + γs∗ ∂ J2 (t, γs θ (t)) g(t) a.e. t ⎪ ⎪ ⎪ ⎩u(0) = u , u (0) = v , θ (0) = θ . 0 0 0 We refer to [8] for a detail explanation of the notation used for this system. Similarly to our description in Section 3.2, the system of inclusions (3.13) is related to the system of the hemivariational inequality of hyperbolic type for the displacement

52


and the parabolic hemivariational inequality for the temperature. The existence and uniqueness of solutions to (3.13) is proved in [5].

3.5 Bilateral Frictional Contact Problem in Viscoelasticity In the following, under the notation of Section 3.2, we consider a dynamic viscoelastic model with the equation of motion (3.1), the Kelvin-Voigt constitutive law (3.2), the displacement and traction boundary conditions (3.3) and (3.4), the initial conditions (3.6) and the following boundary conditions on the contact surface: ⎧ ⎨uν (t) = 0 (3.14) ⎩−σ (t) ∈ μ (t) p(|Rσ (t)|) ∂ j(t, u (t)) on Γ × (0, T ). τ ν C τ Here μ ∈ L∞ (ΓC × (0, T )), μ ≥ 0 a.e. on ΓC × (0, T ) is the coefficient of friction and R ∈ L (H −1/2 (Γ ); L2 (Γ )) is a regularization operator. The symbol ∂ j stands for the subdifferential of the contact superpotential j(x,t, ·) which is now assumed to be convex. We do not state here various possible hypotheses on the friction function p (we refer to [7]). Instead we comment on examples. In (3.14) the potential j : ΓC × (0, T ) × R d → R can be given by j(x,t, ξ ) = ξ for a.e. (x,t) ∈ ΓC × (0, T ) and all ξ ∈ R d . Since the subdifferential of the function ϕ (ξ ) = ξ is the unit vector in the direction of ξ when ξ = 0, and is the whole unit ball when ξ = 0, in this case we obtain the Coulomb law of friction, and the contact boundary condition −στ (x,t) ∈ μ (x,t) p(x, |Rσν (x,t)|) ∂ uτ (x,t) on ΓC × (0, T ) is equivalent to ⎧ ⎪ στ ≤ μ p(|Rσν |) with ⎪ ⎪ ⎨ στ < μ p(|Rσν |) =⇒ uτ = 0, ⎪ ⎪ ⎪ ⎩ σ = μ p(|Rσ |) =⇒ ∃ λ ≥ 0 : σ = −λ u on Γ × (0, T ). τ ν τ C τ

(3.15)

If p is a known function which is independent of |Rσν |, i.e. p(x, r) = h(x) with h ∈ L∞ (ΓC ), h ≥ 0, then the conditions (3.15) become the Tresca friction law (cf. Section 2.6 of [39] for a detailed discussion). If p(x, r) = |r|, then (3.15) reduces to the usual regularized Coulomb friction boundary condition which was extensively used in the literature (cf. e.g. [11, 13, 14, 22, 26, 37, 39, 40, 41, 43]). If p(x, r) = |r|(1 − δ |r|)+ with (·)+ = max{·, 0}, where δ is a small positive coefficient related to the wear and hardness of the surface, then we obtain a modification of the Coulomb law of friction. Such a modification, called the SJK model, consists of the factor (1 − δ | · |)+ and was derived in [42] from the thermodynamical considerations. It leads to the condition


53

⎧ ⎪ ⎪στ ≤ μ |Rσν |(1 − δ |Rσν |)+ with ⎪ ⎨ στ < μ |Rσν |(1 − δ |Rσν |)+ =⇒ uτ = 0, ⎪ ⎪ ⎪ ⎩ σ = μ |Rσ |(1 − δ |Rσ |) =⇒ ∃ λ ≥ 0 : σ = −λ u on Γ × (0, T ). τ ν ν + τ C τ For the discussion of the SJK generalization of the Coulomb law, we refer to [42, 39, 40]. Finally, if the potential is given by j(x,t, ξ ) = α1 |ξ1 | + . . . + αd |ξd |, where αi ≥ 0, we are lead to the orthotropic friction. In this case the relations −στi ∈ μ p(|Rσν |) αi ∂ |uτi | can be interpreted as follows ⎧ ⎪ |στi | ≤ μi p(|Rσν |) with ⎪ ⎪ ⎨ στi < μi p(|Rσν |) =⇒ uτi = 0, ⎪ ⎪ ⎪ ⎩ στi = μi p(|Rσν |) =⇒ στi = −λi uτi with λi ≥ 0 on ΓC × (0, T ) for μi = μ αi . Since j(x,t, ·) is convex, the viscoelastic contact model (3.1), (3.2), (3.3), (3.4), (3.6) and (3.14) is described by a hyperbolic variational inequality. The results on the existence and uniqueness of weak solutions to this problem when the friction coefficient is sufficiently small can be found in Theorem 12 of [7].

3.6 Bilateral Contact Problem for Viscoelastic Piezoelectric Materials with Adhesion In this section we describe the viscoelastic problem of piezoelectricity with adhesion. We present its classical and variational formulations. We complete the notation of Section 3.2. We assume that the set Ω is occupied by a viscoelastic piezoelectric body that remains in adhesive contact with an insulator obstacle, the so-called foundation. The contact is supposed to be bilateral, so during the process there is no gap between the body and the foundation. We consider now two partitions of Γ = ∂ Ω . A first partition is given (as in Section 3.2) by three disjoint measurable parts ΓD , ΓN and ΓC such that m(ΓD ) > 0, and a second one consists of two disjoint measurable parts Γ1 and Γ2 such that m(Γ1 ) > 0 and ΓC ⊂ Γ2 . In addition to the notation of Section 3.2, for simplicity, we assume free electric charges. Due to the adhesive contact, we suppose a nonmonotone possibly multivalued law between the shear and the tangential displacement. This law depends also on a bonding field, denoted by β , which describes the pointwise fractional density of active bonds on ΓC . Following [12], the evolution of the bonding field is governed by an ordinary differential equation depending on the displacement and considered on contact surface. When β = 1 at a point of the contact part, the adhesion is complete and all

54


the bonds are active, when β = 0 all bonds are inactive and there is no adhesion, when 0 < β < 1 the adhesion is partial and a fracture β of the bonds is active. The dynamical model for the process under consideration is as follows: find a displacement field u : Q → Rd , an electric potential ϕ : Q → R and a bonding field β : ΓC × (0, T ) → [0, 1] such that u (t) − divσ (t) = f1 (t) in Q

(3.16)

divD(t) = 0 in Q σ (t) = A ε (u (t)) + B ε (u(t)) − P E(ϕ (t)) in Q

(3.17) (3.18)

D(t) = DE(ϕ (t)) + P ε (u(t)) in Q

(3.19)

u(t) = 0 on ΓD × (0, T ) σ (t)ν = f2 (t) on ΓN × (0, T )

(3.20) (3.21)

ϕ (t) = 0 on Γ1 × (0, T ) D(t) · ν = 0 on Γ2 × (0, T )

(3.22) (3.23)

uν (t) = 0 on ΓC × (0, T ) −στ (t) ∈ ∂ j(β (t), uτ (t)) on ΓC × (0, T )

(3.24) (3.25)

β (t) = F(t, u(t), β (t)) on ΓC × (0, T ) β (0) = β0 on ΓC u(0) = u0 , u (0) = v0 in Ω .

(3.26) (3.27) (3.28)

We shortly comment on the above model. Equations (3.16) and (3.17) are the equation of motion for the stress field and the Gauss equilibrium equation for the electric displacement field, respectively. Recall that div stands for the divergence operator for vector valued functions, i.e. divD = ∂i Di . The relations (3.18) and (3.19) represent the electroviscoelastic constitutive law of the material in which A , B, D and P are respectively the (fourth order) viscosity tensor, the (fourth order) elasticity tensor, the (second order) electric permittivity tensor and the (third order) piezoelectric tensor. The equation (3.18) describes the converse effect and (3.19) models the direct effect of piezoelectricity. Furthermore, D : Q → Rd , D = {Di } denotes the electric displacement field and P is the tensor transposed to P. The decoupled state (purely viscoelastic and purely electric deformations) can be obtained by setting the piezoelectric tensor P = 0. The electric field–potential relation is given by E(ϕ ) = {−∂i ϕ } in Q. The equations (3.20) and (3.21) are the displacement and traction boundary conditions, respectively, while (3.22) and (3.23) represent the electric boundary conditions. Condition (3.24) shows that the contact is bilateral and (3.25) is the subdifferential boundary condition with a nonconvex nonsmooth superpotential j. This relation says that the tangential traction στ depends on the intensity of adhesion β and the tangential displacement uτ . The evolution of the adhesion (bonding) field β is governed by the ordinary differential equation (3.26) on the contact surface and (3.27) represents a given initial bonding field (cf. [12]). The adhesive evolution rate function F depends on both the bonding field and the displacement and may change sign. This allows for rebonding after debonding took


55

place, and it allows for possible cycles of debonding and rebonding (cf. [39, 29] for examples of F). Finally, the initial values for the displacement and the velocity are prescribed in (3.28). in Section 3.2 and introducing the space Φ = Using1 the notation introduced ϕ ∈ H (Ω ) | ϕ = 0 on Γ1 , the problem (3.16)–(3.28) in its weak form is formulated as follows: find u ∈ V with u ∈ W , ϕ ∈ C(0, T ; Φ ) and β ∈ W 1,∞ (0, T ; L2 (ΓC )) such that ⎧ u (t), v + a(u(t), v) + b(u(t), v) + e(ϕ (t), v)+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j0 (x, β (t), uτ (t); vτ ) dΓ ≥ f (t), v for a.e. t, all v ∈ V + ⎪ ⎪ ⎨ ΓC (3.29) d(ϕ (t), ψ ) = e(u(t), ψ ) for a.e. t ∈ (0, T ), all ψ ∈ Φ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β (t) = F(t, u(t), β (t)) on ΓC , for a.e. t ∈ (0, T ) ⎪ ⎪ ⎪ ⎪ ⎩ β (0) = β0 on ΓC , u(0) = u0 , u (0) = v0 in Ω . This problem is a system coupled with the evolution hemivariational inequality for the displacement, the time dependent stationary equation for the electric potential and the ordinary differential equation for the bonding field. The existence of a weak solution to (3.29) can be found in [29]. The other frictional contact problems in piezoelectricity have been studied in [16, 17, 23, 34].

3.7 Comments on Other Nonconvex Inequality Models In this section we give further comments on results for contact problems with nonsmooth and nonconvex superpotentials. 1. A general approach to study viscoelastic contact problems with subdifferential boundary conditions has been introduced in [20, 26]. This approach allows to unify several methods for models considered in contact mechanics and obtain new existence results which are not available in the literature. The existence results for the dynamic hemivariational inequalities of hyperbolic type can be found in [21, 22, 27, 36]. 2. Optimal control problems for dynamic hemivariational inequalities have been investigated in several papers, cf. e.g. [8, 4, 6]. The results on inverse and identification problems for such inequalities can be found in [31]. 3. For the contact problems with slip dependent friction, we refer to [24] and the references therein. The hemivariational inequalities modeling contact with wear and adhesion have been studied in [1, 2, 29]. 4. Homogenization results for stationary elasticity problems described by hemivariational inequalities are presented in [19, 18].

56


5. A class of inequality problems for the stationary Navier-Stokes type operators related to the model of motion of a viscous incompressible fluid through a tube or channel have been studied in [25]. 6. Vanishing viscosity method and asymptotic behavior of solutions to hemivariational inequalities have been considered in [28, 18]. 7. The quasistatic hemivariational inequalities which describe the contact problems have been treated by a vanishing acceleration method in [30] while the antiplane frictional contact problems have been considered in [33]. Acknowledgements. The authors would like to express their gratitude to Prof. M. Kuczma and Prof. K. Wilmanski for their invitation to deliver a Keynote Lecture and to Prof. P. Litewka and Prof. A. Zmitrowicz for an invitation to participate in the Minisymposium on Computational Contact Mechanics at the 18th International Conference on Computer Methods in Mechanics held in Zielona Gora, Poland, May 18–21, 2009. Research supported in part by the Ministry of Science and Higher Education of Poland under grant no. N201 027 32/1449.

References [1] Bartosz, K.: Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear. Nonlinear Anal. 65, 546–566 (2006) [2] Bartosz, K.: Hemivariational inequalities modeling dynamic contact problems with adhesion. Nonlinear Anal. 71, 1747–1762 (2009) [3] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Interscience, New York (1983) [4] Denkowski, Z., Migórski, S.: Sensitivity of optimal solutions to control problems for systems described by hemivariational inequalities. Control Cybernet. 33, 211–236 (2004) [5] Denkowski, Z., Migórski, S.: A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60, 1415–1441 (2005) [6] Denkowski, Z., Migórski, S.: On sensitivity of optimal solutions to control problems for hyperbolic hemivariational inequalities. Lect. Notes Pure Appl. Math. 240, 145–156 (2005) [7] Denkowski, Z., Migórski, S., Ochal, A.: Existence and uniqueness to a dynamic bilateral frictional contact problem in viscoelasticity. Acta Appl. Math. 94, 251–276 (2006) [8] Denkowski, Z., Migórski, S., Ochal, A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems. Control Cybernet. 36, 611–632 (2007) [9] Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003) [10] Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003) [11] Duvaut, G., Lions, J.-L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972) [12] Frémond, M.: Adhérence des solides. J. Mécanique Théorique et Appliquée 6, 383–407 (1987)


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[13] Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society, International Press, Providence (2002) [14] Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987) [15] Kulig, A.: Evolution Inclusions and Hemivariational Inequalities for Unilateral Contact in Viscoelasticity. PhD Thesis, Jagiellonian University, Krakow (2009) [16] Li, Y., Liu, Z.: Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction. Nonlinear Analysis, Theory, Methods and Applications 71, 1414–1424 (2009) [17] Liu, Z., Migórski, S.: Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete Contin. Dyn. Syst. Ser. B 9, 129–143 (2008) [18] Liu, Z., Migórski, S., Ochal, A.: Homogenization of boundary hemivariational inequalities in linear elasticity. J. Math. Anal. Appl. 340, 1347–1361 (2008) [19] Migórski, S.: Homogenization technique in inverse problems for boundary hemivariational inequalities. Inverse Problems in Eng. 11, 229–242 (2003) [20] Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669–699 (2005) [21] Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim. 31, 505–533 (2005) [22] Migórski, S.: Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput. Math. Appl. 52, 677–698 (2006) [23] Migórski, S.: Hemivariational inequality for a frictional contact problem in elastopiezoelectricity. Discrete Contin. Dyn. Syst. 6, 1339–1356 (2006) [24] Migórski, S., Ochal, A.: Hemivariational inequality for viscoelastic contact problem with slip dependent friction. Nonlinear Anal. 61, 135–161 (2005) [25] Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier-Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005) [26] Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elasticity 83, 247–275 (2006) [27] Migórski, S., Ochal, A.: Existence of solutions for second order evolution inclusions with application to mechanical contact problems. Optimization 55, 101–120 (2006) [28] Migórski, S., Ochal, A.: Vanishing viscosity for hemivariational inequality modeling dynamic problems in elasticity. Nonlinear Anal. 66, 1840–1852 (2007) [29] Migórski, S., Ochal, A.: Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal. 69, 495–509 (2008) [30] Migórski, S., Ochal, A.: Quasistatic hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009) [31] Migórski, S., Ochal, A.: An inverse coefficient problem for a parabolic hemivariational inequality. Appl. Anal. (in press, 2009) [32] Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008) [33] Migórski, S., Ochal, A., Sofonea, M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal. 70, 3738–3748 (2009) [34] Migórski, S., Ochal, A., Sofonea, M.: Weak solvability of a piezoelectric contact problem. Eur. J. Appl. Math. 20, 145–167 (2009) [35] Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York (1995)

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[36] Ochal, A.: Existence results for evolution hemivariational inequalities of second order. Nonlinear Anal. 60, 1369–1391 (2005) [37] Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985) [38] Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993) [39] Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer, Berlin (2004) [40] Sofonea, M., Han, W., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman-Hall/CRC Press, New York (2006) [41] Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Springer, New York (2009) [42] Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact, friction and wear. Internat. J. Solids Structures 33, 1817– 1836 (1996) [43] Wriggers, P.: Computational Contact Mechanics. Wiley, Chichester (2002)

Chapter 4

Quadrature for Meshless Methods John E. Osborn

The results in this paper are joint work with I. Babuška, U. Banerjee, Q. Li, and Q. Zhang. Abstract. In this paper we discuss quadrature for meshless methods. We present results of two analyses. The first of these was developed in [1] and the second was developed in [2]. Using a simple example we show that a natural quadrature scheme that satisfies no special properties, leads to severe inaccuracies in the approximate solution. We then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature. In the second analysis we show that if the quadrature scheme satisfies Green’s formula, then the effect of the quadrature is controlled.

4.1 Introduction Meshless Methods (MM) have been the focus of considerable interest in recent yeas, especially in the engineering community. This interest was mainly stimulated by difficulties with mesh generation with the usual Finite Element Method (FEM), which is delicate in many situations; for example, when the domain has complicated geometry or when the mesh changes with time, as in crack propagation, and remeshing is required at each time step. It was recognized early (see, e.g., [5], [6]) that the important problem of creating effective quadrature schemes for MM is more challenging that that for FEM. The FEM was completely analyzed 30 year ago (see, e.g., [7]). The major feature of the FEM is that the shape functions are piecewise polynomials of degree p, and hence their k-th order derivatives vanish on each element for k ≥ p + 1. This permits the John E. Osborn Department of Mathematics, University of Maryland, College Park, MD 20742 e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 59–73. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

60

J.E. Osborn

exact calculation of the stiffness matrix for PDEs with constant coefficients. With the MM, the shape functions are generally not polynomials, and their k-th orderderivatives grow with k, and essentially no quadrature scheme will be accurate. In this paper we discuss two developments, referred to below as the First Analysis and the Second Analysis, in the creation of effective quadrature schemes for MM. These analyses are fully developed in [1] and [2], respectively. Several engineering papers ([3], [4], [5], [6]) discuss quadrature for MM, concentrating mainly on implementational issues, but to the best of our knowledge, a careful mathematical analysis of the effect of quadrature in MM was first reported in [1]. It is shown in [1] that MM with standard quadrature—assuming nothing special about the quadrature scheme—is erratic and practically no reasonably accuracy is obtained, and the row sum condition is identified and shown to be the key hypothesis in creating quadrature schemes that perform well. Specifically, it is shown that if the stiffness matrix satisfies the zero row sum condition, the MM reproduces polynomials of degree k = 1, and certain other conditions are satisfied, then the energy norm of the error in the approximate solution is O(h + η ), where h is the standard discretization parameter related to the diameters of the supports of the shape functions and η is a parameter indicating the accuracy of the underlying numerical quadrature. Furthermore, a simple “recipe” is identified for correcting the elements of the stiffness matrix so that zero row sum condition is satisfied. In [2], our second analysis is presented. The major hypotheses in this analysis are that the MM reproduce polynomials of degree k ≥ 1 and that the quadrature satisfies a Green’s formula. Under these and certain other conditions it is shown that the energy norm of the error in the approximate solution is O(hk−1 (h + η )), where η is a parameter related to the accuracy of the numerical quadrature and h is the standard discretization parameter. We also indicate how to obtain numerical quadrature schemes satisfying these conditions. Thus MM does not yield optimal order of convergence for η = O(h). Certainly if η = O(h), we have the optimal order of convergence. It is important to note that the parameter η (see (4.16)) associated with the quadrature scheme usually used in the FEM, namely the Gauss rule, is O(h). We mention that the numerical integration used in this paper yields a nonsymmetric stiffness matrix. But this does not pose a serious problem since nonsymmetric linear systems can be solved efficiently by iterative methods. Both analyses are for MM approximation of a second order Neumann boundary value problem. In both analyses, we see that MM does not yield optimal convergence if η , respectively, η does not equal O(h). The outline of this paper is as follows: In Section 4.2, we state the main results of the First Analysis, and in Section 4.3 we present the main results of the Second Analysis. In Section 4.4 a comparison of the results of the two analyses is presented. Computational results (plots) are included in Sections 4.2 and 4.3. The author wishes to thank Wiley InterScience for permission to use Figures 4.1 and 4.2 in this paper; they were published in International Journal for Numerical Methods in Engineering. And to thank Elsevier Science for permission to use Figures 4.3–4.5 in this paper; they were published in Computer Methods in Applied Mechanics and Engineering.

4 Quadrature for Meshless Methods

61

4.2 First Analysis The results stated in this section are fully developed and proved in [1].

4.2.1 Preliminaries Throughout the paper Ω will be a bounded domain in Rd with boundary Γ = ∂ Ω , H m (Ω ) will be the usual Sobolev space with norm and seminorm, um,Ω and |u|m,Ω , respectively. And uL2(Ω ) , uL∞ (Ω ) , uL2 (Γ ) , and uL∞ (Ω ) will denote the usual norms on L2 (Ω ), L∞ (Ω ), L2 (Γ ), and L∞ (Γ ), respectively, whereas uE = 1/2 2 denotes the energy norm on H 1 (Ω ). Ω |∇u| dx We consider the model problem, −Δ u = f in Ω , (4.1) ∂u ∂ n = g on Γ = ∂ Ω with variational formulation, u ∈ H 1 (Ω ) , B(u, v) = L(v), ∀v ∈ H 1 (Ω )

(4.2)

where B(u, v) =

Ω

∇u · ∇v dx ,

L(v) =

Ω

f v dx +

Γ

gv ds.

If the compatibility condition, L(1) = 0, is satisfied, the solution exists and is unique up to a constant. We assume, in addition to the compatibility condition, that Γ , f , and g are such that u is in H 2 (Ω ). Consider first an MM based on uniformly distributed particles and associated shape functions:

x1 − j1 h x2 − j2 h h h x j = ( j1 h, j2 h), φ j (x1 , x2 ) = φ , , with φ given, h h where j = ( j1 , j2 ) ∈ Z2 , with Z the integer lattice, and 0 < h. Let Vh = span{φ hj : j ∈ Nh }, where Nh is an index set and {φ hj : j ∈ Nh } is a basis for Vh . Then we consider the Galerkin method corresponding to the approximation space Vh . Our MM approximation uh to u is characterized by uh ∈ Vh . (4.3) B(uh , v) = L(v), ∀v ∈ Vh If we let

γi,h j =

Ω

∇φih · ∇φ hj dx =

ωih ∩ω hj

∇φih · ∇φ hj dx

62

J.E. Osborn

and lih =

Ω

f φih dx +

Γ

gφih ds =

f φih dx +

ωih

Γ ∩ω hi

gφih ds

be the stiffness matrix and the right-hand side vector, and we write uh (x) = ∑ j∈Nh uh, j φ hj (x), then uh satisfies (15.3) if and only if

∑ γi,h j uh, j = lih , for all i ∈ Nh .

j∈Nh

We also consider the quadrature version of these integrals:

γi,h∗j = −

ωih ∩ω hj

∇φi · ∇φ j dx and lih∗ = − f φih dx + − ωih

Γ ∩ω hi

gφih ds,

where − is a quadrature version of . Our MM quadrature approximation u∗h to u is characterized by ∗ uh ∈ Vh (4.4) B∗ (u∗h , v) = L∗ (v), ∀v ∈ Vh , where

B∗ (uh , vh ) =

∑

γi,h∗j uh,i vh, j and L∗ (vh ) =

i, j∈Nh

∑ lih∗ vh,i .

i∈Nh

If we write u∗h = ∑ j∈Nh u∗h, j φhj (x), then u∗h satisfies (4.4) if and only if ∑ j∈Nh γi,h∗j u∗h, j = lih∗ , for all i ∈ Nh . We will see that u − u∗h E behaves erratically and is not small, whereas u − uh E ≤ O(h). Later we will consider a corrected stiffness matrix, γi,h∗∗ j , and corrected right-hand side vector, lih∗∗ , and the corresponding quadrature approximate solution, u∗∗ h , and show that it provides an accurate approximation to u.

4.2.2 An Example / Numerical Results

Consider

−u = cos x, x ∈ Ω ≡ (0, π ) , u (0) = u (π ) = 0

with variational formulation, u ∈ H 1 (0, π ) B(u, v) = F(v), ∀v ∈ H 1 (0, π ),

π

π

where B(u, v) = 0 u v dx and F(v) = 0 cos x v dx. This is a one-dimensional version of the model problem (4.1). To construct a meshless method, we let xhj = jh, for j = . . . , −1, 0, 1, . . . , where h = π /n, for n = 1, 2, . . . , be a family of uniformly distributed particles. And we


63

use the Reproducing Kernel Particle (RKP) construction, with respect to a window function, to construct associated shape functions that are reproducing of order 1, i.e., that satisfy (4.5) ∑ (xhj )i φ hj (x) = xi , ∀x and i = 0, 1. j∈Nh

Plot of ||u−u* || /||u|| for



Trapezoid Rule, m=10



h E

E

h E

0.2 0

E

2 1 0

0.5

1 1.5 2 h Plot of ||u−u* || /||u|| for h E

3

h E

0.4

||u−u* || /||u||

||u−u*h||E/||u||E

E h E

||u−u* || /||u||

0.6

0

E

0.5

1

E

||u−u* || /||u||

h E

0

0.2 1.5

0.4 0.2 0

E

2

h E

E

15

h E

10 5 0

0

0.5

1 h

1 2 h Plot of ||u−u* || /||u|| for E

quad, tol=10−4 ||u−u* || /||u||

||u−u*h||E/||u||E

E h E

||u−u* || /||u||

0.4

E

0.6

quad, tol=10−2

0.6

1 1.5 h Plot of ||u−u* || /||u|| for

0.8

0

1 2 h Plot of ||u−u* || /||u|| for h E

0.5

h E

0.2

E

0.8

1 h

0

Gauss Rule, p=10

0.4

quad, tol=10−1

0.5

0.05

E

0.6

h Plot of ||u−u* || /||u|| for

0

0.1

0

1.5

0.8

0

1.5

h E

1 h E

||u−u*h||E/||u||E

||u−u*h||E/||u||E

2

0

0.5

0.2

Gauss Rule, p=4

4

0

E

0.15

h Plot of ||u−u* || /||u|| for

Gauss Rule, p=3

0

h E

4

0.8

0

E

1.5

0.8 0.6 0.4 0.2 0

0

1 h

2

u−u∗

Fig. 4.1 The plot of uh E with respect to h for various quadrature schemes. We observe E that the behavior of the relative error is erratic and that practically no reasonable accuracy was obtained. This figure is from [1], and has been used with permission of the publisher.

Note that (15.5) implies that ∑ j∈Nh φ hj (x) = 1, ∀x, i.e., {φ hj } is a partition of unity. Specifically, we use the window function

1 w(x) = exp 2 , −r < x < r, with r = 1.1. x − r2

64

J.E. Osborn u−u∗

In Fig. 4.1, we plot uhE E with respect to h for various quadrature schemes: the m-panel Trapezoid Rule, the p-point Gauss Rule, and MATLAB’s quad (adaptive Simpson quadrature) with various tolerances, tol. We observe that the behavior of the relative error is erratic, and that practically no reasonable accuracy is obtained. On the other hand, when we use exact integration, we get good accuracy. What feature distinguishes the stiffness matrix for exact integration from that for quadrature? Here is one possibility: With exact integration the Row Sums of the stiffness matrix are 0; while with quadrature, they differ from 0. For exact integration, this is seen from the calculation,

∑ γi,h j = ∑

j∈Nh

j∈Nh Ω

∇φih · ∇φ hj dx = =

Ω

Ω

∇φih · ∇(

∑ φ hj ) dx

j∈Nh

∇φih · ∇1 dx dy = 0;

(4.6)

and for quadrature, it is seen by examining the calculated stiffness matrix. This observation suggests a possibility when using quadrature: We consider a corrected stiffness matrix, γi,h∗∗ j , formed by modifying just the diagonal elements; h∗∗ h∗ we set γi,i = − ∑ j=i γi, j , so that ∑ j γi,h∗∗ j = 0, ∀i.

4.2.3 Theoretical Results We suppose Vh ⊂ H 1 (Ω ) is a one parameter family of finite dimensional approximation subspaces, and suppose {φ hj (x) : j ∈ Nh } is a basis for Vh . The φ hj (x) are our shape functions. We impose certain assumptions (Axioms) on the spaces Vh and on the quadrature schemes. We mention a few of the axioms: • Axiom 4.1 (Reproducing Polynomial Property; cf. (15.5))

∑ p(xi )φih (x) = p(x) for x ∈ Ω , ∀ polynomials of deg ≤ 1.

i∈Nh

• Axiom 4.2 There is a constant C, independent of h, such that

∑

i, j∈Nh

|γi,h j |(vi − v j )2 ≤ C

∑

(−γi,h j )(vi − v j )2 , ∀v =

i, j∈Nh

∑ vi φih ∈ Vh .

i∈Nh

This axiom is easily seen to be true for certain finite element spaces, but is hard to verify in general. • Axiom 4.3 (Row Sum Condition)

∑ γi,h∗j = 0.

j∈Nh


65

• Axiom 4.4 h h γi,h∗j = γi,h j + ηi,h j , fih∗ = fih + εih , and gh∗ i = gi + τi , with

|ηi,h j | ≤ η max(|γi,h j |, ν hd ), |εih | ≤ ε max(| fih |, ν hd f L∞ (Ω ) ), and |τih | ≤ τ max(|ghi |, ν hd−1 gL∞ (Γ ) ), where φih L∞(Ω ) ≤ ν and d = dimension. These are the relative and absolute quadrature error assumptions, with error parameters η , ε , and τ . Theorem 4.1 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1–4.4 and certain other axioms. Then for small η , there is a constant C > 0, independent of u, η , ε , τ , and h such that uh − u∗hE ≤ C η uE + ε f L∞ (Ω ) + τ gL∞(Γ ) , ∀h. (4.7) Theorem 4.2 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1–4.4 and certain other axioms. Then for small η , there is a constant C > 0, independent of u, η , ε , τ , and h such that u − u∗hE ≤ C hu2,Ω + η uE + ε f L∞ (Ω ) + τ gL∞(Γ ) , ∀h. (4.8)

Correction Process Axiom 4.3 (the Row Sum Condition) is unlikely to be satisfied. To see this note that the Row Sum Condition is satisfied for γi,h j (cf. (4.6)), and that ∑i, j γi,h∗j , which differs from ∑i, j γi,h j due to the quadrature error, is unlikely to be 0, and so we do not, in fact, have an estimate for u − u∗hE . We thus consider a corrected stiffness matrix γi,h∗∗ j , redefined by modifying the diagonal elements: h∗∗ γi,i =−

∑

j∈Nh , j=i

γi,h∗j .

