Numerical Shape Optimization in Structural Acoustics

Numerical Shape Optimization in Structural Acoustics (Numerische Formoptimierung in der Strukturakustik)

Vom Fachbereich Maschinenbau an der Technischen Universit¨at Darmstadt zur Erlangung des Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte

Dissertation

vorgelegt von Dipl.-Ing. Joachim B¨ os, M.S./SUNY aus Fulda

Berichterstatter: Mitberichterstatter: Tag der Einreichung: Tag der m¨ undlichen Pr¨ ufung:

Prof. em. Dr.-Ing. Dr.-Ing. E.h. Franz Gustav Kollmann Prof. Dr. rer. nat. Michael Sch¨afer 27. Januar 2004 20. April 2004

Darmstadt 2004 D17

Numerical Shape Optimization in Structural Acoustics

Dissertation

approved by the Faculty of the Department of Mechanical Engineering, Darmstadt University of Technology, Germany, in fulfillment of the requirements for the degree of Ph.D. in Mechanical Engineering (Dr.-Ing.)

by Dipl.-Ing. Joachim B¨ os, M.S./SUNY from Fulda, Germany

Advisor: Co-advisor: Date of submission: Date of defense:

Prof. em. Dr.-Ing. Dr.-Ing. E.h. Franz Gustav Kollmann Prof. Dr. rer. nat. Michael Sch¨afer January 27, 2004 April 20, 2004

Darmstadt 2004 D17

Forschungsberichte Mechatronik & Maschinenakustik

Joachim Bös

Numerical Shape Optimization in Structural Acoustics

.

D 17 (Diss. TU Darmstadt)

Shaker Verlag Aachen 2004

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the internet at http://dnb.ddb.de. Zugl.: Darmstadt, Techn. Univ., Diss., 2004

.

Umschlaggestaltung Veronika Monz © TU Darmstadt Fachgebiet Mechatronik und Maschinenakustik

Copyright Shaker Verlag 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers. Printed in Germany.

ISBN 3-8322-2926-4 ISSN 1616-5470 Shaker Verlag GmbH • P.O. BOX 101818 • D-52018 Aachen Phone: 0049/2407/9596-0 • Telefax: 0049/2407/9596-9 Internet: www.shaker.de • eMail: [email protected]

The scientist describes what is; the engineer creates what never was.∗ Theodore von K´arm´an (1881–1963)

∗

quoted in: A. L. Mackay, A Dictionary of Scientific Quotations (Adam Hilger, Bristol, 1991)

Abstract This dissertation presents a numerical method to optimize the thickness distribution of threedimensional structures with respect to various vibrational and structural properties. A combination of a commercially available finite element software package and additional user-written programs is used to modify the shape (but not the number of nodes and elements) of finite element models of the structures to be optimized. This is done iteratively and without manual intervention so as to achieve significant improvements of the objective function. The optimization process continues automatically until some predefined convergence criterion is met or until some prespecified maximum number of iterations is reached. The design variables are the structure’s local thickness values at selected surface nodes. Possible objectives of the optimization include the minimization of the mean level of structure borne sound (a measure of the vibrational sensitivity of a structure), the minimization of the structural mass, the maximization of the fundamental frequency, and the maximization of the difference between two arbitrarily chosen natural frequencies. In addition, suitable constraints must be specified that restrict the number of acceptable solutions to the optimization problem. Possible candidates for constraints are the structural mass, the mean level of structure borne sound, and the fundamental frequency. Furthermore, the allowable range of design variable values is restricted by prescribed upper and lower limits. The optimization procedure is tested on the finite element models of four different three-dimensional structures made of steel, namely, a rectangular plate, two rectangular plates joined at 90◦ , a half-cylinder, and a gearbox. Two different derivative-free optimization algorithms are tested against each other, namely, the deterministic COBYLA algorithm and the genetic algorithm PIKAIA, which contains various stochastic features. Depending on the structure to be optimized and on the constraints, COBYLA achieves objective function improvements of up to 10.2 dB (−90.5%) for the minimization of the mean level of structure borne sound, up to 44.9% for the mass minimization, up to 91.4% for the fundamental frequency maximization, and up to 2440% for the maximization of the difference between two natural frequencies. Although COBYLA is only a local optimization algorithm, i.e., not designed to find the global optimum, it turns out to be much more effective and efficient than the global PIKAIA algorithm, which (in general) takes many more iterations to achieve inferior results. However, PIKAIA enables the objective function values of several different design proposals to be calculated simultaneously on several computers, whereas COBYLA uses a sequential optimization strategy. The computation time required to obtain an optimized design depends not only on the size of the finite element model and the corresponding computation time consumed for every single iteration (each iteration includes a finite element analysis), but also on the number of design variables, which influences the necessary number of iterations. Therefore, an approach to reduce the number of design variables by means of spline functions is presented. A spline surface, defined by just a small number of key points that serve as design variables, is used to model and vary the thickness distribution of the structure. The optimization results indicate that there seems to be a proportional relation between the number of design variables and the number of iterations (and therefore computation time). For example, a reduction of the number of design variables by 90% by means of splines yields a computation time decrease of 88%. Significant computation time savings can also be achieved by terminating the optimization procedure prior to reaching the formal mathematical convergence criterion. If tiny constraint violations of the order of 0.01% are deemed acceptable, the computation time can be reduced by up to 78%. i

Zusammenfassung In dieser Arbeit wird ein Verfahren zur numerischen Optimierung der Wandst¨arkeverteilung dreidimensionaler Strukturen bez¨ uglich verschiedener Schwingungs- und Struktureigenschaften vorgestellt. Ein kommerzielles Finite-Elemente-Programm wird mit selbstgeschriebenen Programmen kombiniert, um die Geometrie der zu optimierenden Finite-Elemente-Modelle iterativ und ohne a¨ußere Eingriffe so zu ver¨andern, dass die Zielfunktion deutlich verbessert wird. Dieser automatische Optimierungsprozess l¨auft so lange weiter, bis entweder ein vorgegebenes Konvergenzkriterium oder eine vorgegebene maximale Anzahl von Iterationen erreicht ist. Entwurfsvariablen sind die lokalen Wandst¨arken an ausgew¨ahlten Oberfl¨achenknoten. M¨ogliche Zielfunktionen sind die Minimierung des mittleren K¨orperschallpegels (ein Maß f¨ ur die Schwingfreudigkeit einer Struktur) oder der Strukturmasse oder die Maximierung der ersten Eigenfrequenz oder des Abstandes zwischen zwei beliebigen Eigenfrequenzen. Zus¨atzlich m¨ ussen geeignete Restriktionen festgelegt werden, die die Anzahl m¨oglicher L¨osungen des Optimierungsproblems einschr¨anken. Restriktionen k¨onnen der mittlere K¨orperschallpegel, die Strukturmasse oder die erste Eigenfrequenz der Struktur sein. Desweiteren wird der zul¨assige Bereich der Entwurfsvariablen auf das Intervall zwischen einem vorgegebenen Mindest- und einem vorgegebenen H¨ochstwert begrenzt. Das Optimierungsverfahren wird an den Finite-ElementeModellen von vier dreidimensionalen Strukturen aus Stahl, n¨amlich an einer Rechteckplatte, einem L-f¨ormigen Plattenstoß, einem Halbzylinder sowie einem Getriebegeh¨ause, erprobt. Zwei verschiedene ableitungsfreie Optimierungsalgorithmen werden miteinander verglichen, n¨amlich der deterministische COBYLA-Algorithmus und der genetische PIKAIA-Algorithmus, der einige stochastische Elemente enth¨alt. Je nach zu optimierender Struktur und den Restriktionen erzielt COBYLA Zielfunktionsverbesserungen von bis zu 10,2 dB (−90, 5%) bei der Minimierung des mittleren K¨orperschallpegels, bis zu 44,9% bei der Massenminimierung, bis zu 91,4% bei der Maximierung der ersten Eigenfrequenz und bis zu 2440% bei der Maximierung des Abstandes zwischen zwei beliebigen Eigenfrequenzen. Dabei zeigt sich, dass der lokale COBYLA-Algorithmus in der Regel sehr viel effektiver und effizienter arbeitet als der globale PIKAIA-Algorithmus, der viel mehr Iterationen ben¨otigt und letztlich doch nur ein schlechteres Ergebnis erzielt als COBYLA. PIKAIA bietet jedoch die M¨oglichkeit, die Zielfunktionen von mehreren modifizierten Entw¨ urfen parallel auf verschiedenen Computern auszuwerten, w¨ahrend COBYLA eine streng sequentielle Optimierungsstrategie darstellt. Die Rechenzeit, die f¨ ur eine Optimierung ben¨otigt wird, ist nicht nur von der Gr¨oße des Finite-Elemente-Modells und der damit verbundenen Rechenzeit je Iteration abh¨angig, da jede Iteration eine Finite-Elemente-Berechnung erfordert, sondern auch von der Anzahl der Entwurfsvariablen, die die Anzahl der ben¨otigten Iterationen beeinflusst. Deshalb wird in dieser Arbeit auch ein Ansatz vorgestellt, die Entwurfsvariablenanzahl mit Hilfe von Splinefunktionen zu reduzieren. Dabei wird die Wandst¨arkeverteilung einer Struktur durch eine Splinefl¨ache modelliert und modifiziert, die nur durch eine geringe Anzahl von St¨ utzstellen definiert ist, welche als Entwurfsvariablen dienen. Hier scheint es eine in etwa proportionale Beziehung zwischen der Anzahl der Entwurfsvariablen und der ben¨otigten Iterationszahl bzw. Rechenzeit zu geben. Zum Beispiel f¨ uhrt eine Reduzierung der Entwurfsvariablenanzahl mittels Splinefunktionen um 90% zu einer Rechenzeitersparnis von 88%. Eine weitere M¨oglichkeit der Rechenzeiteinsparung besteht darin, den Optimierungsprozess gezielt abzubrechen, obwohl das formale Konvergenzkriterium noch nicht erreicht ist. Wenn man bereit ist, winzige Restriktionsverletzungen in der Gr¨oßenordnung von 0,01% hinzunehmen, kann man die Rechenzeit um bis zu 78% verk¨ urzen. ii

Acknowledgments The research described in this dissertation was done while I was a Ph.D. student and research assistant at the Department of Mechatronics and Machine Acoustics at Darmstadt University of Technology, Germany, from May 1998 through April 2003. I would like to thank my advisor and former department chair, Prof. em. Dr.-Ing. Dr.-Ing. E.h. Franz Gustav Kollmann, as well as the current department chair, Prof. Dr.-Ing. Rainer Nordmann, for continuous advice and support. I am much obliged to my co-advisor, Prof. Dr. rer. nat. Michael Sch¨afer, chair of the Department of Numerical Methods in Mechanical Engineering at Darmstadt University of Technology, who agreed to be my co-advisor at short notice. Thanks are due to my former colleagues at the Department of Mechatronics and Machine Acoustics, in particular to the head of the research group Machine Acoustics, Dr.-Ing. Rainer Storm, and to my former office mate, Dipl.-Ing. Marcus Stein, for fruitful discussions and a nice and pleasant working atmosphere. This also includes all secretaries and staff of the department. I am indebted to three former students of mine who greatly contributed to this work by means of their Bachelor and Master theses and as student research assistants. These former students are (in alphabetical order) cand. ing. Andreas Friedmann [Friedmann 2002], Dipl.-Ing. Mike Robers [Robers 2003], and Dipl.-Ing. Alexander Wesp [Wesp 2002]. I am particularly grateful to our computer administration assistant, cand. phys. Christian Tacke, for his invaluable support and help concerning anything that is computer-related: computer hardware, software, licensing, programming, LATEX, etc. I also want to acknowledge the useful advice and support of Prof. Dr. Peter Spellucci (research group Numerical Analysis and Scientific Computing, Department of Mathematics, Darmstadt University of Technology), Prof. Dr. Ulrich Reif (research group Differential Geometry and CAGD, Department of Mathematics, Darmstadt University of Technology), and Dr.-Ing. habil. Frank Ihlenburg (ISKO engineers AG, Munich) who all provided many helpful suggestions. The support and advice of numerous proofreaders is gratefully acknowledged as well. Both the English language and the content of this work were considerably improved by suggestions and contributions from (in alphabetical order) Ms. Marit Adjei, B.A. (Oregon, USA), my wife Sonja B¨os, M.A., Dr. Jeffrey L. Cipolla (ABAQUS, Inc., Rhode Island, USA), cand. ing. Andreas Friedmann, my father-in-law Dipl.-Ing. Manfred Lodahl, Dipl.-Ing. Stefan Rosenberger, Dr.-Ing. Rainer Storm, and Prof. Robert C. Wetherhold (State University of New York at Buffalo, USA). Last-minute e-mail support was provided by Ms. Amy F. Hieter, M.B.A. (IBM Corporation, New York, USA), and Dr. Evan Mellander (California, USA). Despite this invaluable help and support by these proofreaders, I am solely responsible for any remaining mistakes and errors. A big “Thank you!” goes to the open source and free software community for providing great software tools and programs such as Linux [Linux 2004], LATEX [LATEX 2004], OpenOffice.org [OpenOffice 2004], GTK+ [GTK+ 2004], GIMP [GIMP 2004], xmgr [xmgr 2004], xfig [xfig 2004], etc., free of charge. It is amazing how much one can do without spending a single cent. Thanks are also due to my parents for continuous mental and financial support, which enabled me to pursue my ideas and goals. Last but not least I would like to express my gratitude to my wonderful wife Sonja for her incredible patience and her never ending love and support.

Darmstadt, January 25, 2004

Joachim B¨os iii

Dedicated to my Daughter Jana Anna Maria, Born April 27, 2004

iv

Contents Abstract

i

Zusammenfassung

ii

Acknowledgments

iii

List of Symbols and Acronyms

ix

1 Introduction 1.1 Background and Motivation . . . . . . . . . . . . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . 1.2.1 Optimization in (Structural) Acoustics . . . . . . 1.2.2 Numerical Optimization . . . . . . . . . . . . . . 1.2.3 Structural Optimization . . . . . . . . . . . . . . 1.2.4 Topology Optimization and Fully Stressed Design 1.2.5 Active Control . . . . . . . . . . . . . . . . . . . 1.2.6 Other Topics . . . . . . . . . . . . . . . . . . . . 1.3 Scope and Objectives of this Study . . . . . . . . . . . . 1.3.1 Issues that are Objectives . . . . . . . . . . . . . 1.3.2 Issues that are not Objectives . . . . . . . . . . . 1.3.3 Further Considerations . . . . . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Structural Acoustics 2.1 Fundamental Equation of Machine Acoustics . . . . . . . . . . . . . . . . . . . 2.2 Level of Structure Borne Sound LS . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mean Level of Structure Borne Sound MLS . . . . . . . . . . . . . . . . . . . . .

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3 Optimization Algorithms 3.1 General Aspects of Numerical Optimization Methods . . 3.2 COBYLA Algorithm . . . . . . . . . . . . . . . . . . . . 3.2.1 Motivation for Using COBYLA . . . . . . . . . . 3.2.2 The Simplex Algorithm . . . . . . . . . . . . . . 3.2.3 Description of the COBYLA Algorithm . . . . . 3.3 PIKAIA Algorithm . . . . . . . . . . . . . . . . . . . . . 3.3.1 Motivation for Using PIKAIA . . . . . . . . . . . 3.3.2 Basic Concepts of a General Genetic Algorithm . 3.3.3 PIKAIA—A Specific Implementation of a Genetic

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4 Optimization Procedure 4.1 Optimization Procedure in General . . . . . . . . . . . . . 4.1.1 Start of Optimization Procedure . . . . . . . . . . 4.1.2 Calculation of Objective Function and Constraints 4.1.3 Evaluation of Modified Design . . . . . . . . . . . 4.2 Optimization Procedure using COBYLA . . . . . . . . . . 4.2.1 Objective Function and Constraints . . . . . . . . v

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4.2.2 Convergence Criterion . . . . . . . . . . . . . . . . . 4.2.3 Feasible Results and Tolerable Constraint Violations Optimization Procedure using PIKAIA . . . . . . . . . . . 4.3.1 Calculation of Fitness . . . . . . . . . . . . . . . . . 4.3.2 Convergence Criterion . . . . . . . . . . . . . . . . . 4.3.3 Control Parameter Setting . . . . . . . . . . . . . .

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5 Finite Element Models 5.1 Common Properties . . . . . . . . . . . . . . . . . . 5.2 Rectangular Plate . . . . . . . . . . . . . . . . . . . . 5.2.1 Refined Discretization . . . . . . . . . . . . . 5.2.2 Coarse Discretization . . . . . . . . . . . . . . 5.2.3 Discretization Error and Numerical Pollution 5.3 Two Plates Joined at 90◦ . . . . . . . . . . . . . . . . 5.4 Half-Cylinder . . . . . . . . . . . . . . . . . . . . . . 5.5 Gearbox . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Employing Spline Functions to Reduce 6.1 General Aspects . . . . . . . . . . . . 6.2 Bicubic Spline Surface . . . . . . . . . 6.3 Hermite Spline Surface . . . . . . . . . 6.4 Tensor Product B´ezier Surface . . . .

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7 Optimization Results: Rectangular Plate 7.1 Original Rectangular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Refined Discretization . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Coarse Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Rectangular Plate Optimized with COBYLA . . . . . . . . . . . . . . . 7.2.1 Minimization of MLS (Refined FE Mesh) . . . . . . . . . . . . . 7.2.2 Minimization of MLS (Coarse FE Mesh) . . . . . . . . . . . . . . 7.2.3 Minimization of Mass . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Maximization of the Fundamental Frequency . . . . . . . . . . . 7.2.5 Maximization of the Difference between Two Natural Frequencies 7.3 Rectangular Plate Optimized with COBYLA Using Splines . . . . . . . 7.3.1 Minimization of MLS (Coarse FE Mesh) . . . . . . . . . . . . . . 7.3.2 Minimization of MLS (Refined FE Mesh) . . . . . . . . . . . . . 7.3.3 Minimization of Mass . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Rectangular Plate Optimized with PIKAIA: Minimization of Mass . . . .

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8 Optimization Results: Two Plates Joined at 90◦ 115 8.1 Original Two Plates Joined at 90◦ . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.2 Two Plates Joined at 90◦ Optimized with COBYLA . . . . . . . . . . . . . . . 116 8.2.1 Minimization of MLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.2.2 Minimization of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.2.3 Minimization of Mass with Fundamental Frequency Constraint . . . . . . 122 8.2.4 Maximization of the Difference between Two Natural Frequencies . . . . 126 8.3 Two Plates Joined at 90◦ Optimized with PIKAIA: Maximization of the Difference between Two Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . 128 vi

9 Optimization Results: Half-Cylinder 9.1 Original Half-Cylinder . . . . . . . . . . . . . . . . . . . . . . 9.2 Half-Cylinder Optimized with COBYLA . . . . . . . . . . . . 9.2.1 Minimization of MLS . . . . . . . . . . . . . . . . . . . 9.2.2 Minimization of Mass . . . . . . . . . . . . . . . . . . . 9.2.3 Maximization of the Fundamental Frequency . . . . . 9.3 Half-Cylinder Optimized with COBYLA Using Splines . . . . 9.3.1 Minimization of MLS . . . . . . . . . . . . . . . . . . 9.3.2 Minimization of Mass . . . . . . . . . . . . . . . . . . 9.4 Half-Cylinder Optimized with PIKAIA: Minimization of MLS

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10 Optimization Results: Gearbox 151 10.1 Original Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 10.2 Gearbox Optimized with COBYLA: Minimization of MLS . . . . . . . . . . . . 152 11 Discussion 157 11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 11.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A Some Remarks on the Implementation A.1 Calculation of Mass, Surface Area, Normal Vectors, and Thickness A.1.1 Calculation of the Structural Mass . . . . . . . . . . . . . A.1.2 Calculation of the Surface Area . . . . . . . . . . . . . . . A.1.3 Calculation of the Normal Vectors . . . . . . . . . . . . . A.1.4 Calculation of the Local Thickness . . . . . . . . . . . . . A.2 (Pseudo-)Parallelizing PIKAIA . . . . . . . . . . . . . . . . . . . A.3 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . A.4 Visualization of Optimization Results . . . . . . . . . . . . . . .

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B Verification of Optimization Results

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C Nonlinearity and Multimodality of the Objective Function

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List of Figures

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List of Tables

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Bibliography

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Resume of the Author

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vii

viii

List of Symbols and Acronyms Latin Symbols A A B B∗ E F F (x) Fˆ (x) J L L LF LSh2t LSh2t LW Lσ Na Ng Np Ns NT P S S(x, y, z) T T Tmeas Vel Vstruc Za c(x) ceq (x) cineq (x) c ca ci (x) cmin (x) cˆi (x) d ei f (x) f  (x)

surface area of an element face [m2 ] arbitrary positive number that serves as basis for an exponential function bending stiffness of a rectangular plate [N m] auxiliary quantity [s/m2 ] Young’s modulus, elastic modulus [N/m2 ] force, excitation force [N] objective function (simplex algorithm, COBYLA algorithm) linearized objective function (COBYLA algorithm) Jacobian matrix level [dB] characteristic length of the computational domain [m] excitation force level [dB] (reference value F0 = 1 N) level of structure borne sound (LS) [dB] (reference value S0 h2t0 = 2.5 · 10−15 m4 /(N2 s2 )) mean level of structure borne sound (MLS) [dB] sound power level [dB] (reference value P0 = 10−12 W) level of radiation efficiency [dB] (reference value σ0 = 1) shape function (FEM) number of generations (PIKAIA algorithm) population size (PIKAIA algorithm) number of cycles (SA algorithm) number of iterations before temperature reduction (SA algorithm) radiated sound power [W] surface area [m2 ] spline function value at grid point (x, y, z) period of a periodic oscillation [s] fictitious temperature (SA algorithm) measurement time [s] volume of a single element [m3 ] material volume of a structure [m3 ] specific impedance of air (≈ 414 N s/m) vector of constraints vector of equality constraints vector of inequality constraints speed of sound [m/s] speed of sound in air [m/s] constraint smallest constraint linearized constraint (COBYLA algorithm) distance [m] coordinate vector, basis vector objective function, fitness function preliminary fitness function ix

f f1 fmax fmin f∗ h h ht m n nin nn nout tp t t(0) (x, y, z) t(i) (x, y, z) v v vrms v⊥ wc x x x(0) x(i) x(∗) xi x, y, z

frequency [Hz] fundamental frequency [Hz] upper limit of a frequency range [Hz] lower limit of a frequency range [Hz] specified frequency [Hz] (∆f maximization) admittance [m/(N s)] FE mesh size [m] transmission admittance [m/(N s)] mass [kg] unit normal vector number of inner nodes number of surface nodes number of outer nodes local thickness direction vector at node p (unit length) plate thickness, local thickness [m] thickness distribution prior to optimization thickness distribution at iteration i velocity vector [m/s] velocity [m/s] rms velocity [m/s] normal surface velocity [m/s] weighting factor for the constraints vector of design variables global nodal coordinate vector initial vertex of a simplex (start vector) vertex of a simplex new vertex of a simplex design variable, design parameter global coordinates

Greek Symbols ∆ Φ(x) ˆ Φ(x) λ λb µ ν ρ ρ ρa ρbeg ρend ρnew σ ξ, η, ζ

difference between two values merit function (COBYLA algorithm) linearized merit function (COBYLA algorithm) wave length [m] bending wave length [m] weighting parameter for the constraint violation function (COBYLA algorithm) Poisson’s ratio trust-region radius (COBYLA algorithm) material density [kg/m3 ] density of air [kg/m3 ] initial trust-region radius (COBYLA algorithm) final trust-region radius (COBYLA algorithm) new trust-region radius (COBYLA algorithm) radiation efficiency local coordinates x

Subscripts, Superscripts, and Other Symbols  · 0  · W  · a  · max  · min  · t  · rms  · σ  · ⊥  · T  · (0)  · (i)  · max  · min  · (∗) ·  ˆ·   · 2 ∞ N R

standardized reference value power of air maximum value minimum value transmission root mean square value acoustic radiation normal to surface transposed vector or matrix initial value value during iteration i upper limit (predefined prior to optimization) lower limit (predefined prior to optimization) new value mean value, average linearized version of a function
n n 2 L2 norm of a vector, Euclidean norm (x2 = i=1 xi , x ∈ R ) infinity set of integer numbers set of rational numbers

Acronyms and Abbreviations ADS ANC ASAC ASCII AVC BEM CAGD CAO CAOSS COBYLA CPU CRS DFO DFT DLR DOF FE FEA FEAP FEM

Acoustic Design Sensitivity Active Noise Control Active Structural Acoustic Control American Standard Code for Information Interchange Active Vibration Control Boundary Element Method Computer Aided Geometric Design Computer Aided shape Optimization Computer Aided Optimization System Sauter Constrained Optimization By Linear Approximations Central Processing Unit Controlled Random Search Derivative-Free Optimization Discrete Fourier Transform Deutsches Zentrum f¨ ur Luft und Raumfahrt (German Aerospace Center) Degree Of Freedom Finite Element Finite Element Analysis Finite Element Analysis Program Finite Element Method

xi

FFT FRF FS GA GCS GIMP GNU GTK GUI IE IEM IMSL ISO IUTAM LS MLS MRI NAG NSAO OFAC PDS RA RAM rms SA SIAM SLP SPL SQP TL

Fast Fourier Transform Frequency Response Function Finite Strip Genetic Algorithm Generalized Compound Scaling GNU Image Manipulation Program GNU’s Not Unix (recursive acronym) GIMP Tool Kit Graphical User Interface Infinite Element Infinite Element Method International Mathematical and Statistical Library International Organization for Standardization International Union of Theoretical and Applied Mechanics Level of Structure borne sound Mean Level of Structure borne sound Magnetic Resonance Imaging Numerical Algorithms Group Numerical Structural Acoustic Optimization Objective Function And Constraints Parallel Direct Search Reverse Adaptivity Random-Access Memory root mean square Simulated Annealing Society for Industrial and Applied Mathematics Sequential Linear Programming Sound Pressure Level Sequential Quadratic Programming Transmission Loss

xii

1

Introduction

This dissertation deals with numerical structural acoustic optimization of various three-dimensional structures. The following section provides some information on the background of and the motivation for the present study. The literature survey in Sec. 1.2 reviews and summarizes many books and dissertations as well as numerous conference papers and scientific journal articles. Section 1.3 describes the scope and objectives of this research. Finally, the outline in Sec. 1.4 explains how the remainder of this work is organized.

1.1

Background and Motivation

Noise pollution, annoyance, and hearing impairment have been a problem since the beginning of industrialization in the 18th and 19th century. More and faster cars, trains, and airplanes have increased this problem within the last 30 or 40 years. Also, in almost every household, workshop, or factory, there are noisy appliances and loud machines such as blow-dryers, food processors, vacuum cleaners, drills, lathes, punches, presses, conveyor belts, etc. A lot has been done already to protect workers, population, and environment: new environmental laws and regulations; noise barriers alongside roads, highways, railroad tracks, and airports; new developments in car tires, train wheels, engines, and aerodynamics; quieter airplane engines; quieter industrial machines, etc. But there remains much to be done, and the legislative pressure on the manufacturers to make their products quieter increases on an almost annual basis. On the other hand, there is a demand that machines and their components become lighter (lightweight design) in order to save fuel or decrease energy consumption as well as faster in order to produce more parts per unit time or to shorten travel times. Unfortunately, these tendencies are somewhat contradictory to the demand of quiet products, since light structures tend to be noisier than heavy ones, and fast machines incline to be louder than slow ones. It is desirable to include noise-reducing measures into the design process of parts and machines (primary noise reduction at the source) rather than trying to reduce the radiated noise of conventionally designed structures and machines afterwards by damping treatments and encapsulation (secondary noise reduction of radiated sound) [Kollmann 2000]. One way to achieve this is to conduct numerical simulations on virtual computer models during the design phase of a product [Koopmann 1997, Lyon 2000]. Problem zones can be detected at an early stage, and it is possible to modify the design such that the product “sounds good” (e.g., a sports car) or is substantially quieter than before (e.g., trains, airplanes). This approach is called “virtual acoustic prototyping”.

2

1

Introduction

Primary noise reduction can be further subdivided into active as opposed to passive noise control. Active noise control (ANC) uses sensors, controllers, and loudspeakers, which generate a secondary sound field that (partially) cancels the original sound field within a defined volume [Nelson 1992]. Another approach is active structural acoustic control (ASAC) in which structurally radiated sound is directly controlled by active structural inputs [Fuller 1996]. Passive noise control describes procedures that influence the acoustical properties of structures by modifying their shape or thickness, adding stiffeners (ribs, beads, etc.) or masses, or applying additional layers of damping material to the surface. The latter approach motivates the present work. It describes methods to optimize structures numerically with respect to various acoustical and structural properties such as level of structure borne sound (a measure of the vibrational sensitivity of a structure), structural mass, natural frequencies, etc. These techniques can be applied during the design phase of a machine or machine component, since they use a finite element (FE) model of the structure. The shape of the structure, i.e., its thickness distribution, is modified automatically such that the desired goal is achieved without violating given constraints. The procedures can also serve the purpose of lightweight design because, due to the optional definition of suitable constraints, they are able to reduce the level of structure borne sound without applying additional mass. As an alternative and virtually “inverse” objective they can reduce the mass of the structure without increasing the vibrational level.