The Row Sum Condition is now satisfied. Axiom 4.4 also holds, but with modified error parameters cη , c(ε + τ ), and c(ε + τ ). So we have the following error estimates for the corrected approximation. Theorem 4.3 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1-4.4 and certain other axioms. Then for small η , there is a constant C, independent of u, η , τ , η , and h, such that uh − u∗∗ h E ≤ C[η uE + (ε + τ ) f L∞ (Ω ) + (ε + τ )gL∞ (Γ ) ], ∀h.

(4.9)

Theorem 4.4 Suppose our shape functions and our quadrature scheme satisfy Axioms 4.1-4.4 and certain other axioms. Then for small η , there is a constant C, independent of u, η , τ , η , and h, such that u − u∗∗ h E ≤ C[hu2,Ω + η uE + (ε + τ ) f L∞ (Ω ) + (ε + τ )gL∞ (Γ ) ], ∀h. (4.10)

66

J.E. Osborn

4.2.4 Numerical Results u −u∗∗

u−u∗∗

In Figure 2.2 we present log-log plots of the relative errors huhE E and uhE E with respect to h for the one dimensional problem discussed earlier in the section. The stiffness matrix γh∗ is computed with the same quadrature methods, but the matrix is then corrected to satisfy the Row Sum Condition (Axiom 4.3), and denoted by γh∗∗ . The right-hand side vector fih + ghi has been computed exactly (ε = τ = 0).

Loglog Plot of ||uh−u**h ||E/||u||E for



Trapezoid Rule with correction

Gauss Rule with correction

quad with correction

−6 −8 −3

−2

−1 h

0

E

−4

E

tol=10−4

h

−6

−2

−1 h

−6 −8

p=100

−10

0

−12 −3

1

−5

tol=10

tol=10−8 −2

−1 h

0

1

Loglog Plot of ||u−u**|| /||u|| for

Trapezoid Rule with correction

Gauss Rule with correction

quad with correction

h E

0

−3

−2

E

p=10

−3

−1

−1 h

Fig. 4.2 The log-log plot of

0

1 uh −u∗∗ h E uE

−4 −3

−2

−1 h

0

1

−1

tol=10

−2

tol=10 tol=10−4

−2 −3

tol=10−5 −8 tol=10

−4 −3

−2

p=100

m=100 −4 −3

E

−2

E

0

p=2 p=3 p=4

−1

h E

h

m=4

E

||u−u**|| /||u||

m=2 m=3 m=10

−1 −2

E

||u−u** || /||u||E h E

E E

tol=10−2

Loglog Plot of ||u−u**|| /||u|| for

0

h

−2

−4

−8 −3

1

−1

tol=10

Loglog Plot of ||u−u**|| /||u|| for h E

||u−u**|| /||u||

−2

h

m=100

h

h

−4

0

p=2 p=3 p=4 p=10

||u −u**|| /||u||

−2

0 ||uh−u** || /||u||E h E

m=2 m=3 m=10 m=4

E

||u −u**|| /||u||

E

0

−1 h

0

1

u−u∗∗ h E with respect to h with correction for uE u−u∗∗ h E error u first decreases with decreasing h, E

and

various quadrature schemes. The relative and then approaches a constant as h → 0. This figure should be compared with Fig. 4.1. This figure is from [1], and has been used with permission of the publisher.


67

We observe that the relative error uh − u∗∗ h E /uE becomes nearly constant as h → 0; this constant reflects the accuracy of the quadrature (η ). On the other hand, the relative error u − u∗∗ h E /uE first decreases with decreasing h and then levels off, becoming nearly constant as h → 0. These figures and estimate (15.10) indicate that the error has two components: one due to the MM approximation (see the estimate (15.8) for u − uhE when η = ε = τ = 0) and the other due to quadrature (see estimate (15.9)). From (15.10) and the second row in Fig. 4.2 we see that for given quadrature accuracy η , ε , and τ , the error for small h is completely governed by the quadrature accuracy. We have to set η , ε , and τ equal to o(1) if we want the relative error to converge, and we have to set η , ε , and τ equal to O(h) if we want the relative error to be O(h).

4.3 Second Analysis The results stated in this section are fully developed and proved in [2].

4.3.1 Preliminaries We again consider the problem introduced in Section 4.2: u ∈ H 1 (Ω ) B(u, v) = L(v), ∀v ∈ H 1 (Ω ). As said there, if the compatibility condition is satisfied, this problem has a unique solution up to a constant. A standard way of specifying a unique solution is to consider a linear functional Φ : L2 (Ω ) → R with Φ (1) > 0, and seek the unique solution u satisfying Φ (u) = 0. We will use an alternate variation formulation for our problem. Given Φ and letting HΦ = {(v, μ ) ∈ H 1 (Ω ) × R : (v, μ )2HΦ = |v|2H 1 (Ω ) + |Φ (v)|2 + μ 2 < ∞}, we consider the variation formulation, (u, λ ) ∈ HΦ BΦ (u, λ ; v, μ ) = L(v), ∀(v, μ ) ∈ HΦ , where

(4.11)

BΦ (u, λ ; v, μ ) = B(u, v) + λ Φ (v) + μΦ (u),

which can equivalently be written B(u, v) + λ Φ (v) = L(v), ∀v ∈ H 1 (Ω ) μΦ (u) = 0, ∀μ ∈ R.

(4.12)

68

J.E. Osborn

Remark The second equation specifies the constraint Φ (u) = 0, and the first equation is the Euler-Lagrange equation for the constrained extremal problem min J(v),

u∈H 1 (Ω ) Φ (v)=0

where J(v) = 12 B(v, v) − L(v), for which λ is the Lagrange multiplier. Problem (4.12) has a unique solution, whether or not the compatibility condition is satisfied. If it is satisfied, then λ = 0 and u ∈ H 1 (Ω ) satisfies B(u, v) = L(v), ∀v ∈ H 1 (Ω ) Φ (u) = 0. We next define the discrete problem. We define a linear functional on Vh (Vh as defined in Section 4.2) by

Ψ (vh ) =

Φ (1) ∑ vi , ∀vh = ∑ vi φih ∈ Vh, |Nh | i∈N i∈Nh h

where |Nh | = cardinality of Nh , and define the space VΨh = {(vh , μ ) ∈ Vh × R : (vh , μ )2HΦ = |vh |V2 h + |Ψ (vh )|2 + μ 2 < ∞}. Ψ

Then the meshless method approximation (uh , λh ) to (u, λ ) is defined by (uh , λh ) ∈ VΨh BΨ (uh , λh ; v, μ ) = L(v), ∀(v, μ ) ∈ VΨh , where

(4.13)

BΨ (uh , λh ; v, μ ) = B(uh , v) + λhΨ (v) + μΨ (uh ).

Problem (4.13) has a unique solution. It is easily seen that λh = 0 and that uh ∈ Vh satisfies B(uh , v) = L(v), ∀v ∈ Vh Ψ (uh ) = 0. If we write uh = ∑ j∈Nh c j φ hj , (4.13) can be written as

∑ γi, j c j + λhΨ (φih ) + μ

j∈Nh

Ψ (1) |Nh |

∑ c j = lih , ∀i ∈ Nh , μ ∈ R,

j∈Nh

where

γi,h j = and

Ω

∇φ hj · ∇φih dx =

ω hj ∩ωih

∇φ hj · ∇φih dx =

ωih

∇φ hj · ∇φih dx

(4.14)


lih =

Ω

f φih dx +

Γ

69

gφih ds =

ωih

f φih dx +

Γ ∩ω hi

gφih ds = fih + ghi.

Problem (4.14) has a unique solution: λh = 0, and

∑ γi, j c j = lih , ∀i ∈ Nh , Ψ ( ∑ c j φ hj ) = 0.

j∈Nh

j∈Nh

We impose several axioms on the discretization and on the quadrature. First an axiom on the discretization: • Axiom 4.5 (Reproducing Polynomial Property)

∑ p(xi )φih (x) = p(x) for x ∈ Ω , ∀ polynomials of deg ≤ k.

i∈Nh

This should be compared to Axiom 4.1. We next consider the quadrature version of (4.14):

∑ γi,h∗j c∗j + λh∗Ψ (φih ) + μ

j∈Nh

Ψ (1) |Nh |

∑ c∗j = lih∗ , ∀i ∈ Nh , μ ∈ R,

(4.15)

j∈Nh

We note, however, that we have two formulas for γi,h j , and one can base the quadraof them. In Section 4.2, we based the quadrature version on ture versionh on either h dx; here we will base our quadrature version on h · ∇φ h dx. ∇ φ · ∇ φ ∇ φ j i j i ω h ∩ω h ωh j

i

i

So, we define

γi,h∗j = − ∇φ hj · ∇φih dx, lih∗ = − f φih dx + − ωih

ωih

Γ ∩ω hi

gφih ds.

With these values for γi,h∗j and lih∗ we solve (4.15) for u∗h = ∑ j∈Nh c∗j φ hj and λh∗ , and view u∗h as the approximation to u : u ≈ u∗h . Remark As a consequence of this choice for γi,h∗j , we have ∑ j∈Nh γi,h∗j = 0, ∀i ∈ Nh , i.e., the Row Sum Condition is satisfied. We also note that

γi,h∗j = − ∇φ hj · ∇φih dx = − ∇φ hj · ∇φih dx = γ h∗ ji , ωih

ω hj

i.e., the matrix {γi,h∗j } is not symmetric. This does not pose a serious problem since non-symmetric linear systems can be solved efficiently by iterative methods. Furthermore we observe that ∑ fih∗ + ∑ gh∗ i = 0, i∈Nh

i∈Nh

i.e., the compatibility condition with quadrature is not satisfied. This is due to the use of quadrature to evaluate fih and ghi . It is here that we see the advantage of the alternate variational formulation based on the Lagrange multiplier.

70

J.E. Osborn

Next, two axioms on quadrature: • Axiom 4.6 There are small constants η and τ , independent of i and h, such that |

ωih

ρ dx − − ρ dx| ≤ η |ωih |ρ L∞(ωi )

(4.16)

ωih

and |

ω hi ∩Γ

θ ds − −

ω hi ∩Γ

θ dx| ≤ τ |ω hi ∩ Γ |θ L∞(ω h ∩Γ ) i

for appropriate classes of functions ρ and θ . • Axiom 4.7 For each i ∈ Nh , let G∗i : C2 (ω i ) → R be a linear functional given by G∗i (v) = − ∇v · ∇φih dx + − Δ vφih dx − − ωih

ωih

ω hi ∩Γ

∇v · n φih ds.

Then G∗i (p) = 0, ∀polynomials p of degree ≤ k and ∀i ∈ Nh ,

(4.17)

i.e., it is assumed that this Quadrature version of Green’s formula holds for polynomials p of degree ≤ k.

4.3.2 Theoretical Results Theorem 4.5 Suppose the approximating subspace Vh and the quadrature schemes satisfy Axioms 4.1-4.3 and certain other axioms. Then for small η , there is a constant C, independent of u, η , τ , and h, such that |u − u∗h|H 1 (Ω ) ≤ C hk + (η + τ )hk + η hk−1 uW k+1,∞ (Ω ) , ∀h.

4.3.3 Numerical Results We have developed quadrature schemes satisfying Axiom 4.3 and the other axioms. Remark on Axiom 4.7 If k = 1, for (4.17) to hold, the quadrature scheme ωih must satisfy ∂ φih − dx − − n1 φih ds = 0 ωih ∂ x1 Γ ∩ω hi and −

ωih

∂ φih dx − − n2 φih ds = 0, ∂ x2 Γ ∩ω hi


71

where n = (n1 , n2 ). In particular, when Γ ∩ ωi h = 0, / then the quadrature must satisfy − ∇φih dx = 0.

(4.18)

ωih

Consider d = 1, and suppose we have a p-point integration rule on ωi = (αi , βi ): βi αi

p

f dx =

∑ w j f (x j ).

j=1

We then define a corrected rule βi αi

p

f dx =

∑ w j f (x j ),

j=1

where

− ∑ j=1 [φih ] (x j ) p

w j = w j + θi w j [φih ] (x j ),

with θi =

∑ j=1 w j [φih ]2 p

.

Then − [φih ] dx = 0, ωih

when Γ ∩ ω hi = 0. /

Suppose the shape function φih is symmetric on ωih , and we start with the standard Gauss rule on ωih . Then, since the Gauss weights and points are symmetric on ωih , we find that θi = 0, and we do not have to correct the rule. Gauss rule (uncorrected): k=1

−1

10

8−point

−2

*

|u−uh|H1(Ω)

10

−3

10

16−point −4

10

32−point −5

10

−4

10

−3

−2

10

10

−1

10

h

Fig. 4.3 The log-log plot of |u − u∗h |E with respect to h. u∗h is the approximate solution using the p-point standard Gaussian quadrature (symmetric) with p = 8, 16, and 32. This figure is from [2], and has been used with permission of the publisher.

72

J.E. Osborn

In Fig. 4.3 we present the log-log plot of the error |u − u∗h|E with respect to h for the one-dimensional problem (4.1) with exact solution u(x) = ex − (e − 1) using the Gauss rule, which doesn’t have to be corrected. In Fig. 4.4 we present the log-log plot of |u − u∗h|E , when using a non-symmetric Gauss rule; this quadrature rule does not satisfy (4.18), and we see the the results are not satisfactory. Finally, in Fig. 4.5 we present a plot for the corrected non-symmetric Gauss rule. Non−symmetric Gauss rule (uncorrected): k=1

0

10

8−point

16−point

−1

10

32−point

*

|u−uh|H1(Ω)

−2

10

−3

10

64−point

−4

10

−5

10

−4

−3

10

−2

10

10

−1

10

h

Fig. 4.4 The log-log plot of |u − u∗h |E with respect to h. u∗h is the approximate solution obtained using the p-point non-symmetric Gaussian quadrature (uncorrected) with p = 8, 16, 32, and 64. This figure is from [2], and has been used with permission of the publisher. Corrected non−symmetric Gauss rule: k=1

−2

10

8−point −3

10

*

|u−uh|H1(Ω)

16−point

−4

10

32−point 64−point

−5

10

−4

10

−3

−2

10

10

−1

10

h

Fig. 4.5 The log-log plot of |u−u∗h |E with respect to h. u∗h is the approximate solution obtained using corrected non-symmetric Gaussian quadrature with p = 8, 16, 32, and 64 points. This figure is from [2], and has been used with permission of the publisher.


73

4.4 Comparison of the Results of the Two Analyzes • The main hypotheses in the First Analysis: the basis function reproduce polynomials of degree 1; the Row Sum Condition is satisfied. Results when ε = τ = 0 : ∗∗ u − u∗∗ h E ≤ C(h + η ), u − uh E ≤ Ch if η ≤ Ch. • The main hypotheses in the Second Analysis: the basis functions reproduce polynomials of degree k ≥ 1; the Quadrature Green’s formula is satisfied for all polynomials of degree ≤ k. The Row Sum Condition is automatically satisfied. The stiffness matrix is non-symmetric. Results when ε = τ = 0 : u − u∗h E ≤ C(hk + η hk−1 ), u − u∗hE ≤ Chk if η ≤ Ch. • The numerical results support the theorems in both analyzes. The results indicate that increased quadrature accuracy is required as h → 0 to obtain optimal order of convergence.

References [1] Babuška, I., Banerjee, U., Osborn, J., Li, Q.: Quadrature for meshless methods. Int. J. Numer. Math. Engng. 76, 1434–1470 (2008) [2] Babuška, I., Banerjee, U., Osborn, J., Zhang, Q.: Effect of numerical integration on meshlesss methods. Comput. Methods Appl. Mech. Engrg. 198, 2886–2897 (2009) [3] Beissel, S., Belytschko, T.: Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Engrg. 139, 49–74 (1996) [4] Carpinteri, A., Ferro, G., Ventura, G.: The partition of unity quadrature in meshless methods. Computers and Structures 54, 987–1006 (2002) [5] Chen, J.-S., Wu, C.-S., Yoon, S., You, Y.: A stabilized conformal nodal integration for a Galerkin mesh-free method. Int. J. Numer. Meth. Engng. 50, 435–466 (2001) [6] Chen, J.-S., Wu, S., You, Y.: Non-linear version of stabilized conforming nodal integration Galerkin mesh-free methods. Int. J. Numer. Meth. Engng. 53, 2587–6515 (2002) [7] Ciarlet, P.G.: The Finite Element Method. North Holland, Amsterdam (1978)

Chapter 5

Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks ˙ Jan Sokołowski and Antoni Zochowski

Abstract. In the keynote lecture we describe how the asymptotic analysis in singularly perturbed domains can be employed to determine effectively the influence of nucleation of small voids on some shape functionals. To this end the classical shape sensitivity analysis is combined with asymptotic expansions in order to determine the singular limits of shape derivatives which are called topological derivatives of shape functionals. The topological derivatives are determined for elastic bodies weakened by cracks on boundaries of rigid inclusions. On the crack faces the nonpenetration conditions are prescribed, such conditions are non linear and assure that the displacements of the crack lips or surfaces cannot penetrate each other. Small voids are located on the finite distance from the crack, so there is no interaction between the crack and the voids. In such a way nucleations of small voids can be implemented in the numerical procedures of optimum design or solution of inverse problems. A nonlinear model in the framework of damage theory is presented in details for modeling and sensitivity analysis. The example of an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion is considered. The asymptotic analysis which leads to topological derivatives is performed in two and three spatial dimensions. The derived formulas can be used in numerical methods of shape and topology optimization.

Jan Sokołowski Institut Elie Cartan, UMR 7502 Nancy-Université-CNRS-INRIA, Laboratoire de Mathématiques, Université Henri Poincaré Nancy 1, B.P. 239, 54506 Vandoeuvre lés Nancy Cedex, France e-mail: [email protected] ˙ Antoni Zochowski Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 75–98. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

76

˙ J. Sokołowski and A. Zochowski

5.1 Introduction Optimum shape design in structural mechanics is a classical domain of research for many years. One of the goals in the field is to devise numerical methods for shape and topology optimization. In the framework of boundary variations technique, since the shape transformations are smooth and regular in the rigorous mathematical fashion, the topology changes during the process of shape optimization are quite difficult to be included. Therefore, the singular limits of shape or material derivatives of shape functionals turn out to provide a remedy for this difficulty to propose reasonable topology changes for the structure under design procedure. From one point of view, by the singular limit procedure we mean the derivation of expressions, within the framework of matched and compound asymptotic expansions in asymptotic analysis in singularly perturbed domains, for the first nontrivial terms in expansions of associated shape functionals. In another words, in the case of small void which is included in the elastic body, the asymptotic analysis is performed with respect to the size of small void which becomes the small parameter. Such an analysis of the specific shape functional in the optimum design problem under considerations, e.g., the potential energy of the body, results in an expansion of the functional with respect to the parameter, and the first nontrivial term in the expansion is called the topological derivative of the specific shape functional. The topological derivative is a function defined inside the elastic body, depending on the location of the material point which becomes the center of the small void. We point out that such a derivation can also be performed by means of the classical shape sensitivity analysis and an appropriate singular limit passage of the shape derivatives with the size of the void tending to zero. Singular perturbations of elastic bodies are analyzed, and the influence of small defects in the body on the displacement and stress fields as well as on the energy functionals is determined for variational inequalities by a domain decomposition method, combined with asymptotic analysis of the Steklov-Poincaré operator in singularly perturbed domains. In the paper we propose an example illustrating the recent results obtained in the framework of asymptotic analysis in singularly perturbed geometrical domains for the purposes of shape and topology sensitivity analysis [20].

5.1.1 Cracks and Rigid Inclusions The problem associated to cracks in elastic bodies on boundaries of rigid inclusions appears in a vast number of applications in civil, mechanical, aerospace, biomedical and nuclear industries. In particular, some classes of materials are composed by a bulk phase with inclusions inside. When the inclusions are much stiffer than the bulk material, we can treat them as rigid inclusions. In addition, it is quite common to have cracks between both phases. Thus, in this paper we review the mechanical

5 Shape and Topology Sensitivity Analysis

77

modeling as well as the topology sensitivity analysis associated to the limit case of rigid inclusions in elastic bodies with a crack at the interface. The mechanical modeling is based on the assumption of non-penetration conditions at the crack faces between the elastic material and the rigid inclusion, which do not allow the opposite crack faces to penetrate each other, leading to a new class of variational inequalities. For the topological sensitivity analysis, we obtain the topological derivatives of the energy shape functional associated to the nucleation of a smooth imperfection in the bulk elastic material. The paper is organized as follows. The problem formulation associated to cracks in elastic bodies on boundaries of rigid inclusions is presented in Section 5.2. The topological derivatives associated to the energy shape functional are calculated in Section 5.3. We provide some closed formulas for the case of nucleation of spherical holes in 3D and circular elastic inclusions in 2D. In this last case, we present the limit cases in which the elastic inclusion becomes a hole (void) and also a rigid inclusion. The authors are indebted A.M. Khludnev and A. Novotny for scientific collaboration on modeling and sensitivity analysis of problems with rigid inclusions and cracks.

5.2 Problem Formulation Let Ω ⊂ R3 be a bounded domain with smooth boundary Γ , and ω ⊂ Ω be a subdomain with smooth boundary Ξ such that ω ∩ Γ = 0. / We assume that Ξ consists of two parts γ and Ξ \ γ , meas(Ξ \ γ ) > 0, where γ is a smooth 2D surface described as xi = xi (y1 , y2 ), (y1 , y2 ) ∈ D, i = 1, 2, 3, with bounded domain D ⊂ R2 having a smooth boundary ∂ D, and the rank of the matrix ∂∂ xy is equal to 2. Denote by ν = (ν1 , ν2 , ν3 ) a unit outward normal vector to Ξ , see Fig. 5.1. The subdomain ω is assumed to correspond to a rigid inclusion, and the surface γ describes a crack located on Ξ . Domain Ω \ ω corresponds to the elastic part of the body. For the further use we introduce the space of infinitesimal rigid displacements R(ω ) = {ρ = (ρ1 , ρ2 , ρ3 ) | ρ (x) = Bx + C, x ∈ ω }, where

⎛

⎞ 0 b12 b13 B = ⎝ −b12 0 b23 ⎠ , −b13 −b23 0

C = (c1 , c2 , c3 );

bi j , ci = const, i, j = 1, 2, 3.

Denote Ωγ = Ω \ γ . Problem formulation describing an equilibrium of the elastic body with the rigid inclusion ω and the crack γ is as follows. In the domain Ωγ , we


78

Fig. 5.1 Domain Ω with rigid inclusion ω

have to find functions u = (u1 , u2 , u3 ), u = ρ0 on ω ; ρ0 ∈ R(ω ); and in the domain Ω \ ω we have to find functions σ = {σi j }, i, j = 1, 2, 3, such that −divσ = F σ − Aε (u) = 0

in in

Ω \ ω, Ω \ ω, u = 0 on Γ ,

(u − ρ0) · ν ≥ 0

on γ ,

(5.4)

στ = 0, σν ≤ 0 on γ ,

(5.5)

σν (u − ρ0) · ν = 0 on γ ,

(5.6)

+ + +

σν · ρ =

−

F ·ρ

∀ρ ∈ R(ω ).

(5.1) (5.2) (5.3)

(5.7)

ω

Ξ

Here F = (F1 , F2 , F3 ) ∈ L2 (Ω ) is a given function,

σν = σi j ν j νi , στ =

στ 1 2 3 (στ , στ , στ ),

= σ ν − σν ν ,

σ ν = {σi j ν j }i=3 i=1 ,

1 εi j (u) = (ui, j + u j,i), i, j = 1, 2, 3. 2 All functions with two lower indices are assumed to be symmetric in those indices. Summation convention over repeated indices is accepted throughout the paper. Elasticity tensor A = {ai jkl }, i, j, k, l = 1, 2, 3, is given, and it satisfies usual symmetry and positive definiteness properties, ai jkl = akli j = a jikl ,

ai jkl ∈ L∞ (Ω ), i, j, k, l = 1, 2, 3,

ai jkl ξkl ξi j ≥ c0 |ξ |2 ,

∀ ξi j = ξ ji ,

c0 = const.


79

In addition, we consider the isotropic case, namely A = 2mI + l (I ⊗ I) ,

(5.8)

where I and I respectively are the second and fourth order identity tensors and, m and l are the Lamé coefficients, which can be defined in terms of the Young modulus E and the Poisson ratio υ as m=

E 2(1 + υ )

and

l=

υE . (1 + υ )(1 − 2υ )

(5.9)

Relations (5.1) are equilibrium equations, and (5.2) corresponds to the Hooke’s law. Inequality (5.4) describes a mutual non-penetration between crack faces γ ± . The first relation in (5.5) means a zero friction between the crack faces. For simplicity we assume clamping condition (5.3) on Γ . Note that external forces F are applied to Ω \ ω as well as to ω , but there are no equilibrium equations in ω . Influence of these forces is taken into account through (5.7). If we have no crack γ on Ξ , relations (5.4)-(5.6) should be omitted. This specific problem formulation for the particular case F = 0 in ω can be found in [27]. The problem formulation with the crack and non-penetration conditions seems to be new. First of all we provide a variational formulation of problem (5.1)-(5.7). To this end, let us consider the Sobolev space HΓ1,ω (Ωγ ) = {v ∈ H 1 (Ωγ )3 | ε (v) = 0 on ω ; v = 0 on Γ }

(5.10)

and define the set of admissible displacements Kω = {v ∈ HΓ1,ω (Ωγ ) | ε (v) = 0 on ω ; (v+ − v−) · ν ≥ 0 on γ }.

(5.11)

Let (·, ·)Ω \ω be the inner product in L2 (Ω \ ω ). Consider the energy functional 1 Π (v) = (σ (v), ε (v))Ω \ω − (F, v)Ωγ , 2

(5.12)

where the stress field σ (v) = σ is defined in (5.2) for u = v, and introduce the following minimization problem inf Π (v).

v∈Kω

(5.13)

The convex cone Kω is weakly closed in the space HΓ1,ω (Ωγ ), and the functional Π is coercive and weakly lower semi-continuous on the same space. Hence, by the standard result in the calculus of variations problem (5.13) admits a solution satisfying the variational inequality


80

u ∈ Kω , (σ (u), ε (u − u))Ω \ω ≥ (F, u − u)Ωγ

(5.14) ∀ u ∈ Kω .

(5.15)

Since bilinear form is coercive, the solution u of problem (5.14)-(5.15) is unique and Lipschitz continuous with respect to data.

5.2.1 Dual Problem Formulation We introduce the dual formulation of problem (5.14)-(5.15) in stresses. By this approach a solution of dual problem σ = {σi j } in the domain Ω \ ω is defined, and moreover, we show that a solution of dual problem given by stresses σi j coincides with the solution σi j = σi j (u) given by (5.14)-(5.15). Below we provide rigorous explanations of the procedure. First, we need the deformations in terms of stresses, thus we write Hooke’s law (5.2) in the inverted form A−1 σ = ε (u) in Ω \ ω .

(5.16)

Note that the tensor A−1 enjoys the properties similar to those of A, i.e., it is symmetric and positive definite. Consider the space of stresses H = {σ = {σi j } | σi j ∈ L2 (Ω \ ω ), i, j = 1, 2, 3} and the quadratic functional G defined on H, 1 G(σ ) = (A−1 σ , σ )Ω \ω . 2 The set of admissible stresses is a cone in the space H with the elements which satisfy the sign condition for normal stresses on the crack as well as the global equilibrium condition over the inclusion, thus it is defined as follows M = {σ ∈ H | equations (5.1) and conditions (5.5), (5.7) hold}. The above cone is well defined since equations (5.1) in the definition of M are satisfied in the sense of distributions, and conditions (5.5), (5.7) hold in a weak sense. Let us consider now the dual problem in the form of the minimization problem for quadratic functional over a convex and weakly closed cone, inf G(σ ).

σ ∈M

(5.17)

Under our assumptions there exists a unique solution σ 0 of this problem which satisfies the following variational inequality


81

σ 0 ∈ M, −1

(5.18)

(A σ , σ − σ )Ω \ω ≥ 0 ∀ σ ∈ M. 0

0

(5.19)

5.2.2 Passage from Elastic Inclusion to Rigid Inclusion In fact, problem (5.1)-(5.7) can be considered as a limit problem for a family of elasticity problems with the crack γ formulated in the domain Ωγ . This means that we can construct a family of problems depending on a positive parameter λ such that for any fixed λ > 0 the problem describes the equilibrium state of an elastic body occupying the domain Ωγ with the crack γ . We expect that a rigid inclusion ω is obtained for λ → 0 , i.e., for such a limit any point x ∈ ω has a displacement ρ0 (x), ρ0 ∈ R(ω ). In what follows we provide a rigorous proof of the above statement. Introduce the tensor Aλ = {aλi jkl }, i, j, k, l = 1, 2, 3, aλi jkl =

ai jkl in Ω \ ω λ −1 ai jkl in ω ,

and consider the following problem. In the domain Ωγ , we have to find functions uλ = (uλ1 , uλ2 , uλ3 ), σ λ = {σiλj }, i, j = 1, 2, 3, such that −divσ λ = F

in Ωγ ,

(5.20)

σ − A ε (u ) = 0 in Ωγ ,

(5.21)

λ

λ

λ

λ

u =0

on Γ ,

(5.22)

[uλ ] · ν ≥ 0, [σνλ ] = 0, σνλ [u] · ν = 0

on γ ,

(5.23)

σνλ

≤ 0,

στλ

=0

±

on γ .

(5.24)

Here we use notations of the previous section, and [v] = v+ − v− is a jump of v on γ , where ± fit positive and negative crack faces γ ± with respect to the normal vector ν. For any fixed λ > 0 problem (5.20)-(5.24) is well known and admits a variational formulation ([22], [23], [19]). Indeed, introduce the set of admissible displacements K = {v ∈ HΓ1 (Ωγ )3 | [v] · ν ≥ 0 on γ }, where

HΓ1 (Ωγ ) = {v ∈ H 1 (Ωγ ) | v = 0 on Γ }.

There exists a unique solution uλ of the minimization problem 1 inf { (σ λ (v), ε (v))Ωγ − (F, v)Ωγ } 2

v∈K

(5.25)


82

with the stress field σ λ (v) determined by (5.21) for uλ = v. Solution uλ of the minimization problem satisfies the variational inequality uλ ∈ K,

(5.26)

(σ λ (uλ ), ε (u − uλ ))Ωγ ≥ (F, u − uλ )Ωγ ∀ u ∈ K.

(5.27)

By the convexity of the quadratic functional in (5.25) with respect to v, it follows that problems (5.25) and (5.26)-(5.27) are equivalent. Moreover, all relations (5.20)(5.24) can be obtained from (5.26)-(5.27), and conversely, relations (5.20)-(5.24) imply (5.26)-(5.27). Below we justify the limit passage with λ → 0 in (5.26)-(5.27). Substitute u = 0, u = 2uλ as test functions in (5.27), and sum up the obtained relations. It implies the equality (σ λ (uλ ), ε (uλ ))Ωγ = (F, uλ )Ωγ . (5.28) Assuming that λ ∈ (0, λ0 ), from (5.28) we obtain uλ H 1 (Ωγ )3 ≤ c1 , Γ

1 λ

ai jkl εkl (uλ )εi j (uλ ) ≤ c2

(5.29) (5.30)

ω

with constants c1 , c2 being uniform with respect to λ ∈ (0, λ0 ). Choosing a subsequence, if necessary, it can be assumed as λ → 0 uλ → u weakly in HΓ1 (Ωγ )3 . Then by (5.30)

εi j (u) = 0 in ω , i, j = 1, 2, 3.

This means that a function ρ0 exists such that u = ρ0 in ω ; ρ0 ∈ R(ω ). Since uλ converge weakly in HΓ1 (Ωγ )3 , the limit function u satisfies the inequality (u+ − ρ0 ) · ν ≥ 0 on γ . In particular, u ∈ Kω . Let us take any fixed element u ∈ Kω . Then, there exists ρ ∈ R(ω ) such that u = ρ in ω , and u can be taken as a test function in (5.27). In such a case, inequality (5.27) implies (σ λ (uλ ), ε (u − uλ ))Ωγ ≥ (F, u − uλ )Ωγ . (5.31) By using the equality Aλ = A in Ω \ ω , we can pass to the limit in (5.31) as λ → 0 which implies


83

u ∈ Kω , (σ (u), ε (u − u))Ωγ \ω ≥ (F, u − u)Ωγ ∀ u ∈ Kω , what is precisely (5.14)-(5.15). Hence a passage from the elastic inclusion to the rigid inclusion is shown. We formulate the obtained result as follows Theorem 1. The solution uλ of problem (5.26)-(5.27) weakly converge in HΓ1 (Ωγ )3 to the solution u of problem (5.14)-(5.15). Observe that there is no limit in ω for the stress tensor σ λ with λ → 0.

5.3 Topological Asymptotic Analysis The topological derivative introduced in [37] quantifies the sensitivity of a given shape functional with respect to the introduction of a non-smooth perturbation (hole, inclusion, source term, for instance) in a ball Bδ (x0 ) ⊂ Ω of radius δ > 0 and center at x0 ∈ Ω , that is Bδ (x0 ) = {x ∈ R3 : x − x0 < δ }, Bδ (x0 ) is the closure of Bδ (x0 ). Therefore, this derivative can be seen as a first order correction on the shape functional J (Ω ) to estimate J (Ωδ ), where Ωδ is the perturbed domain. Thus, we have the following topological asymptotic expansion for functional J , J (Ωδ ) = J (Ω ) + f (δ )DT (x0 ) + o( f (δ )) ,

(5.32)

where f (δ ) is a positive function that decreases monotonically such that f (δ ) → 0 when δ → 0+ and the term DT (x0 ) is defined as the topological derivative of J . Then, from (5.32) we have that the classical definition of the topological derivative is given by [31, 38] DT (x0 ) = lim

δ →0

J (Ωδ ) − J (Ω ) 1 d = lim J (Ωδ ) . f (δ ) δ →0 f (δ ) d δ

(5.33)

We point out, that even if formula (5.32) looks very different from the classical shape derivatives currently used in shape optimization, however its nature is the same, since it is defined by [37] in the form of a singular limit of shape derivatives evaluated on boundaries of small voids with respect to the radius of the voids δ → 0. In this way the topological derivative is a generalization of the classical shape derivative in smooth case to the singular boundary perturbations. We refer the reader to [28] for the asymptotic analysis in singularly perturbed domains by means of the matched and compound asymptotic expansions which leads to the topological derivatives of shape functionals in elasticity with complete proofs in general case. On the other hand, from recent developments of shape optimization in fluid dynamics for the drag functional presented in [34], it turns out that the shape derivative of the drag functional can be derived in the form of a singular limit for the appropriate volume integrals, so in the framework of classical shape sensitivity analysis the


84

same principle of singular limit passage can be applied in order to identify the shape gradients. We recall here, with few references, that the topological derivative has been successfully applied in the context of • topology optimization: [2], [1], [6], [10], [11], [32], [33], • inverse problems: [3], [8], [26], • image processing: [4], [5], [16], [24]. Concerning the theoretical development of the topological asymptotic analysis, the reader may refer to [28], for instance. The review of available techniques used for asymptotic analysis with respect to the size of small cavities for the elastic energy functional is presented in [25]. In our particular case, we consider a regular perturbation of the domain given by the nucleation of a small elastic inclusion with Young modulus Eη = η E, where E is the Young modulus of the bulk material and η ∈ [0, ∞) represents the contrast. We assume that there is a small elastic inclusion Bδ (x0 ) in the elastic region Ωω = Ω \ ω . If the elastic inclusion becomes a cavity, it is denoted by ωδ = Bδ (x0 ). The cavity can be obtained from the elastic inclusion by the limit passage η → 0, in the limit case we have a singular perturbation of the domain. In the case of elastic inclusion the elastic region Ωω is decomposed into two disjoint parts Ωω \ Bδ (x0 ) and Bδ (x0 ) with different material properties, namely E and η E, respectively. The other limit passage with the contrast η → ∞ results in the small rigid inclusion ωδ = Bδ (x0 ). See Fig. (5.2). We are also interested in the topological asymptotic expansion of the energy shape functional of the form 1 1 Πδ (v) = (σ (v), ε (v))Ωω \B (x0 ) + (σ (v), ε (v))Bδ (x0 ) − (F, v)Ωγ , δ 2 2

(5.34)

where we have to find function v = uδ such that −divσ = F

in

σ − Aε (uδ ) = 0 σ − Aη ε (uδ ) = 0 [uδ ] = 0 [σ ]ν = 0 uδ = 0

in in

Ω \ ω,

Ωω \ Bδ (x0 ), Bδ (x0 ), on ∂ Bδ (x0 ) on ∂ Bδ (x0 ) on Γ ,

(5.38) (5.39) (5.40) (5.41)

στ = 0, σν ≤ 0 on γ ,

(5.42)

σν (uδ − ρ0 ) · ν = 0 on γ ,

(5.43)

+ +

σν · ρ = Ξ

(5.36) (5.37)

(uδ − ρ0 ) · ν ≥ 0 on γ , +

−

(5.35)

F ·ρ ω

∀ρ ∈ R(ω ).

(5.44)


d

85

x0

R

x0

Fig. 5.2 Domain Ω with rigid inclusion ω and an elastic inclusion Bδ (x0 ).

with A such as before and Aη = η A (since Eη is the Young modulus of the inclusion).

5.3.1 Domain Decomposition Since the problem is non-linear, let us introduce a domain decomposition given by ΩR = Ωω \ BR (x0 ), where BR (x0 ) is a ball of radius R > δ and center at x0 ∈ Ω , that is BR (x0 ) = {x ∈ R3 : x − x0 < R}, BR (x0 ) is the closure of BR (x0 ), as shown in Fig. (5.2). For the sake of simplicity, we assume that F = 0 in BR (x0 ). Thus, we have the following linear elasticity system defined in BR (x0 ) with an inclusion Bδ (x0 ) inside −divσ = 0

in

BR (x0 ),

(5.45)

σ − Aε (wδ ) = 0 σ − Aη ε (wδ ) = 0 wδ = v [wδ ] = 0 [σ ]ν = 0

in

BR (x0 ) \ Bδ (x0 ),

(5.46)

in Bδ (x0 ), on ∂ BR (x0 ),

(5.47) (5.48)

on ∂ Bδ (x0 ), on ∂ Bδ (x0 ).

(5.49) (5.50)

We are interested in the Steklov-Poincaré operator on ∂ BR , that is Aδ : v ∈ H 1/2 (∂ BR ) → σ (wδ )ν ∈ H −1/2 (∂ BR ) .

(5.51)

Then we have σ (uR )ν = Aδ (uR ) on ∂ BR , where uR is solution of the variational inequality in ΩR , that is uR ∈ Kω : aΩR (uR , ϕ − uR) ≥ (F, ϕ − uR)Ωγ \BR (x0 )

∀ϕ ∈ Kω

(5.52)


86

and the bilinear form aΩR is such that aΩR (u, ϕ ) =

σ (u) · ε (ϕ ) +

ΩR

∂ BR

Aδ (u) · ϕ .

(5.53)

Finally, in the disk BR (x0 ) we have

BR \Bδ

σ (w) · ε (w) +

σ (w) · ε (w) = Bδ

∂ BR

Aδ (w) · w ,

(5.54)

where w = wδ is the solution of the elasticity system in the disk (5.45)-(5.50) or equivalently solution of the following variational problem wδ ∈ W :

BR \Bδ

σ (wδ ) · ε (ϕ ) +

Bδ

σ (wδ ) · ε (ϕ ) = 0 ∀ϕ ∈ W0 ,

(5.55)

with W and W0 such that W = {w ∈ H 1 (BR )3 | [w] = 0 on ∂ Bδ , w = v on ∂ BR } ,

(5.56)

W0 = {ϕ ∈ H (BR ) | [ϕ ] = 0 on ∂ Bδ , ϕ = 0 on ∂ BR } .

(5.57)

1

3

5.3.2 Shape Sensitivity Analysis of the Energy Functional Let us introduced the energy-based shape functional defined in the disk BR (x0 ), that is 1 1 Eδ (wδ ) := σ (wδ ) · ε (wδ ) + σ (wδ ) · ε (wδ ) . (5.58) 2 BR \Bδ 2 Bδ We need to calculate d E (w ) = dδ δ δ

BR \Bδ

+ BR \Bδ

σ (wδ ) · ε (w˙ δ ) + Σ (wδ ) · ∇V +

Bδ

Bδ

σ (wδ ) · ε (w˙ δ )

(5.59)

Σ (wδ ) · ∇V ,

which was obtained using the Reynold’s transport theorem and the concept of material derivatives of spatial fields ([14, 41]). Some of the terms in (5.59) require explanation. Vector V represents the shape change velocity field defined on the disk BR (x0 ) and such that V = 0 on ∂ BR and V = ν on ∂ Bδ . Thus, w˙ δ ∈ W0 is the material (total) derivative with respect to δ . Finally, the Eshelby energy-momentum tensor Σ takes the form ([7, 15]) 1 Σ (wδ ) := σ (wδ ) · ε (wδ )I − (∇wδ )T σ (wδ ) . 2

(5.60)


87

Since w˙ δ ∈ W0 and considering that Σ (wδ ) is a free-divergence tensor field (divΣ (wδ ) = 0), the shape derivative of the energy functional becomes d E (w ) = − dδ δ δ

∂ Bδ

[Σ (wδ )]ν · ν .

(5.61)

5.3.3 Topological Derivatives Calculation The topological derivative can be evaluated by a singular limit of shape derivatives. Singular limit in such a case means that the support of of the speed velocity field tends to a point. In other words the shape derivative of a given functional is concentrated on the boundary ∂ Bδ of small void and the passage to the limit with δ → 0 is performed. Since in asymptotics for displacement and stress fields there are specific coefficients in function of the small parameter δ > 0, the precise analysis of asymptotic expansions for the fields provides the factor in (5.62) such that the limit is finite and non trivial. Therefore, the pure shape sensitivity analysis in the framework of the speed method is not sufficient in order to determine the limit which we call the topological derivative. The necessary ingredient for such derivation is the method of matched and compound asymptotic expansions in singularly perturbed domains [28], however the resulting formulas are of the same nature as the shape derivatives in smooth case, which is an important mathematical result. We refer the reader to [34, 35, 36] in the case of compressible fluids and singular limits which govern the shape derivatives of the drag functional. We turn back to the evaluation of topological derivatives for the elasticity boundary value problems with respect to nucleation of small voids or cavities. By introducing (5.61) in (5.33), we have DT (x0 ) = − lim

1

δ →0 f (δ )

∂ Bδ

[Σ (wδ )]ν · ν .

(5.62)

We shall consider here several particular cases: energy functional and spherical cavity, general stress functional and cavity of arbitrary shape, and finally energy functional and inclusion of arbitrary shape.

5.3.3.1 Topological Derivative of the Energy Functional in Three Spatial Dimensions for a Small Cavity In this case we assume that the system corresponds to three dimensional isotropic elasticity and we are interested in the energy change due to the nucleation of a spherical cavity. Thus, for the convenience of the reader we recall here the results derived in [12], [17], [33] for the three dimensional elasticity case.


88

Theorem 2. Let us consider the contrast η → 0. Thus, the elastic inclusion degenerates to a spherical cavity with homogeneous Neumann boundary condition. In this case, the energy shape functional admits for δ → 0 the following topological asymptotic expansion

Πδ (uδ ) = Π (u) + πδ 3DT (x0 ) + o(δ 3 ) ,

(5.63)

with the topological derivative DT (x0 ) given by DT (x0 ) = Hσ (u(x0 )) · ε (u(x0 )) ∀x0 ∈ Ω \ ω ,

(5.64)

where u is solution of the variational inequality (5.14)-(5.15) and H is a forth-order tensor defined as

1−υ 1 − 5υ H= 10I − I⊗I . (5.65) 7 − 5υ 1 − 2υ 5.3.3.2 Topological Derivative of the General Stress Functional for Anisotropic Elasticity and a Small Cavity Let us consider the elasticity problem written in the matrix/column form L u = D(−∇x ) AD(∇x )u = 0 N

Ω ω

u = D(n) AD(∇x )u = g

N u = D(n) AD(∇x )u = 0

Ω

in Ωδ ,

(5.66)

on Ω ,

(5.67)

on ωδ ,

(5.68)

where A is a symmetric positive definite matrix√of size 6 × 6, consisting of the elastic material moduli (the Hooke’s matrix) α = 1/ 2 and D(∇x ) is 6 × 3-matrix of the first order differential operators (ξi = ∂ /∂ xi ), ⎡ ⎤ ξ1 0 0 0 αξ3 αξ2 D(ξ ) = ⎣ 0 ξ2 0 αξ3 0 αξ1 ⎦ (5.69) 0 0 ξ3 αξ2 αξ1 0 u is displacement column, n = (n1 , n2 , n3 ) is the unit outward normal vector on ∂ Ωδ . We assume here that the origin O of the coordinate system lies in ω , O ∈ ω . Then ωδ denotes ω scaled by δ , that is ωδ = δ · ω , ω1 = 1 · ω . In this notation the strain and stress columns are given respectively by ε (u) = D(∇x )u and σ (u) = AD(∇x )u, which gives √ √ √ ε (u) = ε11 , ε22 , ε33 , 2ε23 , 2ε31 , 2ε12 √ √ √ σ (u) = σ11 , σ22 , σ33 , 2σ23 , 2σ31 , 2σ12


89

The load gΩ is supposed to be self equilibrated in order to assure the existence of a solution to the elasticity problem,

∂Ω

where

d(x) gΩ (x)dsx = 0 ∈ IR6

(5.70)

⎡

⎤ 1 0 0 0 − α x3 α x2 0 − α x1 ⎦ d(x) = ⎣ 0 1 0 α x3 0 0 1 − α x2 α x1 0

(5.71)

represents rigid body motion. The theory presented here can be applied to a broad class of shape functionals. Let us consider the functional J1δ (u) =

Ωδ

σ (u; x) B(x)σ (u; x)dx ,

(5.72)

Functional (5.72) looks like the elastic energy functional but can contain a certain symmetric 6 × 6–matrix function B. In the case of constant, diagonal matrix B functional (5.72) is related to square of the L2 (Ω )–norm of the stress tensor or of its components. On the other hand, if A(x)−1 B(x)A(x)−1 becomes a constant diagonal matrix with our choice of B, then in (5.72) the similar strain norms are obtained. From condition (5.70) follows that both problems, problem (5.66)-(5.68) in the body Ωδ with the cavity ωδ , and the first limit problem in the entire body Ω , D(−∇x ) AD(∇x )v = 0 in Ω ,

(5.73)

Ω

D(n) AD(∇x )v = g in ∂ Ω , admit the solutions u(δ , x) ∈ C2,α (Ωδ )3 and v ∈ C2,α (Ω )3 , respectively, under the loading gΩ ∈ C1,α (∂ Ω )3 . Freedom in selection of such solutions up to the rigid motions has no influence on functional (5.72) and therefore can be neglected (using additional conditions we can pass to uniquely solvable problems). The adjoint state w ∈ C2,α (Ω )3 has the form D(−∇x ) AD(∇x )w = − 2D(∇x ) BAD(∇x )v in Ω , D(n)AD(∇x )w =

(5.74)

2D(n)AD(∇x ) BAD(∇x )v on ∂ Ω . Furthermore, we define the special functions z j solving the exterior elasticity problem D(−∇ξ ) AD(∇ξ )z j = 0 in G = IR3 \ ω1 ,

(5.75)

D(n(ξ )) AD(∇ξ )z j = g j on ∂ ω1

(5.76)

with the special right hand sides


90

g j (ξ ) = −D(n(ξ )) Ae j ,

(5.77)

where j = 1, ..., 6 and e j = δ j,1 , ..., δ j,6 is an element of the canonical basis in IR6 . Theorem 3. The following formula holds true, [28] J1ρ (u) = J10 (v) + δ 3 {ε 0 (v) ABAε 0 (v)|ω1 |+ + (ABAD(∇ξ )zε 0 (v), D(∇ξ )zε 0 (v))G + + (ε 0 (w) − 2BAε 0 (v)) mω ε 0 (v)}+

(5.78)

+ O(δ 3+ε ) , where ε 0 (v) = D(∇ξ )v(O) and ε 0 (w) = D(∇ξ )w(O) are strain columns evaluated at the point x = O for the solutions of problems (5.73) and (5.74); mω is the polarization matrix of size 6 × 6 for the cavity ω in the elastic space with the Hooke’s matrix A, and z = (z1 , ..., z6 ) is the row of energy components of the special solutions to homogeneous exterior elasticity problem (5.75)-(5.76). The term standing at δ 3 corresponds to topological derivative. The 6×6 polarization matrix may be computed explicitly using the result given below. Theorem 4. The following integral representation holds true mωjk = AD(∇ξ )z j , D(∇ξ )zk + A jk |ω1 | . G

(5.79)

The results remain true for operators with variable coefficients. For further developments on the shape sensitivity analysis for the polarization matrices which are integral attributes of small cavities required in derivation of topological derivatives, we refer the reader to [30, 29].

5.3.3.3 Topological Derivative of the Energy Functional in Two Spatial Dimensions for a Small Inclusion In two spatial directions we derive also an exterior expansion for the solutions of the variational inequality. Therefore, the result obtained is more precise compared to the general case of the cavity in three spatial dimensions. First, we repeat the model description, and then we develop the asymptotic analysis in linear elasticity to derive the equivalent form of perturbation of the bilinear form. Since in this Section we are dealing with a two dimensional elasticity problem, then the domain Ω ⊂ R2 . Thus, all indices introduced in the Section 5.2 take values from 1 to 2, instead of 1 to 3. In the particular case of plane stress, the Lamé coefficient l = l ∗ , where υE . l∗ = 1 − υ2


91

2

r d

q

1

x0

R

x0

Fig. 5.3 Domain Ω with rigid inclusion ω and an elastic inclusion Bδ (x0 ).

In addition, the crack γ is represented now by a smooth 1D curve described as xi = xi (y),

y ∈ D, i = 1, 2,

with bounded domain D ⊂ R. The space R(ω ) of infinitesimal rigid displacements is redefined simply by setting

0 b and C = (c1 , c2 ); b, ci = const, i = 1, 2. B= −b 0 The displacement field u = (u1 , u2 ); u = ρ0 in ω ; ρ0 ∈ R(ω ); and in the domain Ω \ ω we have to find the stress tensor components σ = {σi j }, solution of (5.1)(5.7) in Ω ⊂ R2 for i, j = 1, 2. Hence, all definitions and results presented in the previous Sections hold. We use the existence of the asymptotic expansions for wδ , solution of the elasticity system (5.45)-(5.50) now defined in the disk BR (x0 ) ⊂ R2 , in the neighborhood of Bδ (x0 ), namely wδ (x) = w0 (x) + w∞ (x) + o(δ ) . (5.80) In addition, w∞ is proportional to δ , w∞ R2 = O(δ ), on the surface ∂ Bρ of the ball. The expansion of σ (wδ ) corresponding to (5.45)-(5.50) has the form

σ (wδ (x)) = σ ∞ (w0 (x0 ), x) + O(δ ) .

(5.81)

where σ ∞ is the stress distribution around a circular inclusion in an infinite medium and w0 is solution of the elasticity system (5.45)-(5.50) defined in the disk BR (x0 ) ⊂ R2 for δ = 0. Thus, σ ∞ can be calculated explicitly and it is given in a polar coordinate system (r, θ ) by:


92

• for r ≥ δ

1−η δ 2 1−η δ 2 1−η + b 1 − 4 1+ σrr∞ (r, θ ) = a 1 − 1+ 2 ηα r ηβ r2 + 3 1+ηβ 1−η δ 2 1−η δ 4 σθ∞θ (r, θ ) = a 1 + 1+ ηα r2 − b 1 + 3 1+ηβ r4 cos 2θ 1−η δ 2 1−η δ 4 sin 2θ σr∞θ (r, θ ) = −b 1 + 2 1+ − 3 2 4 ηβ r 1+ηβ r

δ4 r4

cos 2θ (5.82) (5.83) (5.84)

• for 0 < r < δ

σrr∞ (r, θ ) = 2 1+ηαηα 1−a υ + 4 1+ηβηβ

b 3−υ

cos 2θ

(5.85)

σθ∞θ (r, θ ) = 2 1+ηαηα 1−a υ − 4 1+ηβηβ

b 3−υ

cos 2θ

(5.86)

σr∞θ (r, θ ) = −4 1+ηβηβ

b 3−υ

sin 2θ

(5.87)

In the above formulas, coefficients a and b are given respectively by 1 1 a = (σ1 + σ2 ) and b = (σ1 − σ2 ) , 2 2

(5.88)

where σ1,2 are the eigenvalues of tensor σ (w0 (x0 )). In addition, constants α and β are respectively given by

α=

1+υ 1−υ

and β =

3−υ . 1+υ

(5.89)

The jump condition of the stress field σ (wδ ) can be written as [σ (wδ )]ν = 0 ⇒ [σrr (wδ )] = 0

and [σrθ (wδ )] = 0 on ∂ Bδ .

(5.90)

In the same way, the continuity condition of the displacement field wδ implies [wδ ] = 0 ⇒ [εθ θ (wδ )] = 0 on ∂ Bδ .

(5.91)

The Eshelby tensor flux through the boundary of the inclusion is given by 1 σ (wδ ) · ε (wδ ) − σ (wδ )ν · (∇wδ ) ν 2 1 = (σθ θ (wδ )εθ θ (wδ ) − σrr (wδ )εrr (wδ ) 2

Σ (wδ )ν · ν =

+ σrθ (wδ ) ∂θ wrδ − ∂r wδθ

(5.92)

.

From the jump and continuity conditions on the boundary ∂ Bδ given by (5.90, 5.91) and considering the constitutive relation (5.36)-(5.37) for l = l ∗ , the jump of the Eshelby tensor flux in the normal direction results in (see, for instance, [13])


1 ([σθ θ (wδ )]εθ θ (wδ ) − σrr (wδ )[εrr (wδ )] 2 + 2(1 − δ )σrθ (wδ )εrθ (wδ )) .

[Σ (wδ )]ν · ν =

93

(5.93)

Finally, considering (5.94) in (5.62) and also formulas (5.82)-(5.86) we can calculate the integral on ∂ Bδ explicitly, which allows to identify function f (δ ) = πδ 2 . Then, after calculate the limit δ → 0, we obtain the following result: Theorem 5. The energy shape functional admits for δ → 0 the following topological asymptotic expansion

Πδ (uδ ) = Π (u) + πδ 2DT (x0 ) + o(δ 2 ) ,

(5.94)

with the topological derivative DT (x0 ) given by DT (x0 ) = Hη σ (u(x0 )) · ε (u(x0 )) ∀x0 ∈ Ω \ ω ,

(5.95)

where u is solution of the variational inequality (5.14)-(5.15) in Ω ⊂ R2 and Hη is a fourth-order tensor defined as Hη =

1 (1 − η )2 4 1+βη

1+β α −β 2 I+ I⊗I . 1−η 1 + αη

(5.96)

Corollary 1. Let us consider the contrast η → 0. Thus, the elastic inclusion degenerates to a circular cavity with homogeneous Neumann boundary condition and the tensor H0 becomes H0 =

1 (2(1 + β )I + (α − β )I ⊗ I) . 4

(5.97)

Corollary 2. Let us consider the contrast η → ∞. Thus, the elastic inclusion degenerates to rigid one and the tensor H∞ takes the form

α −β 1+β 1 2 H∞ = − I− I⊗I . (5.98) 4 β αβ Remark 1. From equality (5.54) we observe that the result given by theorem 5 represents the topological derivative of the Steklov-Poincaré operator (5.51). In addition, since solution u ∈ Kω of the variational inequality (5.14)-(5.15) in Ω ⊂ R2 is a H 1 (Ωγ )2 function, then it is convenient to compute the topological derivative from quantities evaluated on the boundary ∂ BR . In particular, we have the following exact analytic representation for the strain tensor ε (u(x0 )) ([40])


94 0.6

0.15

after correction

0.5 0.1

0.4

exact value

uncorrected value 0.05

0.3

after correction

uncorrected value 0.2

0

0.1 −0.05

0

exact value −0.1

0

10

20

30

40

50

60

−0.1

0

10

20

30

40

50

60

Fig. 5.4 Comparison of strains along some section of elastic body: ε11 (left), ε12 (right) – exact values, for k = 0 and with correction corresponding to k = 0.

ε11 + ε22 =

1 π R3

ε11 − ε22 =

1 π R3

2ε12 =

1 π R3

(u1 x1 + u2 x2 ) , (5.99)

12k (1 − 9k)(u1x1 − u2x2 ) + 2 (u1 x31 − u2 x32 ) (5.100) R ∂ BR

12k (1 + 9k)(u1x2 + u2x1 ) − 2 (u1 x32 + u2 x31 ) (5.101) R ∂ BR ∂ BR

where

l∗ + m . l ∗ + 3m In Fig. 5.4 we see the graphs of strains along some, irrelevant here, crossection of the elastic body. They give numerical illustration of the performance of the above formulas. The naive approach (k = 0, uncorrected values in figures) is shown to lead to substantial errors. The terms with k = 0 are needed, because u1 , u2 satisfying elasticity equations are not harmonic. Once the above integrals are evaluated e.g. numerically, then we can use the constitutive relation (5.36) to compute the stress tensor σ (u(x0 )). Finally, these results can by used to compute the topological derivative through formula (5.95). k=

5.3.4 Approximation of Solutions for Variational Inequalities We define a variational inequality for the crack problem with a perturbed bilinear form. The bilinear form is defined in the whole domain of integration, it is bounded and coercive on the energy space for the crack problem without any inclusion, and provides the first order topological sensitivity for the solutions of nonlinear elasticity boundary value problem with the nonlinear crack.


95

Approximation of crack problem in Ωδ . We determine the modified bilinear form as a sum of two terms, as it is for the energy functional, the first term defines the elastic energy in the domain Ω , the second term is a correction term, determined in Section 5.3.3. The correction term is quite complicated to evaluate, and we provide its explicit form, such a form is actually defined by the formulas in Section 5.3.3. The values of the symmetric bilinear form a(δ ; ·, ·) are given by the expression a(δ ; v, v) = a(u, u) + δ 2b(v, v) .

(5.102)

The derivative b(v, v) of the bilinear form a(δ ; v, v) with respect to δ 2 at δ = 0+ is given by the expression b(v, v) = −2π ev (0) −

2π m l ∗ + 3m

σII δ1 − σ12 δ2 ,

(5.103)

where all the quantities are evaluated for the displacement field v according to formulas in Section 5.3.3, where we provide the line integrals which defines all terms in (5.99), (5.100) and (5.101). Hence, we can determine the bilinear form a(δ ; v, w) for all v, w, from the equality 2a(δ ; v, w) = a(δ ; v + w, v + w) − a(δ ; w, w) − a(δ ; v, v) . In the same way the bilinear form b(v, w) is determined from the formula for b(v, v). The convex set is defined in this case by Kδ = {v ∈ HΓ1 (Ωδ )2 | [v]ν ≥ 0 on γ } .

(5.104)

Let us consider the following variational inequality which provides a sufficiently precise for our purposes approximation uδ of the solution u(Ωδ ) to crack problem defined in singularly perturbed domain Ωδ , uδ ∈ Kδ :

a(δ ; u, v − u) ≥ (F, v − u)Ωδ

∀v ∈ Kδ .

(5.105)

The result obtained is the following, for simplicity we assume that the linear form L(δ ; ·) is independent of δ . Theorem 6. For δ sufficiently small we have the following expansion of the solution uδ with respect to the parameter δ at 0+, uδ = u(Ω ) + δ 2 q + o(δ 2) in H 1 (Ω )2 ,

(5.106)

where the topological derivative q of the solution u(Ω ) to the crack problem is given by the unique solution of the following variational inequality q ∈ SK (u) = {v ∈ (HΓ1 (Ωγ ))2 | [v] · ν ≥ 0 on Ξ (u) , a(0; u, v) = 0} a(q, v − q) + b(u, v − q) ≥ 0 ∀v ∈ SK (u) .(5.107)


96

The coincidence set Ξ (u) = {x ∈ γ | [u(x)] · ν (x) = 0} is well defined ([9]) for any function u ∈ H 1 (Ω )2 , and u ∈ K is the solution of variational inequality (5.104) for δ = 0. For the proof of theorem we refer the reader to [39]. For the convenience of the reader we provide the explicit formulas for the terms in b(v, v) defined by (5.103), we refer to section 5.3.3 and to [39], [40] for details. We have

2π ev (0) =

π (l ∗ + m) π 2 R6

ΓR

2 (v1 x1 + v2 x2 ) ds

(5.108)

2 12k m 3 3 (1 − 9k)(v x − v x ) + (v x − v x ) ds 1 1 2 2 1 1 2 2 π 2 R6 R2 ΓR

2 m 12k 3 3 (1 + 9k)(v1x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds + 2 6 , π R R ΓR +

with m σII = π R3

σ12 =

m π R3

12k 3 3 (1 − 9k)(v1 x1 − v2 x2 ) + 2 (v1 x1 − v2 x2 ) ds, R ΓR 12k 3 3 (1 + 9k)(v1x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds, R ΓR

and 9k δ1 = π R3

δ2 =

9k π R3

4 3 3 (v1 x1 − v2 x2 ) − 2 (v1 x1 − v2 x2 ) ds, 3R ΓR 4 (v1 x2 + v2 x1 ) − 2 (v1 x32 + v2 x31 ) ds. 3R ΓR

Acknowledgements This research is partially supported by the Brazilian-French research program CAPES/COFECUB under grant 604/08 between LNCC in Petropolis and IECN in Nancy and by the Brazilian agencies CNPq under grant 472182/2007-2 and FAPERJ under grant E-26/171.099/2006 (Rio de Janeiro). The work is also supported by the grant N51402132/3135 Ministerstwo Nauki i Szkolnictwa Wyzszego: Optymalizacja z wykorzystaniem pochodnej topologicznej dla przeplywow w osrodkach scisliwych in Poland. These supports are gratefully acknowledged.