1.2

Literature Review

The present dissertation deals with a range of different disciplines including acoustics (particularly structural acoustics), optimization methods, the finite element method (FEM), geometric design, etc. Numerous textbooks, dissertations, and scientific journal papers are available for each of these topics. Thus, although this literature review is relatively detailed and exhaustive, it can present just an overview of the available background literature in some of the fields. Nevertheless, it is intended to serve other researchers in these fields as a guide and entry point for their own literature search. 1.2.1

Optimization in (Structural) Acoustics

There are only a limited number of textbooks that deal with numerical optimization in acoustics or structural acoustics. Koopmann and Fahnline [Koopmann 1997] were the first to publish a textbook solely on designing quiet structures by means of a sound power minimization approach. A chapter in the textbook on machine acoustics by Kollmann [Kollmann 2000] focuses on structural acoustic optimization. Finally, Munjal [Munjal 2002] edited a book that contains 18 papers presented at the IUTAM Symposium on Designing for Quietness in 2000. To the author’s knowledge, the paper by Olhoff [Olhoff 1970] on the optimal design of vibrating circular plates can be regarded as the first one in the field of numerical optimization in acoustics. He maximized the fundamental frequency of rotationally symmetric plates by optimizing the shape of the plates for a given volume, diameter, and Poisson’s ratio. Depending on the boundary conditions, a fundamental frequency increase of up to 544% is reported. In a successive article, Olhoff [Olhoff 1974] applied the same methods to simply supported rectangular plates of various aspect ratios for a given volume, size, and Poisson’s ratio. The plate thickness function was optimized to maximize the fundamental frequency, which increased by up to 19%.

1.2

Literature Review

3

Lyon, Mark, and Pyle Jr. [Lyon 1973] conducted theoretical and numerical studies to reduce the noise radiated from helicopter rotor tips. The shape of the rotor blades was optimized in order to minimize the radiated sound power subject to prescribed upper and lower bounds on the allowable longitudinal section area using a steepest descent method. A sound power reduction of approximately 10 dB was achieved in some frequency bands up to 5500 Hz. The optimal acoustic design of sandwich panels was the objective of Lang and Dym [Lang 1975]. They used a pattern search algorithm to optimize various combinations of sandwich panel skin and core thickness and density, subject to a constraint on the total panel surface mass. The A-weighted average sound transmission loss (TL) over the frequency range from 1 to 4 kHz was increased by up to 46%. Lang and Dym cite an earlier paper by Ford, Lord, and Walker [Ford 1967] who compared the TL of three sandwich panels with different core stiffnesses. In this context they use the term “optimize”, but they considered only three specific design variations rather than performing numerical optimization. The computer optimized design of engine structures is the topic of an article by Lalor [Lalor 1980]. He used static deflection techniques to simplify the FE calculations. The overall noise level was reduced by up to 5 dBA subject to weight, space, and strength constraints. However, it is not clear if numerical optimization techniques were used in this process. Rather, it appears as if FE analyses just helped to identify regions of the engine block that should be modified in order to achieve reduced noise levels. The actual modifications seem to be applied “manually”. In another paper by Wilcox and Lalor [Wilcox 1987], a simple univariate algorithm with four design variables and a fixed step length line search reduced the noise of an engine by changing the shape of the crankcase. The obtained noise reduction of 2.1 dBA was verified by experiments in which a noise reduction of 2.0 dBA in a frequency band from 500 to 3200 Hz was measured. Yildiz and Stevens [Yildiz 1985] optimized the thickness of unconstrained viscoelastic damping layer treatments for plates. Their objective was the maximization of the system loss factor by means of varying the mechanical properties of the plate and the damping layer as well as the layer/plate thickness ratio. An increase of the system loss factor by more than 100% is reported. A method for sensitivity analysis and optimization of nodal point locations in connection with vibration reduction was developed by Pritchard, Adelman, and Haftka [Pritchard 1987]. The sizes of added lumped masses on helicopter rotor blades were used as design variables to move nodal positions to a preselected location. The objective function that was to be minimized was the sum of the added masses. A potential application of nodal point placement is the reduction of overall vibration response by generalized force minimization. Lamancusa [Lamancusa 1988] addressed the geometric optimization of internal combustion engine induction systems. He used the IMSL ZXMWD optimization subroutine to perform global optimization but did not provide further information as to how this algorithm works. He just states that “this routine seems to function well and achieves convergence.” The aim was to control the low frequency inlet noise of a 4-cylinder automobile engine by optimizing the pipe lengths of an intake manifold system using up to three design parameters. The average exit sound pressure in the frequency range 50–250 Hz was reduced by up to 20 dB, primarily by shifting the major response peaks out of the frequency range under consideration. The analytically predicted noise reductions were verified experimentally. Optimum vibrating shapes of beams and circular plates were investigated by Thambiratnam and Thevendran [Thambiratnam 1988]. They optimized the thickness to maximize the fundamental frequency, keeping the volume of the structure constant, or to minimize the volume and shape of the structure for a given minimum allowable fundamental frequency. Various boundary

4

1

Introduction

conditions were employed, and the Complex algorithm [Box 1965] was used, incorporating the constraints into the objective function as a penalty function. The optimization results varied with the boundary conditions. The authors report a fundamental frequency increase of up to 350% and a volume decrease of up to 75%. In a series of papers by various combinations of the authors Belegundu, Constans, Cunefare, Lamancusa, Koopmann, Naghshineh, Salagame, and St. Pierre Jr., the “weak radiator” concept was introduced and applied to several problems. A weak radiator is a mode that radiates sound very inefficiently due to a low net volume velocity. This is achieved by pressure cancellations at the surface of the plate (“acoustic short circuit”), leaving very little energy left to be radiated by the plate’s edges. The sound power output from a structure was minimized by changing a mode shape or several mode shapes of the structure into a weak radiator using material tailoring [Naghshineh 1992], placing strategically sized masses at specific locations [St. Pierre Jr. 1993, St. Pierre Jr. 1995, Constans 1998], optimizing the thickness distribution [Lamancusa 1991, Lamancusa 1993, Belegundu 1994], or using active control forces [Naghshineh 1994]. Cunefare [Cunefare 1991b] developed a technique for deriving the optimal surface velocity distribution of a finite baffled beam that minimizes the radiation efficiency of the beam. Lamancusa and Koopmann [Lamancusa 1991] employed four different strategies to minimize the radiated sound power from a rectangular plate, namely, minimization of radiated sound power at a single frequency, minimization of radiated sound power over a frequency band, minimization of the radiation efficiency over a number of modes, and forcing the plate to vibrate with a mode shape that is a weak radiator. They used a method of feasible directions to optimize the thickness distribution of the plate and obtained a radiated sound power reduction of 9.1 dB in the first case (800 Hz), an improvement of 2.2 dB in the second case (0–800 Hz), and a decrease of 10.2 dB in the third case (first six modes). Results for the fourth case were not provided. Naghshineh, Koopmann, and Belegundu [Naghshineh 1992] employed material tailoring to achieve a minimum radiation condition. In a first step, a surface velocity distribution with minimum radiation condition, a so-called weak radiator velocity profile, must be obtained by some FE scheme. In a second step, a distribution of Young’s modulus and a density distribution is generated by structural FE modeling and linear programming techniques such that the structure exhibits the weak radiator profile as one of its mode shapes. A finite baffled beam was chosen to illustrate this approach. Lamancusa [Lamancusa 1993] found optimal thickness distributions of rectangular, flat plates with clamped edges that minimize the acoustic response to point forces at a single frequency or over a wide frequency bandwidth. He states that appropriate objective functions and constraints are critical to optimization success and lists some possible candidates: total weight, placement of structural natural frequencies, mode shape, mean square velocity, radiation efficiency, total radiated sound power, and matching of a predetermined weak radiator mode shape. An optimization program based on the method of feasible directions is used to optimize the thickness of rectangular plates for various of the abovementioned objective functions subject to appropriate constraints. Distributed point masses were used by St. Pierre Jr. and Koopmann [St. Pierre Jr. 1993] to minimize the sound power radiated from baffled rectangular plates. The placement of strategically sized masses at specific locations on the surface produced weak radiator mode shapes. Sound power reductions of more than 10 dB were achieved and verified by acoustic intensity measurements. A second paper by St. Pierre Jr. and Koopmann [St. Pierre Jr. 1995] extends this approach to a wider frequency range and presents some more results. Belegundu, Salagame, and Koopmann [Belegundu 1994] minimized the radiated acoustic power of a baffled plate by optimization of the thickness distribution and produced weak radiator mode shapes as well. They calculated the design sensitivity coefficients

1.2

Literature Review

5

for gradient-based optimization analytically and applied their approach to the optimization of an engine timing chain cover plate. The total power radiated from the first three modes was reduced by 12 dB while, interestingly, the weight is decreased as well by 32%. A modal tailoring approach was used by Constans, Koopmann, and Belegundu [Constans 1998] to minimize the sound power radiated from a vibrating shell structure. They employed a simulated annealing (SA) algorithm to place two small masses (35.8 g) at optimal FE node positions on a half-cylindrical aluminum shell in such a way that the radiated sound power (considering the first five modes) was minimized. The four design variables were the axial and circumferential coordinates for both masses. As a result of the optimization, the structure was converted to a weak radiator, leading to a sound power reduction of 9.5 dB at the first three modes. These findings were confirmed by experimental measurements. A paper presented by Sivakumar, Sung, and Nefske [Sivakumar 1991] describes the noise reduction of engine component covers. No numerical optimization was performed, but rather the forced vibration response was predicted by means of FEM, and “manual” design variations were employed to reduce the radiated sound power. The predicted noise reduction of 2.7 dB through the addition of stiffening ribs was surpassed by a 4.5 dB reduction measured in experiments. When the material of an engine front cover was changed from steel to laminated steel, a decrease of 6.2 dB (predicted) and 8 dB (measured) was achieved. Cunefare and Koopmann [Cunefare 1992] developed an acoustic design sensitivity (ADS) analysis technique by means of a boundary element method (BEM) formulation of the Helmholtz integral equation and partial differentiation of a quadratic power expression with respect to known surface velocities. This procedure leads to a sensitivity distribution, which quantifies the incremental change in radiated sound power from three-dimensional structures due to incremental changes in the surface normal velocity. Regions with high sensitivity would be likely candidates for the placement of active sources or for design modifications. The method was successfully applied to a rectangular box and a right circular cylinder. A generalized compound scaling (GCS) algorithm, presented by Grandhi, Bharatram, and Venkayya [Grandhi 1992a], optimized the thickness of large scale plate structures to minimize the weight subject to multiple frequency constraints and bounds on the design variables. Despite the actual objective to minimize the weight, nonstructural masses had to be added to the structures in order to satisfy the imposed frequency constraints. A review paper on structural optimization methods with frequency constraints by Grandhi [Grandhi 1992b] lists 186 references. Sensitivity analysis, constraint approximations, and optimization algorithms are discussed. According to Grandhi, the most common problem in frequency optimization seems to be the switching of vibration modes due to structural size modifications, causing convergence difficulties to the optimizer. Four classes of optimization methods were addressed, namely, mathematical programming, optimality criteria, optimal control, and others. ¨ In another series of papers, Hinton, Ozak¸ ca, Rao, and Sienz focus on the free vibration analysis and shape optimization of axisymmetric plates and shells, variable thickness plates, ¨ prismatic folded plates, and curved shells. Hinton, Ozak¸ ca, and Sienz [Hinton 1993] optimized simply supported and clamped plates as well as conical, spherical, and branched shells with respect to certain vibration characteristics. They used cubic splines to define the overall geometry and thickness variation. A semi-analytical sensitivity analysis was combined with a sequential quadratic programming (SQP) optimization algorithm. In the first part of their ¨ ¨ article, Ozak¸ ca and Hinton [Ozak¸ ca 1994a] derived new FE element formulations, forming a family of variable thickness, curved C(0) Mindlin-Reissner axisymmetric elements that include

6

1

Introduction

shear deformation and rotatory inertia effects. Accuracy, convergence, and efficiency were tested by free vibration analyses and comparison with other numerical and analytical methods. ¨ In the second part [Ozak¸ ca 1994b], these new elements were used for shape optimization of axisymmetric plates and shells. Shape and thickness of conical shells, circular plates, branched shells, and bells were optimized either to maximize certain natural frequencies or to minimize the material volume. Cubic splines were used to define shape and thickness. A combination of semi-analytical sensitivity analysis and mathematical programming (SQP from the NAG ¨ library) served as optimization algorithm. In the following year, Hinton, Ozak¸ ca, and Rao again published a two-part paper. The first part [Hinton 1995a] describes the free vibration analysis of variable thickness plates, prismatic folded plates, and curved shells carried out by using curved, variable thickness finite strips based on Mindlin-Reissner shell theory. The finite strip (FS) method combines the use of Fourier expansions and one-dimensional finite elements. Accuracy and effectiveness were illustrated on square plates, variable thickness plates in various shapes, cylinders with interior partitions, and a two-cell right box-girder bridge. In the companion paper [Hinton 1995b] the FS method was applied to shape optimization of square plates, a cylindrical shell, and a box-girder bridge. A computational tool combines FS analysis, cubic spline geometry definition, semi-analytical sensitivity analysis, and mathematical programming. Shape and thickness were the design variables, objectives were the maximization of the fundamental frequency subject to a volume constraint or the minimization of volume (or weight) subject to frequency constraints. Inoue, Townsend, and Coy [Inoue 1993] optimized the design of a simple model gearbox to minimize the overall vibration energy by varying the finite element thickness subject to lower and upper bounds on the thickness and to constant weight. A modal analysis is performed to derive the sensitivities after which a gradient projection method and a unidimensional search procedure is used to calculate the optimal design. The vibration energy is decreased by about 81% in the frequency band from 500 to 1500 Hz by optimizing just the top plate. In his Ph.D. dissertation, Broschart [Broschart 1994] calculated sound pressure, sound intensity, and sound power of vibrating structures based on BEM and compared the results with measurements performed on rectangular plates and boxes. He reduced the computation time by up to 99% with multigrid methods, applied additional actively controlled forces to reduce the surface velocity level by up to 7 dB, and employed structural optimization to reduce the vibration level, increase the fundamental frequency, or reduce the mass subject to various constraints. Design parameters were the plate thickness and the height and width of additionally applied ribs. Hambric [Hambric 1995] presented various approximation techniques for broadband acoustic radiated noise design optimization. Low-order series approximations replace full numerical systems to save computation time, thus making the models suitable for global search methods such as SA, which usually require a large number of function evaluations. The methods were tested for effectiveness, efficiency, and generality on three test cases in which shell thickness, shell loss factors, and rib stiffener locations served as design variables to minimize weight and manufacturing costs while lowering broadband radiated noise levels below a specified limit. The three test cases were the single octave band, two design variable study and the multiple octave band, four design variable study of a simply supported cylindrical shell with end caps as well as the multiple aspect, multiple octave band, six design variable study of a rib-stiffened cylindrical shell with end caps. In a sequel paper [Hambric 1996], the influence of sensitivities on the convergence characteristics was investigated for one of the abovementioned three test cases by varying the design variable step sizes and the frequency resolution.

1.2

Literature Review

7

Keane [Keane 1995] investigated passive vibration control via unusual geometries. A genetic algorithm (GA) optimized the position of 36 joints of a network consisting of 40 coupled beams such that the frequency averaged vibrational energy level at one end of the structure was minimized. The frequency range of interest was 150–250 Hz. The results depended on the strictness of the constraints to the design variables: When the joint positions were allowed to vary by up to ±5%, a reduction of 23 dB was achieved, while the vibrational energy was decreased even by 60 dB for a ±25% limit. In a second paper [Keane 1996], Keane and Bright describe experiments, which were conducted to verify the above optimization results. Two aluminum alloy structures (scaled down from 10 m to 2.6 m length) representing the original and the optimized structure were built and tested. The detailed behavior showed some deviations from the theoretical design, but good overall agreement between simulation and experiments was observed. McMillan and Keane [McMillan 1996] “manually” applied five concentrated masses to a thin rectangular plate in order to shift resonances from a frequency band, thus reducing the eigenvalues within certain bounds. Three different analytical approaches were developed for calculating the eigenfunctions, eigenvalues, and frequency response of the mass loaded plates. The results were obtained relatively easily and compared well with those obtained by using FEM. In a subsequent paper by the same authors [McMillan 1997], these methods are utilized to place 50 small point masses (totaling 10% of the plate mass) at optimal positions in such a way that a thin rectangular plate has no (or reduced) resonance peaks in its vibration transmission characteristics in a given frequency band. For the optimization, a sequential positioning method was compared with a GA. The former method achieved the greatest suppression of vibration, but the latter performed almost as well. A group of researchers led by Marburg has published quite a number of articles on acoustic optimization during the period 1997–2003. Marburg et al. [Marburg 1997a] introduced the concept of acoustic influence coefficients for the optimization of a vehicle roof whose geometry was described by 6 parameters. Optimal shell curvatures, which decrease the sound pressure at the driver’s ear, are calculated by a coupled FEM/BEM procedure with one-way structurefluid interaction, i.e., the structure excites the fluid but not vice versa. Various combinations of objective functions and constraints yielded sound pressure reductions between 3.2 dB for tight constraints and 53 dB for an optimization run without constraints on the design variables. The same approach was used by Marburg et al. [Marburg 1997b] in a case study in order to investigate the effects of stiffening a vehicle roof model by additional beams and ribs. An optimization of the geometry based parametric roof model with the objective to minimize the sound pressure at a specified point inside the car resulted in a decrease of 8 dB mainly due to a reduction of the number of natural frequencies in the frequency range of interest. Marburg and Hardtke [Marburg 2000] introduced a weighting function that helps the optimizer to focus on the reduction of the high level peaks rather than on the low-level parts of the objective function. Further, they used acoustic influence coefficients and a semi-analytical method for the sensitivity analysis. In another paper, Marburg and Hardtke [Marburg 2001] describe the shape optimization of a vehicle hat-shelf. The curvature of the hat-shelf was optimized to maximize the fundamental frequency and, ideally, to shift it out of the frequency range of interest. A multigrid strategy using four different FE discretizations was applied: The coarsest mesh provided initial parameter sets, which were preoptimized with either a refined linear element mesh or a refined quadratic element mesh. An even more refined mesh with a reduced number of parameter sets was then used for the actual optimization. The fundamental frequency was increased from 32 to 101 Hz, thus, out of the frequency range

8

1

Introduction

of interest (0–100 Hz). The corresponding sound pressure level (SPL) was reduced by up to 13.9 dB. In a two-part article, a general concept for the design modification of shell meshes used for acoustic optimization was described. The first part by Marburg [Marburg 2002a] focused on the formulation of the concept. A discussion of the advantages and disadvantages of geometry based models led to a division of the domain under investigation into a modification domain and its complement and to the introduction of global modification functions (defined everywhere in the modification domain) and local modification functions (defined only in local modification subdomains). In the second part by Marburg and Hardtke [Marburg 2002b], they applied the methods developed in the first part to the curvature optimization of a vehicle floor panel. One global and four local modification functions were defined, totaling 33 design variables. The optimization algorithm was a combination of a random iteration and a gradient-based method. The sound pressure level at the driver’s ear was decreased by 2 dB. Marburg et al. [Marburg 2002c] experimentally verified results of structural acoustic design optimization. The structure under investigation was a steel box with the dimensions 1 m × 1.1 m × 1.5 m. An experimental modal analysis was performed, and the sound pressure level was measured at three positions inside the box. Both an original and a numerically optimized design of the box were built and tested, and the agreement between simulation and experiment was found to be satisfactory. Another two-part publication describes the efficient optimization of a noise transfer function by modification of a shell structure geometry. The first part by Marburg [Marburg 2002e] reviews the theory and concepts of structural and acoustic analysis, structure-fluid coupling, objective function, and sensitivity analysis introduced in earlier papers. A concrete description of the optimization technique used is missing. In the second part by Marburg and Hardtke [Marburg 2002f], the concepts described in the first part were applied to the optimization of a vehicle dashboard. The sound pressure at the driver position was to be minimized by optimizing the curvature of the dashboard using various objective functions. Improvements of up to 3.8 dB are reported. Marburg [Marburg 2002g] wrote an exhaustive 80-page review article on structural acoustic optimization for passive noise control, which lists 344 references. First, he presented an overview of structural acoustic simulation techniques, various optimization strategies, and problems in structural acoustic optimization. Then, he elaborated on the suitable choice of objective functions and design variables before he provided a survey of optimization results. He concluded his paper with a description of open problems. The most recent paper in this series was published by Fritze, Marburg, and Hardtke [Fritze 2003]. They reduced the frequency averaged radiated sound power (in the range 0–250 Hz) of plates and shells by local modification of geometry using a gradient-based method. Curvature modifications change the local stiffness without increasing the mass. Parameter studies provided appropriate initial parameters to enhance the efficiency. The position, orientation, and depth of a bead was varied, resulting in six design variables. Sound power reductions of up to 4 dB were achieved. A review article on analysis and optimization in structural acoustics by Christensen, Sorokin, and Olhoff [Christensen 1998a] lists 39 references and focuses on existing numerical methods for solving problems of structural acoustic coupling with an emphasis on analysis, design sensitivity, and optimization. In a companion paper by the same authors [Christensen 1998b], some optimization results with respect to the directivity of sound emission were presented for both a flat and a conical circular shell. The structures were submerged in water (heavy fluid) in order to study the effects of structural acoustic coupling, which usually can be neglected for light fluids such as air. The radial positions and the mass of 29 circular dead ring masses added to the plate or shell served as design variables subject to appropriate constraints. A method of sequential linear programming (SLP) was used for the optimization, a simple finite difference

1.2

Literature Review

9

approach provided the sensitivities, and a harmonically varying point force of 10 N at 1850 Hz excited the center of the plate. The objective was to achieve a uniformly distributed emission of sound pressure in all directions. Finally, the sensitivity of the obtained optimal design with respect to other, preassigned design parameters that had not been used as actual design variables (damping coefficient, excitation frequency, value of largest mass) was investigated. In a third paper, Christensen and Olhoff [Christensen 1998c] performed a directivity optimization of a loudspeaker diaphragm by coupled FEM/BEM calculations. The objective was to uniformize the loudspeaker’s directivity properties by optimizing the masses and radial positions of 32 concentric dead ring masses attached to the diaphragm (similar to [Christensen 1998b]) or the shape of the diaphragm’s mid-surface by using B-splines (up to 10 design variables). In his Ph.D. dissertation, Hibinger [Hibinger 1998] described the optimization of various three-dimensional model structures with respect to vibrational levels, mass, or natural frequencies. He used a revised version of the simplex method of linear programming (first published by Dantzig [Dantzig 1963] and completely unrelated to Nelder and Mead’s simplex algorithm [Nelder 1965] as described in Sec. 3.2.2) to optimize the thickness of shell elements or the height of additionally applied ribs. He also defined appropriate constraints such as maximum allowable mass, maximum allowable vibration level, minimum and maximum plate thickness or rib height, and maximum allowable thickness or height difference for adjacent elements (continuity condition). Hibinger performed experimental measurements for some of the structures, thus verifying the numerical simulation and optimization results. The vibration level was reduced by up to 11.6 dB, the mass was decreased by up to 35%, and the fundamental frequency was increased by up to 52%. Lumsdaine and Scott [Lumsdaine 1998] dealt with shape optimization of unconstrained viscoelastic layers. According to the authors they were the first to use continuum finite elements for optimization. They reduced the peak displacement of simple, symmetric beam and plate structures by up to 98% in the frequency range from 100 to 1300 Hz (and simultaneously improved the system loss factor by up to 5270%), using an SQP algorithm and the commercial FE program ABAQUS. Vibrational optimization of a mass-loaded stepped plate was performed by Moshrefi-Torbati, Simonis de Cloke, and Keane [Moshrefi-Torbati 1998]. They minimized the integral of the frequency response (i.e., deflection in m per unit force in N) of simply supported stepped plates in the frequency range from 55 to 65 Hz by optimally placing one, two, five, ten, or twenty point masses with a total mass of 5 kg, which was about 10% of the plate’s weight. A GA yielded frequency response improvements by up to a factor of 35. Ratle and Berry [Ratle 1998] also used a GA for the vibro-acoustic optimization of a pointloaded and an acoustically excited plate, both carrying point masses. The objective function was either the mean square velocity of the plate or the far field sound pressure level (SPL) in a prescribed direction, both in the frequency band 200–250 Hz. The goal was to determine the optimal positions of five additional point masses (each one representing 20% of the plate mass). Due to the discretization of the plate, there would have been 64 × 64 = 4096 possible positions for placing a single point mass. For the five masses problem, however, about 9.6×1015 possible positions exist, making an exhaustive search impossible. Thus, a GA with a large population size (100) and a large number of generations (200) was used. The “intuitive” solution (i.e., placing all masses on the excitation point) did not yield the optimal placement (56 dB reduction of the mean square velocity level, 28 dB SPL reduction for point force excitation, and even an SPL increase of 4 dB for plane wave excitation), since there were still two peaks in the frequency range of interest. The true optimal placement shifted all natural frequencies

10

1

Introduction

out of the frequency range of interest, resulting in a reduction of the mean square velocity level by 70 dB, an SPL reduction of 58 dB for point force excitation, and an SPL decrease by 4 dB for plane wave excitation. The authors also investigated the influence of the correct choice of objective function. When they used the vibratory criterion for the noise reduction, they obtained only a 38 dB SPL reduction as opposed to a 58 dB SPL reduction for the noise radiation criterion. Vice versa, employing the acoustic criterion for the vibration reduction yielded only 56 dB as opposed to 70 dB using the vibration criterion. Noise radiation of principle gearbox housings with stiffening ribs was investigated by Wender [Wender 1998]. Parameter studies conducted on twelve different rib configurations (parallel to the edges, diagonal, star-shaped) of various heights showed that the radiated sound power depends on the arrangement and the height of the ribs. An attenuation of up to 11.5 dB was achieved, but the mass increased by up to 100%. Tinnsten et al. published a series of papers as well. The wall thicknesses at the FE nodes on the top of a rectangular box were optimized subject to a mass constraint by Tinnsten, Esping, and Jonsson [Tinnsten 1999] using a method of moving asymptotes. Four different cases were considered with the sound intensity level at a prescribed position above the box being the objective function in all four cases. The sound intensity level was reduced by up to 56 dB. Tinnsten [Tinnsten 2000] compared numerical optimization results with experimental measurements. He optimized the shell thickness of the aluminum top plate of a closed steel cylinder such that the sound intensity level at specified points outside the cylinder was minimized subject to weight and thickness constraints. In case of a free top edge, the sound intensity level was reduced by 4.0 dB in the simulation and by 2.3 dB in the measurements, whereas in case of a clamped top edge it decreased by 14.7 dB in the computation and by 19.5 dB in the experiments. Carlsson and Tinnsten [Carlsson 2002] developed a material model for softwood for violin top plates and employed SA to adapt the natural frequencies of a violin top plate made of “artificial wood” to that of a real one. The thickness distribution of the top plate was optimized using a total of 68 design variables to minimize the weighted sum of the differences between the natural frequencies of the original and the modified top plate, considering the first three eigenmodes. The new thickness values were constrained to ±10% of the original values. The optimal design yielded a maximal natural frequency deviation of 0.048%. In the paper by Tinnsten and Carlsson [Tinnsten 2002a], the methods and results presented in the previous paper [Carlsson 2002] are described in more detail. Additionally, the arch height of the violin top plate served as an alternative design variable instead of the thickness distribution. This new choice of design parameters led to a slightly inferior optimization result than the old one. Tinnsten, Carlsson, and Jonsson [Tinnsten 2002b] presented a numerical and experimental comparison of stochastic optimization results. The same structure as in [Tinnsten 2000], i.e., a closed steel cylinder with an aluminum top plate, with the same discretization and the same objective function and constraints, was investigated. The SA optimization algorithm achieved a sound intensity level reduction of 18.7 dB, whereas a reduction of 24.1 dB was measured in the experiment. The numerical optimization results using SA were also compared with the ones obtained from a gradient-based method in [Tinnsten 2000]. The SA algorithm reached an optimum where the intensity level was 4 dB lower and the weight of the top plate was 14% lower than that achieved with the gradient-based method. An engineering toolbox for structural acoustic design was developed and presented in Kessels’ Ph.D. dissertation [Kessels 2001]. He applied the toolbox to the reduction of sound power radiated from MRI scanners (MRI: Magnetic Resonance Imaging) and obtained an attenuation of up to 13 dB by optimizing glue modulus and carrier layer thickness.