97

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[21] Khludnev, A., Sokolowski, J.: Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. European Journal on Mechanics A/Solids 19, 105– 119 (2000) [22] Khludnev, A., Sokolowski, J.: On differentation of energy functionals in the crack theory with possible contact between crack faces. Journal of Applied Mathematics and Mechanics 64(3), 464–475 (2000b) [23] Khludnev, A., Sokolowski, J.: Smooth domain method for crack problem. Quarterly Applied Mathematics 62(3), 401–422 (2000c) [24] Larrabide, I., Feijóo, R., Novotny, A., Taroco, E.: Topological derivative: a tool for image processing. Computers & Structures 86(13-14), 1386–1403 (2008) [25] Lewinski, T., Sokołowski, L.: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40(7), 1765–1803 (2003) [26] Masmoudi, M., Pommier, J., Samet, B.: The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Problems 21(2), 547–564 (2005) [27] Morassi, A., Rosset, E.: Detecting rigid inclusions, or cavities, in an elastic body. Journal of Elasticity 72, 101–126 (2003) [28] Nazarov, S., Sokołowski, J.: Asymptotic analysis of shape functionals. Journal de Mathématiques Pures et Appliquées 82(2), 125–196 (2003) [29] Nazarov, S.: Elasticity polarization tensor, surface enthalpy, and Eshelby Theorem. Journal of Mathematical Sciences 159(2), 133–167 (2009) [30] Nazarov, S., Sokołowski, J., Specovius-Neugebauer, M.: Asymptotic analysis and polarization matrices (to appear) [31] Novotny, A., Feijóo, R., Padra, C., Taroco, E.: Topological sensitivity analysis. Computer Methods in Applied Mechanics and Engineering 192(7-8), 803–829 (2003) [32] Novotny, A., Feijóo, R., Padra, C., Taroco, E.: Topological derivative for linear elastic plate bending problems. Control and Cybernetics 34(1), 339–361 (2005) [33] Novotny, A., Feijóo, R., Taroco, E., Padra, C.: Topological sensitivity analysis for threedimensional linear elasticity problem. Computer Methods in Applied Mechanics and Engineering 196(41-44), 4354–4364 (2007) [34] Plotnikov, P.I., Sokołowski, J.: Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations. Journal des Mathématiques Pure et Appliquées 92(2), 113–162 (2009) [35] Plotnikov, P.I., Sokołowski, J.: Inhomogeneous boundary value problems for compressible Navier-Stokes equations, well-posedness and sensitivity analysis. SIAM Journal on Mathematical Analysis 40(3), 1152–1200 (2008) [36] Plotnikov, P.I., Sokołowski, J.: Shape derivative of drag functional (submitted, 2009) ˙ [37] Sokołowski, J., Zochowski, A.: On the topological derivatives in shape optmization. SIAM Journal on Control and Optimization 37(4), 1251–1272 (1999) ˙ [38] Sokołowski, J., Zochowski, A.: Topological derivatives of shape functionals for elasticity systems. Mechanics of Structures and Machines 29(3), 333–351 (2001) ˙ [39] Sokołowski, J., Zochowski, A.: Modeling of topological derivatives for contact problems. Numerische Mathematik 102(1), 145–179 (2005) ˙ [40] Sokołowski, J., Zochowski, A.: Topological derivatives for optimization of plane elasticity contact problems. Engineering Analysis with Boundary Elements 32(11), 900– 908 (2008) [41] Sokołowski, J., Zolésio, J.: Introduction to shape optimization. Shape sensitivity analysis. Springer, New York (1992)

Chapter 6

A Boundary Integral Equation on the Sphere for High-Precision Geodesy Ernst P. Stephan, Thanh Tran, and Adrian Costea

Abstract. Spherical radial basis functions are used to approximate the solution of a boundary integral equation on the unit sphere which is a reformulation of a geodetic boundary value problem. The approximate solution is computed with a corresponding meshless Galerkin scheme using scattered data from satellites. Numerical experiments show that this meshless method is superior to standard boundary element computations with piecewise constants. If we increase the element order, BEM might be competitive but then we also have to approximate appropriately the surface otherwise the convergence rate will be spoiled.

6.1 Boundary Integral Equation We consider the linearized Molodensky problem in geodesy for the disturbing gravity potential u. For a gravity g given at scattered points find u satisfying:

Δu = 0 in Ω = R3 \Ω˜ ∂u 2 = g on the boundary ∂ Ω˜ − u− r ∂r Ernst P. Stephan Institute for Applied Mathematics, Leibniz University Hannover, Welfengarten 1, Hannover, Germany e-mail: [email protected] Thanh Tran School of Mathematics and Statistics, Sydney 2052 , Australia e-mail: [email protected] Adrian Costea Institute for Applied Mathematics, Leibniz University Hannover, Welfengarten 1, Hannover, Germany e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 99–110. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

(6.1)

100

E.P. Stephan, T. Tran, and A. Costea

where ∂ Ω˜ denotes the telluroid and r denotes the radius of x ∈ R3 . In the following we consider the model where the telluroid coincides with the surface of the earth which is given by S, the boundary of the unit ball Ω˜ = B1 (0). As described in the paper by Heck [2] a single layer potential ansatz leads to a pseudodifferential equation (a second kind Fredholm integral equation) on S which will be solved numerically in the following by use of radial basis functions. By inserting the approximation of the density back into the single layer potential we thus obtain an approximation for the gravity potential u, the solution of the above geodetic boundary value problem GBVP (6.1). In detail we proceed as follows: We write u as single layer potential S with unknown density μ for X ∈ /S u(X) = S μ (X) =

1 4π

μ (y) ds(y) |X − y| S

(6.2)

and compute 1 1 (grad S μ (x))+ = − μ (x)nx − p.v. 2 4π

x−y μ (y)ds(y) 3 S |x − y|

where + denotes the limit on the surface S from the exterior domain Ω and n is the normal on S pointing into Ω . Furthermore

∂ (S μ ) 1 x 1 (x) = gradS μ (x)· = − μ (x)cos (nx , x)− p.v. ∂r |x| 2 4π and

2 2 1 S μ (x) = r |x| 4π

(x − y) · x μ (y)ds(y) |x||x − y|3 S

μ (y) ds(y). |x − y| S

Inserting these expressions into the boundary condition of (6.1) yields (see [2])

∂ (S μ ) 1 2 (x) = μ (x)cos (nx , x)+ − S μ (x) − (6.3) r ∂r 2 (x − y) · x 2 1 − μ (y)ds(y) + p.v. 3 4π |x||x − y| S |x||x − y| 1 = μ (x) cos(nx , x)+ 2 |x|2 − |y|2 − 3|x − y|2 1 μ (y)ds(y) + p.v. 4π 2|x||x − y|3 S = g(x)

6 A Boundary Integral Equation on the Sphere for High-Precision Geodesy

101

This becomes

μ (y) 1 1 1 dsy + μ (x) + Lμ (x) : = 2 − 4π S ||x − y|| 2 4π −3μ (y) 1 1 dsy = μ (x) + p.v. 2 4π S 2||x − y|| = g(x)

(x − y) · x μ (y)dsy 3 S ||x − y|| (6.4)

Now we observe that the action of the above pseudodifferential operator L can be rewritten via its discrete symbol Lˆ and the Fourier coefficients μˆ l,m of μ with respect to spherical harmonics Yl,m of degree l as Lμ =

∞

l

∑ ∑

ˆ μˆ l,mYl,m L(l)

l=0 m=−l

ˆ = l−1 , and μˆ l,m = μ ,Yl,m L (S) ( compare (6.15) below). As a with symbol L(l) 2l+1 2 pseudodifferential operator of order zero, L maps H s into itself for any s ∈ R where the Sobolev space H s is defined by $ % Hs =

∞

v : S → R| ∑

l

∑

(l + 1)2s |vˆl,m |2 < ∞ .

l=0 m=−l

Let us abbreviate the pseudodifferential equation (6.4) as Lμ = g We observe that

on S.

(6.5)

ˆ = l − 1 = 0 for l = 1. L(l) 2l + 1

So ker (L) = span {Y1,m , m = −1, 0, 1}. Therefore to ensure unique solvability of (6.5) we must impose side conditions :

γ jμ = aj

( j = 1, 2, 3)

(6.6)

with given a j ∈ R and a unisolvent set of linear functionals {γ j }, i.e., for any v ∈ ker(L), if γ j v = 0, j = 1, 2, 3, then v = 0. Application of classical Riesz-Schauder theory gives the following theorem. Theorem 1. Equations (6.5) and (6.6) have a unique solution if g, Φ = 0 ∀Φ ∈ kerL. Next we comment on the above side conditions. First let us rewrite the single layer potential as μ (y) 1 S μ (X) = ds(y) (6.7) 4π S 2 sin ψ2

102


where ψ is the angle between X and y (c.f. [2]). We expand the disturbing potential outside the boundary sphere S into solid spherical harmonics as

n+1 1 ∑ r un(x), n=0 ∞

u(X) = where

X = r · x, r > 1,

(6.8)

n

un (x) =

∑

uˆn,mYn,m (x).

m=−n

Correspondingly we expand the functions g(x) and μ (x) in surface spherical harmonics as ∞

g(x) =

∑ gn(x),

n=0

∞

∑ μn (x).

(6.9)

n+1 1 un (x). r

(6.10)

μ (x) =

n=0

Note that due to (6.2) 1 4π

μn (y) ψ ds(y) = 2 S sin 2

Furthermore with (6.7) the integral equation (6.4) becomes 1 1 μ (x) + 2 4π

−3μ (y) ψ ds(y) = g(x). S 4 sin 2

(6.11)

Then inserting (6.9), (6.10) and (6.8) and equating the coefficients we obtain

μn (x) − 3r−n−1un (x) = 2gn (x)

(6.12)

Setting r = 1 we have

μ0 (x) − 3u0(x) = 2g0 (x) for n = 0, μ1 (x) − 3u1(x) = 2g1 (x) for n = 1. Therefore with u1 (x) = ∑1m=−1 uˆ1,mY1,m (x) and μ1 , g1 correspondingly we obtain

μˆ 1,m − 3uˆ1,m = 2gˆ1,m ,

m = −1, 0, 1.

(6.13)

ˆ Next we comment on the discrete symbol L(l) of the pseudodifferential operator L. This can be computed by simply inserting the expansion for u and g into the boundary condition of the GBVP (6.1). From

2 ∂u − u− =g r ∂r S


103

we have

∞ ∞ 2 ∞ 1 n+1 1 − ∑ un (x) + ∑ (n + 1) n+2 un (x) = ∑ gn (x) r n=0 r r n=0 n=0 or

∞

∞ (n − 1) un (x) = ∑ gn (x). n+2 n=0 r n=0

∑

Equating the coefficients and then inserting in (6.12) gives with r = 1 3 gn (x) = 2gn (x) n−1

(6.14)

3 2n + 1 gn (x) = gn (x) n−1 n−1

(6.15)

μn (x) − hence

μn (x) = 2 +

from which we deduce the result on the discrete symbol Lˆ of the pseudodifferential operator L ( compare Heck [2]).

6.2 Meshless Galerkin Method with Boundary Integral Equations The solutions of (6.5), (6.6) are approximated by spherical radial basis functions. Let $ (1 − r)m+2 , 0 < r ≤ 1, , m = 0, 1, 2 ρm (r) = 0, r > 1. √ We define φ : [−1, 1] → R by φ (t) = ρm ( 2 − 2t). This is called the Wendland function. For a set of data points {x1 , x2 , . . . , xN } on the sphere we define a set of sperical radial basis functions as:

Φi (x) :=

∞

n

∑ ∑

φˆ (n)Yn,m (xi )Yn,m (x) = φ (x · xi )

n=0 m=−n

with Fourier-Legendre coefficients

φˆ (n) = 2π

1 −1

φ (t)Pn (t)dt,

and Legendre polynomials Pn of degree n. It is shown in [5, Proposition 4.6] that for m = 0, 1, 2 c1 (1 + n2)−m−3/2 ≤ φˆ (n) ≤ c2 (1 + n2)−m−3/2 for all n = 0, 1, 2, . . ., where c1 and c2 are positive constants.

104


Now the numerical scheme under consideration is the Galerkin method (G): Find μ˜ N = μ˜ 1 + μ˜ 0 with μ˜ 1 = ∑Ni=1 ci Φi∗ Lμ˜ 1 , Φk∗ L2 (S) = g, Φk∗ L2 (S) ,

1≤k≤N

where Φi∗ is obtained from Φi by deleting the Fourier terms for n = 0 and n = 1. The part μ˜ 0 of the Galerkin solution μ˜ N must satisfy the side conditions

γ 0 μ˜ 0 =

S

μ˜ 0Y0,0 = a0 ,

γ 1 μ˜ 0 =

S

μ˜ 0Y1,−1 = b−1 , (6.16)

γ 2 μ˜ 0 =

S

μ˜ 0Y1,0 = b0 ,

γ 3 μ˜ 0 =

S

μ˜ 0Y1,1 = b1 .

Note that the term for n = 0 in the expansion for Φi is dropped to make all Fourier modes in the entries of the Galerkin stiffness matrix positive. The side conditions γ 1 , γ 2 , γ 3 are just the side conditions (6.6), whereas the side condition γ 0 is needed since we dropped the expansion term for n = 0 in Φi . Theorem 2. For N sufficiently large the Galerkin scheme (G) is uniquely solvable and the Galerkin solution μ˜ N = μ˜ 1 + μ˜ 0 converges to the exact solution μ ∈ H s , 0 ≤ s ≤ m + 3, ||μ − μ˜ N ||H 0 = O(hs ), where H 0 = L2 (S). Proof. The convergence of the Galerkin scheme follows due to the fact that the pseudodifferential operator L can be written as identity plus a compact operator. Therefore it is a strongly elliptic operator in the sense of Wendland [9]. Due to the general convergence results in Stephan and Wendland [6] strong ellipticity together with the side conditions guarantee convergence of general Galerkin schemes including the case of radial basis functions considered here.The convergence estimate follows by applying a coresponding approximation result of functions in the Sobolev space H s by radial basis functions from [4] together with the quasi optimality of the Galerkin error. The latter quasi optimality follows from the analysis in [6]. Next we comment on the computation of the Galerkin stiffness matrix and the right hand side. Note that LΦi∗ , Φ ∗j L2 (S) = =

∞

n−1 & |φ (n)|2Yn,m (xi )Yn,m (x j ) 2n + 1 n=2 m=−n n

∑ ∑

n−1 & |φ (n)|2 Pn (xi · x j ), 4π n=1,n=2

∑

where we have used '∗ )n,m = φˆ (n)Yn,m (xi ) (Φ i

(6.17)


105

and the addition theorem Pn (x · y) =

n 4π Yn,m (x)Yn,m (y). ∑ 2n + 1 m=−n

For the right hand side we have g, Φk∗ L2 (S) =

∞

n

∑ ∑

' φˆ (n)Yn,m (xk ). (g) n,m

(6.18)

n=0 m=−n

6.3 Numerical Example In the following we compute approximations of the Galerkin solution μ˜ N of (G) by truncating the series expansion of Φi∗ at n = 500. We consider u(X) =

1 , ||X − p||

p = (0, 0, 0.5)

and compute the right hand side via g = −2u −

∂u ∂n

on S.

In the side conditions (6.16) we choose √ √ a0 = 2 π , b1 = b−1 = 0, b0 = 3π . This gives

μ˜ 0 = a0Y0,0 + b−1Y1,−1 + b0Y1,0 + b1Y1,1 ( ( √ 1 √ 3 3 =2 π + 3π cos θ = 1 + cos θ . 4π 4π 2 In Table 6.1 we have listed |(uN (q) − u(q))| for q = (1.10227, 1.10227, 0.9) with uN (q) :=

1 4π

μ˜ N (y) ds(y) S ||q − y||

where μ˜ N is the Galerkin solution of our Galerkin system (G) computed with the Wendland radial basis functions [8] mentioned in Section 6.2, namely √ φ (t) = (1 − 2 − 2t)m+2 .

106


We denote this Galerkin approximation “meshless” in Table 6.1 to distinguish from the standard boundary element solution computed with piecewise constants on the uniform mesh of Fig. 6.2. In the tables below we list the respective experimental orders of convergence for the pointwise error of the gravity potential at the point q outside of the unit sphere S. Table 6.1 Meshless Galerkin approximation uN of gravity potential u; single layer density computed by spherical radial basis functions centered at N Saff points

m=0

m=1

m=2

N =number of Value points

|(uN (q) − u(q))|

EOC

200 500 1000 2000 4000 8000

0.69442 0.67349 0.65323 0.62413 0.62213 0.62198

0.07305 0.05212 0.03186 0.00276 0.00076 0.00061

0.36844 0.71009 3.52900 1.86060 0.31691

200 500 1000 2000 4000 8000

0.79285 0.62781 0.62731 0.62326 0.62228 0.62197

0.17148 0.00644 0.00594 0.00189 0.00091 0.00059

3.58177 0.11660 1.65208 1.05445 0.60814

200 500 1000 2000 4000 8000

0.71406 0.62638 0.62603 0.62283 0.62192 0.62156

0.09269 0.00501 0.00466 0.00146 0.00055 0.00019

3.18439 0.10448 1.67436 1.40846 1.53343

The numerical experiments are performed on a uniform grid of Saff points c.f. Fig.6.1. The errors are plotted in Fig.6.3. For comparison we present here also the numerical experiments when solving approximately the integral equation (6.4) with standard BEM when using piecewise constant basis functions on triangles which approximate the surface c.f. Fig.6.2. Table 6.2 shows the corresponding results ( again q = (1.10227, 1.10227, 0.9)). In Fig.6.3 we see that radial basis functions give better convergence than standard BEM. Since data are not available at the poles we must use for the standard BEM with piecewise constants the grid shown in Fig. 6.4 with holes at the poles. Table 6.3 and Table 6.4 and Fig. 6.4 show clearly that scaled radial basis functions give much better results than standard boundary elements.


107

Fig. 6.1 Uniformly distributed Saff points on S (N = 1000 Saff points), c.f. [1]

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8 1 −1 1

0.5 0.8

0.6

0 0.4

0.2

0

−0.5 −0.2

−0.4

−0.6

−0.8

−1 −1

Fig. 6.2 Boundary element mesh consisting of triangles with vertices at N = 1000 Saff points

108


Table 6.2 Standard BEM Galerkin approximation uN of gravity potential u; single layer density computed by pw. constants on triangles with N Saff points N =number of points

Value

|(uN (q) − u(q))| EOC

500 1000 2000 4000 8000

0.61808 0.61932 0.62010 0.62028 0.62051

0.00329 0.00205 0.00127 0.00109 0.00086

0.68246 0.69080 0.22050 0.34192

0.1 m=0 m=1 m=2 Saff pw.const.

|(u_n-u)(q)|

0.01

0.001

0.0001 100

1000 Number of points

10000

Fig. 6.3 Pointwise error |(uN (qq) − u(qq))| μ˜ N computed with radial basis functions (m = 0, 1, 2) and piecewise constants in the Saff points in Fig 6.1 and Fig 6.2. Table 6.3 Meshless Galerkin approximation with spherical radial basis functions at scattered points on S Number of points

Value

|(uN (q) − u(q))| EOC

m=0 scale=20.5

2133 3458 4108 7663 10443

0.62932 0.62507 0.62288 0.62254 0.62167

0.00795 0.00369 0.00151 0.00117 0.00030

1.75153 6.74972 0.40891 1.40226

m=0 scale=20

2133 3458 4108 7663 10443

0.62915 0.62471 0.62241 0.62203 0.62112

0.00778 0.00334 0.00104 0.00066 0.00024

1.58290 5.18667 0.74527 3.20844


109

Table 6.4 BEM Galerkin approximation with pw. constants on triangles with vertices at scattered points Number of points

Value

|(uN (qq) − u(qq))| EOC

2133 7699 10643

0.67059 0.64751 0.63797

0.04922 0.02614 0.01660

0.49303 1.40226

0.1 m=0,scale=20 m=0,scale=20.5 pw

|(u_n-u)(q)|

0.01

1 0.8 0.6

0.001

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

0.0001 1000

10000 Number of points

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

100000

Fig. 6.4 Pointwise error |(uN (qq) − u(qq))| for μ˜ N computed on N=3458 scattered points with scaled radial basis functions or piecewise constants

110


6.4 Conclusion We have shown that for geodetic boundary value problems the reduction to boundary integral equations leads to a fast numerical method when the Galerkin scheme is performed with spherical radial basis functions. It is a meshless method and therefore it can be applied efficiently to problems where the satellite data are given at scattered points. Here standard boundary elements are not appropriate. As we have shown in [3] the approach can be extended to spheroids and is not restricted to the unit sphere as might be expected since in our method the density of the integral equation and also the Galerkin solution are expanded in spherical harmonics. As numerical experiments in [7] have shown the overlapping additive schwarz method can be applied as an efficient preconditioner to reduce the computing time for the numerical method.

Acknowledgements The authors Ernst P. Stephan and Adrian Costea would like to acknowledge the financial support provided by Excellence Cluster QUEST (Centre for Quantum Engineering and Space-Time Research), Leibniz University Hannover.

References [1] Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices of the AMS 51, 1186–1194 (2004) [2] Heck, B.: Integral Equation Methods in Physical Geodesy. In: Grafarend, E.W., Krumm, F.W., Schwarze, V.S. (eds.) Geodesy - The Challenge of the 3rd Millenium, pp. 197–206. Springer, Heidelberg (2002) [3] Le Gia, Q.T., Tran, T., Stephan, E.P.: Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions. Advances in Computational Mathematics (submitted) [4] Le Gia, Q.T., Tran, T., Sloan, I.H., Stephan, E.P.: Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis. Applied Numerical Mathematics (in print), http://dx.doi.org/10.1016/j.apnum.2008.12.033 [5] Narcowich, F.J., Ward, J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math Anal. 33, 1393–1410 (2002) [6] Stephan, E.P., Wendland, W.L.: Remarks to Galerkin and Least Squares Methods with Finite Elements for General Elliptic Problems. Manuscripta geodaetica 1, 93–123 (1976) [7] Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Preconditioners for Pseudodifferential Equations on the Sphere with Radial Basis Functions. Numer. Math. (to appear) [8] Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Advances in Computational Mathematics 4, 389–396 (1995) [9] Wendland, W.L.: Asymptotic accuracy and convergence. In: Brebbia, C.A. (ed.) Progress in Boundary Element Methods, vol. 1, pp. 289–313. Pentech Press, London (1981)

Chapter 7

Unresolved Problems of Adaptive Hierarchical Modelling and hp-Adaptive Analysis within Computational Solid Mechanics Grzegorz Zboiński

Abstract. In this chapter of the book we present some chosen problems of adaptive hierarchical modelling and adaptive hp-analysis of problems of computational solid mechanics. We consider simple and complex structures, i.e. structures described by one mechanical model or more than one mechanical model, respectively. We are interested in three fundamental problems of solid mechanics: the equilibrium (static) problem, eigenvalue (free vibration) problem, and stationary forced vibration problem as well. In the context of static analysis, we consider problems faced while applying: the hierarchical models, hp-approximations, residual error estimation methods, and three- or four-step non- or iterative adaptive strategies, oriented on the target admissible value of the modelling and approximation errors. We also address possibilities and difficulties of generalization of the mentioned techniques onto free and forced vibration analyses. In this contribution we are interested mainly in still unresolved or open problems. We will show some ways, either potentially available or checked by us numerically, to cope with all the mentioned issues.

7.1 Introduction In this book chapter we would like to discuss some theoretical and implementation difficulties of the adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics. We present these difficulties in the context of our uniform methodology that can be applied to the equilibrium, free vibration and stationary forced vibration problems. This methodology [19] can be applied to both, simple or complex structures, i.e. structures of simple or complex mechanical Grzegorz Zboiński Polish Academy of Sciences, Institute of Fluid Flow Machinery, ul. Fiszera 14, 80-952 Gdańsk, Poland e-mail: [email protected] University of Warmia and Mazury, Faculty of Technical Sciences, ul. Oczapowskiego 11, 10-736 Olsztyn, Poland


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description, based on one or more mechanical models, respectively. These two descriptions can be applied within both, simple and complex geometries. Our simple geometries, as well as different parts of complex geometries, can be of either solid, shell, or transition character. Note that our approach is much more general than the approaches of the predecessors, who usually consider one problem of solid mechanics, single mechanical model, and one type of the applied geometry. In other words, we search for one generalizing approach, suitable for a wide range of problems, rather than for specialized approaches, assigned for the specific mechanical problems.

7.1.1 Considered Problems of Solid Mechanics As mentioned above, we consider three fundamental problems of solid mechanics: the static equilibrium problem, and the dynamic problems of free and stationary forced vibrations.

7.1.1.1 Static Equilibrium Problem Let us start with the equilibrium problem. In this case we search for the static solution in displacements u = u(x). The local formulation within the elastic body V under consideration (Fig. 7.1) is

σi j, j + f0i (x) = 0, εi j = 1/2(ui, j + u j,i ), σi j = Di jkl εkl , x ∈ V,

(7.1)

where Di jkl , i, j, k, l = 1, 2, 3 is the elastic constants tensor, σi j and εkl are stress and strain tensors, f0i is the vector of static body forces, while ui is the ith component of the vector u. The above set has to be completed with the traction and kinematic boundary conditions on parts SP and SD respectively, of the body surface S

σi j n j = p0i , x ∈ SP , ui = d0i , x ∈ SD ,

(7.2)

with n j denoting the vector normal to this surface, p0i standing for the static surface tractions, and d0i being the components of the prescribed surface displacements. In order to obtain the finite element equations of the problem we have to take advantage of the variational formulation corresponding to the above local formulation. The former formulation reads Di jkl vi, j uk,l dV = V

vi f0i dV + V

vi p0i dS, SP

(7.3)

7 Unresolved problems of hierarchical modelling...

113 f

0

SD SP

V

M M

Fig. 7.1 The elastic body under consideration (the static case)

p

0

with vi standing for the admissible values of the trial displacement functions, conforming to the kinematic boundary conditions. The vector form of the finite element formulation, obtained from (7.3) after the discretization and the introduction of the hpq-interpolation shape functions, is Kqhpq = F0 ,

(7.4)

where K is the global nodal stiffness matrix, F0 is the global nodal static forces vector, while qhpq stands for the global nodal vector of the displacement dofs correM sponding to the hpq finite element formulation (see the next subsections).

7.1.1.2 Free Vibration Problem In the case of the free vibration problem (eigenproblem) of the elastic body, the solution, we search for, takes the form u = a(x)e jω0t , where a(x) represents free vibration amplitudes, ω0 is the natural frequency of the body, t is the time variable, while j is the imaginary unit. The local formulation reads now ..

σi j, j = ρ ui , εi j = 1/2(ui, j + u j,i ), σi j = Di jkl εkl , x ∈ V,

(7.5)

..

jω t with ρ standing for the material density and ui = −ω02 ai 0 being the second time derivatives of the displacements (the acceleration components). The usual boundary conditions for the free vibration problem are

σi j n j = 0, x ∈ SP , ui = 0, x ∈ SD ,

(7.6)

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The corresponding variational formulation, necessary for derivation of the finite element equations, and expressed through the amplitudes, is

V

Di jkl vi, j ak,l dV − ω02

ρ vi ai dV = 0

(7.7)

V

In the above relation, the values vi represent the kinematically admissible values of the amplitudes, conforming to the boundary conditions (7.6) expressed through the amplitudes. Note, that now we search for the all eigenpairs (eigenvalues and eigenvectors), ω02 and a(x), fulfilling (7.7). In order to obtain the set of the finite element method relations, resulting from the above variational formulation, one has to perform division of the body into finite elements and introduce the hpq-interpolation. This leads to the discretized eigenproblem of N degrees of freedom. The characteristic equation, necessary for deter2 , n = 1, 2, ..., N, is mination of N eigenvalues ω0n det(K − ω02 M) = 0

(7.8)

where M stands for the global mass (inertia) matrix. The dynamic equilibrium equation, necessary for determination of the nth eigenvector of amplitudes is 2 (K − ω0n M)qn

hpq

hpq T hpq qn Mqn

= 0,

=1

(7.9)

So as to obtain the unique values of the eigenvectors, the first equation (7.9) has been completed with the normalization condition (the second equation (7.9)), here corresponding to the M-orthonormal normalization. In the above relations the amplitude hpq dofs vector qn has been employed. Note that the displacements corresponding to nth mode of vibration can be calculated form the amplitudes and the natural frejω t quencies as qhpq n e 0n .

7.1.1.3 Forced Vibrations We consider here the stationary forced vibration problem for which the superposition theorem is valid. In the case of a single force of vibration the local formulation reads .

..

σi j, j + α ρ ui + fi (x,t) = ρ ui , εi j = 1/2(ui, j + u j,i ), . σi j = Di jkl εkl + β ε i j , x ∈ V,

(7.10)

where the solution vector for displacements is u = a(x)e j(ω t+φ +ϕ ) , while the given body force is of the form f(x,t) = f0 (x)e j(ω t+φ ) , with f0 being the force amplitude vector, ω and φ standing for the given force frequency and force phase angle, and


115

ϕ denoting unknown phase angles of the vibration amplitudes. The second and first time derivatives of the displacement vector (the acceleration and velocity vectors) .. . are .ui = −ω ai e j(ω t+φ +ϕ ) , ui = jω ai e j(ω t+φ +ϕ ) , while the strain velocity is equal to ε i j = jω εi j . The coefficients α and β characterize the material dumping due to viscous internal friction and can be treated as equivalent to Rayleigh dumping coefficients. In the case of forced vibrations, the form of the surface tractions is p(x,t) = p0 (x)e j(ω t+φ ) , with p0 standing for the traction amplitude vector. The related traction boundary conditions, as well as the kinematic boundary conditions, are σi j n j = pi (x,t), x ∈ SP , ui = 0, x ∈ SD ,

(7.11)

The variational formulation for the stationary forced vibration problem, corresponding to (7.10) and (7.11), is Di jkl vi, j ak,l dV + jω V

(β Di jkl vi, j ak,l + α ρ vi ai ) dV − ω 2 V

vi f0i e− jϕ dV +

= V

ρ vi ai dV V

vi p0i e− jϕ dS,

(7.12)

SP

with vi representing the kinematically admissible field of the displacement amplitudes. Note that the solution to (7.12) is searched in the complex functions domain. The physical unknowns of the problem are the amplitudes and their phase angles. After performance of the discretization and hpq-interpolation of the filed of unknowns, the matrix form of the finite element equations, corresponding to the above variational formulation, and assigned for any of m (m = 1, 2, ..., M) independent forces of vibration, is (K + jωm C − ωm2 M)qm = F0m e− jϕm , hpq

(7.13)

where F0m is the global vector of the amplitudes of the mth force, while the global dumping coefficient matrix C is composed of two components, i.e. C = β K + α M. The total solution in displacements can be obtained with the superposition theorem, hpq taking advantage of m consecutive solutions for nodal amplitudes qm and phase hpq j(ωm t+φm +ϕm ) . angles ϕm . The mth displacement contribution is qm e

7.1.2 Considered Types of Complexity within the Elastic Bodies In order to retain the general character of the analysis, we allow geometrical and mechanical complexity of the elastic bodies, analyzed in three solid mechanics problems described in the previous subsections.