1.2

Literature Review

11

Kaneda et al. [Kaneda 2002] presented an optimization approach for reducing sound power radiated from a vibrating plate by its curvature design. They employed an optimization procedure based on a genetic algorithm (GA), a shape representation technique using B-splines, vibration analysis (FEM), and acoustic radiation analysis (Rayleigh integral formulation). The curvature of a simply supported square aluminum plate was optimized by varying the positions of six control points in such a way that the radiated sound power is minimized in the frequency band from 10 to 1200 Hz (10 Hz step length). An attenuation of 22 dB was achieved mainly due to shifting the fundamental frequency towards higher values, thus enlarging the quasi-static frequency range. The paper concludes with an interesting investigation, namely, that of the sensitivity and robustness of the optimal design when exposed to small perturbations of the design parameters, which may occur in real life due to unavoidable scatter of properties and boundary conditions caused by manufacturing and assembly tolerances or by thermal expansion and contraction of real systems. Hence, in certain situations it may be more important to focus on the robustness of improved solutions rather than on nominal optimality. Zopp and R¨omer [Zopp 2002] acoustically optimized the powertrain suspension of a car with respect to the sound pressure distribution inside the vehicle considering requirements of vehicle dynamics. They combined a substructuring method based on frequency response functions (FRF), which is computationally much less expensive than FEM analyses, a weighted sum strategy to meet several design goals simultaneously, and an SQP algorithm in order to reduce the SPL at ear positions by up to 10 dB. Problems of analysis and optimization of plates and shells were addressed by Awrejcewicz and Krysko [Awrejcewicz 2003]. They derived equations for calculating the frequency spectra of shells with transverse deformation and rotary inertia and presented finite approximation solution methods, which were compared with the FEM. The paper by Bregant and Puzzi [Bregant 2003] deals with the optimization of free layer damping treatments to reduce vibration levels of plates. The objective was to maximize the damping level while minimizing the amount of added material. Due to the high number of control variables and multimodal solution spaces with many local optima, a GA was employed, and its performance was compared with that of an SQP method. The amount of material added to a rectangular, simply supported steel plate was not allowed to be more than 10% of the plate’s weight; at least 20% of the predefined patches had to be covered. Fuse et al. [Fuse 2003] investigated optimal curvature or rib attachment design for a vibrating aluminum plate. The objective was to reduce the radiated sound power in a frequency band from 50 to 300 Hz using FEM and a GA. The sound power was approximated by the sum of the sound pressure levels computed by the Rayleigh integral at 13 points on a hemisphere above the plate. Nine bending positions were defined along the longer side of the plate, the heights of which were varied by up to ±1 mm. There were only four peaks in the sound power spectrum of the initial design in the 50–300 Hz frequency band of interest. The optimization process shifted the fundamental frequency beyond the upper frequency limit, thus leading to a quasi-static behavior of the plate within the frequency range of interest. The same or a similar result probably could have been achieved easier by maximizing the fundamental frequency of the plate. An optimized rib design (the total mass of added ribs was not more than 20% of the plate mass) was not able to shift the fundamental frequency beyond the upper frequency limit, thus resulting in a smaller reduction of the radiated sound power than the optimal curvature design. Michot, Piranda, and Trivaudey [Michot 2002] presented a method to optimize simplified models meshed with finite triangular plate elements. Large FE models of a car body typically

12

1

Introduction

have about 1 000 000 degrees of freedom (DOFs). The objective was to simplify the mesh by employing some substructuring method, thus reducing the number of DOFs, while preserving the dynamic properties of the structure, i.e., the natural frequencies. A gradient-based algorithm was used to reduce the DOF number of two model structures by a factor of up to 20. The mesh simplification took about two hours of computation time on an HP workstation, but an FE analysis of the simplified model took only 10 seconds instead of 230 seconds for the original model, thus making it suitable for optimization calculations, which usually require a large number of analyses. Moshrefi-Torbati et al. [Moshrefi-Torbati 2003] applied a GA to achieve passive vibration control of a satellite boom structure. The satellite boom was 4.5 m long and constituted a truss made of 93 beams. The geometry of the structure, i.e., the position of the joints, was optimized such that a maximum mid-frequency vibration isolation was achieved. The joints were allowed to move by up to 20% of the length of each bay, i.e., by up to ±9 cm in all directions. Both the original and the optimized model were built and experimentally tested. The numerical optimization yielded a theoretical vibration isolation of 31 dB within the frequency band 150– 250 Hz, which agreed excellently with the experimental results (30 dB attenuation). Bai and Liu [Bai 2004] optimized panel speakers to achieve omni-directional responses at high efficiencies. A GA was used to search for the positions of exciters and the delay of input signals that render optimal performance. A coupled model of panel speakers was developed that incorporated electrical, mechanical, and radiation impedance matrices. The optimization results were verified by experiments according to ISO 3745 in an anechoic room.

1.2.2

Numerical Optimization

According to an introductory book on operations research by Domschke and Drexl [Domschke 1998], optimization procedures or “operations research” were developed in the years just before and during World War II. In Great Britain and the USA various possibilities were investigated to find an optimal composition of ship convoys to cross the Atlantic Ocean. Also in World War II, minimum weight optimum design of basic aircraft structural components (columns, stiffened panels, etc.) was developed [Kirsch 1993]. Since then, optimization methods have been successfully applied to a number of fields such as economics (profit maximization, effort minimization), structural design (lightweight design, high stiffness, comfort), or traffic flow optimization (highways, aircraft, railways). Particularly, the continuous and still ongoing development of cheaper and faster computers facilitates the analysis and optimization of large and complex structures or systems within a reasonable amount of time. Numerous scientific journals focus on numerical optimization. Examples include (in no particular order) Applied Mathematics and Computation, Journal of Global Optimization, Computational Optimization and Applications, Journal of Optimization Theory and Applications, Optimization and Engineering, Fuzzy Optimization and Decision Making, Journal of Combinatorial Optimization, SIAM Journal of Optimization, SIAM Journal of Control and Optimization, Optimization, Optimization Methods and Software, Structural and Multidisciplinary Optimization, Applied Mathematics and Optimization, Advanced Modeling and Optimization, and the website http://www.optimization-online.org/. There are also some textbooks on evolutionary and genetic algorithms (e.g., [Rechenberg 1971, Rechenberg 1973, Schwefel 1975, Schwefel 1977, Goldberg 1989, Davis 1991, B¨ ack 1996,

1.2

Literature Review

13

Mitchell 1996, Michalewicz 1997]) as well as on deterministic optimization techniques (e.g., [Dantzig 1963, Nocedal 1999]) and on operations research (e.g., [Domschke 1998, Neumann 2002]). Therefore, an exhaustive literature review cannot be offered in this section, rather a few selected examples are presented that are directly relevant to this work. The mode of operation of a GA is described in detail in Sec. 3.3.2. A paper often cited by others was published by Rosenbrock [Rosenbrock 1960]. He developed an automatic method for finding the greatest or least value of a function. His technique is based on a modified method of steepest descent. Calculations of gradients or derivatives are not necessary, and the variables can be restricted to a given region. The paper arose from the need to design chemical processes for most economical results, employing five parameters to be optimized. It contains some difficult test cases that later were often used by others (e.g., [Spendley 1962, Fletcher 1963, Powell 1964, Box 1965, Corana 1987, Powell 1994]) as benchmark tests for their algorithms. Powell, Fletcher and Reeves [Powell 1962,Fletcher 1963,Fletcher 1964,Powell 1964] presented several optimization algorithms for finding local minima of functions of several (up to 100) variables, including proofs of convergence. In a review paper, Fletcher [Fletcher 1965] compared several derivative-free function minimization techniques. He concluded that Powell’s method [Powell 1964] was the most efficient. The first simplex algorithm was developed by Spendley, Hext, and Himsworth [Spendley 1962] as a technique for empirical optimization. It can be described as a rudimentary steepest descent or ascent method subject to continuous review. Any point violating a constraint is declared worse than any other point and is eliminated at first opportunity. The arithmetic involved is relatively trivial, since the direction of steepest descent is never explicitely calculated, hence suited for manual application—the authors even supply a record sheet for hand calculation. Box [Box 1965] published an algorithm called the Complex (constrained simplex) method, which is efficient compared to other methods when the required optimum lies on one or more (inequality) constraints. The Complex method was compared with Rosenbrock’s method [Rosenbrock 1960] and with the simplex algorithm [Spendley 1962]. It was shown that Rosenbrock’s method was more efficient for unconstrained problems than the simplex or the Complex method. The simplex method introduced by Spendley, Hext, and Himsworth [Spendley 1962] was revised, extended, and improved to its well-known and often applied form by Nelder and Mead [Nelder 1965]. Whereas the former authors assumed that the relative steps to be made in varying the factors were known, thus making their strategy rather rigid for general use, the latter ones developed a simplex that adapts itself to the local “landscape” of the objective function. Further details on the simplex algorithm by Nelder and Mead [Nelder 1965] can be found in Sec. 3.2.2. Price [Price 1977] introduced a controlled random search (CRS) procedure for global optimization. Random search and mode-seeking routines were combined into a single, continuous process, which is effective in searching for global optima of a multimodal function with or without constraints. Kirkpatrick, Gelatt Jr., and Vecchi [Kirkpatrick 1983] were the first to present the simulated annealing (SA) concept. They developed a combinatorial optimization procedure in a discrete domain and successfully applied it to the traveling salesman problem with 400 cities (see p. 35), to the logic design of an IBM computer chip, and to the optimal placement of computer components (98 chips were to be placed on 100 positions) such that signal propagation times or distances were minimized. SA is a stochastic method, which starts from a user-defined starting

14

1

Introduction

point and searches for a minimum in a random direction. If the move was downhill, the new design is accepted and the search proceeds in another random direction. In case the move was uphill, however, it might still be accepted anyway with a certain probability, depending on the size of the uphill step and a parameter called temperature T . That way, the algorithm may escape local minima and finally find the global minimum [Constans 1998]. Thus, the SA method couples random function evaluations with a gradually reduced search radius to find the global optimum [Hambric 1995]. The algorithm was named after the annealing process. In melted or hot metal, the metal atoms are relatively free to move in random directions. When the temperature is reduced, the atoms are more and more restricted to their positions until they finally “freeze” [Constans 1998]. A slow, careful cooling brings the material to a highly ordered, crystalline state of lowest energy, whereas rapid cooling yields defects and glass-like intrusions inside the material [Corana 1987]. The algorithm was adapted to problems with continuous variables by Corana et al. [Corana 1987]. The adapted version was tested against an adaptive random search method and the Nelder and Mead simplex algorithm [Nelder 1965], using some of Rosenbrock’s test functions [Rosenbrock 1960]. The SA technique proved to be more reliable than the other methods, but it was quite costly in terms of function evaluations. Goffe, Ferrier, and Rogers [Goffe 1994] applied the SA procedure to the global optimization of statistical functions and performed tests on four econometric problems with up to 62 parameters, comparing the results with those of three conventional algorithms from the IMSL library. SA was very successful in finding optima, whereas the other algorithms failed. The very large number of function evaluations required by the SA algorithm was somewhat compensated by the fact that the other algorithms had to be restarted several times and still were not able to locate the global optimum. A Fortran implementation of the SA algorithm as used by Goffe, Ferrier, and Rogers can be found in [SIMANN 1997]. The paper on a direct search optimization method by Powell [Powell 1994] contains a detailed description of the COBYLA algorithm used in this dissertation. The COBYLA algorithm is explained in great detail in Sec. 3.2.3. Its Fortran source code is available from [COBYLA 1994]. Powell also presented a review paper on derivative-free optimization methods [Powell 1998]. He considered advantages, limitations, and theory of several techniques and addressed line search methods, discrete grids, geometric simplices, conjugate direction procedures, trust-region algorithms (linear or quadratic approximations to the objective function), and SA algorithms. Conn, Scheinberg, and Toint [Conn 1998] tested the performance of their derivative-free optimization (DFO) algorithm, which was inspired by the COBYLA algorithm [Powell 1994]. The objective function was approximated by quadratic polynomial interpolation within a trustregion framework. The DFO algorithm was compared with a parallel direct search (PDS) method, with the nonlinear optimization package LANCELOT, and with COBYLA. Public-domain software for black-box global optimization was tested and compared by Mongeau et al. [Mongeau 2000]. Techniques such as integral global optimization, GAs, SA, clustering, random search, continuation, Bayesian, tunneling, and multilevel methods were tested on practical problems such as least median of squares regression, protein folding, and multidimensional scaling. Spellucci’s overview article on nonlinear (local) optimization [Spellucci 2001] contains 120 references. He addressed some theory, unconstrained minimization, bound constrained problems, general linearly constrained problems (active set methods), linearly constrained problems (interior-point methods), and nonlinearly constrained problems (transformation into an only bound constrained problem, linearization methods, modifications of the SQP method, adaptation of interior-point methods, homotopy methods for the KTC-system).

1.2

Literature Review

15

An invaluable source of information on optimization algorithms is the continuously updated “Decision Tree for Optimization Software” by Mittelmann and Spellucci [Mittelmann 2004] on the internet. The source code of numerous optimization algorithms can be downloaded for free, and detailed background information is given for each group of problems as well as for suitable solution techniques. 1.2.3

Structural Optimization

The focus of the literature review in this section is on structural optimization in general where other than acoustic objectives are used. Only a few selected papers that are relevant to the present work are summarized in this section. For detailed information on structural optimization the reader is referred to textbooks by, e.g., Kirsch [Kirsch 1993], Baier, Seeßelberg, and Specht [Baier 1994], and Bendsøe [Bendsøe 1995]. The use of spline functions to reduce the number of design variables as introduced in the present work was first inspired by the paper of Braibant and Fleury [Braibant 1984]. They employed B´ezier and B-spline curves to define design elements, making it possible to describe complex geometries by a small set of design variables and just a few design elements. Weck and Steinke [Weck 1984] smoothed stress peaks caused by notch effects at unloaded holes. A substitute functional was discretized by means of splines and formulated as a nonlinear constrained optimization problem, which was solved with a quasi-Newton method. A new dual method using mixed variables was introduced by Fleury and Braibant [Fleury 1986]. Their technique employs mixed direct/reciprocal design variables in order to get conservative, first-order approximations to the objective function and to the constraints. The primary optimization problem was replaced with a sequence of explicit subproblems that are convex and separable. The algorithm produces a sequence of steadily improving feasible designs. Kibsgaard [Kibsgaard 1992] regarded sensitivity analysis as the basis for optimization. He conducted a numerical study on discrete semi-analytical design sensitivity analysis and described dissimilarities of sensitivities due to mesh refinement. In a paper by Zhang and Belegundu [Zhang 1992] a systematic approach for generating velocity fields in shape optimization was presented. Velocity fields represent the sensitivity of grid points with respect to a design variable and can either be obtained from deformation fields using fictitious auxiliary structures and loads or from a dynamic mode shape analysis. Schmit Jr. and Lai [Schmit 1994] performed structural optimization based on preconditioned conjugate gradient analysis methods. They used their method, which has the potential for parallel computation, to achieve minimum weight truss designs. A review article on the optimal design of mechanical engineering systems was compiled by Papalambros [Papalambros 1995] with an emphasis on partitioning, decomposition analysis and decomposition synthesis, and topology design of systems of structural components. In his opinion, the best general purpose optimization codes are based on SQP. Chickermane and Gea [Chickermane 1996] developed a local function approximation algorithm, which generates a sequence of convex, separable subproblems. Harzheim, Graf, and Liebers [Harzheim 1997] created a program called “Shape200” to create basis vectors for shape optimization using Solution 200 of MSC.Nastran. Previously, deformed shapes were generated using different load cases, and the resulting displacements were used as shape basis vectors. The authors’ method uses Bernstein polynomials to define a “shape box”. Sensitivity analysis for sizing optimization using ABAQUS code was the title of a paper by Zhang and Domaszewski [Zhang 1998]. They treated ABAQUS as a black box and interfaced

16

1

Introduction

its sensitivity analysis capabilities with an optimization algorithm. Information on the element stiffness matrix and the shape functions are not necessary. Hern´andez [Hern´andez 2000] introduced post-optimal procedures for structural optimization. He used a tunneling method to restart the optimization after a (local) optimum was found in order to find the global one. An efficient shape representation concept for shape optimal design of structures was presented by Kegl [Kegl 2000]. It is based on a design element technique in which a design element is defined as a rational B´ezier body. Fish and Ghouali [Fish 2001] performed multiscale analytical sensitivity analysis for composite materials and investigated the sensitivity of global structure behavior (e.g., deformation or vibration modes) with respect to local characteristics (e.g., material constants of microconstituents). Analytical gradient computation was compared with a central finite difference approximation, which is highly sensitive to the step size. Kim and Choi [Kim 2001] dealt with the direct treatment of a max-value cost function in parametric optimization. In earlier studies, the max-value cost function had been replaced with an artificial design variable leading to an additional equality constraint. However, the direct treatment is more efficient and stable than the transformation treatment and up to 50% faster. In a paper by Marcelin [Marcelin 2001], genetic optimization of stiffened plates and shells was described. The method is based on a GA and uses a backpropagation neural network or the Ritz method for creating function approximations. 1.2.4

Topology Optimization and Fully Stressed Design

In this section some selected papers and books on topology optimization and biological design are covered because the author initially considered employing some of these interesting approaches. However, the author did not see a way to define some local quantity that can be used as an indicator for material addition or removal for a frequency dependent objective function or constraint such as the mean level of structure borne sound or some natural frequency. Instead, some global quantity had to be used requiring a more classical structural optimization procedure as described in the previous sections. A fascinating approach to structural optimization was presented by Mattheck and coworkers [Mattheck 1989, Mattheck 1990, Baumgartner 1991, Mattheck 1991]. They investigated the mechanisms of biological growth in trees and bones and derived rules for optimal structural design. Biological structures such as trees or bones seem to grow into a shape of constant surface stress by adding material to high stress regions and removing material from areas with low stress in order to reduce or avoid high local notch stresses. This discovery was transformed into an FE procedure that automatically adds material to high stress regions by virtually simulating a fictitious thermal expansion in which the thermal expansion coefficient is set proportionally to the von Mises stress, leading to a “swelling” in areas with high stress and, thus, to a homogenization of stress at a lower level. The CAO (computer aided shape optimization) method is based on the unproved but plausible assumption that biological load carriers can survive only if they are mechanically optimized, lightweight, and fatigue resistant. These findings are summarized in an interesting book by Mattheck [Mattheck 1997, Mattheck 1998]. Numerous examples demonstrate the enormous increase in durability and lifetime for various structures. Bubenhagen and Harzheim [Bubenhagen 1998] successfully applied this method to the optimization of a fillet at a rear axle differential of a car and achieved a stress reduction of 28%.

1.2

Literature Review

17

Various techniques for the optimization of structural topology, shape, and material were introduced in a book by Bendsøe [Bendsøe 1995] with a focus of topology optimization. Bakhtiary et al. [Bakhtiary 1996] presented a new interface between the FE program MSC.Nastran and CAOSS (computer aided optimization system Sauter), which is based on fully stressed design as well. Duysinx and Bendsøe [Duysinx 1998] described a procedure to solve optimal material distribution problems with stress constraints by means of topology optimization of continuum structures. A method called reverse adaptivity (RA) was invented by Reynolds et al. [Reynolds 1999]. Low stress regions of the FE mesh are refined by element subdivision and low stress subdivided elements are physically removed from the data structure. This technique is computationally highly efficient, since the initial design can be a relatively coarse FE model of the structure, which is iteratively refined during the optimization. In a subsequent paper by Reynolds et al. [Reynolds 2001], the RA procedure is extended to allow adding of material instead of only removing it. This way, improved structural designs may be found outside a predetermined domain. A detailed review article by Eschenhauer and Olhoff [Eschenhauer 2001] lists 425 references to papers on topology optimization of continuum structures. It gives an overview of developments within the material technique (microstructure) and geometrical technique (macrostructure) with a special emphasis on optimum topology and layout design of linearly elastic two- and three-dimensional continuum structures. 1.2.5

Active Control

Some references related to active control of sound and vibration are cited here as well, even though only passive vibration control by means of structural optimization is considered in the present work. This is done because they employ some of the methods described in the previous sections. Introductory and advanced textbooks on active noise or vibration control were written by Nelson and Elliott [Nelson 1992], Fuller, Elliott, and Nelson [Fuller 1996], Hansen and Snyder [Hansen 1996], Snyder [Snyder 2000], Hansen [Hansen 2001], and Preumont [Preumont 2002]. Cunefare and Koopmann [Cunefare 1991a] considered both surface and far-field effects when they applied active noise control to achieve a global minimum of sound power radiated from a box structure with active sources. The sound power of the [2,1] mode of the box’s top surface was reduced by up to 12.9 dB. In the far field, the control sources seem to increase the apparent complexity of the noise source, making a monopole appearing as a dipole or a dipole as a quadrupole. This leads to a less efficient or weak radiator as described in Sec. 1.2.1. Giordano, Cunefare, and Koopmann [Giordano 1993] compared numerical results for the optimization of active source strengths with experimental measurements. They used the same rectangular box as described in [Cunefare 1991a] and [Cunefare 1992] and achieved a sound power reduction of 10.6 dB. Elliott [Elliott 1994] presented an overview of global vs. local control and feedforward vs. feedback control and described adaptive filters for feedforward control. Good passive design can minimize the complexity of the active controller. Global noise control with a reasonable number of loudspeakers in an enclosure the size of a car is only possible up to an upper frequency limit of 250 Hz.

18

1

Introduction

Naghshineh and Koopmann [Naghshineh 1994] developed an active control strategy for achieving weak radiator structures. They employed an interesting twist on the weak radiator concept by using active vibration control to force a beam to vibrate as a weak radiator. A set of control forces that resulted in minimum radiated sound power was found by means of numerical optimization. The effect of the number and location of the actuators on sound power level reductions was studied as well. “Natural algorithms”, i.e., several genetic and simulated annealing algorithms, were developed by Baek and Elliott [Baek 1995] and tested as methods of finding optimal secondary loudspeaker positions in an active noise control system. They compared the performance of these algorithms with that of a simple random search. The best GA and the best SA algorithm exhibited a similar convergence speed. The numerical results agreed well with experimental results. The methods seemed to be reasonably robust for slight frequency changes. Bein et al. [Bein 2000] performed active noise reduction on circular saws. By applying piezoelectric patches and anti-phase excitation they achieved a noise reduction of up to 12 dB at 1010 Hz. An add-on, long life adaptive vibration compensation system for a miniature mechanical cryocooler on a satellite (intended to cool infrared detectors) was developed at the DLR (Deutsches Zentrum f¨ ur Luft- und Raumfahrt, German Aerospace Center) by Melz, Melcher, and Matthias [Melz 2000]. The vibration levels were reduced by 53 dB, and long life testing revealed no performance degradation after 108 cycles. In his very recent Ph.D. dissertation, Oude Nijhuis [Oude Nijhuis 2003] described the development, validation, and application of efficient analysis tools for active structural acoustic control (ASAC) with an emphasis on the use of piezoelectric patches as control actuators. The structural model (reduced FE model) and the acoustical model (Rayleigh integral method) were successfully validated with experiments performed on a clamped plate setup with surface bonded piezoelectric patches. A GA was applied for finding the optimal placement of independent direct velocity feedback controllers. Another recent Ph.D. dissertation by Vogl [Vogl 2003] applied vibro-acoustic system analysis as a tool for realizing adaptive structural systems. The focus was on the development of vibroacoustic sensor concepts as a basis for adaptive vibro-acoustic measures to reduce the sound radiation of vibrating structures into the acoustic free field. Li et al. [Li 2004] investigated the combinatorial optimal design of the number and positions of actuators in actively controlled structures using GAs. They considered a building subjected to earthquake excitation and controlled by active tendon actuators. Finding the optimal actuator placement is a discrete nonlinear optimization problem, and the number of sensors and actuators was not known in advance. The authors employed a multilevel optimization approach in which the first level implemented the optimal control (continuous objective function), the second level sought the optimal configuration of actuators (objective function: top floor displacement response), and the third level optimized the number of actuators (objective function: control cost and control performance).

1.2

Literature Review

1.2.6

19

Other Topics

The following sections list some additional references on more general topics such as acoustics, FEM, numerical methods, or splines. Acoustics and Vibration in General Numerous textbooks deal with more or less general aspects of acoustics and vibration. A (not necessarily complete) list of relatively recent books includes the works by Cremer and Heckl [Cremer 1988,Cremer 1996], Beranek [Beranek 1993,Beranek 1996], Leissa [Leissa 1993a,Leissa 1993b], Pierce [Pierce 1994], Schirmer [Schirmer 1996], Veith [Veith 1996], Harris [Harris 1998], Kollmann [Kollmann 2000], Lyon [Lyon 2000], Raichel [Raichel 2000], Henn, Sinambari, and Fallen [Henn 2001], Mechel [Mechel 2002], Barron [Barron 2003], Bies and Hansen [Bies 2003], Cremer and M¨oser [Cremer 2003], and M¨ uller and M¨oser [M¨ uller 2004]. Ochs and Snowdon [Ochs 1975] designed true simple supports for rectangular and square plates by means of spring-steel flanges. The first 20 measured resonant frequencies deviated less than 0.55% from the computed values, which can be considered extremely good agreement. A report by M¨ uller et al. [M¨ uller 1983] introduced simple approximation methods for acoustical calculations. In his Ph.D. dissertation, Rautert [Rautert 1990] both theoretically and experimentally determined bearing loads in spur gears and bevel gears that cause structure borne sound. A Ph.D. dissertation by Angert [Angert 1992] describes investigations of the acoustical behavior of machine structures. The author conducted experimental modal analyses and narrow-band intensity measurements to determine the vibrational and radiational behavior of rectangular plates, of a rectangular box, and of an industrial gearbox. Sch¨osser [Sch¨osser 1996] investigated the vibrational and radiational behavior of gearboxes with the objective of active noise reduction. Zopp’s Ph.D. thesis [Zopp 2000] introduced an FE formulation of aluminum foam structures suitable for the calculation of structure borne sound. Zopp included an approach to optimize the material density distribution of an aluminum foam plate such that all natural frequencies were maximized or, alternatively, such that the first three natural frequencies were minimized while the next three natural frequencies were to be maximized. In the first case the fundamental frequency was increased by almost 25%, whereas in the second case the three lower natural frequencies were reduced by up to 4% and the three upper natural frequencies were increased by up to 14%. Finite/Infinite Element Method and General Numerical Methods Since the finite element method (FEM) can nowadays be considered a standard tool for engineers to simulate and predict the static and dynamic behavior of structures and systems, this numerical method is not described in the present thesis. Instead, the reader is referred to standard textbooks on FE such as the exhaustive three volume set by Zienkiewicz and Taylor [Zienkiewicz 2000] or the book by Bathe [Bathe 1996, Bathe 2002]. Giljohann [Giljohann 1996] introduced FE methods for calculating the sound radiation of arbitrary three-dimensional structures and compared the results of a newly developed method for the calculation of sound radiation into the free field with those of well-known methods. He applied the numerical techniques to an industrial gearbox and achieved good agreement between numerically predicted and experimentally measured values.

20

1

Introduction

A book and some papers by Ihlenburg [Ihlenburg 1995,Ihlenburg 1998,Ihlenburg 2002,Ihlenburg 2003] deal with the discretization and maximum mesh size required for reliable results of acoustic FE calculations. According to these references, one needs an extremely refined mesh for medium and high wavenumbers in order to avoid “numerical pollution” effects caused by the fact that “numerical waves” are dispersive also in nondispersive media, i.e., the discrete wavenumber in an FE model depends on the frequency due to discretization. Therefore, the “rule of thumb” of ten elements per wavelength leads to inaccurate results due to meshes that are too coarse. The infinite element method (IEM) is a relatively new technique to calculate the sound pressure distribution in the surroundings of a structure. A survey article by Astley [Astley 2000] presents a review of various infinite element schemes for wave problems and contains a comparison and assessment of performance. Tests of different conjugated and unconjugated formulations applied to a spherical radiator showed that conjugated formulations are more effective for far fields, whereas unconjugated schemes are more effective for near-field calculations. Landfester [Landfester 2001] was the first to implement conjugated infinite elements (IEs) with ellipsoidal coordinates and compared them with unconjugated elements in ellipsoidal coordinates [Burnett 1998]. General numerical methods and algorithms are described in detail, for instance, in the book by Engeln-M¨ ullges and Uhlig [Engeln-M¨ ullges 1996] and in the famous “Numerical Recipes” books by Press et al. [Press 1992, Press 1996]. All three books contain numerous computer routines in Fortran source code.

Parametric Geometry and Splines There are also quite a number of textbooks on splines, parametric geometry, and computer aided geometric design (CAGD) available. Examples include the books by de Boor [de Boor 1978], Sp¨ath [Sp¨ath 1978,Sp¨ath 1990,Sp¨ath 1991], Hoschek and Lasser [Hoschek 1992,Hoschek 1993], Engeln-M¨ ullges and Uhlig [Engeln-M¨ ullges 1996], and Farin [Farin 1994, Farin 2001]. Schmidt [Schmidt 1988] described Hermite splines for shape preserving interpolation, which avoid all extrema except those dictated by the data (key points) while preserving smoothness. A cubic chain was constructed and defined in terms of its first derivative at each node. The derivative is set to zero whenever a data point has an extreme value compared to its neighbors while the mean-squared second derivative discontinuity is minimized. Dietz [Dietz 1995, Dietz 1998] generated smooth surfaces from measured data points and performed geometry reconstruction from measured data points using smooth B-spline surfaces. Eck and Hoppe [Eck 1996] automatically reconstructed B-spline surfaces of arbitrary topological type by converting data points of laser scanners into useful geometric models. Els¨asser [Els¨asser 1998] investigated approximation with rational B-spline curves and surfaces, and Schickentanz [Schickentanz 1999] considered interpolation und approximation by means of B-spline surfaces with rational offsets. El-Abbasi, Meguid, and Czekanski [El-Abbasi 2001] developed a new strategy for the smooth representation of two-dimensional contact surfaces, which are modeled by using cubic splines that interpolate the FE nodes. The splines provide unique surface normal vectors and do not require prior knowledge of surface tangents and normals. The new method results in a significant improvement in the prediction of contact stresses and contact areas.

1.2

Literature Review

21

Finite Element Mesh Refinement, Coarsening, and Adaptation Initially, it was also considered to employ mesh adaptation techniques to reduce the number of design variables and computation time. It turned out, however, that, as for the topology optimization and fully stressed design techniques, it was not possible to derive criteria to determine at which locations of a structure the FE mesh is to be refined or coarsened. This is due to the frequency dependence of computed quantities and to the various local thickness modifications that occur while the optimization is in progress. Furthermore, “[m]esh refinement is not suited in this case since this would cause discontinuities of the objective function” [Fritze 2003]. Thus, after each mesh adaptation, the optimization procedure would have to be restarted, which is impractical. Campagna and Kobbelt [Campagna 1998] wrote an informative article on three-dimensional graphics for computer games and virtual reality applications, in which surfaces were modeled by triangles. Things that seem to appear in some distance from the viewer can be modeled with a significantly coarsened mesh, which reduces computation time. An automatic adaptive meshing technique for Delaunay triangulations was introduced by Xu et al. [Xu 1998]. FE model regions can be refined, coarsened, or kept unaltered. The method automatically identifies nodes that are candidates for deletion. Kobbelt, Campagna, and Seidel [Kobbelt 1998] presented a general framework for mesh decimation. They iteratively modified the original discretization without changing the topology. For an FE simulation small details can be important, whereas they are unimportant when rendering remote objects. The incremental mesh reduction is based on the half-edge collapse technique and consists of three fundamental ingredients, namely, a topology operator, a distance measure, and a fairness criterion. A paper by de Cougny and Shephard [de Cougny 1999] addressed parallel refinement and coarsening of tetrahedral meshes. Mesh coarsening relied on an edge collapsing tool, refinement used edge-based subdivision templates, and mesh optimization maintained the quality of the adapted meshes. Doege and Meinen [Doege 1999] investigated adaptive remeshing for FE simulations of bulk metal forming processes in which highly distorted meshes can occur causing numerical instability. In this case a new mesh with regular elements was generated. Hattangady [Hattangady 1999a, Hattangady 1999b] applied coarsening of mesh models for representation of rigid objects in finite element analysis to contact problems (sheet or bulk forming) as well as automatic remeshing in three-dimensional analysis of forming processes. Naylor [Naylor 1999] filled space with tetrahedra, thus creating three-dimensional meshes. This process is not as trivial as it is in two dimensions, since regular tetrahedra cannot be fitted together to fill space. Instead, cubes were taken to fill the space and were then subdivided into five or six tetrahedra. Christiansen and Pedersen [Christiansen 2001] addressed automatic mesh refinement in limit analysis, and Vaz Jr. and Owen [Vaz 2001] examined aspects of ductile fracture and adaptive mesh refinement in damaged elasto-plastic materials.

22

1

1.3

Introduction

Scope and Objectives of this Study

This section describes the scope and objectives of this dissertation (Sec. 1.3.1). Issues that are not objectives are discussed in Sec. 1.3.2. Finally, some further considerations are presented in Sec. 1.3.3. 1.3.1

Issues that are Objectives

The objectives of the present research, which can be derived partially from the previous literature review, can be described as follows: • As mentioned in Sec. 1.1, the main objective of this study is the numerical optimization of the thickness distribution of three-dimensional structures with respect to vibrational or structural properties. For that purpose a tool called Numerical Structural Acoustic Optimization (NSAO) has been developed. It is a combination of commercial FE software, user-written Fortran routines, and Unix shell scripts that automatically optimizes the thickness in an iterative manner with respect to various objective functions and subject to various constraints. Possible objective functions include the mean level of structure borne sound (MLS), the structural mass, the fundamental frequency, or the difference between two natural frequencies. Possible candidates for constraints include upper and lower bounds on the design variables (i.e., local thickness values), structural mass, MLS, or fundamental frequency. • In order to reduce computation time, an optimization algorithm should be used that does not require the calculation of sensitivities, which are the gradients (derivatives) of the objective function with respect to the design variables. “The gradient computation is known as a sensitivity analysis, and is typically the most computationally intensive component of design optimization” [Hambric 1995]. Therefore, a derivative-free optimization algorithm was selected. In fact, two different techniques, namely, a deterministic and a stochastic method, were applied to various optimization problems. • Almost all researchers used finite plate or shell elements to model their structures, the only exception (to the author’s knowledge) being Lumsdaine and Scott [Lumsdaine 1998]: “[P]revious optimization studies have not used continuum based finite elements to model the structure, as is done here.” However, even they only investigated simple symmetric beam and plate structures in the x-y plane and held the damping layer elements at constant height. Thus, they had only one design variable per element, and the elements were not distorted. For the present study it is assumed that the use of solid continuum elements facilitates the creation of realistic three-dimensional FE models, particularly of complex structures such as gearboxes, which can hardly be modeled solely by plate and shell elements. Furthermore, the elements are arbitrarily oriented in three-dimensional space, and the thickness can vary within an element. The latter has the advantage that no additional continuity conditions (constraints) have to be defined, which limit the maximum thickness difference between adjacent elements to ensure manufacturability as was done by Hibinger [Hibinger 1998]. A drawback is the more complicated calculation of the thickness because now an element does not have a constant thickness, but the thickness can vary from node to node.

1.3

Scope and Objectives of this Study

23

• In many of the former publications the symmetry of the FE models was exploited, i.e., only one half or even one quarter of the actual structure was modeled, thus significantly reducing computation time and the number of design variables. In the present work, however, no such simplifications are made. Though the structures investigated do show various symmetries, these cannot be exploited due to off-centered excitation forces. It is believed that these are more realistic and challenging test cases for real-life applications than simple symmetric models. • Most of the previous studies used a rather moderate number of design variables ranging from four to eight. Hardly ever was this number greater than ten. Marburg and Hardtke [Marburg 2002f] consider 44 design parameters a “large number” or, at another place in the same paper, even “a huge number of parameters”. In this dissertation, the smallest number of design variables is 77 for the rectangular plate (see Sec. 5.2). The half-cylinder has 165 design variables (see Sec. 5.4), the two plates joined at 90◦ have 220 (see Sec. 5.3), the rectangular plate with a refined discretization has 345 (see Sec. 5.2), and the gearbox has 821 (see Sec. 5.5). It is possible to reduce the number of design variables by means of spline functions (see Chap. 6). However, even then the models still have at least 35 design parameters. Therefore, the present optimization problems constitute challenging test cases for any optimization algorithm due to their great number of design parameters. • Many other researchers chose a relatively narrow frequency range of interest for their optimization calculations, which drastically reduces computation times. Extreme cases with a frequency bandwidth of 100 Hz or less include, for instance, [Keane 1995]: 150– 250 Hz, [McMillan 1997]: 100–110 Hz, [Moshrefi-Torbati 1998]: 55–65 Hz, [Ratle 1998]: 50–70 Hz (but also 0–600 Hz), [Marburg 2002b]: 0–100 Hz, or [Moshrefi-Torbati 2003]: 150–250 Hz. Examples for medium frequency bandwidths between 100 and 1000 Hz can be found in [Lamancusa 1988]: 50–250 Hz, [Lamancusa 1991]: 0–800 Hz and 0– 1000 Hz (but also 200–2000 Hz), [Inoue 1993]: 500 Hz–1500 Hz, [Lamancusa 1993]: 0– 1000 Hz, [Marburg 1997b]: 0–200 Hz, [Ratle 1998]: 0–600 Hz, [Marburg 2002f]: up to 200 Hz, [Fritze 2003]: 0–250 Hz (the authors call it a “[. . . ] relatively large frequency band [. . . ]”), or [Fuse 2003]: 50–300 Hz (ten lowest modes). For most of the optimization runs presented in this dissertation, a significantly wider frequency band of interest ranging from 0 to 3000 Hz is chosen, again constituting more challenging test cases. The only exception is the gearbox model as described in Sec. 5.5 for which the frequency range is restricted to 1000 Hz in order to avoid excessive computation time. A disadvantage of using such high frequencies is the fact that the FE models must be discretized by a very refined mesh so as to avoid “numerical pollution” errors caused by a too coarse discretization [Ihlenburg 1998]. Unfortunately, refined FE meshes lead to longer computation times. For a detailed discussion on this issue the reader is referred to Sec. 5.2.3. • Most of the previously published papers do not contain statements concerning the number of iterations required by the optimization algorithm to reach convergence or at least to achieve an improved design. Exceptions include, for instance, [Lamancusa 1993], [Hambric 1995], and [Hinton 1995b]. Some other researchers, e.g., [Pritchard 1987], [Grandhi 1992a], [Lamancusa 1993], [Moshrefi-Torbati 1998], [Tinnsten 1999], [Tinnsten 2000], [Kaneda 2002], or [Michot 2002], stated the number of iterations implicitly by provid-

24

1

Introduction

ing iteration history plots, i.e., line graphs showing the objective function or some other quantity vs. the number of iterations. Often, it is not clear if the reported number of iterations is the true number of objective function evaluations, i.e., including the number of function evaluations required for the sensitivity analysis, or just the much lower number of combined objective function and sensitivity calculation steps. For genetic algorithms, [McMillan 1997], [Bregant 2003], and [Li 2004] presented iteration history plots in terms of generations. This form of representation obscures the true number of design or function evaluations somewhat (one generation can stand for hundreds of function evaluations) and may therefore appear more advantageous. Bregant and Puzzi [Bregant 2003] did not even give the number of individuals per generation, thus making it impossible to assess the performance and efficiency of their approach. Likewise, the CPU time required for the optimization procedure was hardly ever reported in previous publications. [Lamancusa 1993] gave the total CPU time, and [Fleury 1986], [Schmit 1994], [McMillan 1996], [Conn 1998], and [Marburg 2002c] presented the required computation time per iteration. Keane [Keane 1995] reported, that “[. . . ] some of the runs discussed here take more than five days of computer time to carry out.” Marburg and Hardtke [Marburg 2002b] wrote: “[This technique] is only recommended, if several processes are run [in] parallel. In the present case, about 10 processes have been manually controlled and continuously run. [. . . ] We can hardly estimate how often the objective function has been computed in the entire optimization process. It is estimated that between 4000 and 5000 computations suffice to improve the objective function by about 1.5 dB. However, to achieve an additional half decibel, ten times more computations become necessary. Hence, we assume that 2 dB improvement require about 50,000 calculations of the objective function. Recalling that a single computation of the objective function takes about 2.5 min, it is estimated that the entire process will take about 12 weeks, a continuous run provided. Using the above described manually controlled parallel technique, this could be reduced to a 2 weeks lasting operation.” For the present study, all iteration histories and computation times were thoroughly recorded and reported in detail. Obviously, the procedures presented in this thesis are rather academic in nature and cannot readily be applied to real-life, industrial applications where time is at a premium. Nevertheless, the CPU time consumption can be assessed here, which was seldom the case in previous studies—probably because it would have looked too disastrous as the above example shows. • Some people claim that mathematical optimization calculations were not really necessary because solutions could be found faster and easier by applying traditional engineering expertise. Ratle and Berry [Ratle 1998] called this approach the “intuitive solution”. This might be true for simple optimization problems such as the minimization of vibrational levels or the minimization of structural mass. In most of these cases, the optimization algorithms indeed concentrate the material on and near the excitation point, thus increasing the input impedance, what can be considered the intuitive solution. However, Ratle and Berry [Ratle 1998] showed, that this intuitive solution most often is not the optimal solution because the frequency band of interest might still contain some response peaks. If the material is distributed in such a way that not only the peak levels, but also the number of peaks in the frequency band of interest is reduced by shifting natural frequencies out of that frequency range, an even higher reduction of the objective function can be achieved.

1.3

Scope and Objectives of this Study

25

It can be doubted that anybody can easily find an intuitive solution for less trivial optimization problems such as minimization of mass subject to an additional fundamental frequency constraint or the maximization of the difference between two arbitrarily chosen natural frequencies as presented in this work. Obviously, numerical optimization methods are able to find unconventional designs for such unconventional problems. However, trivial problems can be considered test cases for an optimization procedure, since they should yield designs that resemble the intuitive solution. • Quite often, particularly in industrial applications, a limited number of design variations that yield some improvement is called an optimization. Papalambros [Papalambros 1995] observed that “[. . . ] one will find that many such ‘optimized designs’ have little to do with locating local (certainly not global) minima. Rather, they refer to efforts at improving some measure of design performance by changing some design variables—often in a trial and error manner.” He also criticized the “[. . . ] nonrigorous and sometimes na¨ıve use of the term ‘optimization’ by many practicing design engineers [. . . ].” In this dissertation, mathematical optimization techniques in the true sense of the word “optimum—best” were employed to find the best solution of all possible solutions. Nonetheless, some restrictions apply. Although theoretically possible, it is very difficult in practice to determine the global optimum. Instead, numerical algorithms will most often find just a local optimum. However, since the computational effort to obtain a global optimum can be extremely high, it is considered sufficient to find a local optimum that yields a significant improvement compared to the initial design. Marburg et al. [Marburg 1997a] are of the opinion that “[, u]sually, achieving a decrease of 3 dB is considered successful.” In another paper [Marburg 2002f], Marburg and Hardtke admitted that “[. . . ] we can be sure that we will not find the [global] optimum. However, for technical requirements a significant improvement of the objective function in a certain period of time is more important than a long or almost infinite search for the global optimum.”

1.3.2

Issues that are not Objectives

The following issues are not objectives of the present research project: • The Numerical Structural Acoustic Optimization (NSAO) method introduced in the present study does not employ active noise or vibration control strategies, but is solely based on design modifications of passive structures. Keane [Keane 1995] stated that “[active vibration control measures] are inevitably expensive to install and maintain, and passive solutions would be preferable if they could be found.” Kaneda et al. [Kaneda 2002] expressed their opinion that “[. . . ] passive measures such as use of plate thickness, added masses, damping materials, and ribs or other stiffeners [. . . ] are still an attractive alternative [compared to active noise control] for reason of economy, simplicity, and stability.” • No experimental measurements were performed to verify the numerical results. The prime reason for that is simply lack of time. In the present case, performing experiments properly is considered quite costly in terms of both time and money. The optimized designs would have to be manufactured by means of some computer controlled milling technique, and an appropriate test stand would have to be designed and built.

26

1

Introduction

Since the correct modeling of the boundary conditions would be rather complicated, the measured results would have to be considered somewhat dubious. Neither a true simple support nor a true clamped edge condition can be realized easily in practice. Instead, some intermediate boundary condition would have to be created in the FE model of the structure in order to model the real boundary conditions, probably by using some sort of torsion spring elements as was done by Hibinger [Hibinger 1998]. Another problem is the correct modeling of damping as it is frequency dependent. It is relatively easy to determine the modal damping coefficients experimentally at the natural frequencies. However, it is rather complicated and time-consuming to ascertain the damping coefficients at other frequencies. For these reasons only numerical investigations but no experiments were conducted as is emphasized in the title of this thesis. However, in App. B an attempt to verify the numerical results by comparing them with results obtained in a previous study by Hibinger [Hibinger 1998] is presented. • Neither material strength nor manufacturability issues are considered. It should be checked if the optimized structures can withstand given static and dynamic loads. Of course, designs optimized with respect to certain vibrational or structural properties are worthless if they fail when subjected to operating conditions. Likewise, it should be investigated and ensured that an optimal design can be manufactured at reasonable cost. If it is not manufacturable at all or only at very high costs, it is impractical as well. • As mentioned before, only structure borne sound is considered but not air borne or radiated sound, e.g., the sound pressure level at some predefined location or the total sound power radiated from the structure (cf. Sec. 2.1). • It is not an objective to find optimal realistic designs, but rather to create design proposals that may be redesigned and adapted to real applications. The design proposals can also serve to gain some insight in the behavior of a structure. This approach is ¨ supported by a statement by Hinton, Ozak¸ ca, and Rao [Hinton 1995b]: “Although some of the optimal shapes of the structures obtained may look impractical, they serve as a guide to designing practical shapes and as an educational tool.” 1.3.3

Further Considerations

Since Hibinger’s Ph.D. dissertation [Hibinger 1998] can be regarded as the immediate predecessor of the present dissertation, the following list is intended to highlight some of the major improvements and differences between the two: • Hibinger used the revised simplex method of linear programming [Dantzig 1963] as the optimization algorithm. This method requires the calculation of sensitivities (gradients of the objective and constraint functions with respect to the design variables), which can be achieved by some finite difference scheme or by using analytical or semi-analytical approaches (see [Kibsgaard 1992]). In contrast to this, two different derivative-free optimization algorithms are employed in the present paper (see the item “sensitivities” on the list of objectives in Sec. 1.3.1 (p. 22) or the description of the two algorithms in Secs. 3.2 and 3.3). • Hibinger modeled the structures under investigation with combined shell/membrane elements, whereas hexahedral continuum elements are used in this study. For a discussion of

1.4

Outline

27

advantages and drawbacks see the item “elements” on the list of objectives in Sec. 1.3.1 (p. 22). • An approach to reduce the number of design variables (while still using a refined FE mesh) by means of spline functions is introduced in the present work. This technique is described in detail in Sec. 6. • From Hibinger’s thesis it is often not clear how many iterations were required to find the optimal designs. Even when the number of iterations is given, Hibinger did not specify if it includes the function evaluations required for the sensitivity analysis or not. In particular, neither iteration history plots nor CPU times were provided. For the present study, all iteration numbers and computation times were thoroughly recorded, and iteration numbers, iteration history plots, and CPU times are reported (cf. item “number of iterations/CPU time” on the list of objectives in Sec. 1.3.1, p. 23). • Hibinger optimized the thickness distribution and rib heights of two example structures, namely, a 90◦ joint of two rectangular plates (called “L-shaped plate joint” in [Hibinger 1998]) and a rectangular box. The optimization methods introduced in the present thesis were first tested on a simple rectangular plate (Sec. 5.2) and then applied to more complex three-dimensional structures such as two plates joined at 90◦ (Sec. 5.3) or a half-cylinder (Sec. 5.4). Finally, the optimization of a three-dimensional model of a gearbox (Sec. 5.5) can be considered the most challenging test case in this thesis. • Hibinger used a noncommercial, somewhat experimental FE code called FEAP (Finite Element Analysis Program [FEAP 2004]) and implemented the aforementioned combined shell/membrane elements as well as combined beam/truss elements himself. In the present study, however, the well-known commercial FE program ABAQUS [ABAQUS 2001a] with its standard FE element types is used. Furthermore, it should be mentioned that Hibinger’s results were summarized in a paper by Hibinger and B¨os [Hibinger 1999]. Likewise, some aspects and intermediate results of the present dissertation were published in various conference papers by B¨os and Nordmann [B¨os 2002a, B¨os 2002b, B¨os 2003b, B¨os 2003c, B¨os 2003d] as well as in a peer reviewed journal article by B¨os and Nordmann [B¨os 2003a].

1.4

Outline

The remainder of this dissertation is organized as follows: In Chap. 2 some theory on structural acoustics is provided. Section 2.1 introduces the fundamental equation of machine acoustics. The calculation of the level of structure borne sound is described in Sec. 2.2. Then, Sec. 2.3 shows how the mean level of structure borne sound is computed. Chapter 3 first gives a general overview on numerical optimization methods (Sec. 3.1) before it focuses on the two optimization algorithms used in this study, namely, the COBYLA algorithm (Sec. 3.2) and the PIKAIA algorithm (Sec. 3.3). A detailed description of the optimization procedure in general (Sec. 4.1) and using either of the two optimization techniques in particular (Secs. 4.2 and 4.3) is given in Chap. 4. Chapter 5 introduces the finite element models investigated, namely, a rectangular plate (Sec. 5.2), two plates joined at 90◦ (Sec. 5.3), a half-cylinder (Sec. 5.4), and a gearbox (Sec. 5.5).

28

1

Introduction

The spline functions used to reduce the number of design variables, i.e., bicubic and Hermite spline surfaces and tensor product B´ezier surfaces, are described in Chap. 6. Chapters 7 through 10 present various optimization results for the rectangular plate (Chap. 7), the two joined plates (Chap. 8), the half-cylinder (Chap. 9), and the gearbox (Chap. 10). Finally, Chap. 11 summarizes the findings of this research project and provides some conclusions as well as suggestions for future work. Appendix A explains some details of the implementation of the methods and procedures described in the preceding sections. Appendix B presents a verification the optimization results while App. C presents a visualization the nonlinearity and multimodality of the objective function.

2

Structural Acoustics

This chapter provides some theory on structural acoustics required for the optimization calculations presented in the subsequent chapters. First, the fundamental equation of machine acoustics is introduced in Sec. 2.1. Section 2.2 describes the calculation of the level of structure borne sound from the rms velocity vectors obtained from an FE analysis. Finally, Sec. 2.3 shows how the mean level of structure borne sound is determined.

2.1

Fundamental Equation of Machine Acoustics

A reliable and well-known measure for the noise emitted from some structure or machine part is the level of radiated sound power (or sound power level for short) LW (f ), which is a function of frequency f , P (f ) LW (f ) = 10 lg dB , (2.1) P0 where P (f ) is the radiated sound power and P0 = 10−12 W is a standardized reference value. The radiated sound power P (f ) in turn can be calculated by [Kollmann 2000] 2 P (f ) = ρa ca S v⊥rms (f ) σ(f ) .

(2.2)

Here, ρa and ca are the density and the speed of sound of the surrounding fluid (in this case 2 air), respectively, S is the area of the sound radiating surface, v⊥rms (f ) is the mean squared rms normal velocity of the surface averaged over the radiating surface, and σ(f ) is the radiation efficiency. The quantity Za = ρa ca is the so-called specific impedance of air. The rms value of some arbitrary quantity, e.g., x(t), that is periodic in time with some period T is defined by  xrms =

1 T





T

x2 (t)

dt ≈

0

1 Tmeas



Tmeas

x2 (t) dt ,

(2.3)

0

where Tmeas  T is the measurement time in practical applications. By introducing the so-called mean squared transmission admittance [Kollmann 2000] h2t (f ) =

2 v⊥rms (f ) , 2 (f ) Frms

where Frms (f ) is the rms excitation force, Eq. (2.2) can be rewritten as

(2.4)

30

2

Structural Acoustics

2 P (f ) = ρa ca σ(f ) S h2t (f ) Frms (f ) ,

(2.5)

which is considered the fundamental equation of machine acoustics [Kollmann 2000]. In level notation, Eq. (2.5) becomes LW (f ) = Lσ (f ) + LSh2t (f ) + LF (f ) ,

(2.6)

where Lσ (f ) is the level of radiation efficiency, LSh2t (f ) is the so-called level of structure borne sound (LS), and LF (f ) is the level of the excitation force. The level of the specific impedance ρa ca in Eq. (2.5) vanishes because the standardized reference value for the calculation of the level is equal to the specific impedance of air at normal temperature and air pressure (ρa ca )0 = ρa ca ≈ 414 N s/m. Since ρa ca and its reference value (ρa ca )0 are identical, the argument of the logarithm becomes unity and, thus, the logarithm itself becomes zero. From Eq. (2.6) it can be seen that the radiated sound power level LW (f ) can be interpreted as the sum of three contributing levels, namely, Lσ (f ), LSh2t (f ), and LF (f ). Hence, the radiated sound power can be reduced if all three contributing levels are reduced or if one or two of the contributing levels are decreased as long as the remaining ones do not overcompensate this by a drastic increase. Whereas measures to reduce the excitation forces (LF ) generally do not influence the vibrational (LSh2t ) or radiational behavior (Lσ ) of a structure, measures to lessen the vibrational level always do have an influence on the radiation efficiency and vice versa [Kollmann 2000]. There are cases where the vibrational and the radiational behavior exhibit contrary tendencies, i.e., a decrease of structure borne sound can sometimes lead to increased acoustic radiation and vice versa. However, it has been shown (e.g., [M¨ uller 1983]) that measures that influence the vibrational behavior, i.e., structure borne sound, in a positive way have the greatest potential to reduce radiated sound power levels [Kollmann 2000]. This is because there is an upper limit on the frequency dependent radiation efficiency σ(f ), namely, the radiation efficiency of a monopole radiator.1 In contrast, there is no equivalent lower bound on the level of structure borne sound, so it can be further reduced even when the radiation efficiency is at its upper limit. Therefore, only structure borne sound is considered in this thesis. This has the advantageous side-effect that fast, simple dynamic structural FE calculations suffice to compute the level of structure borne sound, whereas more complicated and quite time-consuming boundary element method (BEM) or infinite element method (IEM) computations would be necessary to determine the radiated sound power.

2.2

Level of Structure Borne Sound LS

The level of structure borne sound LSh2t (f ) in Eq. (2.6), which is from now on referred to as LS for convenience, can be interpreted as a measure of the vibrational sensitivity of a structure when subjected to some excitation. It can be calculated from [Kollmann 2000] LS = LSh2t (f ) = 10 lg

S h2t (f ) dB , S0 h2t0

(2.7)

where S0 h2t0 = 2.5 · 10−15 m4 /(N2 s2 ) is a standardized reference value. From Eq. (2.7) the LS can be interpreted as the level of the mean squared transmission admittance h2t (f ) in Eq. (2.4) 1 Theoretically, in certain cases the radiation efficiency of a rectangular plate at the coincidence frequency can exceed that of a monopole radiator by up to 5 dB. For practical applications, however, this is irrelevant.

2.2

31

Level of Structure Borne Sound LS

that is weighted with the surface area of the sound radiating surface S before taking the logarithm. In order to calculate the mean squared transmission admittance h2t (f ) according to Eq. (2.4), which is an ingredient for the LS in Eq. (2.7), the mean squared rms normal velocity of the 2 surface averaged over the radiating surface v⊥rms (f ) must be determined. The rms velocity vector v rmsi (f ) at some point i on the sound radiating surface can be obtained either from an experimental measurement using accelerometers or a laser vibrometer, or numerically from a dynamic FE analysis. Either way, the component of the rms velocity vector v rmsi (f ) that is normal to the surface v⊥rms i (f ) is then calculated by v⊥rms i (f ) = v rmsi (f ) · ni ,

(2.8)

where ni is the unit normal vector on that particular surface point i (see Sec. A.1.3 in App. A for details). Once the rms normal surface velocity distribution at the nodal points of some measurement mesh on the structure’s surface or of some FE discretization is known, the mean squared rms normal surface velocity averaged over the radiating surface 2 v⊥rms (f ) =

nn 1  v2 (f ) , nn i=1 ⊥rms i

(2.9)

where nn is the number of measurement points or FE nodes on the surface of the structure, can then be calculated, provided that the measurement or FE mesh is rather uniform. Thus, the LS in Eq. (2.7) can be determined from the nodal rms surface velocity vectors v rmsi (f ) by combining Eqs. (2.8), (2.9), and (2.4). A detailed description of the experimental measurements or of the FEM used to obtain the nodal rms surface velocity vectors v rmsi (f ) is beyond the scope of this work and can be found elsewhere in the literature (e.g., [Bathe 1996,Bathe 2002,Beranek 1993,Bies 2003,Cremer 1988,Cremer 1996,Harris 1998,Kollmann 2000,Zienkiewicz 2000]). It is worth noting, however, that in the present dissertation the velocity vectors v rmsi (f ) are solely calculated by means of the FEM using a modal superposition technique, which is admissible for lightly damped structures (see item “material” in Sec. 5.1). This means that each FE analysis consists of two consecutive steps: The first step is a numerical modal analysis, which determines the natural frequencies and the mode shapes of the structure. This step is called “natural frequency extraction” in the FE software ABAQUS [ABAQUS 2001a]. In the second step, called “modebased steady-state dynamic analysis” in ABAQUS, the velocity vectors v rmsi (f ) are obtained by superposition of the previously computed mode shapes using scalar mode participation factors [Zienkiewicz 2000, ABAQUS 2001b]. The mode participation factors can be regarded as weighting factors, which show the proportions of each mode occurring [Zienkiewicz 2000]. The numerical modal analysis in the first step ensures that all natural frequencies and mode shapes in the frequency band of interest are automatically taken into account and none can be missed when the vibrational response of the structure subject to some force excitation is calculated in the second step. Previous studies used constant frequency increments of 10 Hz [Inoue 1993, Kaneda 2002] or 2 Hz [Marburg 1997a], which contains the risk of missing some resonance peaks and requires long computation times due to the refined frequency resolution. The technique employed in the present work is explained in more detail in App. A, Sec. A.4.

32

2.3

2

Structural Acoustics

Mean Level of Structure Borne Sound MLS

The LS in Eq. (2.7) constitutes a spectrum, i.e., it is a function of frequency f . To obtain some single global measure of the vibrational behavior of a structure in a given frequency range of interest, the mean level of structure borne sound averaged over that frequency band LSh2t , known hereafter as MLS,  fmax [S h2t (f )/S0 h2t0 ] df f MLS = LSh2t = 10 lg min dB (2.10) fmax − fmin is calculated [Zopp 2000], where fmin and fmax are the lower and upper bounds of the frequency range under consideration, respectively. The MLS is the area beneath the LS spectrum (see, for instance, Fig. 7.6 on p. 84) divided by the width of the frequency band and can be computed numerically by means of, e.g., the trapezium integration rule [Zopp 2000]. It can be considered a quantity characterizing the vibrational energy contained in the given frequency range and can serve as objective function or constraint in the optimization problems presented in this dissertation. In the chapters describing the optimization results, i.e., Chaps. 7 through 10, the reduction of the MLS or of the LS is often additionally given as a percentage. These percentage changes refer to absolute values, not to levels given in dB. If the MLS or LS is reduced from, say, X dB to Y dB then the percentage difference is given by ∆% =

10Y /10 − 10X/10 · 100% . 10X/10

(2.11)

This percentage difference can also be expressed directly in terms of the MLS or LS reduction ∆L = Y − X, namely,   ∆% = 10∆L /10 − 1 · 100% . (2.12)

3

Optimization Algorithms

First, this chapter describes in Sec. 3.1 a general optimization problem as well as various possible classification categories to which optimization algorithms can be assigned. Then, the following two sections introduce and describe the two optimization algorithms used in this study, namely, the deterministic COBYLA algorithm (Sec. 3.2) and the stochastic PIKAIA algorithm (Sec. 3.3), and provide some rationale for preferring these two to other numerical optimization techniques.