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solid geometry

transition geometry mid-surface shell geometry top surface mid-surface

bottom surface Fig. 7.2 Different types of the geometry

7.1.2.1 Geometrical Complexity Let us start with exemplary simple geometries. Three of such geometries, the solid, shell, and transition ones, are shown in Fig. 7.2. The solid geometry is bounded with surfaces. The shell geometry is based on the mid-surface and thickness concepts. The transition geometry is partly defined with the equations of the bounding surfaces and partly described by means of the mid-surface equation and the thickness vector normal to this surface. The numerical representation of the simple shell geometry (the plate example) is shown in Fig. 7.4. Let us pass to complex geometry of the body. We deal with such geometry when more than one type of geometry is necessary for the description of the shape of the body. Such a situation is presented in Fig. 7.3, where the solid, shell, and transition geometries are employed within one body. The numerical representation of the complex geometry can also be seen in Fig. 7.5. A thorough mathematical description and the specific definitions of the above mentioned simple and complex geometries can be found in our work [13].

7.1.2.2 Mechanical Complexity The structure of simple mechanical description is characterized with the application of one mechanical model. The complex mechanical description is a result of the necessity of application of more than one mechanical model within the structure. The application of one or more than one model refers to both, simple and complex geometries, of course. The typical situation for solid mechanics problems, where the solid, shell, and transition mechanical models are employed, is presented in


117

solid geometry transition geometry

shell geometry 3D-model

Fig. 7.3 Complex mechanical description

transition model shell model

Fig. 7.3. It can be seen from the figure that the division of the structure into zones (domains) of the different mechanical description is independent of the division into geometrical parts (members). It is obvious, however, that some models do not appear in certain parts, e.g. shell models are not suitable for solid parts of the body. The admissible neighbourhood of the different models and the appearance of the models in the different geometrical parts are explained in detail in [13]. The relations between the simple and complex geometries and the simple and complex mechanical descriptions are also illustrated in Fig. 7.4 and Fig. 7.5, where the complex mechanical models are applied within the simple and complex geometries, respectively. The first of these two figures shows the numerical representation of the symmetric quarter of the clamped plate, while in the second figure the numerical idealization of the symmetric quarter of the square floor supported by four columns is displayed. The models applied in both examples are: the theory of three-dimensional elasticity (3D) or the hierarchical shell models (MI) of the transverse order I (I = 1, 2, 3, ...), the first-order Reissner-Mindlin shell model (RM), and the solid-to-shell transition models, (3D/RM) or (MI/RM). The description of these models can be found in our work [13].

7.2 Assessed Methodology In this section we would like to present the assessed methodology for the adaptive modelling and adaptive analysis of the simple and complex structures within solid mechanics. Our methodology covers the 3D-based hierarchical modelling and the hierarchical hp-, hpq-, and hpq/hp-approximations, which all allow generating the hierarchy of numerical finite element models of the local character. The methodology includes also the error estimation with the equilibrated residual method (ERM), and the error-controlled adaptivity, based on the three- or four-step adaptive strategies, with possible iterations within the h- and p-steps.

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model

3D MI

260 250 180

280

170 100 90 20

270 200 190

120

10

300

110 40

290 220 210

140

30

320 310 240

130 60

RM/3D RM/MI

230 160

50

150 80

RM

70

z x

y

Fig. 7.4 Numerical representation of the complex mechanical description within simple geometry

7.2.1 3D-Based Hierarchy of Numerical Models In this subsection we discuss three major components necessary for generation of our 3D-based hierarchy of numerical models assigned for the adaptive hierarchical modelling and hpq-adaptive analysis of the simple and complex structures within computational solid mechanics. The first component is the 3D-based approach which lies in the application of the same 3D degrees of freedom for any mechanical model. The second component is the hierarchical modelling, including such different mechanical models as: the 3D-elasticity, the higher-order hierarchical shell models, the first-order shell model, and the solid-to-shell or shell-to-shell transition models. The third component is the set of hpq hierarchical approximations applied to all models of the hierarchy of mechanical models. The combination of theses three components leads to the mentioned hierarchy of numerical models. Note that the hierarchical numerical models can be applied locally, i.e. on the finite element level. The idea is illustrated in Fig. 7.6, where in each element e of the vole

e

ume V , a different model M from the set of mechanical models M, the different size e e e h, and the different longitudinal and transverse approximation orders, p and q, can be chosen.


119

model

3D MI

160

150

140

180

170

130 100 90

RM/3D RM/MI

120 20 110 10

RM

z x

y

Fig. 7.5 Numerical representation of the complex mechanical description within complex geometry

SD SP

e

e

e e

M∈M, h, p, q e

V

Fig. 7.6 The idea of hierarchical numerical modelling

7.2.1.1 3D-Based Approach The idea of the 3D-based approach lies in application of only one type of degrees of freedom within all mechanical models included into the hierarchy of models. In our case only the three-dimensional degrees of freedom (three displacements at a point of the body) are employed, regardless of the model type. The conventional mid-surface displacements and rotations of the first-order shell model, as well as the generalized mid-surface dofs of the higher-order shell models are replaced with the equivalent through-thickness dofs. One more difference between both approaches is the direction of the dofs. In the conventional case we usually apply the local

120

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thickness

thickness

top

ξ' 3 1

d' 2j d' 0j

x'3

d' 3j

top

d'3n middle

x'2

d'2n

middle

u' j

d' 1j

ξ' 30 d' 0j ξ'31d' 1j

0

ξ'32 d' 2j

d'1n

bottom

u' j (ξ' 3 )

ξ'33 d' 3j x'1

-1

bottom − d' 1j

− d' 3j

d' 2j

d' 0j

Fig. 7.7 Mid-surface dofs (left) and the displacement field (right) of the conventional approach

directions, normal and tangent to the mid-surface, while in the case of the throughthickness dofs the global directions are employed. The equivalence of the local displacement fields uj , j = 1, 2, 3 in the cases of the mid-surface and through-thickness dofs is shown in Fig. 7.7 and Fig. 7.8. This equivalence can also be expressed through the relation uj =

I

∑ ξ3

n=0

n n d j

I

=

∑ fn (ξ3 ) u j

n

(7.14)

n=0

In the figures and the equation (7.14) the local directions are denoted as xj , j = 1, 2, 3, while the global ones as xi , i = 1, 2, 3. In the case of the third local direction we also introduce the auxiliary dimensionless coordinate ξ3 = 2x3 /t, where t is the thickness of the shell. The corresponding nth mid-surface dof in the jth local direction is d nj , while the nth through-thickness dof in the local direction j and the global direction i are u nj and uni , respectively. Note that the numbering of dofs in both cases is n = 0, 1, 2, ..., I, where I is the order of the transverse displacement field, equivalent to the order of the applied shell theory. More details on the 3Dbased approach can be found in our works [13, 21]. 7.2.1.2 Hierarchical Modelling Our methodology utilizes the 3D-based adaptive hierarchical models M for complex structures, including the first-order Reissner-Mindlin (RM) shell theory, the hierarchical higher-order shell theories MI of order I, the three-dimensional theory of elasticity (3D), the solid-to-shell transition model (3D/RM), and the shell-to-shell transition models (MI/RM) as well. The complete set of the models is defined as:


x1

1

top

x'3

u' 3j f1u' 1j

u' 2j

x3

top

u3n

u'3n middle

u1n

ξ' 3

thickness

thickness

121

u' 2n

middle

f 0u' 0j + f 3u' 3j

0

f 0u' 0j u' j (ξ' 3 )

x'2

f 2u' 2j

u2n

bottom u'1n

f 3u' 3j

x2

u' 1j

-1

bottom

u' j

u' 0j

x'1

Fig. 7.8 Through-thickness dofs (left) and the displacement field (right) of the 3D-based approach

M ∈ M,

M = {3D, MI, RM, 3D/RM, MI/RM}

(7.15)

The subset MI of the hierarchical shell models as well as the corresponding subset MI/RM of the transition models are MI = { M2, M3, M4, ...}, MI/RM = {M2/RM, M3/RM, M4/RM, ...}

(7.16)

The elements of our hierarchy can be ordered with respect to the order I (or J), I = 1, 2, 3, ..., ∞ of the shell theory , i.e. M = RM ⇒ I = 1, J = I M = M2/RM, M3/RM, M4/RM ⇒ I = 2, 3, 4, ..., J = 1 M = M2, M3, M4, ... ⇒ I = 2, 3, 4, ..., J = I M = 3D/RM ⇒ I → ∞, J = 1 M = 3D ⇒ I → ∞, J = I is

(7.17)

The fundamental property of our hierarchy of the 3D-based mechanical models lim lim uI/J(M) U,V = u3D U,V , (7.18) J=1,I

I→∞

where the global norm of the solutions uI/J(M) , equal to the strain energy U within the body volume V , is uI/J(M) U,V =

1 2

σ T (uI/J(M) ) ε (uI/J(M) ) dV, V

(7.19)

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G. Zboiński

with σ and ε being the vectors of stresses and strains, respectively. Note that the subsequent solutions of the models of the hierarchy tend in the limit to the solution u3D of the model of 3D-elasticity, i.e. the highest model of the hierarchy. More information on the definitions of the models and the properties of our hierarchy can be found in our works [21, 13].

7.2.1.3 Hierarchical hpq-Approximations For each of the mentioned models we introduce the two-dimensional, three-dimensional, or mixed (corresponding to the shell, solid and transition models, respectively) hpq-adaptive approximations, where h is the characteristic dimension of an element, while p and q represent longitudinal and transverse orders of approximation within the element, respectively. In other words, the subsequent solutions uI/J(M) to our subsequent models M of the hierarchy M are now approximated with the solutions uq(M),hp , in which the transverse order of approximation q is equivalent to the order I of the hierarchical model M, i.e. q ≡ I. Taking into account the specific character of the approximations for the cases of the first-order shell, hierarchical shell, and 3D-elasticity models, we can denote the specific solutions, corresponding to these models, in the following way M = RM, I ≡ J = 1 ⇒ uq(M),hp = uhp M ∈ MI/RM, I ≥ 2, J = 1 ⇒ uq(M),hp = uhpq/hp M ∈ MI, I ≡ J ≥ 2 ⇒ uq(M),hp = uhpq M = 3D/RM, I → ∞, J = 1 ⇒ uq(M),hp = uhpp,hp M = 3D, I ≡ J → ∞ ⇒ uq(M),hp = uhpp

(7.20)

In the above notation we simply skip q = 1 for the first-order shell model, and assume q = p for the model of 3D-elasticity. In the case of the transition models we perform accordingly. The evolution of the concept and the corresponding definitions of the hierarchical hpq-approximations for the structures of complex mechanical description can be investigated in our subsequent works [19, 16, 21, 13]. The main feature of the introduced approximations is that, with the increasing p and the increasing inverse of h, the numerical solutions tend in the limit to the exact solution uI/J(M) of the corresponding model M, i.e. lim uq(M),hp

1/h,p→∞

)e

U, V

= uI/J(M) U,V

(7.21)

Combination of the proposed hierarchy of mechanical models and the hierarchical approximations for these models leads to the hierarchy of adaptive numerical models for complex structure analysis (see [13]). The useful property of the solutions within this hierarchy is


lim uq(M),hp lim lim

J=1,I I→∞

1/h,p→∞

)e

U, V

123

= lim

J=1,I

lim uI/J(M) U,V = u3D U,V ,

I→∞

(7.22) which means that with the increasing discretization parameters p and 1/h and the increasing order I of the hierarchy of mechanical models we reach the solution corresponding to the exact solution of the highest model of the hierarchy.

7.2.1.4 Hierarchy of the hp-, hpq- and hpq/hp-Adaptive Finite Elements The hierarchy of the numerical models, introduced in the previous subsection, is encoded into a hierarchy of the adaptive prismatic finite elements. The hierarchy includes: the hpp solid element, acting also as the hpq hierarchical shell elements, a family of the hpp/hp solid-to-shell transition elements, acting also as the hpq/hp shell-to-shell transition elements, and the hp first-order shell element. The local (element-level) model-, h-, as well as p- and q-adaptive capabilities of the hierarchy of elements results from the application of: the hierarchy of 3D-based mechanical models, the constrained approximation (hanging nodes) idea, as well as the hierarchical shape functions associated with the corresponding incremental degrees of freedom, respectively. These three ideas are presented in detail in our work [19]. It should be mentioned here that our works on the constrained approximation and hierarchical shape functions were inspired and took advantage of the works by the predecessors [7, 8]. Application of these concepts on the element level is elucidated in our work [20] for the solid and hierarchical shell elements, [23] for the transition elements, and [21] for the first-order shell element. The hierarchical character of the all elements is explained in [19, 22]. As the all above aspects of the assessed methodology are documented very thoroughly in the all mentioned works, we skip the corresponding details in this presentation.

7.2.2 Error Estimation The applied error-controlled model- and hpq-adaptivity is based on the estimated values of the modelling and approximation errors, obtained from the residual equilibration method (ERM) [2, 3]. The method is applied twice, firstly for estimation of the approximation error and secondly for the modelling or total error. The method provides the upper bounds of the global approximation error for all the applied models as well as the upper bound of the global modelling error for the hierarchical shell models (thus also the upper bound of the global total error for the latter models can be proved). In the case of the first-order shell elements, and the corresponding solidto-shell transition elements, one cannot prove the upper bound of the modelling and total errors. As an undesired consequence, only the global modelling error indicator can be obtained from the proposed approach. In the case of 3D elasticity model, the

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G. Zboiński

total and approximation errors are equivalent, and the upper bound of the errors can be shown. The starting point for the equilibrated residual method is the variational formulation in which the global error functional in the energy norm is employed. The error energy norm is defined as the strain energy of the difference of the exact and numerical solutions. Then, the functional is decomposed into the local (element-level) functionals for each finite element. In this step the upper bound property can be proved, based on the observation that the sum of the minimized potential energies from the decomposed local (element-level) variational problems is always grater than the global minimized potential energy of the body as a whole. As a consequence, the global strain energy of the error, for the whole body, is always smaller than the sum of the strain energies of the errors from the decoupled local problems. Finally, we employ this global strain energy as an exact value of the error, and the element sum of the errors as the estimator to this exact error. This way the element residual approach to error estimation is formed. Details of our implementation of the method can be found in [19] for the whole hierarchy of our adaptive finite elements, and in [17, 18, 14] for the first-order shell element. Note that, from the implementation point of view, the obtainment of the solutions to the local (element-level) variational problems is very important, as these solutions contribute to the global estimator of the error. The local problems to be solved can be presented in the following form e

Q(M),HP Q(M),HP

B(u

,v

e

)− L(vQ(M),HP ) −

T e

e

e

vQ(M),HP r(uhp ) dS = 0,

S\(SP ∪SD )

(7.23) where the bilinear B and linear L forms represent the virtual strain energy and the virtual work of the external forces of the body, both restricted to the element e. The solution and trial functions from the proper space of the element kinematically admissible displacements are denoted as uQ(M),HP and vQ(M),HP , with H, P, and Q standing for the element size, and longitudinal and transverse orders of approximation in the discretized local problem. The last left-hand side term of (7.23) represents the virtual work of the interelement stress reactions. The equilibrated version e of these reactions is denoted as r . The equilibration means that the external and internal forces of the element are in equilibrium, and this is done with the vectors e

ef

e

α of the splitting functions, to be determined on the common faces of the element e and any adjacent element f . Our definitions of these functions can be found in [2, 3, 19]. The formal definitions of the terms of the above relation are

7 Unresolved problems of hierarchical modelling... e

Q(M),HP

B(uQ(M),HP, v

)=

e

125 e

e

T e e

e

e

T e

e

T e

ε T (vQ(M),HP )Dε (uQ(M),HP ) dV = v Q(M),HP k q Q(M),HP,

V e

L(vQ(M),HP ) =

e

T

e

v Q(M),HP f dV +

V

T

e

e

v Q(M),HP p dS = v Q(M),HP (fM + fP ),

SP e

S\(SP ∪SD )

v

e

Q(M),HP T e

r(uq(M),hp ) dS = v Q(M),HP fR

(7.24)

These terms can be also expressed in the language of the finite elements, by means e

of the element stiffness matrix k (defined with the elasticity matrix D and the strain e

e

e

vector ε ), and the element forces vectors fM , fP , and fR due to the body, surface, e and element reaction loads f, p, and r , respectively. Denoting the searched nodal e Q(M),HP displacements as q , the relation (7.23) can be equivalently written as e e

e

e

e

k q Q(M),HP = fM + fP + fR

(7.25)

We will take advantage of the above form of the local problems later on in this chapter of the book.

7.2.3 Adaptive Strategy Our adaptive strategy is based on Texas three-step strategy [9]. The original strategy lies in solution of the global problem thrice, on the initial, intermediate, and final meshes. The intermediate and final meshes are obtained through the local hrefinements (h-step) and local p-enrichments (p-step), respectively. In principle, the method leads to the solution of the assumed accuracy, as the error level is related to the discretization parameters through convergence theories. The original approach is enriched by us with the possible two or three iterations within the h- and p-steps. Additionally, we introduce one additional step proceeding the h- and p-steps, called modification one (see Fig. 7.9), in order get rid of such unpleasant phenomena as: • the improper solution limit of 3D elasticity model for q = 1 and thin structures, • numerical locking (for low values of p and 1/h within thin structures), • and boundary layers. In our additional step some global modifications (the global change of the model, the global p-enrichment, and the introduction of exponential mesh subdivision in the direction normal to the boundary) within the initial mesh are performed. This additional step needs special tools for detection of the mentioned three phenomena. The proposed tools take advantage of the algorithms of the applied error estimation method. They allow qualitative assessment of two local solutions obtained for the models for which these phenomena exist or not in accordance with the theory.

126

G. Zboiński Start Reading data i=1 Initial mesh generation

Formation and solution of the problem equations Detecion of three phenomena (i=1) or error estimation (i=2,3,4) Final mesh generation Initial mesh modification Intermediate mesh generation Yes Yes

One of three phenomena detected?

i=2 i=1

i=2

i=3

No

i=2

No

i=2

No Yes

No

i=3

i=3 No

No

Enough intermediate error iterations? Yes

No

Enough final error iterations?

Yes

i=4

Yes

Target error achieved? Yes

Printing results Stop

Fig. 7.9 Adaptivity control with four-step iterative strategy

7.2.4 Possibility of Other Approaches One should be aware that the proposed approach is not the only option in adaptive analysis within computational solid mechanics. Within the modelling and approximation methods we can mention the methodologies based on one mechanical model, either the first-order shell [4], higher-order shells [1, 5], or 3D elasticity [11, 12]. Possible alternatives concerning the error estimation and adaptive strategies are: the goal-oriented adaptivity [10] and the so called automatic hp-adaptive strategy based on the two-mesh paradigm [6, 12], respectively. Even though, these methodologies were developed as an answer to the problem of poor effectiveness of other techniques, they do not answer to all questions concerning the adaptive modelling and analysis within solid mechanics. For example, the application of the automatic hpadaptive strategy to the problems of complex mechanical description and problems with boundary layers is still an open question. Also, the goal-oriented approach, useful for practical applications, still does not give all answers to the problem of upper and lower boundedness of the errors.


127

7.3 Chosen Problems to Be Resolved and Some Remedies In this section we would like to present some chosen unresolved problems within the assessed methodology. Before the presentation of the problems and the related remedies we would like to state what follows. • Our choice of the problems to be presented is rather subjective. We consider difficulties faced by us during implementation of our specific methodology. • The order in which we present these problems corresponds to the subsequent steps of the algorithm, not to the significance of the problems. • We search for the solution of these problems within the presented methodology, taking advantage of the potential hidden within the applied techniques. We do not consider the escape to other techniques (we do not want to introduce other quantities of interest, apply the goal-oriented adaptivity, or take advantage of the two-mesh paradigm). The short list and the characterization of the main problems, we would like to address in this chapter, is as follows. • Geometry modelling within 3D-based hierarchy of models is important because of its influence on the orthogonality of the modelling and approximation errors. • Excessive growth of the number of dofs in 3D can appear (especially in the case of corner or edge high solution gradients – they may need application of the 2-irregular constrained approximations). • Our approach does not give formal upper bounds for the free and forced vibration problems, however the effectiveness is practically identical as in the case of the equilibrium problem. • Poor or worse effectivity of the ERM estimator can be observed in the case of elongated elements and the meshes of locally varying order and size (the uniform meshes provide higher error but better effectivity than the adapted ones). • Changing convergence rates disturb the obtainment of the target error.

7.3.1 Hierarchical Modelling Issues In the context of the hierarchical modelling, we chose the specific questions of the robust formulations of the hierarchical shell models and transition models as well. One of the key issues within our 3D-based formulation is geometry modelling based on the middle surface and thickness concepts. Such geometry modelling is necessary for orthogonal decomposition of the total error into the modelling and approximation error components. The problem of the shell geometry modeling is illustrated in Fig. 7.10, where two systems of local coordinates are introduced. The first one corresponds to any arbitrary point of the shell, and is based on the assumption that the two local directions, determined by the vectors w1 and w2 , are tangent to the mid-shell surface,

128

G. Zboiński any point of the shell

x3

point j

ξ3 w

ξ3

j 3

top surface

x'3 w3 x'2

w 1j w 2j

w2 x2

x1

ξ2

w1 ξ1

x'1

bottom surface shell mid-surface

ξ2 ξ1

Fig. 7.10 The proper shell geometry definition

or to any other shell surface determined by the shell natural coordinates ξ1 and ξ2 , and additionally that the third direction, determined by w3 , is perpendicular to the mentioned surface. The second system of coordinates, corresponding to any point j of the lateral boundary of the shell, is defined in a different way. Here, the vector j w3 is defined first, so as to coincide with the direction x3 , i.e. to come through the corresponding points of the top and bottom surfaces of the shell. The surface determined with the two remaining vectors, w1j and w2j , is now defined as perpendicular to the first vector. Note that when the first system of coordinates is generated in the point j, both the systems are not identical, unless the mid-shell surface is perpendicular to the direction x3 . There is no problem with fulfillment of this requirement in the case of analytical description, provided that the shell geometry is defined as the one of the symmetric thickness in the direction perpendicular to the mid-surface. Note, however, that in the finite element approximation, the interpolated (polynomial) representation of the shell mid-surface may not be perpendicular to the lateral boundary of the shell. Fortunately, the higher the interpolation order, the better coincidence between the directions x3 and ξ3 at any point j can be observed. Summing up, if one wants to take advantage of all the features of the shell theories, one has to construct geometry of the 3D shell body so as the lateral boundary of the shell is perpendicular to its mid-surface. In the finite element approximation of such shells, careful interpolation of their geometry is required.


129

7.3.2 Problems within hp-Approximation What makes the implementation of the hp-adaptive approach difficult is the constrained approximation necessary for the local (element) h-refinements. In the case of mechanical systems of the complex mechanical description, the transition models and approximations make the issue even more complicated. Because of that below we discuss the chances of getting rid of these two techniques. In this subsection we also address the problem of the excessive growth of the dof number in the case of 3D meshes. 7.3.2.1 The Constrained Approximation In this case we deal with the broken elements equipped with the so called hanging nodes. The general answer to the question if one can get rid of the hanging nodes is no. In the general h- or hp-approach, based on element refinement (as opposed to the remeshing technique), it is not possible. Note that the specific approach called Rivara’s refinement, where the division by four or two is led through the existing vertices of the rectangular faces of the elements, is applicable only to such faces. The approach requires the implementation of two types of elements within one mesh, with the rectangular and triangular faces in the undivided and divided elements, respectively. Note also that the original Rivara’s approach can be applied once. This is because, when one wants to repeat the division, he deals with the triangles, and the avoidance of hanging nodes requires the division by two of the element we want to divide and the adjacent element as well. In other words, in the case of triangular faces such an approach does not work unless we divide also the adjacent element. This makes things much more complicated and leads to over-divided meshes. 7.3.2.2 Necessity of the Transition Approximations In the case of the transition approximations, resulting from the application of the transition models joining the basic models together, the transition elements corresponding to such transition models are necessary. As an example of such a situation within solid mechanics we can mention the solid-to-shell transition elements acting between the 3D-elasticity model (or the higher-order shell models) and the firstorder Reissner-Mindlin shell model. The general answer to the question if one can get rid of the transition elements is no, again. There exist, however, some prosthesis approaches. For example, in the case of the bending-dominated shells, the ReissnerMindlin model can be replaced by the 3D-elasticity model of the transverse order q = 1, with changed elastic constants. In the case of the membrane-dominated shell structures, the Reissner-Mindlin model can be changed to the 3D-elasticity model of q = 1, without any modifications of the elastic constants. In both cases, the transition model and the corresponding transition elements are not necessary. The problem with such approaches is that, in the majority of technical applications, the character of the strain dominance (either bending or membrane one) is not known a priori.

130

G. Zboiński

There are also some structures where a balance between the bending and membrane strains exists. Then, the only correct model is the Reissner-Mindlin one, which unfortunately requires the transition model, when combined with the 3D-elasticity one. As we search for a general approach to hierarchical modelling, we will not consider here the imperfect approaches mentioned above, suitable for the specific states of strains only.

7.3.2.3 Growth of the Dofs Number in 3D Problems There are at least three reasons for the excessive growth in the dofs number for 3D problems. • Application of only 1-irregular hanging nodes due to the constrained approximation. • Poor effectivity of the estimation (overestimation). • Wrong h- or p-convergence exponents in the adaptivity control procedures. Here we would like to address the first problem, while the remaining two will be discussed in the section concerning adaptivity control. So as to make the first problem less severe, we propose to extend the idea of the constrained approximation onto 2irregular meshes. Note that the problem is especially important for large 3D structures with high edge or corner solution gradients. Application of 2-irregular meshes allow the avoidance of the excessive growth of dofs occurring when only 1-irregular subdivisions are possible. Such a limitation in mesh generation causes that 1-irregular mesh penetrates deeply into the regions that could have been undivided, if a sequence of 1irregular subdivisions had not been necessary due to solution continuity reasons. The example of the excessive growth of the number of dofs in the h-adapted, 1-irregular mesh, both in the interior and in the vicinity of the boundary, is shown in Fig. 7.11. The difference between the hanging nodes of 1- and 2-irregular meshes is illustrated in Fig. 7.12, where the node of the first type is marked with 1, while the node of the second type is denoted with the number 2. In order to retain solution continuity, the displacements of the node 1 of the element f have to be equal to the interpolated displacements of the undivided, adjacent element e, at the location e

ξ 1 = 1/2 of the node 1 in the element e f

e

e

q i1 = u i (ξ 1 ),

e

ξ 1=

1 2

(7.26)

In the case of the element g, obtained through a double division, the displacements dofs at the node 2 can be expressed with either the interpolated displacements of the element e1 , that could have been obtained through a single division of the element e, or better directly with the interpolated displacements of the undivided element e, e

at the location ξ 2 = 3/4 of the node 2 in the element e g

e

e1

e

e

q i2 = u1 i (ξ 2 ) = u i (ξ 2 ),

e1 1 ξ 2= , 2

e

ξ 2=

3 4

(7.27)

7 Unresolved problems of hierarchical modelling... p,q

131

28645883 2884 2874 2343 28545863 2183 5903 5843 2323 2223 2383 2203 2363 2023 2663 2163 862 6023 5683 2063 5823 2643 2043 2703 2503 2003 852 3574 3264 2683 5703 5663 3184 6063 6043 2483 3604 2543 3254 3584 3204 3284 3194 2523 25445523 692 652 3594 3504 3104 5643 6003 3274 2564 36633174 3494 2554 6603 702 3524 5503 3124 4064 3683 3114 2743 662 5183 2704 2534 332 2783 3514 2624 863 370336233094 5543 5483 2763 5223 6623 2694 2724 2723 382 2644 4084 3603 5203 3643 2634 342 463 4054 2464 883 5163 732 5463 2714 4074 5043 372 3823 32232614 1264 782 3843 2484 4143 3243 2474 50235003 5443742 3863 1424 1284 5063 326331832454 1254 4144 963 3463 4183 3163 823 3203 1434 1444772 3803 4983 4123 141412741184 4163 2863 33834803 4164 1204 903 1194 4963 39833403 4154 1174 4134 4783 3423 6743 4003 4023 6463 4823 4643 4623 3003 6123 3363 4663 2823 3963 6383 703 3704 4763 3023 2983 6443 6143 1474 2963 4603 2843 3724 6363 763 5303 6543 3714 3694 6283 1803 1783 1723 1703 2803 5283 6523 6163 1823 1863 1743 6843 4463 4303 1763 1903 1683 1883 1843 6303 6183 1643 1623 6823 4503 4343 1663 4483 4443 4323 4283 1603

8

7

6

5

4

3

2

1

z x

y

Fig. 7.11 Edge and interior excessive growth of dofs number (the first mode of free vibration)

element f

1

element g

2

element e 2 1 2

1

1

2 element g

element f element e Fig. 7.12 1- and 2-irregular constrained nodes for vertical (left) and horizontal (right) subdivisions

132

G. Zboiński

As it can be seen, the procedure of the obtainment of the continuity conditions for the 1- and 2-irregular nodes is generally the same. As the interpolated displacements of the undivided element e can be expressed with the active displacement dofs of this element, then also the constrained dofs of the 1- and 2-irregular nodes can be expressed by them, and the corresponding constraint coefficient matrices can be defined. The general and detailed information on how to construct such matrices can be found in [8, 19, 14].

7.3.3 A Posteriori Error Estimation In the error estimation problems, we would like to pay our attention to two issues. The first group of problems is theoretical and concerns the obtainment of the upper bound property of the residual estimators in the free and forced vibration problems. The second, implementation issue concerns poor or worse effectivity of the estimation.