3.1

General Aspects of Numerical Optimization Methods

Optimization is defined as the minimization or maximization of a function subject to constraints on its variables [Nocedal 1999]. Usually, the following notation is applied: • The variables xi (i = 1, 2, . . . , n; i, n ∈ N), which are varied in order to find the minimum or maximum, are the design variables, also called design parameters. These n design parameters form the vector of design variables x. • The function of the vector of design variables f (x) that is to be minimized or maximized is called the objective function or cost function. • The individual restrictions cj (x) (j = 1, 2, . . . , m; j, m ∈ N) that are placed on the variables x are called the constraints, which form the vector of constraints c(x). The set of points x that satisfies all of the m constraints cj (x) is called the feasible region, and any vector of design variables x that lies within the feasible region is called a feasible design. A general optimization problem can then be formulated as [Nocedal 1999] minimize f (x), x ∈ Rn , ceq (x) = 0 subject to cineq (x) ≥ 0 ,

(3.1a) (3.1b)

where ceq (x) are equality constraints and cineq (x) are inequality constraints. The objective function f (x) can be maximized by minimizing −f (x). Likewise, inequality constraints of the form cineq (x) ≤ 0 can be written as −cineq (x) ≥ 0. Optimization methods can be categorized by various criteria. The following list is certainly not complete, but it gives an impression of the numerous existing categories.

34

3

Optimization Algorithms

• deterministic vs. stochastic: Deterministic methods create new designs or trial solutions based entirely on the results and the success of previous iterations by interpolation, extrapolation, calculation of gradients, etc. Stochastic optimization techniques involve at least some degree of randomness and probability, ranging from a minor or moderate influence of heuristic procedures on parameters controlling the optimization process to completely random searches.

f(x)

• local vs. global : Local optimization algorithms may get trapped in a local optimum and hence fail to find the global optimum of an optimization problem. Remedies against this shortcoming include skillfully choosing a suitable initial design that is close to the assumed or actual global optimum, or restarting the optimization procedure several times in succession with various different initial designs. If most or all of the successive runs result in the same optimum it can be assumed that the global optimum was found. Global optimization procedures are designed always to find the global optimum. They contain mechanisms and strategies that enable them to escape local optima and explore other regions of the design space. Most of the global optimization methods (at least all of the methods known to the author) are stochastic methods. Figure 3.1 illustrates the concept of local and global extrema by means of some arbitrary function f (x) of just one variable x. In this example, the global minimum is located right at the upper limit of the feasible design space, whereas the global maximum is situated somewhere in the middle of the feasible interval.

local minimum global minimum local maximum global maximum plateau 0

0

x

Figure 3.1: Examples of local and global extrema of some arbitrary function f (x). • unconstrained vs. constrained : Unconstrained optimization problems are of rather theoretical and mathematical nature. Unconstrained means that there are no restrictions at all to the design parameters or to the objective function values—they can take any value they want to. Most real-life optimization problems are constrained meaning that one or more restrictions apply. For instance, the cross-sectional area of a beam cannot become negative, some physical properties such as mass or density are always nonnegative, the total cost of a product shall not exceed a certain limit, etc.

3.1

General Aspects of Numerical Optimization Methods

35

• linear vs. nonlinear : If the objective function and all of the constraints are linear functions of the design parameters, the optimization problem is said to be a linear one. Numerous specialized and highly efficient algorithms exist for this particular class of problems. General nonlinear optimization problems (in which some or all objective or constraint functions may be nonlinear functions of the design variables) are somewhat more difficult to solve and require some special nonlinear optimization techniques. A special subcase of the nonlinear class is the quadratic optimization problem for which specialized strategies are available as well. • continuous vs. discrete: Continuous optimization problems are characterized by design variables and objective function values that can take any real number that does not violate any constraints. Discrete optimization involves design variables and objective function values that are integer or even natural numbers. Examples for discrete optimization include standardized design variables (e.g., standardized cross-sectional dimensions of steel girders or bolts), the number of parts produced (it is not sensible to take half light bulbs or half cars into account), or the well-known traveling salesman problem, i.e., finding the shortest route to visit a given number of cities and finally returning to the starting point without visiting any city more than once. • analytical vs. numerical : Kirsch [Kirsch 1993] distinguishes between analytical and numerical optimization. Analytical optimization strategies employ the mathematical theory of calculus, variational methods, etc. The optimum solution is theoretically found exactly by solving a system of equations that express the optimality conditions. With numerical procedures a near optimal solution is automatically generated in an iterative manner. An initial guess serves as a starting point for a systematic search for better designs that is terminated when predefined criteria are satisfied. • shape vs. topology: Particularly in structural optimization, one can differentiate between shape and topology optimization [Baier 1994, Bendsøe 1995]. Shape optimization techniques can only modify the outer shape of a structure (thickness, radiuses, curvatures, etc.) but cannot remove any material from the inside of the structure (wholes, branches, struts, etc.). Topology optimization “thins out” and finally removes material on the inside or at the edge of a structure at places where it is not really needed in order to withstand a given load (low stress regions). The result often looks like a truss or framework. Based on this classification scheme of optimization techniques, both algorithms used in the present work can be categorized as constrained, nonlinear, continuous, and numerical, and are used to perform shape optimization. However, one of them, namely, COBYLA, is deterministic and local, whereas the other one, i.e., PIKAIA, is stochastic and global. Both methods are described in detail now.

36

3

3.2

Optimization Algorithms

COBYLA Algorithm

The first optimization algorithm that is used in this study is the COBYLA algorithm by M. J. D. Powell [Powell 1994, Powell 1998]. This section explains the motivation for using COBYLA, introduces the underlying simplex algorithm by Nelder and Mead [Nelder 1965], and finally describes the COBYLA algorithm itself. 3.2.1

Motivation for Using COBYLA

As mentioned before, most of the available optimization algorithms focus on certain types of optimization tasks. Some algorithms are well suited for unconstrained optimization, whereas others are designed for constrained problems. Some methods require the objective function and the constraints to be linear functions of the design variables, others can handle quadratic or generally nonlinear objective functions and constraints. One can also distinguish between local optimization algorithms, which might get trapped in a local extremum, and global techniques, which always find the global extremum. The choice of a particular optimization algorithm was guided by the following considerations: 1. Calculating the LS of a structure is a linear elasto-dynamic problem. However, optimizing a structure with respect to the LS is a nonlinear optimization problem. The same applies to the optimization of structures with respect to mass or to one or several natural frequencies. In any case it is not possible to derive analytical equations that relate a change in the design variables, i.e., the local thickness values, to a change in the objective function. Thus, it is not possible to tell whether the objective function is linear, quadratic, or convex or if it has any other feature making it suitable for a particular type of algorithm—which implies that there is no obvious choice for a suitable optimization algorithm, so a general nonlinear algorithm should be used. 2. It is useful to employ some constraints such as minimum and maximum thickness, maximum mass, maximum LS, minimum fundamental frequency, etc.—hence the need for an algorithm capable of solving constrained optimization problems. 3. The author wanted to avoid algorithms that require the calculation of the first or especially the second derivatives of the objective function with respect to the design variables. The computation of the derivatives takes a lot of time, because the derivative of the objective function with respect to each individual design variable must be calculated for each iteration by means of some finite difference scheme or by some analytical or semianalytical approach. Since the optimization problems described in this thesis involve dozens, even hundreds of design parameters, gradient-based approaches did not seem to be very promising. When confronted with a description of the problem and of the obvious difficulties, Prof. Spellucci suggested trying out the COBYLA algorithm.2 This advice proved to be invaluable, since the algorithm performed very well as the numerical results in Chaps. 7, 8, 9, and 10 show. 2 personal conversation with Prof. Dr. P. Spellucci from the research group Numerical Analysis and Scientific Computing, Department of Mathematics, Darmstadt University of Technology, Germany

3.2

COBYLA Algorithm

3.2.2

37

The Simplex Algorithm

Since the COBYLA algorithm is based on the simplex algorithm by Nelder and Mead [Nelder 1965], it seems sensible to present the underlying principles of the simplex algorithm first. Introductory descriptions of this algorithm can also be found, for instance, in [Press 1992] and [Schwefel 1977]. The simplex algorithm by Nelder and Mead [Nelder 1965] requires only function values but no derivatives. It should not be confused with the simplex method of linear programming [Dantzig 1963] as described, e.g., in Chap. 13 of [Nocedal 1999] or in Sec. 10.8 of [Press 1992]. A simplex is a geometrical figure that consists of n + 1 points (also called vertices) in n dimensions and all the line segments and polygonal faces that connect the vertices. Thus, a simplex is a triangle in two dimensions, a tetrahedron in three dimensions, and a general polyhedron in more than three dimensions. A simplex is called nondegenerate if it encloses a finite inner n-dimensional volume. The initial simplex is defined by n + 1 vertices each of which is a vector of design variables x of length n. If any of the vertices is considered the origin, then the other n vertices can be considered vector directions that span the n-dimensional vector space. Therefore, an initial simplex can be formed by first defining just one initial vertex x(0) (start vector) and by then determining the other n vertices of the simplex x(j) (j = 1, 2, . . . , n) by using, e.g., Eq. (3.9) (see Fig. 3.4 on p. 41).   The algorithm computes the objective function value F x(i) at all vertices x(i) (i = 0, 1, 2, . . . , n). Then, the vertex with the worst (i.e., largest in case of function minimization) objective function is replaced by a new one that results from reflection at the mean value of the remaining n vertices. Furthermore, expansion and contraction can occur (depending on the success of the reflection) to adapt the shape of the simplex in order to maintain its nondegeneracy. These steps are repeated iteratively until some termination criterion is met. In the original version of the simplex algorithm [Nelder 1965] the standard deviation of the objective function values at the vertices of the simplex is compared with some preset value (10−8 in [Nelder 1965]). The iterations are stopped if the standard deviation falls below this predefined value. Figure 3.2 in connection with Tab. 3.1 shows a very simple, completely hypothetical example of some iterations of the simplex algorithm. In two dimensions, a simplex is just a triangle, which can easily be displayed on paper. The first point (filled circle) in the upper left corner of Fig. 3.2 is the starting point of the iterations. The initial simplex is marked by thick lines connecting vertices 1, 2, and 3. A sequence of reflection, expansion, and contraction steps as described in the accompanying Tab. 3.1 creates new vertices (marked by circles) and simplices until the final simplex consisting of points 11, 14, and 15 (again marked by thick lines) is found. The conditions under which expansion or contraction steps can occur are not explained in detail here—they can be found in [Nelder 1965] or [Schwefel 1977]. However, from Tab. 3.1 it can be seen that these steps depend on the success of the previous reflection step, i.e., whether the previous reflection step produced a new best vertex, just a mediocre vertex, or even a new worst vertex. 3.2.3

Description of the COBYLA Algorithm

The COBYLA algorithm (Constrained Optimization BY Linear Approximations) by M. J. D. Powell [Powell 1994, Powell 1998] is a sequential trust-region optimization algorithm based on the simplex algorithm [Nelder 1965] described in the previous section. It is able to solve nonlin-

38

3

Optimization Algorithms

Figure 3.2: Some hypothetical steps of the simplex algorithm (after [Schwefel 1977]). ear, constrained optimization problems, it is purely deterministic (i.e., it does not contain any probabilistic or stochastic elements), and it does not require the calculation of any derivatives (derivative-free method). Instead, gradients are estimated internally by linear interpolation of the objective function and of the constraints. The algorithm tries to maintain a regular-shaped simplex over the iterations. A sequence of iterations is performed with a constant trust-region radius ρ, which is described on p. 41, until the computed objective function reduction is much less than a predicted reduction. Then, the trust-region radius ρ is reduced. This procedure is explained in more detail now, based on [Powell 1994]. Let F (x) be the objective function, which is a function of a vector of design variables x ∈ Rn . The minimization of this objective function subject to m constraints ci (x) ≥ 0 can be expressed in the general form (cf. Eqs. 3.1a and 3.1b) minimize F (x), x ∈ Rn ,

subject to ci (x) ≥ 0, i = 1, 2, . . . , m .

(3.2)

It is assumed that the objective and constraint functions can be calculated for every vector of design parameters x, but there are no smoothness assumptions. The actual objective function F (x) is substituted by a so-called merit function [Powell 1994] Φ(x) = F (x) + µ [max {−ci (x) : i = 1, 2, . . . , m}]+ , x ∈ Rn ,

(3.3)

where µ is a parameter that is adjusted automatically by the algorithm. The subscript + denotes the expression in square brackets (the so-called constraint violation function) is replaced by zero

3.2

39

COBYLA Algorithm

Table 3.1: Some hypothetical steps of the simplex algorithm (after [Schwefel 1977]). Vertices Iteration

Worst

Medium

Best

0

1

2

3

New vertex created by. . . 4 5

1

2

3

5 6

2

3

6

reflection 5 7

8 3

6

5

5

9

reflection 7

10 5

9

reflection 11 partial contraction (outside) 7

11

12 13 15 6

15

reflection expansion (unsuccessful)

7 9

4

(initial simplex) reflection expansion (successful)

14 14

11 11

reflection partial contraction (inside) total contraction (final simplex)

if its value is negative, i.e., if there is not even the slightest constraint violation. Therefore, Φ(x) = F (x) holds whenever x is a feasible vector of design variables. However, if there are one or more constraint violations (ci (x) ≤ 0 for one or some i), the greatest constraint violation is added to the objective function F (x) as a penalty because it is assumed that several constraint violations of, say, 10−6 are usually preferable to a single constraint violation of 10−3 . Hence, this is an elegant way to incorporate the constraints into the objective function. It is recommended that the various constraints be of similar order of magnitude, which can be achieved by appropriate scaling of the constraints. It is assumed that x ∈ Rn is better than y ∈ Rn if and only if the inequality Φ(x) < Φ(y) holds. The next vector of design variables is generated from function values at the vertices x(j) (j = 0, 1, . . . , n) of a nondegenerate simplex in Rn according to Eq. (3.9) or Eq. (3.10). In this case there are unique linear functions Fˆ and cˆi that interpolate the objective and constraint functions at the vertices. Thus, Eq. (3.2) can be approximated by the linear programming problem minimize Fˆ (x), x ∈ Rn ,

subject to cˆi (x) ≥ 0, i = 1, 2, . . . , m ,

(3.4)

which leads to a linearized version of Eq. (3.3), namely, ˆ Φ(x) = Fˆ (x) + µ [max {−ˆ ci (x) : i = 1, 2, . . . , m}]+ , x ∈ Rn .

(3.5)

40

3

Optimization Algorithms

Let g(x) be either the objective function F (x) or any of the constraint functions ci (x) and gˆ(x) its linear approximation. The linear approximation gˆ(x(k) ) at the vertex k (k = 1, 2, . . . , n) is given by n

  (k) (0) g(x(k) ) ≈ gˆ(x(k) ) = g(x(0) ) + aj xj − x j . (3.6) j=1 (k)

The vector a must be determined such that gˆ(x ) = g(x(k) ) (k = 1, 2, . . . , n since the condition is automatically satisfied for x = x(0) ), which is done by solving the system of n linear equations 

    



 (1) (0) (1) (0) x1 − x 1 x2 − x 2   

 

g(x(1) ) − g(x(0) ) (2) (0) (2) (0)  x1 − x 1 x2 − x 2 (2) (0)  g(x ) − g(x )    = .. .. ..   . . .  (n) (0)  g(x ) − g(x ) 



 (n) (0) (n) (0) x2 − x 2 x1 − x1



···



··· ..

(1)

(0)

(2) xn

(0) xn

xn − xn − .. .

.

···

(n)

(0)

xn − x n

             

a1 a2 .. .

    . 

an

(3.7) As the volume of the simplex VS shrinks, the vector a converges towards the gradient of the function g(x) at x(0) , i.e., lim a = ∇g(x)|x(0) . (3.8) Vs →0

Figure 3.3 illustrates an example for a simplex in two dimensions (n = 2), i.e., a triangle in the x1 -x2 plane with the vertices x(0) , x(p) , and x(q) . The curved gray surface represents the objective function F (x). The plane triangle with the objective function values F (x(0) ), F (x(p) ), and F (x(q) ) being its vertices is the linear approximation of the objective function. The same is done for each of the constraint functions ci (x).

F ( x(q))

F ( x(p)) F ( x(0))

x(q) x(p)

x2 x(0) x1

Figure 3.3: Linear approximation of the objective function F (x).

3.2

41

COBYLA Algorithm

On most iterations, the changes to the variables are determined by Eqs. (3.4) and (3.5). Some iterations, however, just modify the shape of the simplex so that the linear interpolation of the objective and constraint functions at its vertices is still acceptable. Changes to the variables are restricted by a trust-region radius ρ (see Fig. 3.4). It remains constant until the computed reduction of the merit function Φ(x) in Eq. (3.3) is much smaller than the reduction ˆ predicted by the linearized merit function Φ(x) in Eq. (3.5) although the simplex has a good shape. If that happens, the trust-region radius ρ is reduced until it reaches some user-defined final value ρend . The user must provide an initial vector of design variables x(0) (which, in general, should not be the zero vector) and an initial value ρbeg for the trust-region radius ρ. Initially, i.e., during the first n + 1 iterations, the parameter µ in Eq. (3.3) is set to zero, yielding Φ(x) = F (x). For each of the indices j = 1, 2, . . . , n a new set of variables x(j) = x(0) + ρbeg ej

(3.9)  is calculated, where ej is the jth coordinate vector. If the condition F x < F x is satisfied, x(j) is exchanged with x(0) before proceeding to the next value of j. This way x(0) becomes the optimal vertex of the initial simplex. Figure 3.4 shows a two-dimensional example of the generation of two new vertices x(p) and x(q) . During the first n+1 iterations the new vertices are generated according to Eq. (3.9). After that, the step to a new vertex can occur in any direction. The step length is restricted by the trust-region radius ρ in order to ensure that Fˆ (x) and cˆi (x) are still acceptable approximations of the objective function F (x) and the constraints ci (x), respectively. If this is not the case, the trust-region radius ρ is reduced (see Eq. (3.13) and its explanation). 

(j)





(0)

x(q)

ρ

x(0)

x(p)

x2 x1

Figure 3.4: Generation of two new vertices x(p) and x(q) . The next step is to find a new vector of design parameters x(∗) that satisfies the condition  (∗)  x − x(0)  ≤ ρ . (3.10) 2  (∗)  This is done by trying to minimize the linear approximation Fˆ x subject to the inequality (3.10) and to the linear constraints   cˆi x(∗) ≥ 0, i = 1, 2, . . . , m . (3.11) If there is more than one solution, the algorithm picks the x(∗) that gives the least value of   x(∗) − x(0)  . 2 If the inequalities (3.10) and (3.11) x(∗) is   defined by minimizing the  are contradictory, greatest of the constraint violations −ˆ ci x(∗) : i = 1, 2, . . . , m subject to the trust-region

42

3

Optimization Algorithms

  bound ρ. If there is  any freedom in x(∗) left, Fˆ x(∗) is minimized and, if some freedom still (∗) (0) (∗) remains, x − x 2 is minimized. x is calculated by increasing ρ continuously from zero to the current value. As mentioned earlier, the parameter µ in Eqs. (3.3) and (3.5) is initially, i.e., during the first n + 1 iterations, set to zero, but is adjusted  automatically    afterwards. It is assumed that it is unreasonable to expect the reduction Φ x(∗) < Φ x(0) in the merit function (3.3) if  (∗)   (0)  ˆ x ˆ x the condition Φ 32 µ ¯, but otherwise µ is increased to 2¯ µ. The trust-region radius ρ should be kept constant until the reductions in the merit function are no longer satisfactory, after which ρ is decreased. However, before decreasing ρ, the current simplex must be “acceptable” in order to avoid a very poor linear approximation (3.4) to the main calculation (3.2). Therefore, ρ is reduced if the simplex is acceptable and if either one of the conditions  (∗)  x − x(0)  ≤ 1 ρ (3.12) 2 2 or          ˆ x(0) − Φ ˆ x(∗) (3.13) Φ x(0) − Φ x(∗) < 0.1 Φ holds. The latter case means that changing the vector of variables from x(0) to x(∗) does not even yield 10% of the improvement in the merit function (3.3) that is predicted by its linear approximation (3.5). As mentioned above, the user has to provide the initial and final values of ρ, namely, ρbeg and ρend . Whereas ρbeg should be large enough to allow a coarse exploration of the design space, ρend should be approximately the distance from the final vector of design variables to the solution of the optimization problem. If the conditions for reducing ρ described above are satisfied, the algorithm proceeds as follows. If ρ ≤ ρend , then the iterative procedure is terminated. The  (∗) (∗) is available but x is preferred instead if Φ x current x(0) is the final vectorof variables,    and satisfies the condition Φ x(∗) < Φ x(0) . If, however, ρ > ρend , the new trust-region radius ρnew is determined from 1 ρ, ρ > 3ρend 2 ρnew = (3.14) ρend , ρ ≤ 3ρend . A detailed definition and thorough discussion of the “acceptability” of a simplex is provided in [Powell 1994]. It can be summarized as follows. A simplex is assumed to be acceptable if the lengths of its edges and its volume are of magnitudes ρ and ρn , respectively, so they are appropriate to the current trust-region radius ρ. Powell’s paper [Powell 1994] contains many further details on the implementation of the algorithm and on the setting and automatic adjustment of some more parameters besides ρ and µ. These details are omitted here for the sake of clarity and brevity. It should be noted, however, that, according to the abstract of [Powell 1994], “the algorithm is easy to use for small numbers of variables” and that “there are no more than 9 variables” in the numerical examples provided in the paper. In contrast, up to 821 design variables were used in the present study—and COBYLA was still able to find reasonable reductions in the objective functions. The Fortran source code of the COBYLA algorithm can be obtained from [COBYLA 1994] at no charge.

3.3

PIKAIA Algorithm

3.3

43

PIKAIA Algorithm

A completely different approach to solving optimization problems like the ones presented here is the use of a so-called genetic algorithm (GA). The PIKAIA algorithm by P. Charbonneau [Charbonneau 1995b, Charbonneau 2002b] is a specific implementation of such a GA. The reasons for trying out an alternative optimization algorithm, the general concepts of a basic GA, and the characteristics of the PIKAIA algorithm are described in the following sections. 3.3.1

Motivation for Using PIKAIA

After working with Powell’s COBYLA algorithm described in Sec. 3.2.3 for quite some time and using it quite successfully (see the optimization results in Chaps. 7, 8, 9, and 10), some drawbacks became evident despite the great performance: 1. The COBYLA algorithm takes n+1 iterations (where n is the number of design variables) to form the initial simplex in n dimensions as can be seen from the iteration history plots (e.g., Fig. 7.17 on p. 93). This process could be considered to be a kind of “orientation phase”, in which the design space is coarsely explored. The “real” optimization process starts only after these n + 1 iterations, i.e., not until then does COBYLA really try to reduce the objective function value. The first n + 1 iterations are therefore “wasted” in the sense that they often do not result in a reduction of the objective function value. Thus, it seemed to be sensible to try out another optimization algorithm that tries to minimize the objective function from the very start of the iterations. Maybe it would be able to gain some ground during the first n + 1 iterations while COBYLA is still busy shaping the initial simplex. 2. The COBYLA algorithm is just a local optimization algorithm. This means that it can get stuck in a local optimum and hence fail to find the global one. A remedy for this could be to restart COBYLA several times in succession, each time with a different start vector of design variables, and to compare the results: If many or most of the optimization runs result in the same solution (i.e., design) despite the various start vectors, then this solution can be considered the global optimum. However, it is impractical to restart the optimization procedure several times in a row since one single optimization run takes quite some time. Therefore, the use of a global optimization algorithm seemed to be worthwhile. 3. The COBYLA algorithm is sequential: Each new set or vector of design variables depends on the results and the success of the previous iteration(s). It is not possible to compute the objective function values of several different designs simultaneously on several computers. Hence, in order to reduce overall computation time, the use of another optimization algorithm that allows parallel computations on several computers seemed to be advantageous. 4. The COBYLA algorithm is a deterministic one, i.e., it is not stochastic. Thus, a comparison with another algorithm that involves some degree of randomness seemed to be interesting. This is particularly important in connection with the use of global optimization algorithms (see 2.) since basically all of these require and feature stochastic procedures.

44

3

Optimization Algorithms

5. Finally, it just seemed to be interesting to try and test an alternative optimization algorithm other than COBYLA to see how it performs in comparison and if it offers any advantages or even superiority over COBYLA. In particular, the concept of a genetic algorithm (GA) with its simulation of biological processes such as breeding, crossing, and mutation seemed to be fascinating and worth a try. Initially, it was considered to use an algorithm based on the simulated annealing (SA) method [Corana 1987,Kirkpatrick 1983], namely, an SA implementation called simann [SIMANN 1997] used in [Goffe 1994]. However, according to the literature [Corana 1987,Goffe 1994], SA should only be used if the objective function evaluation is computationally inexpensive (i.e., very fast). This is due to the fact that it requires numerous function evaluations to reduce some fictitious temperature T just once (see the description of the SA method in the literature review on pp. 13 and 14). The temperature T is reduced repeatedly after Ns · NT · n iterations, where Ns is the number of cycles, NT is the number of iterations before temperature reduction, and n is the number of design variables. Corana et al. [Corana 1987] suggest Ns = 20 and NT = max(100, 5n). This leads to 122 500 iterations for the smallest optimization problem presented in this study (i.e., the rectangular plate using 35 Hermite spline key points as design variables, see Fig. 6.3 on p. 76) and up to 67 404 100 iterations for the largest optimization problem (i.e., the gearbox with 821 design variables, see Fig. 5.11 on p. 71). These numbers apply to just one reduction of the temperature T , which is reduced repeatedly until some convergence criterion is met. If one function evaluation (i.e., an FE analysis combined with the evaluation of its results) takes 8 seconds to 5 minutes of CPU time (depending on the model size and the computer used), it would take more than 11 days of CPU time for each reduction of T in the former case (rectangular plate) and more than 641 years (!) of CPU time in the latter (gearbox). It is not difficult to imagine how long the whole optimization procedure would take to finish if T is to be reduced repeatedly until the stop criterion is met. Coincidentally, Corana et al. [Corana 1987] measure the performance of their implementation of the SA algorithm against the performance of the simplex algorithm on which the COBYLA algorithm is based (see Sec. 3.2.2). They use only two-dimensional and four-dimensional test problems (two of them being some classical test functions taken from [Rosenbrock 1960]) involving just two and four design variables, respectively. Nonetheless, the authors admit that the simplex algorithm is more efficient in terms of the number of function evaluations: “On average, the SA takes from 500 to 1000 times more function evaluations than the simplex method” [Corana 1987]. The outcome of the comparison would probably be worse with significantly more than four design variables as is the case in this work. Therefore, the author decided to try an evolutionary or genetic algorithm. The two methods are very similar and differ basically just in one aspect: Whereas the design variable values are used directly as “genes” in an evolutionary algorithm, they are coded in “chromosomes” in genetic algorithms. An internet search revealed that there are numerous realizations available for both types of algorithms [Mittelmann 2004]. From the various options P. Charbonneau’s PIKAIA algorithm [Charbonneau 1995a, Charbonneau 2002a] was chosen since it is very well documented [Charbonneau 1995b, Charbonneau 2002b, PIKAIA 2004] and its Fortran source code is available for free [PIKAIA 2002].

3.3

PIKAIA Algorithm

3.3.2

45

Basic Concepts of a General Genetic Algorithm

A very compact description of the basic concepts of a GA can be found in [Charbonneau 2002a]. The following is an even more condensed excerpt from that reference. Suppose there is a set (or vector) of design parameters x and a model that relates the parameters x to some measure of quality or fitness f (x) for a particular design, i.e., for a particular vector of design variables x. Now the optimization algorithm has to find the one set of variables x∗ that maximizes the fitness function f (x). A population can be defined as a set of Np realizations (also called individuals or members of the population) of the set of parameters x. A basic GA works then as follows [Charbonneau 2002a]: 1. randomly initialize population and evaluate fitness of its members, 2. breed selected members of current population to produce offspring population (selection based on fitness), 3. replace current population by offspring population, 4. evaluate fitness of new population members, 5. repeat steps 2. through 4. until the fittest member of the current population is deemed fit enough. This concept is illustrated by a very simple example that could be part of a two-dimensional maximization problem in which a function f (x, y) is to be maximized (after [Charbonneau 2002a]): • The “phenotypes” P of the two “parent” individuals P1 and P2 of the population are P(P1): x1 = (x1 , y1 ) and P(P2): x2 = (x2 , y2 ). • x1 and x2 are defined by floating point numbers, for example, P(P1): x1 = 0.1234 and y1 = 0.9876; P(P2): x2 = 0.5678 and y2 = 0.5432 • The “encoding” of the floating point numbers into “chromosomes” is done simply by dropping the decimal point and concatenating the resulting set of integer numbers into “genotypes” G of the parents P1 and P2: G(P1) = 1234 9876 = 12349876, G(P2) = 5678 5432 = 56785432 • The “breeding” process consists of two steps, namely, “crossover”and “mutation”. • For the crossover a cutting point is randomly selected at which the two genotypes are split and the fragments are interchanged. In this example the cutting point is between “genes” 3 and 4, resulting in offspring individuals O1 and O2: G(P1) = 123|49876, G(P2) = 567|85432 ; G(O1) = 123|85432, G(O2) = 567|49876 • The mutation is done by randomly selecting none, one, or several genes of the chromosomes and replacing them by a randomly chosen digit. Here, mutation is applied to gene 6 of individual O1 (replacing 4 by 0), whereas individual O2 remains unchanged: G(O1) = 12385432, G(O2) = 56749876 ; G(O1) = 12385032, G(O2) = 56749876

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Optimization Algorithms

• The resulting genotypes G of the offspring individuals O1 and O2 are then “decoded” again into phenotypes P of O1 and O2, reversing the encoding process: P(O1): x1 = 0.1238 and y1 = 0.5032; P(O2): x2 = 0.5674 and y2 = 0.9876 • The fitness of each of the two new offspring individuals can now be evaluated simply by calculating the function value f (x, y). The procedure described above is just an example of how a GA can work. There are many possible variations. For instance, the encoding/decoding scheme described here is just one of many possibilities. Since the “raw material” for the new offspring individuals stems from the parent individuals but is (slightly or heavily) altered by random processes, this method incorporates the concepts of both inheritance and variability. The available parameter space is explored in a quite broad range and a quite narrow range simultaneously depending on the location of the crossover cut and the location of the possible mutation: These procedures may effect one of the leading or one of the trailing digits of the corresponding floating point numbers. Finally, it can be a great advantage that the fitness function f (x) does not have to be differentiable—it just has to be computable for all x. 3.3.3

PIKAIA—A Specific Implementation of a Genetic Algorithm

The PIKAIA algorithm is a specific implementation of a general GA as described in the previous section. It is written in Fortran and can be downloaded at no cost (public domain software) from the internet [PIKAIA 2002]. A very detailed user’s guide is available as well [Charbonneau 1995b, Charbonneau 2002b]. PIKAIA was first used in the field of astronomy and astrophysics [Charbonneau 1995a] but can be and has been applied to a number of other fields such as engineering design, statistical thermodynamics, aircraft/spacecraft design, and acoustics (vehicle design) [PIKAIA 2004]. The following description is compiled from references [Charbonneau 1995b], [Charbonneau 2002a], and [Charbonneau 2002b]. PIKAIA tries to maximize a user-defined fitness function f (x) in n dimensions maximize f (x) , where x = (x1 , x2 , . . . , xn ) ; xk ∈ [0.0, 1.0] ∀ k; k, n ∈ N .

(3.15)

Since the parameter values xk are restricted to the range [0.0, 1.0], the input parameters of the function to be maximized have to be adequately normalized by the user. At the start of the optimization process (“evolution”), the first population is initialized with random numbers from the interval [0.0, 1.0]. Most of these randomly produced parent individuals will have a quite low fitness, but some of them will be above-average—and this is all that is required for natural selection to operate. Fit individuals are not necessarily good or fit on an absolute scale, but evolution simply produces better-than-average individuals. The fitness value returned by the evaluation of the user-defined fitness function must be a positive definite quantity. Therefore, the original fitness value produced by the original model may have to be transformed to a suitable positive number by the user, e.g., by taking the absolute of the original function value |f (x)|. Likewise, if the user wants to minimize a function, the problem has to be transformed into a maximization problem, e.g., by maximizing the negative −f (x) or the inverse 1/f (x) of the original function (cf. Sec. 4.3.1). There is a total of 12 parameters to be set by the user that control the behavior of the algorithm. Some of these 12 input parameters will be described now. The population size of Np individuals remains fixed during the optimization procedure. The end of the evolution process is defined by a preset user-defined number of generations Ng rather

3.3

PIKAIA Algorithm

47

than by some tolerance criterion. Each individual’s probability of being randomly selected for breeding is proportional to the fitness-based rank of this particular individual within the current population. The user has to specify the number of digits to be retained in the encoding process (default: 5). Also, the crossover rate (probability that crossover actually takes place; default: 0.85) and the mutation rate (probability for each digit that mutation occurs at that digit location; default: 0.005) have to be predefined by the user. Whereas the crossover rate remains constant during evolution, the mutation rate is automatically adjusted between a user-defined lower and upper bound. It is also possible to choose between three different population replacement strategies each of which has its own advantages and drawbacks: • full generational replacement: The whole parent population is completely replaced by the new offspring population in each generation. • steady-state-delete-worst: The least fit member of the population is immediately replaced whenever the fitness of a newly bred offspring exceeds its fitness. • steady-state-delete-random: A randomly selected member of the population (independently of its fitness) is immediately replaced whenever the fitness of a newly bred offspring exceeds the fitness of the least fit member. A nice feature of random-based algorithms such as PIKAIA is the fact that new individuals and iterations do not (or at least not as much as for deterministic algorithms) depend on previous iteration results. Therefore, it is possible to compute the fitness values of many different offspring individuals on several computers in parallel, speeding up the optimization process significantly. Only after a whole generation of new offspring individuals has been created and their fitness values have been determined is the fitness-based ranking carried out on just a single “controlling” computer. This kind of “(pseudo-)parallelization” of the PIKAIA algorithm is described in App. A, Sec. A.2. GA-based optimization typically requires a large number of model evaluations and, thus, is quite CPU time consuming, in particular since most work goes into the fitness evaluation, which can take quite some time. On the other hand, this fitness evaluation can be parallelized as described in the previous paragraph. Furthermore, PIKAIA is a global optimization algorithm unlike the previously described COBYLA algorithm or the quite similar simplex method, which are only local. As Charbonneau states: “The simplex method is (relatively) fast, but local; PIKAIA is slow, but global. There is no such thing as a free lunch” [Charbonneau 1995b]. One final word on the origin of the algorithm’s name PIKAIA (pronounced “pee-kah-yah”), which is not an acronym. According to Charbonneau [Charbonneau 1995b, PIKAIA 2004], Pikaia gracilens was a flattened worm-like creature that existed about 530 million years ago. It was about five centimeters long and crawled in the mud of the seafloor. It is believed to be the founder of the phylum Chordata, i.e., the ancestor of all vertebrates. Thus, it truly can be considered an extremely successful example of an evolutionary process.

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Optimization Algorithms

4

Optimization Procedure

In this chapter, the optimization procedure constituting the Numerical Structural Acoustic Optimization (NSAO) program is described. Section 4.1 provides some information on issues that are common to both optimization algorithms employed. Sections 4.2 and 4.3 focus on aspects of the optimization procedure that are unique to the COBYLA or the PIKAIA algorithm, respectively.

4.1

Optimization Procedure in General

Figure 4.1 gives an overview of the general optimization procedure. Certain components of this flowchart contain subprocedures, which are described in more detail below. The first step (1) is to create an initial FE model of the structure to be optimized. The initial FE models of the structures investigated in the present study are depicted and described in Chap. 5. In step (2) an FE analysis of the initial FE model created in step (1) is performed. This is accomplished by using the commercial FE software ABAQUS [ABAQUS 2001a]. A combination of a numerical modal analysis and a subsequent mode-based steady-state dynamic analysis provides the nodal rms surface velocity vectors v rmsi (f ) (see Sec. 2.2). In the third step (3) the actual optimization procedure is started. This involves the execution of several Unix shell scripts and Fortran programs and is therefore described in a separate section below, namely, Sec. 4.1.1. In step (4) the initial design is evaluated, i.e., the objective function and the constraints are computed and saved to files to allow a continuous comparison between the initial design and the modified designs during optimization. The numerous subprocedures of this step are explained in Sec. 4.1.2. Step (5) invokes the actual numerical optimization algorithm (COBYLA or PIKAIA), which iteratively generates new design variable values based on the previous optimization history and success. The two algorithms are described in Secs. 3.2 and 3.3. Individual aspects of this step are treated below in Secs. 4.2 and 4.3. Steps (6) through (8) constitute the evaluation of the modified design. Step (6) transforms the new design variable values produced in step (5) into a new, modified FE model of the structure. The FE analysis of the modified design in step (7) is carried out in the same way as in step (2) above. Finally, step (8) provides the new objective function value and the constraint values for the modified design as in step (4) described above. The convergence check in step (9) is quite different for the two optimization algorithms and is therefore described in Secs. 4.2.2 and 4.3.2. If the convergence criterion is not yet met, the

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Optimization Procedure

Figure 4.1: Flowchart of the general optimization procedure. procedure branches back to step (5) to generate a new vector of design variables. However, if the convergence criterion is met, the optimization procedure is terminated and the CPU time consumed by the whole process and its various subparts is evaluated and recorded in step (10). The final product of the optimization process is an FE model of the optimized design (step (11)). This optimization procedure is designed and organized such that it can run automatically without any manual intervention for hours, days, or even weeks until the stop criterion is met. It is, however, possible to influence certain process parameters (the maximum number of iterations, the amount of information output to the screen or log file, etc.) while the optimization is running, because they are read from a control parameter file, which can be edited, at least once per iteration. It is not trivial to organize this process in such a way that the right information is available to the various separate Fortran programs and Unix shell scripts at the right time at the right

4.1

Optimization Procedure in General

51

place (various directories, subdirectories, and files, sometimes even on different computers). The whole procedure is rather complex since Unix shell scripts invoke other Unix shell scripts that start Fortran programs that start a number of other Fortran programs that in turn invoke Unix shell scripts that again start other Fortran programs. Some quantities are “static”, which means they need to be calculated or determined only once, i.e., prior to the first iteration at the very start of the procedure, because they do not change from iteration to iteration. Examples include the number of nodes and elements in the FE model and the thickness direction vectors. They are written to files from which they can be simply read again during subsequent iterations. Other quantities are “dynamic”, i.e., they vary from iteration to iteration and are therefore computed and determined anew at each iteration. Examples for “dynamic” quantities are the nodal coordinates, the structural mass, and the results of the FE analyses. 4.1.1

Start of Optimization Procedure

The optimization process in step (3) of the flowchart shown in Fig. 4.1 can be started from a graphical user interface (GUI), which is described in Sec. A.3 in App. A, by pressing the “run optimization” button (see Fig. A.13 on p. 180). A Unix shell script opens two log files. All information and intermediate results produced during optimization are written to the first log file. In the second log file (“time log file”) the real and CPU times required by every FE analysis (steps (2) and (7)) or objective function calculation and constraint evaluation (steps (4) and (8)) are recorded. The names of the initial structure and of the structure to be optimized are saved to files for future use. Likewise, the iteration number is initialized to zero and written to another file where it can be read from or, after an increase, written to again. A second Unix shell script then removes all old job files that might still exist in the current directories and subdirectories from a previous optimization run. After that, a third Unix shell script takes control and enables the CPU time recording. This is necessary because Unix treats the output of the time command, which provides the real and CPU time consumed by some computer process, as an error message to the screen. Therefore, it must be diverted to the time log file in a suitable way. Subsequently, the same Unix shell script first invokes the computation of the objective function and the evaluation of the constraints in step (4), which is described in Sec. 4.1.2, and then starts the actual optimization algorithm, i.e., either COBYLA or PIKAIA, depending on the respective entry in the control parameter file. 4.1.2

Calculation of Objective Function and Constraints

The objective function is calculated and the constraints are evaluated in steps (4) and (8) of Fig. 4.1 by a program module called OFAC (Objective Function And Constraints). The flowchart in Fig. 4.2 depicts the main structure of this procedure. In the first step (1) parameters that control the optimization procedure or that are necessary to define the size of dynamic variable arrays in the various Fortran programs are read from the control parameter file. In step (2) the “dynamic” results of the FE analysis are read from the binary result file, i.e., quantities that change from iteration to iteration such as the nodal coordinates, the nodal rms surface velocity vectors v rmsi (see Chap. 2), the surface area, or the structure’s natural frequencies. Prior to the start of the actual optimization, i.e., when the iteration number is still zero, the “static” data are read from the binary result file and saved to various ASCII files for later use

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Optimization Procedure

Figure 4.2: Flowchart of program module OFAC calculating the objective function and evaluating the constraints (steps (4) and (8) in Fig. 4.1).

(step (3)). During the iterations, i.e., when the iteration number is greater than zero (step (4)), they can then simply be read from those ASCII files. In step (5) the structural mass m is read from the ASCII result file because this quantity is not available from the binary result file.

4.1

Optimization Procedure in General

53

In step (6) the nodal normal vectors at the sound radiating surface ni are calculated at each iteration and saved to files for later use and control purposes. Step (7) computes the thickness direction vectors (see Sec. A.1.4 in App. A) of the initial structure during iteration zero and saves them to an ASCII file. This way, the thickness direction vectors, along which the position of the surface nodes can be varied to optimize the thickness distribution, are available to step (6) of Fig. 4.1 in some other program, which generates new, modified FE models (see Sec. 4.1.3). After that, the local wall thicknesses at the surface nodes are calculated in step (8) at each iteration. Their absolute values as well as the relative changes with respect to the thickness distribution of the initial structure are saved to ASCII files and can be used afterwards to produce the contour plots that visualize the thickness distribution (see Sec. A.4 in App. A). In step (9) the nodal normal rms surface velocities v⊥rms i (f ) are computed according to Eq. (2.8) from the nodal rms surface velocity vectors v rmsi (f ) obtained in step (2) and the nodal normal vectors ni calculated in step (6). Step (10) determines the LS according to Eq. (2.7) from the surface area S read from the binary result file in step (2) and the mean squared transmission admittance h2t (f ) according to Eq. (2.4), which in turn is calculated from the nodal normal rms surface velocities v⊥rms i (f ) computed in step (9) and the rms excitation force Frms (f ) that is read from the binary result file in step (2)). Furthermore, the MLS is computed according to Eq. (2.10), where the lower and upper bounds of the frequency range under consideration, fmin and fmax , are read from the control parameter file in step (1). Finally, possible constraint violations are evaluated in step (11) at the end of the program module OFAC. Constraint violations include exceeding the maximum allowable wall thickness, mass, or MLS, or falling below the minimum allowable wall thickness or fundamental frequency. The evaluation of the constraints is explained in more detail in Sec. 4.2.1 below. Now all the quantities that can serve as either objective function or constraint are available, i.e., natural frequencies (from step (2)), structural mass (from step (5)), wall thicknesses (from step (8)), and MLS (from step (10)). 4.1.3

Evaluation of Modified Design

The new design variable values, which are generated by the optimization algorithm at each iteration in step (5) of Fig. 4.1, are used to create a new FE model of the structure. The flowchart in Fig. 4.3 illustrates this creation of a modified FE model and its evaluation (see steps (6) through (8) in Fig. 4.1). In step (1) of Fig. 4.3 the optimization algorithm hands over the design variable values to the design evaluation program. In the second step (2) the nodal coordinates of the initial structure to be optimized are read from the original FE input file. Step (3) calls optional program routines that compute spline functions if specified in the control parameter file by the user. These spline functions may be used to reduce the number of design variables (see Chap. 6). The thickness direction vectors, which were determined and written to a file at iteration 0 in step (7) of Fig. 4.2, are now read from that file in step (4). In step (5) the surface nodes that are allowed to be varied by the optimization algorithm are moved along the thickness direction vectors to their new position. The distance by which they are moved is determined either directly by the (appropriately scaled) design variable values

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Optimization Procedure

Figure 4.3: Flowchart of design evaluation program module (steps (6) through (8) in Fig. 4.1). provided by the optimization algorithm or, if spline functions are used, indirectly by the spline function values at the surface nodes (see Chap. 6). Step (6) interpolates the nodes at the element edges of the elements at the surface between the respective corner nodes in order to avoid overly distorted elements that would cause the FE analysis to produce inaccurate results or even to fail completely. Then, a new FE input file containing the new nodal coordinates is written (step (7)), the iteration number is increased by one (step (8)), and a new FE analysis using the new input file is invoked by step (9). Upon completion of the FE analysis, the program module OFAC, described in Sec. 4.1.2 and illustrated by the flowchart in Fig. 4.2, is invoked in step (10) to calculate the objective function value and to evaluate the constraints of the new, modified design.

4.2

Optimization Procedure using COBYLA

55

In the final step (11), the new objective function value and the constraints are returned to the optimization algorithm, which starts a new iteration by generating new design variable values if the stop criterion is not yet met.

4.2

Optimization Procedure using COBYLA

The previous section (Sec. 4.1) described features and procedures that are common to both optimization algorithms used in this study. This section and Sec. 4.3 explain issues that are unique to the COBYLA algorithm on the one hand or to the PIKAIA algorithm on the other hand. COBYLA can be regarded as a black-box algorithm that supplies new design variable values and is fed with the corresponding objective function and constraint values. The algorithm does not care how the objective function and constraint values are determined from the design variables. Since COBYLA is basically used unaltered as obtained from [COBYLA 1994] and as described in detail in Sec. 3.2, a detailed description of the optimization algorithm beyond the information provided in Sec. 4.1 and in Fig. 4.1 is not considered necessary. Instead, the focus of this section is on the calculation of the objective function and the evaluation of the constraints (Sec. 4.2.1), on the convergence criterion (Sec. 4.2.2), and on feasible results and tolerable constraint violations (Sec. 4.2.3). 4.2.1

Objective Function and Constraints

The objective function value computed in Sec. 4.1.2 (see Fig. 4.2) is directly used in COBYLA as the objective function value F (x) in Eq. (3.3) on p. 38. If F (x) is to be maximized, −F (x) is minimized instead. Objective functions such as MLS, structural mass, or fundamental frequency are straightforward, but other cases are not as trivial. For instance, if the difference between some specified frequency f ∗ and its two neighboring natural frequencies is to be maximized, the following objective function is used F (x) = − min (∆f1 , ∆f2 ) ,

(4.1)

where ∆f1 and ∆f2 are the differences between said specified frequency f ∗ and the lower and upper neighboring natural frequency, respectively. This finally leads to a solution with ∆f1 = ∆f2 and to a maximum value of ∆f = ∆f1 + ∆f2 . To calculate the merit function Φ(x) in Eq. (3.3), it is necessary to provide the values of possible constraint violations as well. If there is no constraint violation at all, the relation Φ(x) = F (x) holds (see Sec. 3.2.3). According to Eq. (3.2) the constraints have to be of the form ci (x) ≥ 0 (i = 1, 2, . . . , m) and therefore must be formulated appropriately. If some quantity p(x) (e.g., the structural mass) is restricted from above by some upper bound pmax , the respective constraint is defined by ci (x) ≥ pmax − p(x) .

(4.2)

Analogously, if some other quantity q(x) (e.g., the wall thickness or the fundamental frequency) is restricted from below by some lower bound qmin , the respective constraint is specified by ci (x) ≥ q(x) − qmin . Any negative constraint value indicates a violation of that particular constraint.

(4.3)

56

4.2.2

4

Optimization Procedure

Convergence Criterion

The convergence criterion used by COBYLA is defined by the trust-region radius ρ, which is continuously reduced during the optimization (see Sec. 3.2.3). If ρ reaches its final value ρend , which is specified by the user in the control parameter file and set to 10−6 for the present study, the optimization procedure is terminated and the best design variable vector up to that point is returned as the optimized design. Alternatively, the user can specify a maximum number of iterations after which the optimization process is terminated irrespective of the current objective function value F (x) or trust-region radius ρ. 4.2.3

Feasible Results and Tolerable Constraint Violations

As described in the previous section (Sec. 4.2.2), the optimization procedure is terminated if ρ ≤ ρend . At this point, the optimized design is usually feasible, i.e., none of the constraints are violated. However, sometimes it happens that the final design is not feasible in the strict, mathematical sense, i.e., one or some constraints may be slightly violated. Nonetheless, from the engineering point of view it does not matter if the wall thickness at any location is 0.1 µm higher or lower than the respective limit, since this order of magnitude is below the achievable manufacturing tolerance anyway and therefore irrelevant. Likewise, it is not relevant if the structure is too heavy by 0.1 µg or if the fundamental frequency is 10−5 Hz below its lower limit. Also, COBYLA often uses the last hundreds, sometimes even thousands of iterations just to avoid even the slightest constraint violation while the objective function is not significantly improved or even not improved at all. Therefore, it seems to be sensible to define a tolerable constraint violation. In some of the iteration history plots shown in Secs. 7 through 10 (e.g., Fig. 7.2 on p. 81) a maximum allowable constraint violation of 0.01%, which is considered still acceptable, is marked by circles.

4.3

Optimization Procedure using PIKAIA

This section describes some of the features that are unique to the PIKAIA algorithm. Unlike the COBYLA algorithm, PIKAIA is modified and adapted in a number of ways. The most important changes include the usage of 50 parent individuals and 20 offspring individuals in every subsequent generation instead of only two, the (pseudo-)parallelization of the algorithm by performing the time-consuming FE analyses simultaneously on several computers (see Sec. A.2 in App. A) as well as the introduction of a suitable penalty term to take constraints into account (see Sec. 4.3.1 below). Therefore, the optimization procedure employing PIKAIA is illustrated in the flowchart in Fig. 4.4 and explained in some detail now. First of all, the parent population with 50 individuals, i.e., the initial set of 50 individual vectors of design variables, is generated by means of a random number generator in step (1). Step (2) calls a conditioner routine. It translates the design variables provided in the previous step, which are in the range [0.0, 1.0] (see Sec. 3.3.3), into “real world” dimensions, namely, into the range of possible thickness variations [1.0, 10.0] millimeters or rather [−3.0, 6.0] millimeters (since the thickness is varied from its initial value 4 mm) by means of the linear relation xreal = 9 · xpikaia − 3 .

(4.4)

4.3

Optimization Procedure using PIKAIA

57

Figure 4.4: Flowchart of the PIKAIA algorithm. Since the thickness variations are implicitly restricted to the range [−3.0, 6.0] millimeters by means of Eq. (4.4), additional explicit constraints of the type shown in Eqs. (4.2) and (4.3) that take the thickness into account are not necessary when using PIKAIA. After that, all 50 (in subsequent generations 20, see below) FE input files are generated at once in step (3) and sent to the 5 participating computers where the FE analyses are performed in parallel in step (4) as described in detail in Sec. A.2 in App. A. As soon as all the parallel FE jobs are done and their results are available, step (5) evaluates the FE analysis results in rapid succession by calculating the objective function values and evaluating the constraints. This is basically done in the same way as for COBYLA, i.e., as described in Secs. 4.1.2, 4.1.3, and 4.2.1, and as illustrated in the previous flowcharts. From the objective function and constraint values obtained in step (5), suitable fitness values are determined in step (6). This process is described in Sec. 4.3.1 below. For the first generation, all 50 individuals are kept and used as the “mating pool” for the creation of a new offspring generation consisting of 20 members. For all subsequent generations a decision must be made. If the maximum number of iterations specified by the user in the control parameter file is reached in step (7), PIKAIA is terminated, and the best design up to that point is considered the optimum design (step (9)) just as for the COBYLA algorithm (cf. Sec. 4.2.2). If, however, the maximum iteration number is not yet reached (cf. Sec. 4.3.2 below), these of the 20 members of the offspring generation that are deemed fit enough, i.e., which have a high enough fitness value, are inserted into the population in step (8) and replace

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Optimization Procedure

those of the 50 individuals of their parent population that are less fit. Then, a new offspring generation, which consists of 20 individuals, is created by breeding, crossing, and mutation as described in Sec. 3.3.2, and the next loop starts at step (2). It is worth repeating that, unlike in many other publications (cf. item “number of iterations/CPU time” on pp. 23 and 24), each single function evaluation, i.e., FE analysis combined with objective function evaluation, is considered an iteration. This means that the first generation comprises 50 iterations, whereas for each subsequent generation 20 iterations are needed. 4.3.1

Calculation of Fitness

PIKAIA requires the computation of a so-called fitness function, which must be a positive definite quantity (see Sec. 3.3.3). The fitter an individual is, i.e., the higher its fitness value, the higher is its chance of being selected for reproduction in the next offspring generation. The fitness function f (x) is calculated from the actual objective function value F (x) (step (5) of Fig. 4.4), where possible constraint violations are used as a penalty that artificially reduces the fitness. A weighting factor for the constraints wc is introduced, and the smallest constraint cmin (x), which might be negative if the constraint is violated (see Eqs. (4.2) and (4.3)), is determined. If the actual objective function F (x) is to be minimized, the constraint weighting factor wc is chosen to be 103 , and the fitness value f (x) is calculated by f (x) = wc / [F (x) + wc · |min (cmin (x), 0)|] .

(4.5)

If the design is feasible, i.e., if there are no constraint violations and thus min (cmin (x), 0) = 0, the fitness function is just f (x) = wc /F (x), i.e., the lower the objective function value F (x) the higher the fitness f (x). If there are constraint violations, however, the greatest constraint violation, i.e., the largest absolute value of all constraint violations |min (cmin (x), 0)| is added to the actual objective function F (x) in the denominator of Eq. (4.5), thus decreasing the fitness value f (x), i.e., the greater the constraint violation, the lower the fitness. The constraint weighting factor wc = 103 is set rather large in order to avoid the possibility that a high objective function value might compensate a small constraint violation, i.e., any reduction of the greatest constraint violation cmin (x) is “rewarded” with a significant increase of the fitness function. The same factor wc = 103 is used in the numerator of Eq. (4.5) in order to scale the fitness function to a “nice” order of magnitude of 10 or 102 instead of 10−3 or 10−2 . This can be done without loss of generality, since it is not the absolute fitness value that matters but rather just the relative value compared to that of other individuals. If the objective function F (x) is to be maximized, the constraint weighting factor wc is set to 106 , and a preliminary fitness value f  (x) is calculated by f  (x) = F (x) − wc · |min (cmin (x), 0)| ,

(4.6)

thus severely punishing any constraint violation, whereas f  (x) = F (x) if the design is feasible. However, the preliminary fitness f  (x) itself cannot serve as fitness function f (x) because it can (and, at the beginning of the optimization process, definitely will) become negative, which is not allowed for the fitness function. Thus, the actual fitness function is defined as a positive number by f  (x), f  (x) ≥ 1 f (x) = (4.7)  1/ |f (x) − 2| , f  (x) < 1 .

4.3

Optimization Procedure using PIKAIA

This function concatenates the two ranges f  (x) differentiable way. If f  (x) ≥ 1, then basically constraint violations. If there are small or even fitness value f (x) becomes a tiny, yet still positive 4.3.2

59

≥ 1 and f  (x) < 1 in a continuous and f (x) = F (x) except for extremely small significant constraint violations, then the number.

Convergence Criterion

There is no real convergence criterion available in PIKAIA. The optimization procedure is terminated as soon as the maximum number of iterations specified by the user in the control parameter file is reached. As can be seen from the iteration history plots of the optimization runs using PIKAIA in Figs. 7.38 (p. 110), 8.17 (p. 130), and 9.17 (p. 148), the objective function eventually approaches some asymptotic value and does not change significantly even if the optimization continues. In such cases the optimization can be terminated “manually” by reducing the maximum number of iterations in the control parameter file to a suitable value. This can be done any time during the optimization process, since the maximum number of iterations is read from the control parameter file at least once per iteration. If the specified maximum number of iterations is smaller than the current iteration number, the optimization procedure is terminated immediately. The maximum number of iterations used for the PIKAIA optimization runs presented in this study is set to 50 000. 4.3.3

Control Parameter Setting

As mentioned in Sec. 3.3.3, there are 12 parameters that control the behavior of PIKAIA and which have an extreme influence on the convergence rate and efficiency of the algorithm. However, the setting of these parameters is very problem dependent, which means that general recommendations cannot be made in advance. Rather, good parameter values must be found in a trial and error manner. Since each of the optimization runs using PIKAIA, the results of which are presented in Secs. 7.4, 8.3, and 9.4, took between 10 and 27 days, it is obvious that an exhaustive parameter optimization could not be performed prior to the actual optimization run, particularly since the best control parameter set for one problem can have disastrous effects on the efficiency of solving another problem. In a personal conversation with Prof. Spellucci (see footnote on p. 36) he reported, smiling amusedly, that some other researcher proudly told him that his genetic algorithm found the global optimum of some particular problem extremely fast and efficiently. However, when asked how long it took him to fine-tune the control parameters of the algorithm to achieve this efficiency, the researcher had to admit: “About half a year.” Hence, the default values of PIKAIA’s control parameters are basically left unchanged except for the population size, which is set to 50, and the number of new offspring individuals per generation, which is set to 20.

60

4

Optimization Procedure

5

Finite Element Models

In this chapter, the FE models investigated and optimized in this thesis are introduced. Section 5.1 provides some information on features and properties that are common to all the FE models used in the present study. Then, Secs. 5.2 through 5.5 describe the FE models of a rectangular plate, of two plates joined at 90◦ , of a half-cylinder, and of a gearbox, respectively.