7.3.3.1 Error Estimation for the Free and Forced Vibration Problems In the free vibration problem the difficulty results from the fact that the exact solution in frequency (and in the corresponding strain energy) gives the lower bound of the numerical solution, opposite to the static case, where the exact solution in strain energy constitutes the upper bound. In the stationary forced vibration problem the main difficulty arises from the appearance of the phase angles in the acting forces. The solution to such a problem has to be obtained in the domain of complex functions. The phase angles have to be introduced into the formulation of the ERM estimators. Note that when these angles are equal to zero, the solution becomes real and the method suitable for the equilibrium problem can be applied directly. The similarities, differences, and the above mentioned theoretical difficulties within three analyzed problems of solid mechanics, in the frame of the presented methodology, can be summarized as follows. • The equilibrium problem: – the upper bound exists for the approximation, modelling and total errors (3Delasticty and 3D-based hierarchical shell models), and the approximation error (3D-based Reissner-Mindlin model). • The eigenproblem (free vibration): – no formal upper bound of the error can be proved, however effectivity of the estimation of the errors in the energy norm is practically the same as for the equilibrium problem (compare Fig. 7.13 and Fig. 7.14), – the main difficulty results from the lower boundedness of the numerical solution by the exact solution, opposite to the equilibrium (static) case,


133 (X) 1.372 1.372

it

20 10

180 170 40 30

340 330 200 190 60 50

500 490 360 350 220 210 80 70

660 650 520 510 380 370 240 230 100 90

820 810 680 670 540 530 400 390 260 250 120 110

980 X 970 840 830 700 690 560 550 420 410 280 270 140 130

1140 1130 1000 990 860 850 720 710 580 570 x 440 430 300 290 160 150

max

1.208 1.167 1.063 1160 1150 1020 1010 880 870 740 730 600 590 460 450 320 310

avr

0.936 1180 1170 1040 1030 900 890 760 750 620 610 480 470

0.824 1200 1190 1060 1050 920 910 780 770 640 630

0.726 1220 1210 1080 1070 940 930 800 790

0.639 1240 1230 1100 1090 960 950

0.563 1260 1250 0.496 1280 1270 1120 0.436 1110 0.384 0.338 0.298 0.262 0.231 0.203 (x)

0.179 0.179

min

z x

y

Fig. 7.13 Total error effectivities for the equilibrium problem

– the true error is the result of the error in the natural frequency and the error of the mode of vibration (the eigenvalue and eigenmode errors), – our methodology accounts only for the eigenmode error, – our methodology can be converted into the goal-oriented approach, with the normalized strain energy being the quantity of interest, and solutions of the ERM local problems forming the dual solution. • The forced-vibration problem: – the problem is solved for the amplitudes and phase angles – this requires the complex domain, – there is still no formal answer to the question of the upper boundedness of the errors (the work is in progress), – one can expect the effectivity of the estimation to be similar as for the equilibrium and free vibration problems (to be demonstrated numerically).

7.3.3.2 Effectivity of the Estimation The second issue has an implementation character and deals with the poor effectivity of the residual approach in the case of elements elongated in order to resolve the boundary layers. The corresponding exemplary effectivities for the regular meshes

134

G. Zboiński (X) 1.774 1.774

it

max

1.551

20 10

180 170 40 30

340 330 200 190 60 50

500 490 360 350 220 210 80 70

660 650 520 510 380 370 240 230 100 90

820 810 680 670 540 530 400 390 260 250 120 110

980 970 840 830 700 690 560 550 420 410 280 270 140 130

1140 1130 X 1000 990 860 850 720 710 580 570 440 430 300 290 160 150

1160 1150 1020 1010 880 870 740 730 600 590 460 450 320 310

1.356 1.188 1.185 1180 1170 1040 1030 900 890 760 750 620 610 480 470

avr

1.036 1200 1190 1060 1050 920 910 780 770 640 630

0.906 1220 1210 1080 1070 940 930 800 x 790

0.792 1240 1230 1100 1090 960 950

0.692 1260 1250 0.605 1280 1270 1120 0.529 1110 0.462 0.404 0.353 0.309 0.270 0.236 (x)

0.206 0.206

min

z x

y

Fig. 7.14 Total error effectivities for the first mode of free vibration

of the same degrees of freedom, with square and elongated elements, are presented in Fig. 7.15 and Fig. 7.16, where the global effectivities are 1.2 and 11.5, respectively. Worse effectivity appears also for the locally hp-adapted elements, in comparison with the uniform elements. The meshes to be compared, of the similar number of dofs and the same uniform order of approximation, are presented in Fig. 7.15 and Fig. 7.17. The global effectivities are about 1.2 and 1.7, respectively. One of the available remedies, available but not implemented yet in 3D, is the application of the equilibration of the higher order. Such a method might include the equilibration not only at the vertices of the elements. For the first case, when only the vertex equilibration is performed, the directional components i = 1, 2, 3 of the vectors of splitting functions can be expressed through the six (in the case of the applied prismatic elements) vertex splitting coefficients. For the higher-order nodes no equilibration is performed, and the splitting coefficients, equal to 1/2, reflect the averaging of the nodal reactions between the elements. This type of equilibration can be characterized with the relation ef

αi =

6

∑

j=1

ef e 1e α ij χ j + ∑ ∑ χ j,m , j>6 m 2

(7.28)


135 (X) 6.287 6.287

ia

max

5.267 4.412 260

3.697 3.097

250 180

280

170

270

100

2.594

200

2.174

300

1.821 90

190

20

120

10

290 220

110

320 1.526

240

1.278 310 X 1.205 1.071

230

0.897

210

40

140

30

130 60

160

avr

0.752 50 x

150 0.630

80

0.527

70

0.442 (x)

0.370 0.370

min

z x

y

Fig. 7.15 Effectivities for the uniform mesh

e

where χ j represents the shape function of the vertex node j. Note that the non-zero contributions to the splitting functions correspond to four out of six vertex nodes, located on the common face of the elements e and f (see Fig. 7.18). In the higher order equilibration also the higher order mid-edge and/or mid-side nodes are included in the equilibration procedure (see [2, 19]). The respective definition of the splitting function components are ef

αi =

6

∑

j=1 e

ef

e

ef

e

α ij χ j + ∑ ∑ α ij,m χ j,m ,

(7.29)

j>6 m

with χ j,m standing for the shape function corresponding to the higher-order node j, and dof m, defined in this node. Now, the non-zero contributions to the face splitting function correspond to four vertices, four mid-edge nodes and the mid-side node of the common face. The presented idea looks very simple but our numerical tests show that the effective higher-order equilibration is not a trivial task. The results can be even worse than in the case of linear equilibration. The second suggested remedy for poor or worse effectivity of the estimation can be the constraining of the element displacement field while solving local problems of the ERM. This approach is not very common and it needs further theoretical and numerical studies.

136

G. Zboiński (X) 33.069 33.069

ia

max

23.115 260 250 180 170 270 200 x280 190 100 300 120 220 90 290 110 140 210

16.158 11.545 11.295

130

7.895

20 40

5.519

320 240

60

avr

160

3.858

1030

310 2.697 230 50 X

80

150

1.885 1.318 0.921 0.644

70

0.450 0.315 0.220 0.154 (x)

0.107 0.107

min

z x

y

Fig. 7.16 Effectivities for the non-uniform mesh

7.3.4 A Posteriori Detection of the Undesired Phenomena The question on how to detect a posteriori the improper solution limit, the numerical locking, and the boundary layers is still another problem. The first phenomenon leads to the incorrect solution, while the latter two cause varying convergence rates and disturb the desired exponential convergence. Also the problem of coping with these phenomena via automatic, error-controlled, adaptive approach is still open. Below we present our numerical tools [15] for the detection of the mentioned phenomena. These tools take advantage of the algorithms of the residual equilibration method applied to error estimation.

7.3.4.1 The Improper Solution Limit and Numerical Locking The improper solution limit and the locking is illustrated in Fig. 7.19. The first phenomenon appears in thin structures modelled with the 3D model, when the transverse approximation order equals unity (q = 1). Then, the solution is the fraction of the correct solution, represented by the value of 1 in the figure, regardless the structure length to thickness ratio t/l. Note that in the figure the relative value of the solution is calculated as the strain energy U, related to the reference, analytical value Ur of this energy.


137 (X) 8.628 8.628

ia

max

7.326 6.221 114

5.283 4.486

124 94 174

204 184

154 164 134

104 194 334 254

144

354 364 284 264

294 344 274

324 304

3.810 314 3.235

234

2.747

244 214

224 x 2.333 1.981

22

34

12

42 X

32

1.682 1.658 1.428

44 14

24

avr

1.213 1.030

30

0.875 0.743 (x)

0.631 0.631

min

z x

y

Fig. 7.17 Effectivities for the hp-adapted mesh

element f v4

v4 e3

element e v3

v3

e4 s

e2

v1

e1

element f v2

v1

v2

element e Fig. 7.18 Linear (left) and higher-order (right) equilibrations

The numerical locking occurs in thin structures when the longitudinal approximation order is low (p = 1 in the figure) and/or the mesh density is not high ( number of longitudinal edge subdivisions, m, equals 4 in the figure), regardless the value of

138

G. Zboiński 5.0E+0

MI, m=4 q=2, p=1 4.0E+0

q=2, p=4,5,6,7,8 q=1, p=4,5,6,7,8

U/Ur

3.0E+0

2.0E+0

1.0E+0

Fig. 7.19 Illustration of the improper solution limit and numerical locking

0.0E+0 1.00

10.00

l/t

100.00

1000.00

the transverse approximation order q. In such situations, the locked solutions tend to zero in the thin limit. Our numerical tool for the detection of the improper solution limit, lies in the comparison of two solutions obtained from the local problems for the element e chosen from the interior of the thin structure (see Fig. 7.20). The first solution is characterized with the model and discretization data for which the phenomenon may appear, and the second one with the parameters for which the phenomenon does not appear, i.e. e e

e

e

e e Q(RM),HP

e fM

e

k q Q(3D),HP = fM + fP + fR , kq

=

e + fP

Q = 1,

P = 8,

Q = 1,

P=8

e

+ fR ,

(7.30)

The analogous detection strategy is applied to the locking phenomena, either the shear or membrane ones (for the explanation see [19]). The local solutions to be compared are now obtained from the following problems e e

e

e e Q(M),HP

e fM

e

e

k q Q(M),HP = fM + fP + fR , kq

=

e + fP

P = p,

e

+ fR ,

P=8

(7.31)

Note that the same approach can be also utilized for the assessment of the intensity of locking. For this purpose we perform a kind of sensitivity analysis with the changing value of the longitudinal order of approximation P. The information on how to interpret the results from the local problems, concerning the detection of the numerical locking and the improper solution limit, and the


139

element e

Fig. 7.20 Single-element local problem for the detection of the improper solution limit and numerical locking

assessment of the intensity of the locking, can be found in [19, 15]. The main difficulty in this interpretation consists in the dependence of the obtained local solutions on the element dimensions resulting from the density of the global mesh (the quality of the detection is better for dense meshes).

7.3.4.2 The Boundary Layer Phenomenon The phenomenon appears in thin and thick structures, when the analytical solution is the sum of the smooth part, corresponding to the interior of the analyzed domain, and the boundary part of high gradients. In the numerical solution of such problems, the interpolation functions suitable for the smooth part of the analytical solution, may not be appropriate for modelling solution gradients in the vicinity of the boundary. The situation is illustrated in Fig. 7.21, where the convergence curves for three cases are presented. The curves relate the numerical error obtained as a difference of the numerical and reference (exact) values of the strain energy, U and Ur , with the number N of degrees of freedom within the numerical model. The first case corresponds to the plate problem described with the Reissner-Mindlin model (q = 1). This model is not very much prone to the boundary layer phenomenon and the corresponding convergence is very high. Note that the uniform mesh of the division number m = 8 is applied in this case. The second case corresponds to the same plate and the model changed to 3D-elastic one. We apply the same uniform mesh of m = 8. It appears that the convergence is very poor in comparison to the previous case. The reason is the application of the 3D model of the transverse approximation order q = 2, very much prone to the edge effect. In order to resolve the problem of the poor convergence, we introduce the mesh of constant density in the interior of the plate, and varying density in the part adjacent to the boundaries, where the exponential subdivisions towards the boundary are applied. This mesh is denoted

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p=1 p=2

2.0 p=2

p=2

1.0 p=3 p=4 p=3

log (U-Ur)

0.0

p=3

-1.0

-2.0 p=4

t/l=0.33%

-3.0

RM, q=1, m=8

p=4

3D, q=2, m=8 -4.0

3D, q=2, m=4+4

-5.0

Fig. 7.21 Convergence affected by the boundary layer

2.0

2.5

3.0

log N

3.5

4.0

4.5

with the division number m = 4 + 4. The application of this mesh restores the high convergence of the solution, as it can be seen from the third presented curve. Our numerical tool, for the detection of the boundary layers, is based on the comparison of two solutions obtained from the local problems for the chosen pair of elements adjacent to the structure boundary (see Fig. 7.22). The first solution is obtained from the problem characterized with the uniform subdivision of the pair of elements into four smaller elements. For such a subdivision the edge effect may appear. In the second problem we apply the exponential subdivision (see Fig. 7.22, again) and we can expect the solution to be free of the edge effect. Our local problems to be solved are 4 fi f i

4

fi

fi

fi

i=1 4 fi

fi

fi

∑ k q Q(M),HP = ∑ ( f M + f P + f R ), Hn,i = h/2, i = 1, 2, 3, 4,

i=1 4 fi f i Q(M),HP

∑ kq

i=1

=

∑ ( f M + f P + f R ), Hn,1 = Hn,2 = h/10, Hn,3 = Hn,4 = 9h/10,

i=1

(7.32) where Hn,i , i = 1, 2, 3, 4 represent the mesh dimensions of the four smaller elements in the direction normal to the boundary. The element dimensions in the direction tangent to the boundary are kept unchanged and equal Ht = h, with h standing for the dimension of the chosen pair of elements.


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element f4 element f3

element f2

element f1

Fig. 7.22 Four-element local problem for the detection of the boundary layer phenomenon

We would like to add that the similar approach can be utilized for the assessment of the intensity and range of the phenomenon. For this purpose, the performance of a kind of sensitivity analysis is necessary, with the changing value of the elements dimension Hn,i (in the normal direction) in the exponential subdivision of the chosen pair of elements. The information on how to interpret the results from the local problems for the detection and the assessment of the intensity and range of the phenomenon are presented in [19, 15]. The main difficulty in this interpretation is the same as for the detection of the improper solution limit and the numerical locking (the dependence on the global mesh density). The additional difficulty is that in the case of the exponential subdivisions we deal with the elongated elements, and the quality of the boundary layer detection worsens.

7.3.5 Adaptive Procedures Finally, we will address the adaptive strategy issues. In particular, we will discuss the problem of changing convergence rates, which makes achieving the target admissible value of the error difficult within the original three- or our four-step strategies. Having a closer look at the p- or h-convergence curves (Fig. 7.23 and Fig. 7.24, respectively) of the numerical solutions to the problems subject to the numerical locking and boundary layers, one can distinguish three different regions. The first region corresponds to the numerical locking (the horizontal parts of the curves in both figures), the second one to the exponential or algebraic convergence (the middle parts of the curves), and the third region to the lost of regularity due to boundary layers (the third parts of the curves). Though, for each of these three regions some convergence theories exist, the transition from one state to another is unclear and difficult to be determined analytically as a function of the structure thickness t and the discretization parameters h, p, and q. The problem is very similar to that mentioned in Sect. 7.3.2.3, where we dealt with the wrong or unknown values of the exponents of the convergence curves. Note that one of the remedies for the

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problems of this type can be the application of the corrective iterations within the hand p-steps of the adaptive procedure. This idea has already been implemented in our algorithms and programs. 2.00 MI, q=1, t/l=1% m=1

p=1 p=1

p=2 p=3

m=4

p=2

0.00

log (U-Ur)

p=1

p=4

m=7

p=2

p=5 p=3

-2.00 p=3 p=6 p=4

-4.00 p=5

p=4 p=6

Fig. 7.23 Changing pconvergence rates for thin structure problems

p=5 p=6

-6.00 1.00

2.00

3.00

log N

4.00

The significance of the issue can be better understood when the standard relations (see [9, 19]) controlling the h- and p-adaptivity are taken into account. The convergence exponents μ0 and ν0 from the initial mesh, and the estimated values of the element error, η0 and ηI , from the initial and intermediate meshes, influence very much the number nI of the intermediate mesh elements replacing the initial mesh element η 2 EI 2 μ /d+1 nI 0 = 20 2 (7.33) γI u0 U and the value of the approximation order pT in the target (final) mesh pT2ν0 =

p02ν0 ηI2 EI , γT2 u0 U2

(7.34)

where d = 3 is the dimensionality of the 3D problem, γI and γT are the assumed, admissible, relative values of the intermediate and target (final) errors, and EI is the total number of elements in the intermediate and final meshes. Note also that the global strain energy norm of the global solution u0 from the initial mesh is utilized in the above equations.


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2.00 MI, q=1, t/l=1% m=1

m=2

m=4

m=6 m=8

p=1

m=1

p=3

0.00

p=5

m=2

log (U-Ur)

m=1

m=3 m=4

-2.00 m=6 m=8 m=2

-4.00

m=3 m=4 m=5 m=6 m=8

Fig. 7.24 Changing hconvergence rates for thin structure problems

-6.00 1.00

2.00

3.00

log N

4.00

7.4 Conclusions The main problems of the adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics have been presented. This has been done in the context of the proposed methodology, consisted of: the 3D hierarchical modelling, the hierarchical approximations, the a posteriori error estimation with the equilibrated residual method, and the four-step adaptive strategy. We have addressed such problems as: the geometry modelling within 3D-based thin structures, the over-divided meshes, the lack of the upper bound property for the free and forced vibration problems, the poor or worse effectivity of the estimation for the meshes with the elongated or non-uniform elements, the unsatisfactory quality of the detection of three undesired numerical phenomena, and the changing convergence rates for thin structures. The remedies for overcoming these problems have been indicated, basing on the described theoretical premises and our numerical experiments.

References [1] Actis, R.L., Szabó, B.A., Schwab, C.: Hierarchic models for laminated plates and shells. Comp. Methods Appl. Mech. Engng. 172, 79–107 (1999) [2] Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engng. 142, 1–88 (1995) [3] Ainsworth, M., Oden, J.T.: A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65, 23–50 (1993)

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[4] Chinosi, C., Della Croce, L., Scapolla, T.: Hierarchic finite elements for thin plates and shells. Computer Assisted Mechanics and Engineering Sciences 5, 151–160 (1998) [5] Cho, J.R., Oden, J.T.: Adaptive hpq-finite element methods of hierarchical models for plate- and shell-like structures. Comput. Methods Appl. Mech. Engng. 136, 317–345 (1996) [6] Demkowicz, L.: Computing with hp-adaptive finite elements. CRC Press, New York (2007) [7] Demkowicz, L., Bana´s, K.: 3D hp Adaptive Package. Report No. 2/1993. Cracow University of Technology, Section of Applied Mathematics, Cracow (1993) [8] Demkowicz, L., Oden, J.T., Rachowicz, W., Hardy, O.: Towards a universal hp adaptive finite element strategy. Part 1. A constrained approximation and data structure. Comput. Methods Appl. Mech. Engng. 77, 79–112 (1989) [9] Oden, J.T.: Error estimation and control in computational fluid dynamics. The O. C. Zienkiewicz Lecture. In: Proc. Math. of Finite Elements – MAFELAP VIII, pp. 1–36. Brunnel Univ., Uxbridge (1993) [10] Oden, J.T., Prudhome, S.: Goal-oriented error estimation and adaptivity for finite element method. Comp. Math. Appl. 41, 735–756 (2001) [11] Szabó, B.A., Sahrmann, G.J.: Hierarchic plate and shell models based on p-extension. Int. J. Numer. Methods Engng. 26, 1855–1881 (1988) [12] Tews, R., Rachowicz, W.: Application of an automatic hp-adaptive finite element method for thin-walled structures. Comput. Methods Appl. Mech. Engng. (in press) [13] Zboiński, G.: Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 1. Hierarchical modeling and approximations. Comput. Methods Appl. Mech. Engng. (to be published) [14] Zboiński, G.: 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures – Part 2. Application to structural analysis. Int. J. Numer. Methods Engng. 70, 1546–1580 (2007) [15] Zboiński, G.: Numerical tools for a posteriori detection and assessment of the improper solution limit, locking and boundary layers in analysis of thin walled structures. In: Wiberg, N.-E., Diez, P. (eds.) Adaptive Modeling and Simulation 2005. Proceeding of the Second International Conference on Adaptive Modeling and Simulation, Barcelona (Spain), pp. 321–330 (2005) [16] Zboiński, G.: Adaptive modelling and analysis of complex structures with use of 3Dbased hierarchical models and hp-approximations. In: Wiberg, N.-E., Diez, P. (eds.) Adaptive Modeling and Simulation. Proceeding of the First International Conference on Adaptive Modeling and Simulation, p. 50, and: CD-ROM, Göteborg (Sweden), pp. 1–24 (2003) [17] Zboiński, G.: A posteriori error estimation for hp-approximation of the 3D-based first order shell model. Part I. Theoretical aspects. Applied Mathematics, Informatics and Mechanics 8(1), 104–125 (2003) [18] Zboiński, G.: A posteriori error estimation for hp-approximation of the 3D-based first order shell model. Part II. Implementation aspects. Applied Mathematics, Informatics and Mechanics 8(2), 59–83 (2003) [19] Zboiński, G.: Hierarchical modelling and finite element method for adaptive analysis of complex structures (in Polish). Zesz. Nauk. IMP PAN w Gdańsku. Studia i Materiały, 520/1479/2001. IFFM, Gdańsk (2001) [20] Zboiński, G.: Application of the three-dimensional triangular-prism hpq adaptive finite element to plate and shell analysis. Computers & Structures 65, 497–514 (1997)


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[21] Zboiński, G., Jasiński, M.: 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures – Part 1. The model and the approximation. Int. J. Numer. Methods Engng. 70, 1513–1545 (2007) [22] Zboiński, G., Ostachowicz, W.: A family of 3D-based, compatible, shell, transition and solid elements for adaptive hierarchical modelling and FE analysis of complex structures. In: Abstracts of European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, p. 1011 (2000), and: CD-ROM Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona (Spain), pp. 1–20 (2000) [23] Zboiński, G., Ostachowicz, W.: An algorithm of a family of 3D-based, solid-to-shell, hpq/hp-adaptive finite elements. Journal of Theoretical and Applied Mechanics 38, 791–806 (2000)

Part II

Soft Computing and Optimization

Chapter 8

Granular Computing in Evolutionary Identification Witold Beluch, Tadeusz Burczyński, Adam Długosz, and Piotr Orantek

Abstract. The paper deals with the application of the Two–Stage Granular Strategy (TSGS) to the identification problems. Identification of selected parameters of the structures is performed. The identification problem is formulated as the minimization of some objective functionals which depend on measured and computed fields. It is assumed that identified constants and measurements have non–deterministic character. Three forms of the information granularity are considered: interval numbers, fuzzy numbers and random variables. The strategy combines the following techniques: Evolutionary Algorithms (EAs), Artificial Neural Networks (ANNs), local optimization methods (LOMs) and Finite Element Method (FEM). All techniques are appropriately modified to deal with non–deterministic data. The EA is used in the first stage to perform the global optimization. The LOM supported by ANN is used in the second stage. The FEM computations are performed to solve the boundary–value problem. Numerical examples presenting the efficiency of the TSGS in different applications are attached. Witold Beluch Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland e-mail: [email protected] Tadeusz Burczyński Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland Institute of Computer Modelling, Cracow University of Technology, Cracow, Poland e-mail: [email protected] Adam Długosz Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland e-mail: [email protected] Piotr Orantek Department of Strength of Materials and Computational Mechanics, Silesian University of Technology, Gliwice, Poland


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8.1 Introduction In many engineering problems it is necessary to identify some unknown parameters, like material parameters, shape parameters, boundary conditions etc. Identification problems belong to the inverse problems, which are ill–possed ones [4]. To solve such problems it is necessary to collect measurement data (state fields’ values) from the considered structure and compare them with the values of the state fields calculated from the model of a structure. As a result, the identification can be treated as a minimization of the functional J with respect to a design variables vector x: min J(x) x

(8.1)

In order to calculate the state fields value the direct boundary–value problem must be solved. In the present paper the finite element method (FEM) in the granular form is used to solve the direct boundary–value problem. It can be assumed that identified parameters as well as measurements have non– precise character. If it is not possible to determine precisely the parameters of the system, the uncertain parameters which describe granular character of data may be introduced. There exist different models of the information granularity [3]. In the present paper the granularity of information is represented in the forms of the interval numbers, fuzzy numbers or random variables. Selection of the proper model of granularity typically depends on the measurement data [6]. If only a few measurements exist and the measurements have unknown probability density function then the interval approach is suitable. Stochastic approach is convenient if the statistical data exist. If some linguistic description is used to evaluate the parameters of the system, the fuzzy is more appropriate. Different identification tasks are considered: the identification of shape and position of voids in isotropic structures subjected to dynamical load, the identification of boundary conditions in thermo–mechanical problems and the identification of laminate elastic constants.

8.2 Formulation of the Granular Identification Problem The aim of the identification problem is to find a vector x, describing identified parameters. Depending on the kind of granularity each design variable xij consists of: • for interval numbers: 2 values, representing edges of the interval: j j j xi = a(xi ), b(xi )

(8.2)

• for fuzzy numbers: 5 values, representing central value cv and edges of two α – cuts of the trapezoidal fuzzy number (Fig. 8.1):

8 Granular Computing in Evolutionary Identification

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j j j j j j xi = aL (xi ), aU (xi ), cv(xi ), bL (xi ), bU (xi )

(8.3)

• for random variables: 2 values, representing mean value m and standard deviation σ of the stochastic variable (assuming the that the random genes are independent random variables with Gaussian probability density function): xij = m(xij ), σ (xij ) (8.4)

Fig. 8.1 Trapezoidal fuzzy number described by 5 parameters

8.2.1 Identification of Voids in Dynamical Systems Let us consider an elastic isotropic structure containing a void of a circular shape of unknown size and position described by a vector x (Fig. 8.2). The structure is subjected to the dynamic loading.

W

G p( z, t )

Fig. 8.2 Elastic body containing void

sensor points

The vector of displacements u(z,t) is described by equation:

μ ∇2 u + (λ + μ )grad divu + Z = ρ u¨ (z,t),

z ∈ Ω , t ∈ T ∈ [0,t f ]

(8.5)

where: μ , λ – Lame constants, Z – vector of body forces with boundary conditions:

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u(z,t) = u(z,t), z ∈ Γu ≡ ∂ Ωu p(z,t) = p(z,t), z ∈ Γp ≡ ∂ Ω p Γu ∪ Γp = Γ ≡ ∂ Ω , Γu ∩ Γp = 0/

(8.6)

and initial conditions: u(z,t)|t=0 = uo (z),

o u(z,t)| ˙ t=0 = v (z),

z∈Ω

(8.7)

It is assumed that boundary conditions and material parameters have the granular character. The identification of the geometrical parameters of the void is treated as the minimization of the objective functional J, depending on measured uˆ and computed displacements u at n sensor points zi , with respect to x: (u(z,t) − uˆ (z,t))2 δ z − zi d Γ dt

n

min J, J = ∑ x

i=1

(8.8)

T Γ

where: δ – the Dirac function. It is assumed that parameters describing void as well as measurements have granular character.

8.2.2 Identification of Boundary Conditions in Thermo–Mechanical Systems Let us consider an elastic body occupying a domain Ω and having a boundary Γ . The steady–state thermoelasticity problem is considered. It is assumed, that the strain field depends on the temperature field but the temperature field does not depend on the strain field. The governing equations of the linear elasticity and steady–state heat conduction problem is expressed by following equations [9]: G ui, j j +

G 2G(1 − v) u j, ji + α T,i = 0 1 − 2v 1 − 2v

λ T,ii + Q = 0

(8.9) (8.10)

where: α – heat conduction coefficient, λ – thermal conductivity, T – temperature, Q – internal heat source. The boundary conditions are:

ΓT : Ti = T i ,

Γp : pi = pi , Γu : ui = ui Γq : qi = qi , Γc : qi = α (Ti − T ∞ )

(8.11)

where ui , pi , T i , qi , α , T ∞ – known displacements, tractions, temperatures, heat fluxes, heat conduction coefficient and ambient temperature, respectively.


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Separate parts of the boundary satisfy relations:

Γ = Γp ∪ Γu = ΓT ∪ Γq ∪ Γc Γp ∩ Γu = 0/ ΓT ∩ Γq ∩ Γc = 0/

(8.12)

The boundary conditions and measurements are assumed to have the granular character. Identification of the boundary conditions in thermo–mechanical systems can be formulated as a minimization of the funcional J: (q(z) − qˆ (z))2 δ z − zi d Γ

n

min J, J = ∑ x

i=1

(8.13)

Γ

where: qˆ – the measured values of state fields (e.g. displacements, temperatures); q – the values of the same state fields calculated from the solution of the direct problem, z = zi – i-th sensor point, n – a number of sensor points, δ – the Dirac function.

8.2.3 Identification of Elastic Constants in Laminates The identification problem in hybrid multilayered laminates is considered. Hybrid laminates have plies composed of different materials [1]. In present paper interply hybrids are considered. In interply hybrid laminates external plies are made of more expensive material having better properties while core plies are made of ’worse’, but cheaper material. The main reason of using hybrid laminates is looking for a balance between cost and other properties of the laminate. Fibre–reinforced multilayered laminates can be typically treated as 2–dimensional orthotropic structures with 4 independent elastic constants: axial Young modulus E1 , transverse Young modulus E2 , axial–transverse shear modulus G12 and axial– transverse Poisson ratio ν12 . The constitutive equation for a single layer of the laminate in the in–axis orientation has the following form [7]: ⎡ ⎧ ⎫ ⎨ σ11 ⎬ ⎢ ⎢ σ22 = ⎢ ⎩ ⎭ ⎢ ⎣ σ12

E1 ν21 E1 1−ν12 ν21 1−ν12 ν21

ν12 E2 E2 1−ν12 ν21 1−ν12 ν21

0

0

0

⎤

⎧ ⎫ ⎥ ⎨ ε11 ⎬ ⎥ 0 ⎥ ⎥ ⎩ ε22 ⎭ ⎦ ε12 G12

(8.14)

where: σi j – stress vector; εi j – strain vector; ν21 – transverse–axial Poisson ratio.