5.1

Common Properties

The design variables are the positions of specified surface nodes, in other words, the thickness values at these movable surface nodes. If the shape of the surface is varied by means of spline functions (see Chap. 6), then the positions of the spline key points are the design variables. It is assumed that the thickness is not supposed to be changed along the simply supported edges (see below). Several constraints are defined for each structure. The most obvious ones are the lower and upper limits on the local wall thickness at the movable surface nodes, which are 1 mm and 10 mm, respectively. Besides that, a maximum allowable mass or MLS, a minimum allowable fundamental frequency, or any sensible combination of these can be defined as additional constraints. The initial thickness of all FE models, except for the gearbox in Sec. 5.5, is set to 4 mm throughout the structure. For the gearbox, the initial thickness of the plate-like modification domains is set to 6 mm for the lower part of the gearbox and to 4 mm for the upper part. The optimization procedure does not vary the thickness of the flanges and bearing supports. The material of which the structures are made is assumed to be steel with a density of ρ = 7850 kg/m3 , an elastic modulus (Young’s modulus) of E = 2.04 · 1011 N/m2 , and a Poisson’s ratio of ν = 0.3.3 The damping is assumed to be independent of the frequency with a constant damping coefficient of 0.4%. Structures with a damping coefficient of less than 1% can be considered lightly damped so that the modal superposition technique can be applied with negligible error for the FE calculation of the nodal rms surface velocity vectors v rmsi (f ). The support boundary conditions for all structures, except for the gearbox in Sec. 5.5, are chosen to be so-called simple supports, which means that the edges of the structure do not have any translational DOFs (they cannot move up and down or sideways) but they do have rotational DOFs (they are free to rotate about the particular edge under consideration). The gearbox, however, is assumed to be fixed to the ground at its four feet. Further details can be found in the respective sections below. 3

These material properties are taken from [Hibinger 1998].

62

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Finite Element Models

The element type used for all FE models is the 20-node hexahedral solid finite element with quadratic shape functions provided by ABAQUS, i.e., ABAQUS element type C3D20 [ABAQUS 2001a, ABAQUS 2001b]. The excitation force is a harmonically varying point load of 10 N. However, the amplitude 2 of the point load is not relevant because the mean squared rms normal velocity v⊥rms (f ) is 2 divided by the square of the rms excitation force Frms (f ) when the mean squared transmission admittance h2t (f ) is calculated in Eq. (2.4), thus negating the influence of the excitation force’s amplitude. The excitation force always acts perpendicularly to the sound radiating surface at some point on the structure except for the gearbox in Sec. 5.5 where it acts perpendicularly to the surface on the inside of a bearing bore, which is not part of the sound radiating surface. The frequency range of interest is chosen to be 0–3000 Hz for all structures except for the gearbox, for which a smaller frequency range of 0–1000 Hz is used in order to reduce the computation time.

5.2

Rectangular Plate

The same simply supported rectangular plate was discretized with two different FE meshes. Section 5.2.1 describes the FE model of the plate with a refined mesh, whereas the plate with a coarser discretization is introduced in Sec. 5.2.2. The different discretizations are compared and discussed in Sec. 5.2.3. 5.2.1

Refined Discretization

The first FE model studied in this thesis is a simply supported, rectangular plate, shown in Fig. 5.1, with a length of 350 mm, a width of 233.33 mm, and a thickness of 4 mm. The length is discretized with 24 elements, the width with 16 elements, and the thickness with only one layer of elements, totaling 384 elements. Due to the element type used (see Sec. 5.1), the model has 2891 FE nodes and 8673 degrees of freedom (DOFs).

Z Y X

Figure 5.1: Initial FE model of the rectangular plate with 24×16 elements. The excitation force acts at the location denoted by the arrow in Fig. 5.1, i.e., at the coordinates 204.167 mm/175 mm/4 mm, a place where presumably all relevant mode shapes of the structure in the frequency range of interest are excited. The plate is simply supported along its four edges.

5.2

63

Rectangular Plate

The surface nodes, the position of which can be varied along the thickness direction vectors (here: simply the z coordinate) and optimized by the optimization procedure, are marked in Fig. 5.2. As can be seen, only corner nodes of element faces on the surface of the structure are declared movable. The nodes at the edges of the plate are not part of the modification domain. The number of movable surface nodes totals 345. If no spline functions are used to reduce the number of design variables, then the optimization procedure directly varies these node positions, which leads to 345 design variables. If spline functions are used, this number is reduced to 35 in this study (see Sec. 7.3).

Y Z

X

Figure 5.2: Movable nodes at the surface of the rectangular plate with 24×16 elements. The number of constraints depends on the optimization problem. Usually, there are one or two constraints such as a maximum allowable mass or MLS, a minimum allowable fundamental frequency, or a combination of two of these. The thickness at all of the 345 movable surface nodes can only be varied between 1 mm and 10 mm, so there is a lower and an upper bound on the thickness for each movable surface node. This leads to a total of 691 or 692 constraints. 5.2.2

Coarse Discretization

Since a single FE analysis of the rectangular plate with 384 elements takes between 30 and 54 seconds of CPU time (depending on the number of mode shapes of a particular design) and since the numbers of design variables (345) and constraints (691 or 692) are quite high, the same rectangular plate is discretized with a coarser FE mesh. This coarser mesh consists of 12×8 elements, totaling 96, and thus significantly reduces the number of design variables and constraints to 77 and 155 (or 156), respectively. The CPU time per FE analysis is cut down to the range of 7 to 11 seconds, a reduction of 70% to 80%. Figure 5.3 depicts the rectangular plate with the coarse FE discretization. Now the model has only 775 nodes and 2325 DOFs. The excitation force is applied at the same physical location as for the refined discretization.

64

5

Finite Element Models

Z Y X

Figure 5.3: Initial FE model of the rectangular plate with 12×8 elements. The movable nodes of the coarse discretization can be seen in Fig. 5.4. As mentioned above, there are now 77 design variables if the position of the movable nodes is varied directly. If spline functions are used, however, only 35 variables control the shape of the plate’s surface. The number of constraints is 155 or 156, depending on how many additional constraints are used besides the minimum and maximum thickness constraints at each movable surface node.

Y Z

X

Figure 5.4: Movable nodes at the surface of the rectangular plate with 12×8 elements.

5.2.3

Discretization Error and Numerical Pollution

A too coarse discretization can cause numerical errors in acoustical FE analyses as discussed by Ihlenburg in his book and papers [Ihlenburg 1995, Ihlenburg 1998, Ihlenburg 2002, Ihlenburg 2003]. He observed that the classical “rule of thumb” of 6 to 10 linear finite elements per wavelength, which is usually recommended in practice, results in quite large numerical errors. If a small numerical error is desired or required, a drastically refined mesh must be used. This is due

5.2

65

Rectangular Plate

to the so-called numerical pollution effect, which occurs in addition to the FE approximation error and dominates the total error at higher frequencies. The numerical pollution can be interpreted as a numerical dispersion, i.e., due to these numerical errors the speed of sound c seems to become frequency dependent even in media in which this is physically not the case (e.g., in air where c = c(f )). In his papers, Ihlenburg [Ihlenburg 2002, Ihlenburg 2003] provided resolution rules, which, if followed, result in an admissible error for the FE solution of the dynamic Kirchhoff plate equation. He defined some auxiliary quantity B ∗ by   m 12ρ (1 − ν 2 ) B∗ = , (5.1) = B Et2 where m = ρ t is the mass per unit surface area, B is the bending stiffness according to Eq. (5.4), ρ, ν, and E are the material parameters of steel provided in Sec. 5.1, and t is the plate thickness. With the bending wavelength λb (f ) and the mesh size h the resolution rule is  λb (f ) 4 (5.2) > 2π 2πf B ∗ L2 , h where f is the frequency under consideration and L is the size of the computational domain. If f = 3000 Hz, L = 0.35 m (length of the plate), t = 4 mm (thickness of the plate), and B ∗ is calculated according to Eq. (5.1), then Eq. (5.2) yields λb /h > 27.6. This means that at least 28 linear elements per bending wavelength are necessary to achieve an admissible error. The wavelength λb (f ) of a vibrational bending wave within a planar, isotropic rectangular plate of constant thickness t at a certain frequency f can be approximated by [Kollmann 2000]   2π 4 B λb (f ) = , (5.3) f ρt where ρ is the density of the plate material and B is the bending stiffness of the plate B=

E t3 . 12 (1 − ν 2 )

(5.4)

With the material parameters of steel provided in Sec. 5.1, the plate thickness t = 4 mm, and the frequency f = 3000 Hz, Eq. (5.3) yields λb ≈ 114 mm. Therefore, the condition λb /h ≥ 28 leads to a maximum mesh size of hmax ≈ 4 mm for linear elements. In the present study, however, the mesh size of the plate’s refined FE model is h = 14.6 mm and that of the coarse mesh is even h = 29.2 mm, which does not seem to be acceptable at first glance. However, one must recall that quadratic instead of linear elements are used in this thesis. A discretization with n quadratic elements per wavelength always yields more accurate results than a discretization with 2n linear elements per wavelength [Ihlenburg 1998]. Furthermore, numerical experiments by the author of the present study revealed that the error in the highest natural frequency in the range of interest, i.e., 2928.1 Hz, between the analytical and the numerical solution is 8.6% for the 12×8 discretization, 0.5% for the 24×16 discretization, and 0.7% for a 96×64 discretization. Figure 5.5 shows the LS spectra of the rectangular plate for various discretizations. It can be seen that the different mesh sizes effect only the natural frequencies but hardly the LS levels. There is good agreement between the

66

5

Finite Element Models

level of structure borne sound LS [dB]

110 12x8 elements 24x16 elements 36x24 elements

100

90

80

70

60

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 5.5: Comparison of the LS spectra of the rectangular plate for various discretizations. natural frequencies of the three FE models for frequencies up to about 1500 Hz. Beyond that, there is good agreement between the 24×16 and the 36×24 discretization, but the 12×8 discretization clearly deviates from these two. Despite these obvious deviations the author decided to use relatively coarse FE meshes for all the FE models investigated in this dissertation due to several reasons: • The rectangular plate with the coarse FE discretization described above represents a “worst-case scenario”. The mesh size of all other FE models with the same thickness t = 4 mm (two-plate joint, half-cylinder, upper part of the gearbox) is considerably smaller than that of the rectangular plate with the coarse FE mesh. At the lower half of the gearbox (see p. 70) the thickness is t = 6 mm, which means that the mesh size required for an accurate solution could be larger than that for a thickness of 4 mm. Furthermore, Storm [Storm 1980] showed that the bending wavelength of a curved shell (half-cylinder) in axial direction is much greater than the bending wavelength in circumferential direction, which would allow an even larger mesh size for the half-cylinder in axial direction (see p. 68). • A coarse discretization significantly reduces the computation time for every single FE analysis (see the first paragraph in Sec. 5.2.2). Since an optimization run requires hundreds, even thousands of iterations, each second of CPU time saved per iteration shows large effects. • A coarse FE mesh also reduces the number of design variables if the movable corner nodes at the surface are varied directly by the optimization algorithm without using spline functions, thus decreasing the number of iterations required to reach convergence. • The coarse discretizations suffice to show that the optimization procedure works in principle. The optimization results may not be absolutely correct but they are accurate enough to serve as design proposals and to gain some insight in the behavior of the structures.

5.3

5.3

Two Plates Joined at 90◦

67

Two Plates Joined at 90◦

Figure 5.6 shows the FE model of the two plates joined at 90◦ . The front plate is 240 mm long and 190 mm high, whereas the top plate is 240 mm long and 230 mm wide. The former is discretized by 12×10 elements, the latter by 12×12 elements. Both plates have an initial thickness of 4 mm, which is discretized just by a single layer of elements. The region of the actual joint is modeled by 12 elements with a cross section of 4 mm × 4 mm connecting front and top plate. This discretization results in 276 elements with 2110 nodes and 6330 DOFs.

Z Y X

Figure 5.6: Initial FE model of the two plates joined at 90◦ . The arrow denotes the excitation force, which is located at the coordinates 40 mm/80.67 mm/ 194 mm. The two plates are simply supported along their three free edges, the material parameters are given in Sec. 5.1. Figure 5.7 depicts the movable surface nodes. Their location can be influenced by the optimization algorithm. As there are 220 movable surface nodes, this structure’s number of design variables is 220. The number of constraints totals 441 or 442, namely, two thickness constraints, i.e., lower (1 mm) and upper bound (10 mm) on the local thickness at each movable surface node, and one or two additional constraints on mass, MLS, or fundamental frequency. This structure was chosen for two reasons: First, it is a relatively simple yet three-dimensional structure on which the universal applicability of the optimization procedure can be tested. Normal and thickness direction vectors do no longer point directly into z direction as was the case for the rectangular plate in Sec. 5.2 but rather into general three-dimensional space. Likewise, the thickness is not only varied in z direction at the top plate but also in y direction at the front plate. The second reason is the fact that Hibinger [Hibinger 1998] investigated the same structure in his Ph.D. dissertation, thus making a comparison of his optimization results with the present ones possible. This is done in App. B.

68

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Finite Element Models

Z Y X

Figure 5.7: Movable nodes at the surface of the two plates joined at 90◦ .

5.4

Half-Cylinder

The initial FE model of the half-cylinder can be seen in Fig. 5.8. Its mean radius is 100 mm, its length is 200 mm, and the initial wall thickness is 4 mm. There are 16 elements in the circumferential direction, 10 elements along the length and only one element layer in the thickness direction. Thus, the total number of elements is 160 with 1253 nodes and 3759 DOFs. The half-cylinder is made of steel with the material parameters specified in Sec. 5.1. It is simply supported along its two lower edges. The excitation force, marked by an arrow in Fig. 5.8, acts on the structure normally to the surface at the coordinates 39.03 mm/60 mm/94.24 mm. Fig. 5.9 shows the 165 movable surface nodes. The local thicknesses at these nodes serve as design variables. Thus, the optimization problem has 165 design variables if the positions of the movable nodes are directly varied by the optimization algorithm. However, if the half-cylinder’s shape is modified by means of spline functions, the positions of 48 spline key points are used as design variables instead (see Sec. 9.3). The number of 165 movable nodes yields a number of 331 or 332 constraints since at each of the 165 movable nodes both a minimum (1 mm) and a maximum allowable thickness (10 mm) is defined plus one or two additional constraints such as a maximum allowable mass or MLS or a minimum allowable fundamental frequency. This model is even more general than the 90◦ plate joint introduced in Sec. 5.3 because it is not only truly three-dimensional but also curved, so that its surface is not parallel to the x-y, x-z, or y-z plane. Thus, normal and thickness direction vectors point into general threedimensional space and are not parallel to the x, y, or z axis, and the thickness is varied along any direction specified by the thickness direction vectors.

5.4

69

Half-Cylinder

Z Y X

Figure 5.8: Initial FE model of the half-cylinder.

Z Y X

Figure 5.9: Movable nodes at the surface of the half-cylinder.

70

5

5.5

Finite Element Models

Gearbox

Figure 5.10 depicts the initial FE model of the gearbox. All dimensions are given in millimeters. The initial thickness of the plate-like modification domains (see Fig. 5.11) at the lower half of the gearbox is 6 mm, but it is 4 mm at the upper half. The model consists of 2444 elements, resulting in 17 066 nodes and 51 198 DOFs.

Z Y

40

X

80

d)

c) 600 448

346

430

320 208

160

26

146

246

160

292 372

66 166 266 572

a)

b)

Figure 5.10: Initial FE model of the gearbox: a) front view, b) side view, c) top view (wireframe model, all dimensions in mm), d) three-dimensional view. The dark areas in Fig. 5.11 denote the modification domains in which the wall thickness may be varied by the optimization algorithm. These modification domains include 821 corner nodes at the surface of the gearbox. Hence, there are 821 design variables to be controlled by the

5.5

71

Gearbox

algorithm. In addition to the lower (1 mm) and upper bound (10 mm) on the local thickness at the movable nodes, there is a mass constraint for the MLS optimization presented in Sec. 10. Therefore, the model comprises a total of 1643 constraints. The excitation force is located in the middle of the right-hand bearing bore at the coordinates 380.0 mm/30.0 mm/166.0 mm and marked by an arrow in Fig. 5.11. It is assumed that the gearbox is made of steel (see Sec. 5.1 for the material parameters) and that it is fixed to the ground at the underside of its four feet.

Z Y X

Figure 5.11: Initial FE model of the gearbox: arrow at right-hand bearing bore marks excitation point, dark areas denote modification domains. This model is the most challenging test case presented in this dissertation due to its high number of nodes and elements, which leads to long CPU times for every single FE analysis. Furthermore, the high number of design variables results in a high number of iterations required to reach convergence. In various previous publications (e.g., [Hibinger 1998,Inoue 1993,Wender 1998]) simple rectangular boxes were used to approximate a gearbox. In other papers (e.g., [Angert 1992,Cunefare 1991a,Cunefare 1992,Giordano 1993,Tinnsten 1999]) rectangular boxes were investigated without explicitly mentioning that they approximate some other structure. In the present study, however, numerical optimization techniques are applied to a relatively detailed and complex FE model of a gearbox for the first time to the author’s knowledge.

72

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Finite Element Models

6

Employing Spline Functions to Reduce the Number of Design Variables

As has been shown in the previous chapter, the number of design variables can be quite high depending on the structure’s FE model and its discretization. Therefore, an approach to reduce the number of design variables is presented in this chapter. First, some general aspects are discussed in Sec. 6.1. Then, Secs. 6.2 through 6.4 introduce three particular spline function realizations, namely, a bicubic spline surface, a Hermite spline surface, and a tensor product B´ezier surface.

6.1

General Aspects

The aim of the optimization procedures presented in this dissertation is to optimize the thickness distribution of several structures with respect to various objective functions such as MLS, mass, or fundamental frequency. Even for a simple structure like the rectangular plate depicted in Fig. 5.3 the number of design variables, i.e., the number of surface nodes at which the local thickness can be changed, is quite high. The optimization algorithm has to check and control all of these design variables, which takes a lot of computation time. Hence, it is desirable to reduce the number of design variables in order to save CPU time. The technique presented in this chapter was initially inspired by a paper by Braibant and Fleury [Braibant 1984] and is based on suggestions by Prof. Reif.4 It was implemented by Friedmann [Friedmann 2002] and published in [B¨os 2003a]. Therefore, only a brief description is provided here. For further details the reader is referred to the two references just mentioned as well as to the literature on spline functions, i.e., to the references provided in Sec. 1.2.6, in particular to the books [de Boor 1978, Engeln-M¨ ullges 1996, Farin 2001, Hoschek 1993, Press 1992, Sp¨ath 1991]. Instead of modeling the surface of the structure itself by spline functions, only the thickness distribution is modulated by means of spline or polynomial functions. In this way, the FE nodes and the spline key points are completely uncoupled. This method has the advantage that the spline formulation (i.e., the number of key points, the order or type of the spline, etc.) can be changed any time without having to fit the spline surface to the FE surface anew. Conversely, the same spline surface with the same number of key points can be used for different FE discretizations as is done in Secs. 7.3.1 and 7.3.2. 4 personal conversation with Prof. Dr. U. Reif from the research group Differential Geometry and CAGD, Department of Mathematics, Darmstadt University of Technology, Germany

74

6

Employing Spline Functions to Reduce the Number of Design Variables

First, a thickness distribution is calculated by means of a spline surface. Then the FE nodes of the structure are moved such that the thickness of the structure corresponds to this calculated thickness distribution. This procedure is described in more detail now. The thickness distribution of the initial structure t(0) (x, y, z) is determined prior to the start of the iterations. During the optimization process the thickness t(i) (x, y, z) at the grid points (x, y, z) is then modified in iteration i by t(i) (x, y, z) = t(0) (x, y, z) · AS(x,y,z) ,

(6.1)

where S(x, y, z) represents the value of a spline function at the respective grid point and A is a suitable positive number (e.g., the base of natural logarithms e) [Friedmann 2002]. The exponential function in Eq. (6.1) is chosen since its value is always greater than zero for any given value of the exponent as long as its base A is positive, and its value is equal to one (i.e., the thickness does not change) if its exponent is zero. This ensures that the new thickness t(i) (x, y, z) stays within the range 0 < t(i) (x, y, z) < ∞. Thus, the new thickness can never become negative, which would not be sensible from the physical point of view. Figure 6.1 shows a hypothetical example of this procedure. First, the initial local thickness values t(0) (x, y, z) at four surface nodes are determined as described in App. A, Sec. A.1.4 (Fig. 6.1 a). The optimization algorithm modifies the three key point positions (denoted by circles in Fig. 6.1 b) of a spline function S(x, y, z) and this way calculates new values of the modulation function AS(x,y,z) (see Eq. (6.1)) at the node positions. These local modulation function values are then used to move the surface nodes to their new position (Fig. 6.1 c). Thus, three spline key points suffice to influence the thickness distribution at four surface nodes, which does not look like an enormous reduction. In one of the optimization examples presented in the following chapters (see Sec. 7.3.2), however, the positions of 345 surface nodes were controlled by only 35 spline key points. This is a reduction of about 90%, which leads to a CPU time saving of about 88% compared to the same optimization without using splines.

AS(x,y,z)

t1(0)

t2(0) a)

t3(0)

t4(0)

1

t1(i) b)

t3(i)

t2(i)

t4(i)

c)

Figure 6.1: Modification of surface node positions by means of a spline function: a) initial node positions with local thickness t(0) (x, y, z), b) modulation function AS(x,y,z) , c) modified node positions with new local thickness t(i) (x, y, z) (dashed line indicates initial configuration). Three different spline formulations were implemented by Friedmann [Friedmann 2002] and presented in [B¨os 2003a], namely, a bicubic spline surface, a Hermite spline surface, and a tensor product B´ezier surface consisting of Bernstein polynomials. Numerical experiments showed [Friedmann 2002, B¨os 2003a] that the optimization results are very similar for all three spline functions and that none of them can be considered significantly superior to the other two. Only the Hermite spline surface seemed to exhibit slight advantages with respect to the number of iterations needed to reach convergence as well as to the objective function improvement.

6.2

Bicubic Spline Surface

75

Hence, all three spline formulations are briefly introduced in the following three sections (Secs. 6.2 through 6.4) although only Hermite spline surfaces were used for the optimization calculations using splines, the results of which are presented in Secs. 7.3 and 9.3. For the three examples the dimensions of the x-y plane are chosen to be the same as the dimensions of the rectangular plate described in Sec. 5.2, i.e., 350 mm × 233.33 mm. Seven key points are used in the x direction and five key points in the y direction. For demonstration purposes, only two of the key points are out-of-plane and must be interpolated by the spline surfaces. The z coordinates of these two out-of-plane key points were set arbitrarily to the dimensionless values 0.5 and 1.0, which do not have any physical meaning. A rectangular grid is required, but it does not have to be equidistant. No derivatives but only the z coordinates of the key points are supplied by the user.

6.2

Bicubic Spline Surface

An example of a three-dimensional bicubic spline surface that is defined by 35 key points denoted by dots is shown in Fig. 6.2. To calculate the spline surface the first and mixed derivatives at the key points are determined from the given nodal coordinates by means of finite differences. The bicubic spline surface is generated in such a way that it is C2 continuous, which means that not only the nodal coordinates and the gradients (first derivatives) but also the curvatures (second derivatives) of the surface match at the key points.

Figure 6.2: Bicubic spline surface defined by 7×5 key points. As can be seen from the figure, the bicubic spline function reproduces all key points exactly but tends to “overshoot” between the key points. Nevertheless, the surface defined by the key points is smooth and well approximated.

76

6.3

6

Employing Spline Functions to Reduce the Number of Design Variables

Hermite Spline Surface

Figure 6.3 depicts a three-dimensional Hermite spline surface consisting of 35 key points marked by dots. A spline interpolation is called a Hermite spline interpolation if not only the coordinates of the key points but also the first derivatives are given. For the method presented here a bicubic spline surface is generated in a similar way as described in the previous section. However, instead of calculating the derivatives by means of finite differences, the derivatives are preset such that there are no extrema between the key points. This is achieved by setting certain derivatives to zero in a suitable manner and recalculating the spline surface [Schmidt 1988]. In this process it is aspired, though it cannot be guaranteed, that the second derivatives match exactly at the key points. Thus, Hermite splines are only C1 continuous, which means that only the nodal coordinates and the gradients (first derivatives) but not the curvatures (second derivatives) of the surface match at the key points.

Figure 6.3: Hermite spline surface defined by 7×5 key points. A comparison with the bicubic spline surface in Fig. 6.2 reveals that the key points are again reproduced exactly and that the surface is smooth as well. However, the Hermite spline function does not “overshoot” between the key points, so it seems to be less “stiff” than the bicubic spline function in Fig. 6.2. In particular, it can model a plane surface on the basis of only two neighboring in-plane key points, whereas it takes three consecutive in-plane key points to obtain the same effect with bicubic spline functions. This is due to the preset derivative values mentioned above. It can be assumed that this behavior of the Hermite spline surface is the reason for its slight superiority when compared with the two other spline surface types. Therefore, the Hermite spline surface was used for the optimization calculations employing splines shown in Secs. 7.3 and 9.3.

6.4

6.4

Tensor Product B´ezier Surface

77

Tensor Product B´ ezier Surface

An example for a three-dimensional tensor product B´ezier surface, which consists of Bernstein polynomials and is controlled by 35 key points, can be seen in Fig. 6.4.

Figure 6.4: Tensor product B´ezier surface defined by 7×5 key points. The major difference between the B´ezier polynomial surface and the two spline formulations introduced in the previous sections is obvious. The key points are no longer reproduced exactly by the surface but merely serve as a means of controlling the shape of the surface. The tensor product B´ezier surface is very “stiff”, i.e., the position of a key point must be changed quite drastically in order to considerably change the shape of the surface. Another major difference between the B´ezier polynomial surface and the two spline formulations mentioned earlier is the fact that the number of key points in each direction automatically defines the order of the Bernstein polynomial in the respective direction. In order to use a Bernstein polynomial of order m one has to use (m + 1) key points. Therefore, the tensor product B´ezier surface depicted in Fig. 6.4 is defined by 6th order polynomials in x direction and 4th order polynomials in y direction. In contrast to this, the order of the aforementioned spline functions is independent of the number of key points.

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Employing Spline Functions to Reduce the Number of Design Variables

7

Optimization Results: Rectangular Plate

This chapter presents the optimization results for the rectangular plate introduced in Sec. 5.2. First, the FE analysis results of the original plate structures are given in Sec. 7.1. Then, Sec. 7.2 describes the optimization results obtained with COBYLA for the coarse and the refined FE mesh and for various objective functions. Section 7.3 presents results of optimizations employing COBYLA and spline functions to reduce the number of design variables and computation time. Finally, the results of an optimization calculation using the genetic algorithm PIKAIA can be seen in Sec. 7.4.

7.1

Original Rectangular Plate

This section shows the FE analysis results for the original rectangular plate models prior to optimization, first for the refined FE discretization in Sec. 7.1.1, then for the coarse FE mesh in Sec. 7.1.2. This information is intended to provide a means of assessing the effectiveness of the optimization results by comparing them with the properties of the initial structures. 7.1.1

Refined Discretization

The MLS of the initial rectangular plate with a refined mesh (see Sec. 5.2.1) in the frequency range of interest (0–3000 Hz) is 83.9 dB, where the maximum LS level of 107.2 dB can be found in the spectrum at the fundamental frequency 256.4 Hz. The LS spectrum of the original structure is shown as a solid line in various plots such as, e.g., Fig. 7.6 on p. 84. The structural mass of the initial structure can easily be determined analytically. It shows excellent agreement with the number provided by the pre- and post-processing software MSC.Patran and by the FE solver ABAQUS, namely, 2.564 kg. The numerically calculated fundamental frequency of the initial structure (256.4 Hz) deviates only by 0.3% from the analytical value based on Kirchhoff’s plate theory, i.e., 257.2 Hz [Kollmann 2000]. Table 7.1 summarizes the results. The first three of the quantities listed in this table can serve as either objective function or constraint for the optimization calculations. 7.1.2

Coarse Discretization

The initial FE model of the rectangular plate with a coarse discretization (see Sec. 5.2.2) has an MLS of 83.6 dB in the frequency range of 0–3000 Hz and a maximum LS level of 106.8 dB at the fundamental frequency 257.4 Hz. This model’s original LS spectrum can be seen as a solid line in various figures such as, e.g., Fig. 7.12 on p. 88.