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The Poisson ratio ν21 can be expressed as:

ν21 = ν12

E2 E1

(8.15)

Apart from the identification of the elastic constants it is also necessary to identify densities ρ of particular materials in the hybrid laminate. As a result the design variables vector for a hybrid laminate can be written as: 1 2 x = {E11 , E21 , G112 , ν12 , ρ 1 , E12 , E22 , G212 , ν12 , ρ 2}

(8.16)

where superscripts denote the material number. Identification of the laminates’ elastic constants can be formulated as a minimization of the funcional J(x): N

min J, J = ∑ [qi − qî ]2 x

(8.17)

i=1

where: qî – the measured values of state fields; qi – the values of the same state fields calculated from the solution of the direct problem; The identification procedure is usually performed for the data obtained from the structure response (e.g. strain field) to the static load. This attitude is not efficient enough for the laminate structures due to problems with obtaining of the uniform stress and strain fields, boundary effects and scale effect. To avoid mentioned problems and reduce the number of sensor points the modal analysis methods are employed [13]. Two types of dynamic data are considered: i) eigenfrequencies; ii) accelerations in one sensor point in form of the frequency response. It is assumed that identified material parameters and measurements have granular character.

8.3 Two–Stage Granular Strategy In order to solve the identification problem, the Two–Stage Granular Strategy (TSGS) is proposed. Gradient optimization methods are fast and give precise results but they can be used for continuous problems. If the multimodal problems are considered gradient methods usually lead to the local optima. In such cases the evolutionary algorithms, which are the global optimization methods, can be used [10]. As the evolutionary algorithms are time–consuming and they have problems with obtaining the optimum value precisely, it can be convenient to merge both methods in a hybrid algorithm. The TSGS couples the advantages of global and local optimization methods. A Granular Evolutionary Algorithm (GEA) with granular genes is applied in the first stage. The local optimization method supported by an Artificial Neural Network


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(ANN) is used in the second stage to find the precise value of the optimum.The block diagram of the TSGS is presented in Fig. 8.3.

START GRANULAR EA (global optimization)

TRANSITION?

NO

YES

LOCAL OPTIMIZATION with ANN

Fig. 8.3 Block diagram of the Two–Stage Granular Strategy

STOP

8.3.1 The Global Optimization Stage The mean idea of GEA is similar to the classical evolutionary algorithm [2], but some modifications are required due to the granular character of the data. In the GEA the data representation, the evolutionary operators as well as the selection procedure are granular. Each chromosome, being a vector of identified variables (genes), represents one granular solution. The block diagram of the GEA is presented in Fig. 8.4. To solve the boundary–value problem the Granular (interval, fuzzy or stochastic) FEM is employed [8],[11]. Two special evolutionary operators are introduced: i) granular Gaussian mutation operator; ii) granular arithmetic crossover operator [12]. After evaluation of the solution a granular (interval, fuzzy or stochastic) fitness function value of the chromosome is obtained. The granular selection is based on the tournament selection method. The granular fitness function values are compared to select the best individual in the tournament. The better chromosome wins with a probability depending on the introduced parameter β . The cluster of granular solutions is achieved as a result of the first stage of the strategy.

8.3.2 The Local Optimization Stage Assuming that the cluster of points obtained in the previous stage is situated close to the global optimum, the local optimization is performed by means of the steepest descent method. The cluster of points is used to generate the training vector for

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Fig. 8.4 Block diagram of the Granular Evolutionary Algorithm

Stop condition YES STOP

ANN. ANN is used to evaluate the granular fitness function value [5]. The sensitivity of the fitness function is approximated by means of the ANN to avoid problems with the calculation of the objective function gradient. The special multi–level ANN is introduced. The number of levels is dependent on the data representation and is equal to the number of parameters in each gene in the first stage. The number of neurons in the input layer (for each level) equals the number of design variables. The number of neurons in hidden layers depends on the complexity of the identification problem. The single output (at each level of the output layer) represents one parameter of the granular fitness function value. The exemplary ANN for fuzzy representation of the data is presented in Fig. 8.5. The central level of the multilevel ANN corresponds with the central value cv of the fuzzy number (black colour), while the other levels correspond with other parameters of the fuzzy number: the grey levels correspond with the parameters ai , the white levels correspond with the parameters bi .

Fig. 8.5 The scheme of the multilevel ANN (fuzzy case)


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After training the multilevel ANN the local optimization process is performed until the termination condition is satisfied. If not, the point is calculated by using the Granular FEM and added to the training vector.

8.4 Numerical Examples The Two–Stage Granular Strategy is applied to solve some identification problems assuming different types of information granularity: interval numbers, fuzzy numbers or random variables. TSGS is applied to identify: • the position and the radius of the circular void in isotropic structure subjected to the dynamical loading; • the boundary conditions in thermomechanical systems; • the elastic constants in laminate structures.

8.4.1 Identification of Void A square plate 0.2x0.2 m made of an isotropic material (Fig. 8.6) is considered. The aim of the identification is to find the parameters of the circular defect. The chromosome x = [x, y, r] consists of 3 genes representing position x, y and radius r of the void. The plate is loaded by the continuous dynamical loading p(z,t) = p0 (z) H(t) , where p0 (z)=const=10 kN and H(t) is the Heaviside’s function. p ( z, t )

r x,y

sensor point

Fig. 8.6 The structure with identified circular defect

200 time–steps with δ = 1μ s are considered. It is assumed that the Young modulus of the plate material E and loading p0 have granular character. The displacements are measured in 21 sensor points on the boundary. The parameters of the GEA are: the number of chromosomes: psize=20; the number of generations: gen=40; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.4. The number of iterations of the local method

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is assumed to be 200. The variable ranges, actual values and results after the first and second stages for interval and fuzzy cases (for two α –cuts) are collected in Tables 8.1– 8.3 . Table 8.1 Void identification results: interval numbers

Min Max Actual Stage 1 Stage 2

x (m) a

b

y (m) a

b

r (m) a

b

0.005 0.2 0.029 0.0292 0.029

0.005 0.2 0.031 0.0311 0.031

0.005 0.2 0.029 0.0297 0.029

0.005 0.2 0.031 0.0307 0.031

0.005 0.1 0.019 0.0187 0.019

0.005 0.1 0.021 0.0209 0.021

Table 8.2 Void identification results: fuzzy numbers, lower α –cut


x (m) a

b

y (m) a

b

r (m) a

b

0.005 0.2 0.025 0.023 0.025

0.005 0.2 0.035 0.034 0.035

0.005 0.2 0.025 0.028 0.025

0.005 0.2 0.035 0.032 0.035

0.005 0.1 0.015 0.018 0.015

0.005 0.1 0.025 0.023 0.025

Table 8.3 Void identification results: fuzzy numbers, upper α –cut


x (m) a

b

y (m) a

b

r (m) a

b

0.005 0.2 0.025 0.022 0.025

0.005 0.2 0.035 0.037 0.035

0.005 0.2 0.025 0.026 0.025

0.005 0.2 0.035 0.031 0.035

0.005 0.1 0.015 0.015 0.015

0.005 0.1 0.025 0.021 0.025

8.4.2 Identification of Boundary Conditions A box–shaped steel structure presented in Fig.8.7 is considered. The structure is subjected to the thermomechanical loading. One surface of the box is supported


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while the point load is applied at every node on the opposite one. The total load is equal to 224 kN. The temperature T = 10◦C is applied on the supported surface of the structure. The third type thermal boundary condition (convection) is specified on the internal surfaces. The aim of the identification is to find the ambient temperature T ∞ and heat convection coefficient α on the internal surfaces. The material properties are: Young modulus E=2e11 MPa, Poisson ratio ν =0.3, thermal expansion coefficient αT = 12.5 e−61/◦C and thermal conductivity λ = 25W /m◦C.

T u

ts poin ents sor sen placem is of d

0.32

8

sen of t sor po em per ints atu re

a

T

s

1.0

P

0.5

Fig. 8.7 Geometry, boundary conditions and location of the sensor points

To gather measurement data 4 displacement sensors and 4 temperature sensors located on external surfaces of the structures are used. The parameters of the GEA are: the number of chromosomes: psize=30; the number of generations: gen=200; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.1. The number of iterations of the local method is assumed to be 500. The variable ranges, actual values and results after the first and second stages for interval and fuzzy cases (for 2 α –cuts) are collected in Table 8.4– 8.6. Table 8.4 Boundary conditions identification results: interval numbers


α (W /m2◦C) a

b

T ∞ (◦C) a

b

1.00 25.00 3.00 2.61 3.00

1.00 25.00 7.00 6.76 7.00

0.00 105.00 45.00 49.99 45.00

0.00 105.00 52.00 53.33 52.00

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Table 8.5 Boundary conditions identification results: fuzzy numbers, lower α –cut


α (W /m2◦C) a 1.00 25.00 3.00 2.06 3.00

b

T ∞ (◦C) a

b

1.00 25.00 7.00 8.45 7.00

0.00 105.00 45.00 43.54 45.00

0.00 105.00 52.00 54.44 52.00

Table 8.6 Boundary conditions identification results: fuzzy numbers, upper α –cut


α (W /m2◦C) a

b

T ∞ (◦C) a

b

1.00 25.00 4.00 4.98 4.00

1.00 25.00 6.00 6.45 6.00

0.00 105.00 48.00 49.95 48.00

0.00 105.00 51.00 51.93 51.00

8.4.3 Identification of Laminates’ Elastic Constants The aim of the identification is to find the elastic constants of multi–layered, fibre– reinforced laminate. It is assumed, that identified constants are not precise and can represented by different types of information granularity. Dynamical data are collected to perform the identification procedure. A rectangular, symmetrical hybrid laminate plate of shape and dimensions presented in Fig. 8.8a) is considered. Each ply of the laminate has thickness h=0.002 m. The stacking sequence of the laminate is (0/15/-15/45/-45)s. The external plies are made of material M1 , the internal plies are made of the material M2 (Fig. 8.8 b). y

0.2

excitation point

x

M1 { M2

{

symmetry

sensor point 0.5 a)

b)

Fig. 8.8 The hybrid laminate plate: a) shape and dimensions; b) materials location


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Each chromosome in population consists of 10 genes representing identified constants and material densities (8.16). The FEM model consists of 200 4–node plane finite elements.

8.4.3.1 Interval Case It is assumed that each variable xi has granular character represented by interval number. The plate is excited in one point (Fig. 8.8a) by the the sinusoidal signal. The frequency of the excitation varies from 100 Hz to 2000 Hz. 200 samples of the acceleration amplitudes at one sensor point are measured. The parameters of the GEA are: the number of chromosomes: psize=100; the number of generations: gen=400; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.4. The number of iterations of the local method is assumed to be 2500. The variable ranges, actual values and results after both stages for materials M1 and M2 are collected in Tables 8.7– 8.8. Table 8.7 Hybrid laminate identification results: interval numbers, material M1 E1 (Pa) a b Min Max Actual Stage 1 Stage 2

E2 (Pa) a b

G12 (Pa) a b

1.50E10 1.50E10 4.20E9 4.20E9 0.90E9 1.50E11 1.50E11 26.00E9 26.00E9 9.00E9 1.80E11 1.82E11 10.00E9 10.04E9 7.10E9 1.72E11 1.83E11 19.87E9 10.36E9 7.01E9 1.80E11 1.82E11 10.00E9 10.04E9 7.10E9

0.90E9 9.00E9 7.18E9 7.43E9 7.18E9

ν12 a

b

ρ (kg/m3 ) a b

0.190 0.410 0.277 0.273 0.277

0.190 0.410 0.283 0.274 0.283

1.00E3 3.00E3 1.60E3 1.58E3 1.60E3

1.00E3 3.00E3 1.65E3 1.69E3 1.65E3

Table 8.8 Hybrid laminate identification results: interval numbers, material M2 E1 (Pa) a b Min Max Actual Stage 1 Stage 2

E2 (Pa) a b

G12 (Pa) a b

1.50E10 1.50E10 4.20E9 4.20E9 0.90E9 1.50E11 1.50E11 16.00E9 16.00E9 9.00E9 3.82E10 3.90E10 8.23E9 8.31E9 4.10E9 3.74E10 4.12E10 8.21E9 8.22E9 4.01E9 3.82E10 3.90E10 8.23E9 8.31E9 4.10E9

0.90E9 9.00E9 4.18E9 4.54E9 4.18E9

ν12 a

ρ (kg/m3 ) a b

b

0.190 0.410 0.257 0.273 0.257

0.190 1.00E3 0.410 3.00E3 0.263 1.80E3 0.278 1.58E3 0.263 1.80E3

1.00E3 3.00E3 1.81E3 1.69E3 1.81E3

8.4.3.2 Stochastic Case It is assumed that each variable xi and measurements have granular character represented by random variable with normal distribution described by mean value m and

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standard deviation σ . The first 10 eigenfrequencies of the plate are the measurement data. The measurements were repeated 200 times to collect data. The parameters of the GEA are: the number of chromosomes: psize=400; the number of generations: gen=1200; the arithmetic crossover probability pac =0.2; the Gaussian mutation probability: pgm =0.4. The number of iterations of the local method is assumed to be 5000. The variable ranges, actual values and results after both stages for materials M1 and M2 are collected in Tables 8.9– 8.10. Table 8.9 Hybrid laminate identification results: random variables, material M1


E1 (Pa) a b

E2 (Pa) a b

G12 (Pa) a b

ν12 a b

ρ (kg/m3 ) a b

1.20E11 2.50E11 1.80E11 2.01E11 1.80E11

0.80E10 2.00E10 1.00E10 0.89E10 1.00E10

2.00E9 9.00E9 7.14E9 6.89E9 7.14E9

0.00 0.50 0.28 0.31 0.28

1.00E3 3.00E3 1.60E3 1.63E3 1.60E3

0.00E10 0.30E10 0.12E10 0.18E10 0.12E10

0.00E10 0.30E10 0.20E10 0.18E10 0.20E10

0.10E8 0.70E8 0.50E8 0.52E8 0.50E8

0.00 0.10 0.02 0.01 0.02

0.00E2 0.50E2 0.20E2 0.19E2 0.20E2

Table 8.10 Hybrid laminate identification results: random variables, material M2 E1 (Pa) σ m

E2 (Pa) m σ

Min

2.00E10 0.00E9 4.00E9 0.00E9 0 Max 6.00E10 0.30E9 9.00E9 0.30E9 Actual 3.86E10 0.12E9 8.28E9 0.20E9 Stage 1 3.75E10 0.04E9 8.12E9 0.28E9 Stage 2 3.86E10 0.12E9 8.28E9 0.20E9

G12 (Pa) m σ

ν12 m σ

ρ (kg/m3 ) m σ

2.00E9 0.10E8 0.00 0.00 1.00E3 0.00E2 6.00E9 4.14E9 4.21E9 4.14E9

0.70E8 0.50E8 0.61E8 0.50E8

0.50 0.26 0.25 0.26

0.10 0.02 0.04 0.02

3.00E3 1.80E3 1.81E3 1.80E3

0.50E2 0.20E2 0.17E2 0.20E2

8.5 Final Conclusions The Two–Stage Granular Strategy coupling the Granular Evolutionary Algorithm, the multilevel Artificial Neural Networks and local optimization methods has been presented. The application of global optimization method (GEA) in the first stage of the TSGS decreases the possibility of finding the local minimum of the objective function. The gradient method applied in the second stage is supported by ANN. ANN is used to calculate the fitness function gradient and to evaluate the granular fitness function value. The granular version of the FEM is used to solve the boundary–value problems. The identified values and the fitness functions can be represented by different forms of granularity: interval numbers, fuzzy numbers or random variables.


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The strategy gives positive results for the identification of various parameters in isotropic and orthotropic structures, for static and dynamical cases as well as for thermo–mechanical problems with different variants of uncertainties. Acknowledgements. The research is financed from the Polish science budget resources in the years 2008–2011 as the research project.

References [1] Adali, S., et al.: Optimal design of symmetric hybrid laminates with discrete ply angles for maximum buckling load and minimum cost. Compos. Struct. 32, 409–415 (1995) [2] Arabas, J.: Lectures on Evolutionary Algorithms (in Polish). WNT, Warsaw (2001) [3] Bargiela, A., Pedrycz, W.: Granular Computing: An introduction. Kluwer, Boston (2002) [4] Bui, H.D.: Inverse Problems in the Mechanics: An Introduction. CRC PRess, Bocca Raton (1994) [5] Burczyński, T., Orantek, P.: The evolutionary algorithm in stochastic optimization and identification problems. In: Arabas, J. (ed.) Evolutionary Computation and Global Optimization 2007, Warsaw, pp. 309–320 (2006) [6] Burczyński, T., Orantek, P.: Uncertain Identification Problems in the Context of Granular Computing. In: Bargiela, A., Pedrycz, W. (eds.) Human-Centric Inforamtion Processing. SCI, vol. 182, pp. 329–350. Springer, Heidelberg (2009) [7] German, J.: The basics of the fibre-reinforced composites’ mechanics (in Polish). Publ. of the Cracow University of Technology, Cracow (2001) [8] Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. John Wiley & Sons, New York (1992) [9] Maugin, G.A.: The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction. World Scientific, Singapore (1999) [10] Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolutionary Programs. Springer, Berlin (1996) [11] Moens, D., Vandepitte, D.: Fuzzy Finite Element Method for Frequency Response Function Analysis of Uncertain Structures. AIAA Journal 40(1), 126–136 (2002) [12] Orantek, P.: The optimization and identification problems of structures with fuzzy parameters. In: 3rd European conference on computational mechanics ECCM 2006. CDEdition, Lisbon (2006) [13] Uhl, T.: Computer-Aided Identification of Constructional Models (in Polish). WNT, Warsaw (1997)

Chapter 9

Immune Computing: Intelligent Methodology and Its Applications in Bioengineering and Computational Mechanics Tadeusz Burczyński, Michał Bereta, Arkadiusz Poteralski, and Mirosław Szczepanik

Abstract. The aim of this paper is to provide a set of carefully selected problems connected with the current research directions of Immune Computing. This approach belongs to biology inspired methods. Due to the complexity of functioning of the natural immune system, extracting higher level paradigms which could serve as the basis of constructing computational models and algorithmic solutions is made. Applications of this intelligent methodology to bioengineering and computational mechanics problems are presented.

9.1 Introduction Nature inspired computing has proved to be useful in various application areas. Evolutionary methods, neural networks, swarm intelligence and many other approaches have been applied to many technical and engineering problems, such as optimization, learning, data analysis, knowledge engineering and many others. Some of these methods perform better in the given application areas and some work better in others. However, it can hardly be assumed that there exists a problem domain in which nature inspired techniques are not employed, at least as a part of the proposed solution. After decades of development, biologically inspired methods are well Tadeusz Burczyński Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected] Michał Bereta Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland Arkadiusz Poteralski, Mirosław Szczepanik Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland


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established and appreciated tools. On the other hand, many novel approaches arise to solve problems in innovative ways, hopefully more effectively. One of such novel areas is the field of Artificial Immune Systems (AIS) [2][4][6][7]. The question arises as to what AIS can offer as problem solving techniques and whether the paradigms proposed by AIS researchers are in fact novel. The aim of this paper is to provide a set of carefully selected problems connected with the current research directions of Immune Computing.

9.2 Immunology as a Metaphor for Computational Information Processing An immune system, especially this of vertebrates, is a very complicated system of interacting cells, organs and mechanisms, whose purpose is to protect the host body against any danger, either exterior or internal. To achieve that goal the immune system has to decide not only about what is not part of the host but also what can cause a damage. This is a very difficult task, as not all that comes from outside is dangerous. On the other hand, autoimmune diseases are examples of internal threats. To protect the host against all such dangers is not an easy task, especially in a changing environment. It is obvious that the immune system has to develop some sense of self, i.e., the sense of what is part of the host. How the immune system achieves this is hard to explain, as the host body changes its functioning and structure over time. Nonetheless, the immune system is able to perform its task effectively. To deal with such a difficult task the immune system needs the ability to learn new threats, to remember previous experiences and to develop specialized responses to different pathogens. Taking a closer look at all these features one can state that the immune system can be considered as a cognitive system. For that reason the immune system gained an interest of computational sciences. The artificial immune systems [4][6][7] are developed on the basis of a mechanism discovered in biological immune systems. The immune system is a complex system which contains distributed groups of specialized cells and organs. The main purpose of the immune system is to recognize and destroy pathogens - funguses, viruses, bacteria and improper functioning cells. The lymphocytes cells play a very important role in the immune system. The lymphocytes are divided into several groups of cells. There are two main groups B and T cells, both contains some subgroups (like B-T dependent or B-T independent). The B cells contain antibodies, which could neutralize pathogens and are also used to recognize pathogens. There is a big diversity between antibodies of the B cells, allowing recognition and neutralization of many different pathogens. The B cells are produced in the bone marrow in long bones. A B cell undergoes a mutation process to achieve big diversity of antibodies. The T cells mature in thymus, only T cells recognizing non self cells are released to the lymphatic and the blood systems. There are also other cells like macrophages with presenting properties, the pathogens are processed by a cell and

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presented by using MHC (Major Histocompatibility Complex) proteins. The recognition of a pathogen is performed in a few steps (Fig. 9.1). Pathogenreceptor binding Pathogen presentation as MHC signal MHC protein T cell receptor

Pathogen B cell receptor B cell

B cell activation signal

a)

T cell

b) Activated B cell Proliferation

Memory cell Proliferation Memory cell

Memory cell Secreted antibodies

c)

d)

Secreted antibodies

Macrophage

e)

Pathogen with bound antibodies

Fig. 9.1 An immune system, a) B cell and pathogen, b) the recognition of pathogen using Band T-cells, c) the proliferation of activated B-cells, d) the proliferation of a memory cell secondary response, e) pathogen absorption by a macrophage

First, the B-cells or macrophages present the pathogen to a T-cell using MHC (Fig. 9.1b), the T-cell decides if the presented antigen is a pathogen. The T-cell gives a chemical signal to B-cells to release antibodies. A part of stimulated B-cells goes to a lymph node and proliferate (clone) (Fig. 9.1c). A part of the B-cells changes into memory cells, the rest of them secrete antibodies into blood. The secondary response of the immunology system in the presence of known pathogens is faster because of memory cells. The memory cells created during primary response, proliferate and the antibodies are secreted to blood (Fig. 9.1d). The antibodies bind to pathogens and neutralize them. Other cells like macrophages destroy pathogens (Fig. 9.1e). The number of lymphocytes in the organism changes, while the presence of pathogens increases, but after attacks a part of the lymphocytes is removed from the organism. Given all the complexity of functioning of the immune system, it is necessary to extract higher level paradigms which could serve as the basis of constructing computational models and algorithmic solutions. The most important paradigms in the filed of Artificial Immune Systems are a Clonal Selection (CS), a Immune Network Theory (IMT), a Negative Selection (NS) and recently emerged a Danger Theory (DT).

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The question is whether AIS can offer anything really new and/or useful. The Clonal Selection can seem to be another exemplification of evolutionary approach to problem solving. The fact of existing of immune networks in biological immune systems is questioned by biologists. The Negative Selection appears to be truly novel approach, but one could ask whether it is enough to invest time and resources to develop AIS. In the first examinations of the immune ideas, the researchers developed several algorithmic solutions based on immune paradigms, often separately. The Clonal Selection and the Immune Network Theory have been applied to optimization, data analysis or clustering. The Negative Selection has been applied to computer security, anomaly or fault detection. Many of the proposed algorithms have successfully dealt with the tasks appointed to them. However, their usefulness according to their robustness and scalability has been under dispute when compared to other well established computational methods. The broad application areas and the new ideas proposed, show the vitality of the still young research field of Immune Computing in intelligent problem solving techniques. Applications of AIS to bioengineering problems as feature selection and classification of ECG signals are presented. Possibility of a coupling immune computing with the finite element method in solving several 2-D and 3-D optimization problems is also considered.

9.3 Applications of AIS in Computational Bioengineering 9.3.1 Negative Selection In natural immune systems T-lymphocytes are produced in thymus and cannot leave it if they do not pass the test of recognizing none of self-molecules presented to them. If a T-cell recognizes any of self-cells, it is eliminated. T-cells which leave the thymus are capable of recognizing non-self antigens while being neutral to selfcells. This paradigm is called the Negative Selection (NS) and is used in the AIS which deal with the problem of anomaly detection. NS algorithms have two stages: (i) a training stage and (ii) a detection stage (Fig. 9.2). In the training stage, detectors are created only by means of normal samples, no negative samples are presented to candidate T-cells during training. This is a great advantage of negative selection as in many tasks it is hard to prepare a representative collection of negative samples for training purpose. After the training stage, the created T-cells are used for monitoring new samples, which can be categorized as normal (self) or as antigens non-self, anomalies). A sample is detected by a T-cell if it matches to the given T-cell well enough. The matching process depends on the representation of both T-cells, self-samples and antigens. In the case of binary representation there are two manners for matching: (i) checking the number of common bits and (ii) checking the longest common substring between two binary strings.

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Fig. 9.2 Two stages of negative selection: training stage and detection stage

The sample is bound by the detector if the matching is above a given threshold. The matching process is presented in Fig. 9.3.

Fig. 9.3 Matching process in the case of binary representation of T-cells, self-data, and antigens

Assume, that in an anomaly detection task, the normal state of the process over sometime can be represented as one-dimensional signal. Having the records of the normal state of the process, one can create a set of self-cells as a set of binary strings using the method of a moving window (Fig. 9.4).

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Fig. 9.4 Moving window method for binary strings creation from real-valued signals

For each window’s position, one binary string is created. Its length depends on the window’s size and on the number of divisions of the [min, max] range. Each real value in the window is first coded as the number of range in the [min, max], in which it falls. Then, the numbers of range of each real value in the window are concatenated creating one binary string. Created binary strings are the self-samples used during the training stage of negative selection. After learning, while monitoring the process, created T-cells should be capable of recognizing unusual and abnormal states of the process. The negative selection can be also modelled by real-valued representation [1].

9.3.2 Clonal Selection The clonal selection is a process characteristic for B-lymphocytes. B-cells move freely in the body and create clones while they are stimulated. The stimulation of B-cells by antigens can be simulated by means of binary representation in the same fashion as simulating bindings of T-cells, as described earlier. The stronger the binding is, the more clones the given B-cell can be produce. Clones are mutated at very high rates (somatic hyper-mutation) which allows the system to find better lymphocytes. The important mechanisms of the immune systems is the suppression mechanism (crowding mechanism) (Fig. 9.5). B-cells react not only to the presence of the antigens but to the presence other B-cells, too, trying the eliminate them. In this way, the weakest cells are eliminated from the population. The suppression is used in AIS as a mechanisms to eliminate worse solutions represented by less stimulated artificial cells. The proper suppression mechanism plays the important role in the whole process as it is able to keep the tractable population size and necessary population diversity.

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Fig. 9.5 The clonal selection algorithm

9.3.3 Two Level Immune Classification of ECG Signals The concept of two level immune algorithm is used in the proposed system for detectors creation and feature selection (Fig. 9.6). An initial set of subpopulations P is generated randomly. Each subpopulation Pi is characterized by a binary string. This algorithm consists of negative and clonal selections.

Fig. 9.6 The concept of two level immune algorithm

The detailed description of the algorithm is given in [1]. This algorithm is summarized in Fig. 9.7.

9.3.4 Results of Feature Selection and Classification of ECG Signals The immune algorithm was tested in a task of recognizing heart diseases in ECG signals. The signals were taken from the MIT-BIH database [5]. Different types of

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Fig. 9.7 The main steps of two level algorithm

samples representing normal and pathological ECG signals were extracted from this database. Each sample’s length was 128 with middle of QRS complex as the central point. QRS complex is a part of ECG signal (Fig. 9.8) that is supposed to have the most useful medical information for diagnosis.

Fig. 9.8 A fragment of ECG record with QRS complex selected by a rectangular

ECG samples of each type used for learning and testing as well as the explanations of samples’ symbols are contained in Tab. 9.1. Fig. 9.9 presents ECG samples of different types used in the research. The samples were divided into a normal signals set used for creating self-strings and a set of pathological signals. All signals were filtered using window mean method with window size 4 and step size 4 and normalized to have zero mean and

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Fig. 9.9 Examples of ECG signals of normal N type and different pathological types

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Table 9.1 ECG samples used for training and testing Type

Train

Test

Total

Disease

N A E L R V

1226 270 93 236 249 270

23654 4796 119 15898 14261 13914

24880 5066 212 16134 14510 14186

Normal Atrial premature beat Ventricular escape beat Left bundle branch block beat Right bundle branch block beat Premature ventricular contraction

unit variance, resulting in a new set of samples of length 32. Pi were trained to discriminate between normal state and heart diseases. Then Haar wavelets coefficients were counted resulting in 32-dimensional search space. Table 9.2 Results for Haar wavelets coefficients of window -mean filtered ECG signals Type

Train

Recognised Test (% )

Recognised (% )

N (self) A E L R V Non-self

1226 270 93 236 249 270 1120

0 92.2 100 100 99.6 99.3 97.9

41.1 86.4 100 99.4 99.6 98 97.8

23654 4796 119 15898 14261 13914 48988

The results presented in Table 9.2 show that the proposed immune system is able to select the suitable subset of features.

9.4 Applications in Computational Mechanics 9.4.1 Immune Optimization The Clonal Selection mechanism is very useful in global optimization problems, especially in computational mechanics [3]. The unknown global optimum is the most dangerous searched pathogen. The memory cells contain design variables and proliferate during the optimization process. The B-cells created from memory cells undergo a mutation. The crowding (suppression) mechanism forces the diverse between memory cells. The objective functions for B-cells are evaluated by means of the finite element method. The selection process exchanges some memory cells for better B-cells (Fig. 9.10).

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START Creation of memory cells

Memory cells proliferation with hipermutation Evaluation of objective function for B cells Selection

Crowding mechanism STOP

Stop condition CONTINUE

STOP

Fig. 9.10 An artificial immune system for optimization problems

The presented approach is based on the application of the artificial immune system and the finite element method to topology optimization of 2-D and 3-D structures. The fitness function is calculated for each B-cell in each iteration by solving a boundary-value problem by means of the finite element method (FEM). In order to solve the optimization problem the fitness function, design variables and constraints should be formulated.

9.4.2 Formulation of Immune Topology Optimization of Structures Consider a structure which, at the beginning of an immune process, occupies a domain Ω0 (in E d ,d = 2 or 3), bounded by a boundary Γ0 . The domain Ω0 is filled by elastic homogeneous and isotropic material of a Young’s modulus E0 , a mass density ρ0 and a Poisson ratio ν . The structures are considered in the framework of the linear theory of elasticity. During the immune optimization process the domain Ωi , its boundary Γi and the field of mass densities ρ (x) = ρi , x ∈ Ωi can change for each iteration t (for t = 0, ρ0 = const). The immune process proceeds in the environment in which the structure fitness is described by the minimization of the mass of the structure J= ρ dΩ (9.1) Ω

with constraints imposed on displacements of the structure |u(x)| ≤ uad , x ∈ Ω or with constraints imposed on equivalent stresses of the structure

(9.2)

176


σeq (x) ≤ σ ad , x ∈ Ω

(9.3)

In order to solve the formulated problems the finite element models of 2-D and 3-D structures are considered. A 2-D structure domain is divided into shell finite elements and a 3-D structure domain is divided into solid finite elements. In order to solve direct problems for 2-D and 3-D elastic structures the professional program MSC.NASTRAN is used. The distribution of mass density ρ (x), x ∈ Ω , (Fig. 9.11) in the structure is described by a surface Wp (x), x ∈ H 2 (for 2-D) and a hyper surface Wp (x), x ∈ H 3 (for 3-D). The surface (hyper surface) Wp (x) is stretched under H d ⊂ E d , (d = 2, 3) and the domain Ωi is included in H d , i.e. (Ωt ⊆ H d ). The shape of the surface (hyper surface) Wp (x) is controlled by B-cell receptors d j , j = 1, 2, ..., G which create a B-cell B − cell =< d1 , d2 , ..., dG > (9.4) ≤ d j ≤ d max d min j j

(9.5)

where d min , d max - are minimum and maximum values of the B-cell receptor, rej j spectively. B-cell receptors are the values of the function Wp (x) in the control points x j of the surface (hyper surface), i.e. d j = Wp (x j ), j = 0, 1, 2, ..., G.

Fig. 9.11 The illustration of the idea of immune topology optimization for a 2-D structure

The finite element method is applied in analysis of the structure. The domain Ω of the structure is discretized using the finite elements, Ω =

E )

Ωe . The assignation

e=1

of the mass density to each finite element Ωe is performed by the mappings:

ρe = Wp (xe ), xe ∈ Ωe , e = 1, 2, ..., E

(9.6)

It means that each finite element can have different mass density. When the value of the mass density for the e-th finite element is included in:

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• the interval 0 ≤ ρe < ρmin , the finite element is eliminated and a void is created, • the interval ρmin ≤ ρe < ρmax , the finite element remains. In the next step the Young’s modulus for the e − th finite element is evaluated using the following equation Ee = Emax (

ρe r ) ρmax

(9.7)

where: Emax and ρmax - Young’s modulus and mass density for the same material, respectively, r - parameter which can change from 1 to 9 [3]. By means of the proposed method, the material properties of finite elements are changed and some of them are eliminated. As a result the optimal shape, the topology and the material of the structures are obtained. The multinomial interpolation of the hyper surface is expressed as follows ⎤ ⎡ d1 −1 Wρ (x) = Φ X ⊗ Y−1 ⊗ Z−1 ⎣ div ⎦ (9.8) d27 where

Φ = [1,x,x2 ] ⊗ [1,y,y2 ] ⊗ [1,z,z2 ] = [1,z,z2 ,y,yz,yz2 ,y2 , y2 z,y2 z2 ,x,xz,xz2 ,xy,xyz,xyz2 ,xy2 ,xy2 z,xy2 z2 , x2 ,x2 z,x2 z2 ,x2 y,x2 yz,x2 yz2 ,x2 y2 ,x2 y2 z,x2 y2 z2 ]

and X, Y and Z are matrices described as follows ⎤ ⎡ 100 X = Y = Z = ⎣1 1 1⎦ 124

(9.9)

(9.10)

Fig. 9.12 Spacing of control points

The structure is under the immune optimization process inserted into a cube whose edges have the length A=2, B=2, C=2 (Fig. 9.12). The model of the structure is scaled by means of following expressions: x = (A* coordinate x)/length of the model y = (B* coordinate y)/width of the model z = (C* coordinate z)/height of the model

(9.11)

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The final structure obtained after the optimization process has a rough boundary. In order to get a smooth shape of the boundary, a procedure which smoothes it has to be used. The artificial immune system works on the group of B-cells. The operations described above are performed for a single B-cell from the population and lead to the evaluation of the fitness function value (Fig. 9.13).

Fig. 9.13 Operation scheme performed for a single Bcell

9.4.3 Numerical Examples of Immune Topology Optimization Two numerical examples of topology optimization are presented: • a 2-D structure (plane stress) (Example 1), • a 3-D solid body (Example 2), for minimization of the structures mass (9.1) with displacement (9.2) and stress (9.3) constraints. Structures are considered in the framework of the theory of elasticity. Results of topology optimization are obtained by using the artificial immune system with following parameters (Tab. 9.3) Table 9.3 Parameters of the artificial immune system Parameter

Value

number of memory cells number of clones crowding (suppression) factor Gaussian mutation

8 4 25% 20%

9.4.3.1 Example 1 A rectangular 2-D structure (plane stress), of dimensions 100 200 [mm], loaded with the concentrated force P in the center of the lower boundary and fixed on the bottom corners is considered (Fig. 9.14a). Due to the symmetry a half of the structure is considered. The input data to the optimization program and parameters of the

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artificial immune system are included in Tab. 9.3 and 9.4, respectively. The geometry, the distribution of the control points of the interpolation surface is shown in the Fig. 9.14b. The results of the optimization process are presented in the Fig. 9.15. Table 9.4 The input data to the optimization task of for a 2-D structure (Example 1)

σ ad

thickness [mm] P [kN]

range of ρe [g/cm3 ]

80.0

4.0

7.3≤ ρe < 7.5 (FE elimination) 7.5≤ ρe < 7.86 (FE leaving)

2

a)

b)

Fig. 9.14 A 2-D structure (Example 1): a) the geometry; b) the distribution of the control points of the interpolation surface

a)

c)

b)

d)

Fig. 9.15 Results of immune topology optimization of 2-D structure: a) the solution of the optimization task; b) the map of mass densities; c) the map of stresses; d) the map of the displacement

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9.4.3.2 Example 2 A 3-D structure with dimensions and loading is presented in the Fig. 9.16a and 9.16b. The input data to the optimization program are included in Tab. 9.5. The geometry, the distribution of the control points of the interpolation hyper surface is shown in the Fig. 9.16c. The results of the optimization process are presented in Fig. 9.17a and Fig. 9.17b.

Fig. 9.16 Two cases of loading with the hyper surface a) first case (compression), b) second case (tension), c) the distribution of the control points of the interpolation hyper surface

Table 9.5 The input data to the optimization task of for a 3-D structure (Example 2) a

Dimensions [mm] b c

100

100

100

Loading/Constraints Compression (a) Tension (b) Q = −36.3[kN] uad = 0.05[mm] σ ad = 35[MPa]

Q = −36.3[kN] uad = 0.03[mm] σ ad = 55[MPa]

It is worth noticing that different constraints can lead to the total different final solutions of topology optimization of the structure.

9 Immune Computing

Fig. 9.17 Immune optimization of 3-D structure: a) first case (compression), b) second case (tension)

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a)

b)

9.5 Conclusions The Artificial Immune Systems (AIS) as novel areas of computing can be considered as computational systems based on natural immune system metaphors. They are composed by intelligent methodologies and can be regarded as data manipulation, classification, representation and reasoning strategies that follow a biological paradigm - the human immune system. The growing interest for AIS has been observed recently. Biological inspiration provides utility and extension and also improves comprehension of natural phenomena. Examples presented in the paper show that immune computing can be very useful in computational engineering, especially for bioengineering and computational optimization of structures. AIS have also several interesting features, lake: high degree of parallelism, robust-ness with low redundancy and possibility of using heuristics to improve convergence and scope of applications. Immune computing is strongly related to other intelligent approaches as artificial neural networks and evolutionary computing.

References [1] Bereta, M., Burczyński, T.: Comparing binary and real-valued coding in hybrid immune algorithm for feature selection and classification of ECG signals. Engineering Applications of Artificial Intelligence 20, 571–585 (2007) [2] Burczyński, T. (Guest Editor): Special Issue on Artificial Immune Systems. Information Sciences 179(10) (2009) [3] Burczyński, T.: Evolutionary and immune computations in optimal design and inverse problems. In: Waszczyszyn, Z. (ed.) Advances of Soft Computing in Engineering. Springer, Heidelberg (2009) (in press) [4] Castro, L.N., Timmis, J.: Artificial immune systems as a novel soft computing paradigm. Soft Computing 7(8), 526–544 (2003) [5] Goldberger, a.I., Amaral, L.A.N., Glass, L., Hausdorff, J.M., Ivanov, P.C., Mark, R.G., Mietus, J.E., Moody, G.B., Peng, C.K., Stanley, H.E.: PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (2000); circulation Electronic Pages: http://circ.ahajournals.org/cgi/content/full/101/23/e215 [6] Perelson, A.S., Weisbuch, G.: Immunology for physicists. Reviews of Modern Physics 69(4), 1219–1267 (1997) [7] Wierzchoń, S.T.: Artificial Immune Systems. Theory and Applications. Exit, Warsaw (2001) (in Polish)

Chapter 10

Bioinspired Algorithms in Multiscale Optimization Wacław Ku´s and Tadeusz Burczyński

Abstract. The paper is devoted to bioinspired optimization in multiscale problems. The composite modeled as a macrostructure with a local periodic microstructure is considered. The multiscale analysis is performed with the use of the homogenization method. The evolutionary algorithm, the artificial immune system and the particle swarm optimization are used in computations. The objective function evaluation with the use of the parallel homogenization algorithm is considered. The paper contains a description of the evolutionary algorithm, artificial immune system, particle swarm optimization, the homogenization method, the optimization formulation.

10.1 Introduction The paper is an extension of previous authors paper [1]. The multiscale model of a structure is considered. The goal of optimization presented in this paper is minimization of a functional depending on state field in one of a structure scale with respect to design variables described on another scale. The optimization problem is solved by using bioinspired algorithms. Evolutionary, immune system and particle swarm optimizations are used in computations. The objective function is calculated for each set of design parameters on the basis of results of a direct problem analysis. The direct problem is modeled with the use of a computational homogenization. The home made programs based on the Finite Element Method (FEM) [8] are used. Wacław Ku´s Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected] Tadeusz Burczyński Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland Silesian University of Technology, Department of Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland e-mail: [email protected] M. Kuczma, K. Wilmanski (Eds.): Computer Methods in Mechanics, ASM 1, pp. 183–192. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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10.2 The Multiscale Model The multiscale model of the structure is considered. One of the numerical techniques which enables multiscale analysis of structures is the computational homogenization. The detailed description of the computational homogenization can be found in [4]. The local periodicity is assumed. It means that there are areas of the structure with the same microstructure. The example of macrostructure with periodic microstructures is shown in Fig. 10.1.

W Fig. 10.1 Two scale model of macrostructure with periodic microstructure

macroscale

microscale

The microstructure can also be built from the lower scale of a locally periodic microstructure. The goal of the computational homogenization is analysis of the structure taking into account the local periodicity of the microstructure. The main advantage of the computational homogenization is an possibility of performing analysis in a few scales. It allows to use models with at least a few orders of degrees of freedom lower than model created in one scale. The material parameters for each integration point in finite elements depend on the solution of a representative volume element (RVE) in the lower scale. The RVE is a model of the microstructure, voids, inclusions and other properties of the microstructure can be included in the model. The RVE is in most cases modeled as a cube or a square. The numerical method like FEM is used to solve the boundary value problem for RVE. The periodic displacements boundary conditions are taken into account. The strains from the higher level are prescribed as additional boundary conditions. The RVE for each integration point of the higher level model must be created and stored for the next iteration steps if the nonlinear problem with plasticity is considered. The transfer of information both form lower to higher and higher to lower scales is needed in most cases. The one way transfer of results (from lower to higher scales) is possible if the linear problem is considered. The material parameters for the higher scale are obtained on the basis of solving a few direct problems for RVE in the lower scale. The homogenized material parameters depend on average stress values in RVE obtained after applying average strains to RVE. The stress-strain relation obtained using RVE is

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used in the higher level model. Strains in the integration point from the higher level are considered as average strains.

10.3 The Optimization Problem Formulation The design parameters describe the properties of the microscale model, and an objective function value depends on the macroscale model. The objective function is defined as: F(ch) = max{u} (10.1) where ch = [ch1 , ch2 , ...chn ]T is a vector containing design parameters values, u is a vector of reduced displacements in the macroscale. The optimization goal considered in the paper is minimization of maximal displacements in the macroscale level formulated as: min F(ch) ch

(10.2)

The design parameters describe the shape of an inclusion in the microscale. The constraints on design parameters values are imposed in the form chimin ≤ chi ≤ chimax

(10.3)

where chimin is minimal and chimax maximal value of design parameter i. The constraints on a inclusion area RA are used RA ≤ RAmax

(10.4)

where RAmax is maximal area of the inclusion. The objective function value is computed by using multiscale FEM (Fig. 10.2). The microstructure finite element mesh is created on the base of NURBS curves.

10.4 The Bioinspired Optimization Algorithms The evolutionary algorithms (EA) [6] are based on mechanisms taken from biological evolution of species. The selection based on the individual fitness, mutations in chromosomes and individuals crossover are adopted. The evolutionary algorithms operate on a population of individuals. The individuals contain one chromosome in most cases. The evolutionary operators change the chromosomes during an iterative process. The selection is performed in every iteration. The changed by mutation or crossover individuals need fitness function evaluation. The flowchart of the evolutionary algorithm is shown in Fig. 10.3. The detailed description of used EA can be found in [5].

186


design parameters vector Prepare NURBS curve for microscale model

Mesh the microscale model

Compute multiscale problem by using FEM Compute objective function value based on analysis results Objective function value

Fig. 10.2 The objective function evaluation algorithm START Starting population creation Fitness function evaluation for each chromosome

Selection Evolutionary operators Fitness function evaluation for each chromosome STOP


Fig. 10.3 The evolutionary algorithm

STOP

The artificial immune systems (AIS) are developed on the basis of mechanism discovered in biological immune systems. An immune system is a complex system which contains distributed groups of specialized cells and organs. The main purpose of the immune system is to recognize and destroy pathogens - funguses, viruses, bacteria and improper functioning cells. The artificial immune systems [2] take only a few elements from the biological immune systems. The mutation of the B cells, proliferation, memory cells, and recognition by using the B and T cells are applied very


187

frequently. The artificial immune systems have been used to optimization problems, classification and also computer viruses recognition. The cloning algorithm Clonalg presented by von Zuben and de Castro uses some mechanisms similar to biological immune systems to global optimization problems. The unknown global optimum is the searched pathogen. The memory cells contain project variables and proliferate during the optimization process. The B cells created from memory cells undergo mutation. The B cells are evaluated and better ones exchange memory cells. The flowchart of AIS is shown in Fig. 10.4.

START Creation of memory cells

Memory cells proliferation with hipermutation Evaluation of objective function for B cells Selection

Crowding mechanism STOP


Fig. 10.4 The artificial immune system

STOP

The particle swarm optimization (PSO) algorithm is also based on observation done in biology [3]. The algorithm has similar behavior as a birds flocking or fish schooling. The individual bird (in PSO particle) changes velocity and position taking into account neighbor birds. This social behavior shown by some animals can be very efficient in nature. PSO incorporated some biological elements in the numerical algorithm, such as velocity change of the particle on the base of neighbors. The PSO is the iterative algorithm. The positions of the particles in the search space and starting velocities are defined randomly on the beginning of the algorithm. The change of particle velocities are due to the velocities of best ever found, best in neighborhood and previous particle values. The algorithm updates position of all particles in every iteration on the basis of computed velocities. The flowchart of PSO is shown in Fig. 10.5.

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W. Ku´s and T. Burczyński START Randomly choose particles positions Randomly choose starting velocities for particles

Compute objective function value for particles Update particles velocities Update particles positions STOP


Fig. 10.5 The particle swarm optimization

STOP

10.5 Numerical Example The numerical example of optimization of a 2D composite structure is considered (Fig. 10.6). The optimization criterion is to minimize the maximal reduced displacement of the structure.

a)

b)

Fig. 10.6 The macromodel a) geometry, b) finite element mesh


189

The multiscale analysis was performed by using home made software based on FEM package MSC.Nastran [7]. The shape of the macro level structure is shown in Fig. 10.6. The inclusion in the microstructure is described using a NURBS curve (Fig. 10.7. The 8 design variables (g1 − g8 ) represent coordinates of the NURBS curve polygon control points. The size of RVE is 1 by 1. The constraints on design parameters values are shown in Tab. 10.1. The maximum area of inclusion and matrix RAmax was equal to 0.3. Table 10.1 The constraints on design parameters values The design parameter

Minimum

Maximum

g1 g2 g3 g4 g5 g6 g7 g8

0.10 0.10 0.50 0.10 0.50 0.50 0.10 0.50

0.47 0.47 0.90 0.47 0.90 0.90 0.47 0.90

(g5,g6)

NURBS curve NURBS control polygon (g3,g4) NURBS control point

(g7,g8)

(g1,g2)

Fig. 10.7 The Representative Volume Element (micromodel)

The optimization problem was solved by using EA, AIS and PSO. The parameters of the algorithms are shown in Tab. 10.2. The numerical tests were performed 5 times for each algorithm. The best obtained and reference solution are shown in Fig. 10.8. The displacement map for the macroscale is shown in Fig. 10.9. The comparison of the best objective functions obtained by bioinspired algorithms and reference solutions are shown in Tab. 10.3.

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Table 10.2 The bioinspired algorithms parameters Parameter

Value The Evolutionary Algorithm

Number of chromosomes Prob. of Gaussian mutation Prob. of uniform mutation Prob. of simple crossover Selection mechanism

50 90% 10% 90% rank selection

The Artificial Immune System Number of memory cells Number of cloned B cells Prob. of Gaussian mutation

5 50 100%

The Particle Swarm Optimization Number particles in swarm Influence of the best ever found particle Influence of the previous state of particle Influence of the best particle in neighborhood

a)

50 0.33 0.33 0.33

b)

Fig. 10.8 The microstructure: a) starting, b) the best found. The dashed lines represent NURBS curves control polygons Table 10.3 The results of optimization Algorithm

The objective function value for the best found solution

The RA value

Reference solution (Fig. 10.8a) The Evolutionary Algorithm The Artificial Immune System The Particle Swarm Optimization

0.2605 0.2482 0.2433 0.2435

0.30 0.30 0.30 0.30


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Fig. 10.9 The macrostructure reduced displacements map for the best design vector

10.6 Conclusions The paper presents the application of bioinspired optimization techniques in multiscale modeling. The three presented algorithms can be used in optimization procedure. The use of all presented algorithms allows to choose the best solution. The minimal value of the objective function was found by AIS in considered optimization problem. The performance of the algorithms depends on objective function shape and there is not the best general algorithm. The application of different algorithms for optimization problem can increase the probability of finding global optimum. Acknowledgements. The research is financed from the Polish science budget resources in the years 2008-2010 as the research project.

References [1] Burczyński, T., Ku´s, W.: Microstructure optimisation and identification in multi-scale modeling. In: ECCOMAS Multidisciplinary Jubilee Symposium on New Computational Challenges in Material, Structures and Fluids, Computational Methods in Applied Sciences, vol. 14, pp. 169–181. Springer, Heidelberg (2009) [2] de Castro, L.N., Timmis, J.: Artificial Immune Systems as a Novel Soft Computing Paradigm. Soft Computing 7(8), 526–544 (2003)

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[3] Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufmann Publishers, San Francisco (2001) [4] Kouznetsova, V.G.: Computational homogenization for the multi-scale analysis of multiphase materials. Ph.D. Thesis, TU Eindhoven (2002) [5] Ku´s, W.: Grid-enabled evolutionary algorithm application in the mechanical optimization problems. Engineering Applications of Artificial Intelligence 20, 629–636 (2007) [6] Michalewicz, Z.: Genetic algorithms + data structures = evolutionary algorithms. Springer, Berlin (1996) [7] MSC. Nastran, Users guide. (2005) [8] Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics. Elsevier, Oxford (2005)

Chapter 11

Sensor Network Design for Spatio–Temporal Prediction of Distributed Parameter Systems Dariusz Uciński

Abstract. An activation strategy of pointwise sensors used for estimating unknown parameters in models described by partial differential equations is addressed. In contrast to the common approach based on parameter-space criteria, attention is paid here to a criterion in output space, which is of interest if the purpose of parameter estimation is to accurately predict system outputs. The problem is formulated as the determination of the density of gaged sites so as to minimize the adopted design criterion, subject to inequality constraints incorporating a maximum allowable sensor density in a given spatial domain. The search for the optimal solution is performed using a simplicial decomposition algorithm. The use of the proposed approach is illustrated by a numerical example involving sensor selection for a two-dimensional diffusion process.

11.1 Introduction Building models of dynamic systems is a key activity in process engineering. Modern process control frequently demands using very accurate models in which spatial dynamics has to be included in addition to the temporal one. The processes in question are often termed distributed parameter systems (DPSs) and they are described by partial differential equations. One of the basic and most important questions in DPSs is parameter estimation, which refers to the determination from observed data of unknown parameters in the system model such that the predicted response of the model is close, in some well-defined sense, to the process observations made by some suitable collection of sensors termed the measurement or observation system. A major difficulty here is related to the impossibility to measure process variables over the entire spatial domain. Moreover, the measurements are inexact by virtue Dariusz Uciński Institute of Control and Computation Engineering, University of Zielona Góra, ul. Licealna 9, 65–417 Zielona Góra, Poland e-mail: [email protected]


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of inherent errors of measurement associated with transducing elements and also because of the measurement environment. The inability to take distributed measurements of process states leads to the question of where to locate sensors so that the information content of the resulting signals with respect to the distributed state and PDE model be as high as possible. This is an appealing problem since in most applications these locations are not prespecified and therefore provide design parameters. The location of sensors is not necessarily dictated by physical considerations or by intuition and, therefore, some systematic approaches should still be developed in order to reduce the cost of instrumentation and to increase the efficiency of identifiers. An example which is particularly stimulating in the light of the results reported in this note constitutes optimization of air quality monitoring networks. One of the tasks of environmental protection systems is to provide expected levels of pollutant concentrations. But to produce such a forecast, a smog prediction model is necessary, which is usually chosen in the form of an advection-diffusion partial differential equation. Its calibration requires parameter estimation, e.g., the unknown spatially-varying turbulent diffusivity tensor should be identified based on the measurements from monitoring stations whose number can be quite large. Then designers must address the question of how to optimize sensor locations in order to obtain the most precise model. This issue acquires especially vital importance in the context of recent advances in distributed sensor networks [2]. Over the past years, applications have stimulated laborious research on the development of strategies for efficient sensor placement (for reviews, see papers [8, 16] and comprehensive monographs [14, 13, 12]). Nevertheless, although the need for systematic methods was widely recognized, most techniques communicated by various authors usually rely on exhaustive search over a predefined set of candidates and the combinatorial nature of the design problem is taken into account very occasionally [16]. An approach to alleviate problems with the combinatorial nature of sensor selection consists in operating on the spatial density of sensors (i.e., the number of sensors per unit area), rather than on the sensor locations. It is proved reasonable for a sufficiently large number of sensors and potential solutions would be satisfactory for many processes. The underlying idea has its origins in the concept of replication-free designs in spatial statistics, cf. [4], and over the past few years, successful attempts have been made at adapting it for use in problems ranging from maximization of observability [14] to optimization of measurement strategies for scanning observations [14, Ch. 4.1.1]. Consequently, convenient and efficient mathematical tools of convex programming theory made it possible to derive interesting characterizations of optimal solutions. All the same, some generalizations are still expected in this aspect of the sensor location problem, particularly regarding computational procedures which were left without due consideration. Although some simple exchange-type algorithms have been proposed to determine optimal designs [14], they are only locally convergent and still need a thorough refinement. In most existing approaches sensor locations are determined in experiments performed for the most accurate determination of parameter values which may have

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some physical significance. By contrast, the reliability of model predictions is the main focus of interest here. This is because in many applications, especially when a control scheme is to be built, the accuracy of model predictions is more important than the accuracy of model parameters, because the ultimate objective in modeling is the prediction or forecast of the system states. The aim of the research reported here was thus to develop a computational algorithm to determine optimum sensor densities which, while being independent of a particular model of the dynamic DPS in question, would be versatile enough to cope with large-scale monitoring networks. For that purpose, the original problem is reduced to minimization of the mean variance of the prediction over the set of all convex combinations of a finite number of nonnegative definite matrices subject to additional box constraints on the weights of those combinations. Then simplicial decomposition is applied which is a simple and direct method for dealing with large-scale convex optimization problems [7, 9]. The decomposition iterates by alternately solving a linear programming subproblem within the set of all feasible points and a nonlinear master problem within the convex hull of a subset previously generated points. As a result, an uncomplicated computational scheme is obtained which can be easily implemented without resorting to sophisticated numerical software. This extends the results of [15]where D-optimal designs were considered. Notation. Throughout the paper, R+ and R++ stand for the sets of nonnegative and positive real numbers, respectively. We use Sm to denote the set of symmetric m × m matrices, The curled inequality symbol (resp. ) is used to denote generalized inequalities. More precisely, between vectors, it represents a componentwise inequality, and between symmetric matrices, it represents the Löwner ordering: given A, B ∈ Sm , A B means that A − B is nonnegative definite (resp. positive definite). The symbols 1 and 0 denote vectors whose all components are one and zero, respectively. We call a point of the form α1 a1 + · · · + α a , where α1 + · · · + α = 1 and αi ≥ 0, i = 1, . . . , , a convex combination of the points a1 , . . . , a . Given a set of points A, co(A) stands for its convex hull, i.e., the set of all convex combinations of elements ofA. The probability (or canonical) simplex in Rn is defined as Sn = co e1 , . . . , en where e j is the usual unit vector along the j-th coordinate of Rn .

11.2 Optimal Sensor Location Problem 11.2.1 Quantification of Prediction Accuracy Let y = y(x,t; a) denote the scalar state of a given DPS at a spatial point x ∈ Ω ⊂ Rd and time instant t ∈ T = [0,t f ], t f < ∞. Here a represents an unknown constant parameter vector which must be estimated using observations of the system. In what follows, we consider the observations provided by N stationary pointwise sensors, namely

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zmj (t) = y(x j ,t; a) + ε (x j ,t),

t ∈ T,

(11.1)

zmj (t)

where is the scalar output and x j ∈ X stands for the location of the j-th sensor ( j = 1, . . . , N), X signifies the part of the spatial domain Ω where the measurements can be made and ε (x j ,t) denotes the measurement noise. A customary assumption is that the measurement noise is zero-mean, Gaussian, spatially uncorrelated and white [8]. We assume that the parameter estimate aˆ , defined as the solution to the usual output least-squares formulation of the parameter estimation problem, is to provide a basis for prediction of certain variables depending on spatial location and/or time. Let the solution to the prediction problem in context be a scalar quantity q = q(x,t; a). We are interested in selecting the sensor configurations in such a way as to maximize the accuracy of q in a given compact spatio-temporal domain Q = X × T . Clearly, in order to compare different configurations, a quantitative measure of the ‘goodness’ of particular configurations is required. A logical approach is to choose a measure related to the expected accuracy of prediction. For a given (x,t) ∈ Q, the variance of q obtained by a first-order expansion around a preliminary estimate a0 of a has the form var(q(x,t; aˆ )) = E (q(x,t; a) − q(x,t; aˆ ))2 T (11.2) ≈ ∇a q(x,t; a0 ) cov(â)∇a q(x,t; a0 ) 0 T −1 0 ∼ ∇a q(x,t; a ) M ∇a q(x,t; a ) where we write ∇a q for the gradient of q with respect to a. It is customary to choose a0 as a nominal value of a or a result of a preliminary experiment. As for cov(â), we used the fact that it can be approximated by the inverse of the Fisher Information Matrix (FIM) whose normalized version can be written down as [10] M=

1 Nt f

N

∑

tf

g(x j ,t)gT (x j ,t) dt,

(11.3)

j=1 0

/ where g(x,t) = ∇a y(x,t; a)/a=a0 stands for the so-called sensitivity vector. A criterion may now be set up such that the ‘optimal’ sensor positions x j minimize var(q(x,t; aˆ )) over Q. Based on the suggestions of (Fedorov and Hackl, 1997, p.25), in the sequel the following V-optimality criterion is considered:

Ψ [M] = where C=

Q

Q

var(q(x,t; aˆ )) dx dt = trace CM−1

T ∇a q(x,t; a0 ) ∇a q(x,t; a0 ) dx dt

(11.4)

(11.5)

The introduction of an optimality criterion renders it possible to formulate the sensor location problem as an optimization problem: Minimize Ψ M(x1 , . . . , xN ) with respect to x j , j = 1, . . . , N belonging to the admissible set X .

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11.2.2 Problem of Finding Optimal Sensor Densities When the number of sensors N is large, which becomes a common situation in applications involving sensor networks, the optimal sensor location problem becomes extremely difficult from a computational point of view. In order to overcome this predicament, we can operate on the spatial density of sensors (i.e. the number of sensors per unit area), rather than on the sensor locations. The density of sensors over X can be approximately described by a probability measure ξ (dx) on the space (X, B), where B is the σ -algebra of all Borel subsets of X . Feasible solutions of this form make it possible to apply convenient and efficient mathematical tools of convex programming theory. As regards the practical interpretation of the so produced results (provided that we are in a position to calculate at least their approximations), one possibility is to partition X into non-overlapping subdomains Xi of relatively small areas and then to allocate to each of them the number 0 1 Ni = N ξ (dx) (11.6) Xi

of sensors (here ρ ! is the smallest integer greater than or equal to ρ ). Accordingly, we define the class of admissible designs as all probability measures ξ over X which are absolutely continuous with respect to the Lebesgue measure and satisfy by definition the condition

ξ (dx) = 1.

(11.7)

X

Consequently, we replace (11.3) by M(ξ ) =

G(x) ξ (dx),

(11.8)

X

where G(x) =

tf

1 tf

g(x,t)gT (x,t) dt.

0

The integration in (11.7) and (11.8) is to be understood in the Lebesgue-Stieltjes sense. This leads to the so-called continuous designs which constitute the basis of the modern theory of optimal experiments [5, 17]. We impose the crucial restriction that the density of sensor allocation must not exceed some prescribed level. For a design measure ξ (dx) this amounts to the condition ξ (dx) ≤ ω (dx), (11.9) where ω (dx) signifies the maximal possible ‘number’ of sensors per dx [5, 14, 13] such that ω (dx) ≥ 1. (11.10) X

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Consequently, we are faced with the following optimization problem: Problem 1. Find

ξ = arg min Ψ (M(ξ )) ξ ∈Ξ (X)

(11.11)

subject to

ξ (dx) ≤ ω (dx),

(11.12)

where Ξ (X ) denotes the set of all probability measures on X . The design ξ above is then said to be a (Ψ , ω )-optimal design [5]. Let us make the following assumptions: (A1) X is compact, (A2) g ∈ C(X × T ; Rm ), (A3) There exists a finite real α such that ξ : Ψ [M(ξ )] < α = Ξ2 (X ) = 0, / (A4) ω (dx) is atomless, i.e., for any Δ X ⊂ X there exists a Δ X ⊂ Δ X such that Δ X

ω (dx)

Computer Methods in Mechanics: Lectures of the CMM 2009 (Advanced Structured Materials)

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