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Optimization Results: Rectangular Plate

Table 7.1: Properties of the original rectangular plate (refined FE discretization). Property

Value

MLS 83.9 dB Mass m 2.564 kg Fundamental frequency f1 256.4 Hz Max. LS (at f1 = 256.4 Hz) 107.2 dB

The structural mass of the plate’s FE model with the coarse mesh is the same as the one for the refined discretization, i.e., 2.564 kg. Interestingly, the fundamental frequency is now 257.4 Hz, which deviates only by 0.08% from the analytical value of 257.2 Hz. A summary of the results is given in Tab. 7.2. MLS, mass, and fundamental frequency can become either objective function or constraint. Table 7.2: Properties of the original rectangular plate (coarse FE discretization). Property

Value

MLS 83.6 dB Mass m 2.564 kg Fundamental frequency f1 257.4 Hz Max. LS (at f1 = 257.4 Hz) 106.8 dB

7.2

Rectangular Plate Optimized with COBYLA

The following sections show the optimization results for the rectangular plate in case the COBYLA algorithm is used. Section 7.2.1 presents the results of an MLS minimization of the plate’s FE model with the refined mesh, whereas Sec. 7.2.2 provides the results for the coarse mesh. As it turns out, the outcome is very similar, but the number of iterations required for convergence and the corresponding computation time for the model with the coarse mesh are just fractions of the ones for the refined discretization. Therefore, the minimization of the structural mass (Sec. 7.2.3), the maximization of the fundamental frequency (Sec. 7.2.4), and the maximization of the difference between two natural frequencies (Sec. 7.2.5) were all performed on the coarse mesh plate model only. 7.2.1

Minimization of MLS (Refined FE Mesh)

First, the optimization results for the simply supported rectangular plate with the refined mesh as described in Sec. 5.2.1 and depicted in Fig. 5.1 on p. 62 are introduced. The MLS is the objective function that is to be minimized by optimizing the plate’s thickness distribution. The maximum allowable mass of the plate is set equal to the mass of the initial configuration mmax = m(0) = 2.564 kg, i.e., the structure is allowed to become lighter during optimization, but it is not permitted to become heavier. The upper and lower limit on the thickness of the

7.2

81

Rectangular Plate Optimized with COBYLA

mean level of structure borne sound MLS [dB]

structure are specified as tmax = 10 mm and tmin = 1 mm, respectively, whereas the initial thickness was t(0) = 4 mm. Therefore, the number of design variables and constraints for this optimization problem is 345 and 691, respectively (see Fig. 5.2 on p. 63). Figures 7.1 and 7.2 show the iteration history plots without any and with a slight constraint violation (marked by circles), respectively.

92 MLS value during optimization no constraint violation MLS of original structure

90 88 86 84 82 80 78 76 74

0

2000

4000

6000 8000 10000 number of iterations

12000

14000

mean level of structure borne sound MLS [dB]

Figure 7.1: Iteration history for the rectangular plate (COBYLA, MLS minimization, refined FE mesh, circles denote iterations without any mass or thickness constraint violation).

92 MLS value during optimization max. constraint violation < 0.01% MLS of original structure

90 88 86 84 82 80 78 76 74

0

2000

4000

6000 8000 10000 number of iterations

12000

14000

Figure 7.2: Iteration history for the rectangular plate (COBYLA, MLS minimization, refined FE mesh, circles denote iterations with a maximum constraint violation of less than 0.01%).

82

7

Optimization Results: Rectangular Plate

Obviously, COBYLA is not able to create an optimized design that is feasible in the strict mathematical sense as discussed in Sec. 4.2.3 even though the optimization converges (see Fig. 7.1). As can be seen from Fig. 7.2, however, there are only negligible constraint violations of less than 0.01%, which are considered tolerable (see Sec. 4.2.3), even at an early stage of the optimization. The MLS is reduced by 8.6 dB (−86.2%) from 83.9 dB to 75.3 dB. The structural mass of the optimized design is the same as for the initial design, i.e., there is no violation of the mass constraint. Both thickness constraints, however, are slightly violated by 0.002%. It takes 15 037 iterations and 152.75 h of CPU time on an Intel Pentium 4 (2.533 GHz, 1.5 GB RAM, cf. Tab. A.1 on p. 178) to reach convergence. Approximately 85.7% of the CPU time (130.9 h) are taken up by ABAQUS for the FE analyses. Figure 7.2 suggests that an acceptable result can be obtained after about 6000 iterations although COBYLA’s formal convergence criterion is not yet met. At this point, a remarkable MLS reduction of 8.5 dB (−85.9%) from 83.9 dB to 75.4 dB is achieved, which hardly differs from the results obtained at convergence. The computation time, however, can be reduced by 62% to approximately 58.2 h by ignoring the formal convergence criterion. The history of the trust-region radius ρ as described in Sec. 3.2.3 is depicted in Fig. 7.3. The parameter is continuously reduced from ρbeg = 6 to ρend = 10−6 , i.e., until it reaches the convergence criterion. The reductions do not occur at a constant but rather at a quite irregular rate. For instance, it takes 4570 iterations to reduce ρ just once from 1.5 to 0.75, but it takes only 1510 iterations to reduce ρ eight times from 2.34 · 10−2 to 9.16 · 10−5 . 1

10

0

trust−region radius ρ

10

−1

10

−2

10

−3

10

−4

10

−5

10

−6

10

0

2000

4000

6000 8000 10000 number of iterations

12000

14000

Figure 7.3: History of the trust-region radius ρ for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh). A three-dimensional view of the optimized plate design can be seen in Fig. 7.4. The arrow indicates the location of the excitation force. The plate’s structural mass stays constant and is only redistributed to achieve a low MLS. The mass is concentrated at and near the excitation location in order to increase the input impedance, which reduces the vibrational levels. But this is not the only effect. The optimized thickness distribution can also be interpreted as a stiffening rib slightly diagonal across the middle of the plate, which efficiently suppresses

7.2

83

Rectangular Plate Optimized with COBYLA

vibrations as well. The contour plot of the plate’s optimized thickness distribution in Fig. 7.5 is another representation of the optimized design. The values in the spectrum on the righthand side of Fig. 7.5 are in meters, i.e., dark areas indicate a thickness of up to tmax = 10 mm, whereas light zones represent a thickness of down to tmin = 1 mm.

Z Y X

Figure 7.4: Optimized design of the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh, arrow indicates excitation point).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

X

1.8-03

1.0-03

Figure 7.5: Thickness distribution of the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh, values in m).

84

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Optimization Results: Rectangular Plate

Figure 7.6 shows the LS spectra of the original (solid line) and the optimized rectangular plate (dashed line). The Xs denote natural frequencies. The fundamental frequency f1 of the original plate is 256.4 Hz, the one of the optimized structure is increased by 123.6 Hz (+48.2%) to 380 Hz. The LS of the optimized structure is generally lower than the one of the original plate, which leads to the minimized MLS. The maximum LS of the original plate is 107.2 dB at the fundamental frequency, whereas the one of the optimized structure is decreased by 8.6 dB (−86.2%) to 98.6 dB at the new fundamental frequency. This significant decrease dominates the MLS reduction. The greatest LS peak reduction of 19.5 dB (−98.9%) from 100.4 dB to 80.9 dB can be observed for the third mode. Interestingly, the number of natural frequencies within the frequency range of interest stays constant at 15 despite the optimization, which means that the MLS reduction is not caused by shifting natural frequencies out of the frequency range but purely by reducing the LS levels.

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.6: LS spectra of the original and the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh). Table 7.3 summarizes the optimization results for the MLS minimization of the rectangular plate with a refined discretization using COBYLA as described above. 7.2.2

Minimization of MLS (Coarse FE Mesh)

The second optimization example is the MLS minimization of the rectangular plate with the coarse FE discretization as introduced in Sec. 5.2.2 and shown in Fig. 5.3 on p. 64. The objective function and constraints are the same as in the previous section, resulting in 77 design variables and 155 constraints (see Fig. 5.4 on p. 64). A comparison between Fig. 7.7 and Fig. 7.8 reveals that this time the final, optimized design is feasible, but acceptable designs with a tolerably small, negligible constraint violation of less than 0.01% can be obtained considerably earlier in the process. The MLS reduction for the plate with the coarse FE mesh is 7.7 dB (−83%) from 83.6 dB to 75.9 dB, whereas it is 8.6 dB (−86.2%) from 83.9 dB to 75.3 dB for the refined FE discretization (see Sec. 7.2.1). Thus, there is 0.9 dB less reduction for the coarse mesh than for the refined one.

7.2

85

Rectangular Plate Optimized with COBYLA

Table 7.3: Optimization results for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh). Property MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

Initial design

Optimized design

83.9 dB 2.564 kg 4.0 mm 4.0 mm 256.4 Hz 107.2 dB 345 691

75.3 dB (−8.6 dB, −86.2%) 2.564 kg (no constraint violation) 0.99998 mm (0.002% constr. viol.) 10.00002 mm (0.002% constr. viol.) 380 Hz (+123.6 Hz, +48.2%) 98.6 dB (−8.6 dB, −86.2%) 345 691 15 037 6 d 8 h 45 min (152.75 h) ≈ 6000 75.4 dB (−8.5 dB, −85.9%) ≈ 58.2 h (−62%)

mean level of structure borne sound MLS [dB]

The computation time to reach convergence after 4137 iterations is now approximately 10 h on the same Intel Pentium 4 PC as before. This is an iteration reduction of 10 900 (−72.5%) and a CPU time reduction of 142.75 h (−93.5%) compared to the refined mesh (Sec. 7.2.1). About 86.8% of the CPU time (8.7 h) are used by the FE analyses. From Fig. 7.8 it can be seen that acceptable results are available after about 1000 iterations, which take only approximately 2.4 h of CPU time, a reduction of −76% compared to the computation time required for convergence. At this point the MLS is decreased by 7.5 dB (−82.2%) from 83.6 dB to 76.1 dB, which is only 0.2 dB less than the reduction obtained at convergence. 92 MLS value during optimization no constraint violation MLS of original structure

90 88 86 84 82 80 78 76 74

0

500

1000

1500 2000 2500 3000 number of iterations

3500

4000

Figure 7.7: Iteration history for the rectangular plate (COBYLA, MLS minimization, coarse FE mesh, circles denote iterations without any mass or thickness constraint violation).

86

mean level of structure borne sound MLS [dB]

7

Optimization Results: Rectangular Plate

92 MLS value during optimization max. constraint violation < 0.01% MLS of original structure

90 88 86 84 82 80 78 76 74

0

500

1000

1500 2000 2500 3000 number of iterations

3500

4000

Figure 7.8: Iteration history for the rectangular plate (COBYLA, MLS minimization, coarse FE mesh, circles denote iterations with a maximum constraint violation of less than 0.01%). The history of the trust-region radius ρ for the plate with the coarse discretization in Fig. 7.9 is just as irregular as the one for the refined FE mesh in Fig. 7.3. Sometimes 1151 iterations are required to reduce ρ just once from 4.69 · 10−2 to 2.34 · 10−2 , but then only 104 iterations are needed to decrease ρ five times from 1.47 · 10−3 to 4.58 · 10−5 . After 4137 iterations the trust-region radius ρ is equal to the final value ρend = 10−6 , i.e., the convergence criterion is met. 1

10

0

trust−region radius ρ

10

−1

10

−2

10

−3

10

−4

10

−5

10

−6

10

0

500

1000

1500 2000 2500 3000 number of iterations

3500

4000

Figure 7.9: History of the trust-region radius ρ for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh).

7.2

Rectangular Plate Optimized with COBYLA

87

Though not identical, the optimized geometry of the plate model with the coarse FE mesh shown in Figs. 7.10 and 7.11 closely resembles that of the plate with the refined discretization depicted in Figs. 7.4 and 7.5. In particular, the same mass concentration at and near the excitation point, which increases the input impedance, and the same stiffening, slightly diagonal rib, which lessens vibrational levels and increases some natural frequencies, can be observed. As for the model with the refined mesh the structural mass stays constant, and the material is just redistributed to other locations where it can help to reduce the MLS. Nowhere on the plate does the thickness exceed the upper limit tmax = 10 mm (dark areas in Fig. 7.11) or fall below the lower limit tmin = 1 mm (light regions).

Z Y X

Figure 7.10: Optimized design of the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh, arrow indicates excitation point). The LS spectra of the original and the optimized rectangular plate with the coarse discretization can be seen in Fig. 7.12. The LS spectrum of the original structure with the coarse discretization in Fig. 7.12 is very similar to the one with the refined mesh in Fig. 7.6, but the LS spectra of the optimized plates differ due to the slightly different optimized geometries as shown above. As a consequence of the optimization, the fundamental frequency is increased by 146.1 Hz (+56.8%) from 257.4 Hz to 403.5 Hz. The generally lower LS yields a reduced MLS of 75.9 dB. Prior to optimization, the maximum LS peak was 106.8 dB at the fundamental frequency, whereas it is reduced by 8.3 dB (−85.2%) to 98.5 dB at the new fundamental frequency due to the optimization process. This time, the number of natural frequencies in the frequency range of interest is reduced from 14 to 12, so the MLS reduction is not only caused by peak level reductions but also by the fact that two natural frequencies are shifted beyond the upper frequency limit. Table 7.4 presents a summary of the optimization results for the MLS minimization of the rectangular plate with a coarse FE mesh using COBYLA as described above. The table also contains additional lines that compare the optimization results for the coarse mesh with the ones for the refined discretization as summarized in Tab. 7.3.

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Optimization Results: Rectangular Plate

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

1.8-03

X

1.0-03

Figure 7.11: Thickness distribution of the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.12: LS spectra of the original and the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh).

7.2

89

Rectangular Plate Optimized with COBYLA

Table 7.4: Optimization results for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh). Property MLS Compared to refined mesh Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to refined mesh CPU time until convergence Compared to refined mesh Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

7.2.3

Initial design

Optimized design

83.6 dB −0.3 dB 2.564 kg 4.0 mm 4.0 mm 257.4 Hz 106.8 dB 77 155

75.9 dB (−7.7 dB, −83%) +0.6 dB (0.9 dB less reduction) 2.564 kg (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 403.5 Hz (+146.1 Hz, +56.8%) 98.5 dB (−8.3 dB, −85.2%) 77 155 4137 −10 900 (−72.5%) 10 h 2 min (≈ 10 h) −142.75 h (−93.5%) ≈ 1000 76.1 dB (−7.5 dB, −82.2%) ≈ 2.4 h (−76%)

Minimization of Mass

As mentioned in the first paragraph of Sec. 7.2 above, the remaining optimization results in Sec. 7.2 were all obtained exclusively for the rectangular plate with the coarse FE mesh because of the smaller number of design variables and, thus, drastically reduced CPU times compared to the ones for the refined discretization. The minimization of structural mass subject to an MLS constraint can be considered a reverse optimization problem to the MLS minimization subject to a mass constraint described in Sec. 7.2.2. The latter reduces vibrational levels without adding mass, whereas the former produces lightweight design proposals without increasing vibrational levels. Now the structural mass is to be minimized by optimizing the thickness distribution, but the MLS is not permitted to increase. The thickness constraints are the same as before. In the iteration history plot in Fig. 7.13 iterations with a slight constraint violation of less then 0.01% are marked with a circle. The figure shows that acceptable improved designs are available after about 1000 iterations, but it takes 3088 iterations to meet COBYLA’s formal convergence criterion. Although the final optimized design at convergence is feasible, only the history plot with slightly violated constraints is shown here since the first feasible design occurs after 2686 iterations, whereas the first acceptable design with a negligible constraint violation is obtained after only 976 iterations. The structural mass of the optimized plate after 3088 iterations is reduced by 0.990 kg (−38.6%) from 2.564 kg to 1.574 kg compared to the initial configuration. Neither the MLS constraint (MLSmax = MLS(0) = 83.6 dB) nor the usual thickness constraints are violated. The CPU time required to reach convergence after 3088 iterations is approximately 10.1 h on the Intel Pentium 4 PC (see Tab. A.1 on p. 178). The FE analyses using ABAQUS take

90

7

Optimization Results: Rectangular Plate

mass [kg]

2.5

mass during optimization max. constraint violation < 0.01% mass of original structure

2.0

1.5

1.0

0

500

1000 1500 2000 number of iterations

2500

3000

Figure 7.13: Iteration history for the rectangular plate (COBYLA, mass minimization, coarse FE mesh, circles denote iterations with a maximum constraint violation of less than 0.01%). about 8.8 h or 87.1% of the total CPU time. Acceptable results (m = 1.578 kg, −0.986 kg, −38.5%) are available after about 1000 iterations, which take only 3.3 h of CPU time (−67%). The optimized plate with minimal mass is shown in Figs. 7.14 and 7.15. Obviously, material is removed down to the lower thickness limit tmin = 1 mm almost everywhere on the plate except for the excitation point close to which some remaining material is concentrated and where the thickness is at its upper limit tmax = 10 mm. This measure as well as the stiffening “ribs” in the shape of an upside-down Y keep the MLS below its upper limit.

Z Y X

Figure 7.14: Optimized design of the rectangular plate (COBYLA, mass minimization, MLS and thickness constraints, coarse FE mesh, arrow indicates excitation point). The LS spectrum of the mass optimized plate in Fig. 7.16 does not differ much from the spectrum of the initial plate. This is not surprising since their MLS are the same. At some frequencies the LS of the optimized plate is slightly above the one of the original structure, at other frequencies the situation is vice versa. The initial fundamental frequency 257.4 Hz

7.2

91

Rectangular Plate Optimized with COBYLA

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

1.8-03

X

1.0-03

Figure 7.15: Thickness distribution of the optimized rectangular plate (COBYLA, mass minimization, MLS and thickness constraints, coarse FE mesh, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.16: LS spectra of the original and the optimized rectangular plate (COBYLA, mass minimization, MLS and thickness constraints, coarse FE mesh).

92

7

Optimization Results: Rectangular Plate

drops by 20.2 Hz (−7.8%) to 237.2 Hz. Interestingly, the initial maximum LS of 106.8 dB at f1 = 257.4 Hz increases by 1.5 dB (+41.3%) to 108.3 dB at the new fundamental frequency f1 = 237.2 Hz. This is obviously compensated by levels below the original LS in other parts of the spectrum. Due to the lower mass and thickness of the plate, the number of natural frequencies in the frequency range of interest goes up from 14 to 19. Table 7.5 provides an overview of the optimization results for the mass minimization of the simply supported rectangular plate with the coarse FE discretization presented in this section. Table 7.5: Optimization results for the rectangular plate (COBYLA, mass minimization, MLS and thickness constraints, coarse FE mesh). Property Mass m MLS Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result Mass of acceptable design CPU time until acceptable result

7.2.4

Initial design

Optimized design

2.564 kg 83.6 dB 4.0 mm 4.0 mm 257.4 Hz 106.8 dB 77 155

1.574 kg (−0.990 kg, −38.6%) 83.6 dB (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 237.2 Hz (−20.2 Hz, −7.8%) 108.3 dB (+1.5 dB, +41.3%) 77 155 3088 10 h 7 min (≈ 10.1 h) ≈ 1000 1.578 kg (−0.986 kg, −38.5%) ≈ 3.3 h (−67%)

Maximization of the Fundamental Frequency

The same FE model of a rectangular plate as before (see Sec. 5.2.2) is used to optimize its thickness distribution such that the fundamental frequency f1 is maximized. This time, there are not only the usual thickness constraints (tmin = 1 mm and tmax = 10 mm), but also both an MLS constraint (MLSmax = MLS(0) = 83.6 dB) and a mass constraint (mmax = m(0) = 2.564 kg), yielding 77 design variables but 156 instead of only 155 constraints. The iteration history plot in Fig. 7.17 shows that COBYLA’s convergence criterion (see Sec. 4.2.2) is met after 2004 iterations and 5.2 h of CPU time on the Intel Pentium 4, 86.5% of which (4.5 h) were consumed by the ABAQUS FE analyses. The optimized design has a fundamental frequency f1 of 492.6 Hz, which is 235.2 Hz (+91.4%) higher than that of the initial structure (257.4 Hz). None of the constraints are violated. Nonetheless, Fig. 7.17 shows the iterations with a slight constraint violation of less than 0.01% in order to demonstrate that an acceptable design with negligibly small constraint violations and a fundamental frequency of 492.5 Hz (+235.1 Hz, +91.3%) can be found after about 800 iterations and 2.1 h of CPU time (60% less than required to reach convergence). The geometry of the optimized structure with the maximized fundamental frequency is depicted in Figs. 7.18 and 7.19. A massive rib across the plate, which is 10 mm high, has developed, thus maximizing the fundamental frequency due to stiffening effects. At other regions

7.2

93

Rectangular Plate Optimized with COBYLA

of the plate, the thickness is reduced to the minimum allowable 1 mm in order to prevent the structural mass from increasing. Interestingly, the thickness is reduced to the allowable minimum at the excitation point as well, thus actually reducing the input impedance, which appears to be counterproductive. However, the stiffening effects of the rib as well as the local thickening at the left-hand upper edge and at the lower right-hand corner of the plate seem to suffice to keep the MLS from exceeding its upper allowable limit of 83.6 dB. This can be considered the first case where it probably would have been difficult to find an “intuitive solution” as discussed in Sec. 1.3.1 (p. 24) due to the additional MLS and mass constraints, which require special attention.

fundamental frequency f1 [Hz]

500 450 400 fund. freq. f1 during optimization max. constraint violation < 0.01% fund. freq. f1 of original structure

350 300 250 200

0

200

400

600

800 1000 1200 1400 1600 1800 2000 number of iterations

Figure 7.17: Iteration history for the rectangular plate (COBYLA, fundamental frequency maximization, coarse FE mesh, circles denote iterations with a maximum constraint violation of less than 0.01%).

Z Y X

Figure 7.18: Optimized design of the rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints, coarse FE mesh, arrow indicates excitation point).

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7

Optimization Results: Rectangular Plate

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

X

1.8-03

1.0-03

Figure 7.19: Thickness distribution of the optimized rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints, coarse FE mesh, values in m). The maximization of the fundamental frequency f1 is clearly visible in the LS spectra in Fig. 7.20. It is remarkable that there are relatively high LS peaks for the optimized structure throughout the whole frequency range of interest. Particularly, at 1539.7 Hz, 1834.0 Hz, and 2713.1 Hz there are LS peaks that exceed 95 dB. The maximum LS, which was 106.8 dB at the fundamental frequency 257.4 Hz for the original rectangular plate, is now decreased by 6.2 dB (−76.0%) to 100.6 dB, but, surprisingly, it can be found at the third natural frequency, i.e., at 1044.4 Hz instead of at the new fundamental frequency 492.6 Hz. However, the LS peak at the new fundamental frequency is close behind, namely, 100.3 dB. The regions with a low LS, e.g., 600–900 Hz, 1400–1500 Hz, and 1900–2200 Hz seem to compensate the otherwise relatively high levels so that the MLS stays below its upper limit. The number of natural frequencies within the frequency range of interest, namely, 14, is not changed by the optimization procedure. Table 7.6 contains the usual summary of the optimization results, this time for the fundamental frequency maximization of the simply supported rectangular plate as presented in this section. It reflects the fact that there are two additional constraints besides the usual 154 thickness constraints, namely, both an MLS and a mass constraint, thus increasing the total number of constraints to 156.

7.2

95

Rectangular Plate Optimized with COBYLA

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.20: LS spectra of the original and the optimized rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints, coarse FE mesh). Table 7.6: Optimization results for the rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints, coarse FE mesh). Property Fundamental frequency f1 MLS Mass m Minimum thickness tmin Maximum thickness tmax Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result f1 of acceptable design CPU time until acceptable result

7.2.5

Initial design

Optimized design

257.4 Hz 83.6 dB 2.564 kg 4.0 mm 4.0 mm 106.8 dB 77 156

492.6 Hz (+235.2 Hz, +91.4%) 83.6 dB (no constraint violation) 2.564 kg (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 100.6 dB (−6.2 dB, −76.0%) 77 156 2004 5 h 12 min (5.2 h) ≈ 800 492.5 Hz (+235.1 Hz, +91.3%) ≈ 2.1 h (−60%)

Maximization of the Difference between Two Natural Frequencies

In this section, optimization results are reported for the maximization of the difference between two given natural frequencies subject to the same MLS, mass, and thickness constraints as in Sec. 7.2.4. Therefore, there are again 77 design variables and 156 constraints. The differ(0) ence ∆f between the third and the fourth natural frequency, which are f3 = 804.5 Hz and (0) f4 = 901.2 Hz for the initial geometry, is chosen to be maximized by optimizing the thickness

96

7

Optimization Results: Rectangular Plate

minimum frequency difference ∆fmin [Hz]

distribution of the rectangular plate. To accomplish this, some frequency f ∗ = 850 Hz, which is (0) (0) somewhere between f3 and f4 , is specified, and the minimum difference ∆fmin = f4 − f ∗ or ∆fmin = f ∗ − f3 , whichever is smaller during the optimization process, is to be maximized (cf. Eq. (4.1) in Sec. 4.2.1 on p. 55). As described in Sec. 4.2.1, this finally leads to a design with maximized differences ∆f1 and ∆f2 that are symmetrical with respect to f ∗ , i.e., ∆f1 = ∆f2 . Thus, the difference between the two natural frequencies ∆f = ∆f1 + ∆f2 = f4 − f3 is maximized. This approach can be useful if some harmonic excitation force with known frequency f ∗ exists. In order to avoid resonance, one would want those natural frequencies of the structure that are closest to the excitation frequency f ∗ to be as far away from it as possible. Figure 7.21 shows the iteration history of this optimization procedure where the iterations indicated by circles, including the final one, produced feasible designs. COBYLA’s convergence criterion is reached after 3747 iterations, which takes 9.4 h of CPU time on the Intel Pentium 4 PC (see Tab. A.1 on p. 178). The FE analyses alone take 8.2 h or 87.2% of the total CPU time. The smallest difference between the specified frequency f ∗ = 850 Hz and its neighboring natural frequencies for the optimized design is 173.6 Hz, an increase by 128.1 Hz or 281.5% from the initial value 45.5 Hz. Both differences are equal and symmetric with respect to f ∗ , i.e., ∆f1 = ∆f2 = f4 − f ∗ = f ∗ − f3 = 173.6 Hz, yielding a maximum difference between the two natural frequencies of ∆f = ∆f1 + ∆f2 = f4 − f3 = 347.2 Hz compared to 96.7 Hz for the initial plate, an increase by 250.5 Hz or 259.0%. An acceptable result can be obtained after only about 1700 iterations or 4.3 h of CPU time, a CPU time reduction of 54%. The minimum frequency difference is then ∆fmin = 173.2 Hz, an improvement by 127.7 Hz or +280.7%.

150

min. freq. diff. ∆fmin during optimization no constraint violation min. freq. diff. ∆fmin of original structure

100

50

0

0

500

1000

1500 2000 2500 number of iterations

3000

3500

4000

Figure 7.21: Iteration history for the rectangular plate (COBYLA, ∆f maximization, coarse FE mesh, circles denote iterations without any MLS, mass, or thickness constraint violation). Figures 7.22 and 7.23 depict the optimized geometry and thickness distribution, respectively. The thickness at the excitation point is at its upper limit (10 mm) in order to keep the MLS low by means of increasing the input impedance. All other modifications cannot be easily explained and serve to keep the mass below its upper limit (in fact, the plate is even 0.090 kg or 3.5% lighter than before) and to maximize the minimum frequency difference ∆fmin .

7.2

97

Rectangular Plate Optimized with COBYLA

Z Y X

Figure 7.22: Optimized design of the rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, coarse FE mesh, arrow indicates excitation point).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

X

1.8-03

1.0-03

Figure 7.23: Thickness distribution of the optimized rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, coarse FE mesh, values in m). This is the second example where an “intuitive” approach based on engineering expertise as discussed in Sec. 1.3.1 (p. 24) probably would produce significantly inferior results due to the somewhat unusual objective function and additional constraints. Since it is not obvious in advance how a structure needs to be modified in order to drive its third and fourth natural

98

7

Optimization Results: Rectangular Plate

frequencies apart, one might have to employ some trial and error approach to come up with a similar design. The maximized frequency difference can be seen in the LS spectrum of the optimized structure in Fig. 7.24. The specified frequency f ∗ = 850 Hz is marked with a dot-dashed line. The fundamental frequency is increased by 20.6 Hz (+8.0%) from 257.4 Hz to 278.0 Hz. High levels in some parts of the frequency range are compensated by low levels in other regions of the spectrum, thus preventing an increase of the MLS beyond its upper limit of 83.6 dB, which is the MLS of the original plate. The maximum LS of 106.8 dB at the initial fundamental frequency goes up by 0.8 dB (+20.2%) to 107.6 dB at the new fundamental frequency. The number of natural frequencies in the frequency range of interest is reduced from 14 to 11, thus helping to keep the MLS below its upper limit.

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.24: LS spectra of the original and the optimized rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, coarse FE mesh, dot-dashed line marks f ∗ = 850 Hz). A summary of the optimization results for the maximization of the difference between two given natural frequencies is presented in Tab. 7.7. Note that the table (as well as Fig. 7.21) shows the minimum difference ∆fmin between f ∗ = 850 Hz and its closest neighboring natural frequency instead of the actual difference between the natural frequencies ∆f . However, ∆f can easily be derived from ∆fmin by multiplying it by a factor of two, thus yielding ∆f = 2∆fmin = 347.2 Hz for the final, optimized design.

7.3

99

Rectangular Plate Optimized with COBYLA Using Splines

Table 7.7: Optimization results for the rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, coarse FE mesh). Property Minimum frequency difference ∆fmin MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result ∆fmin of acceptable design CPU time until acceptable result

7.3

Initial design

Optimized design

45.5 Hz 83.6 dB 2.564 kg 4.0 mm 4.0 mm 257.4 Hz 106.8 dB 77 156

173.6 Hz (+128.1 Hz, +281.5%) 83.6 dB (no constraint violation) 2.474 kg (−0.090 kg, −3.5%) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 278.0 Hz (+20.6 Hz, +8.0%) 107.6 dB (+0.8 dB, +20.2%) 77 156 3747 9 h 24 min (9.4 h) ≈ 1700 173.2 Hz (+127.7 Hz, +280.7%) ≈ 4.3 h (−54%)

Rectangular Plate Optimized with COBYLA Using Splines

This section shows the results of some optimization calculations for which Hermite spline functions were used to reduce the number of design variables and, hence, the number of iterations as well as the CPU time required to meet COBYLA’s convergence criterion. CPU time reductions of up to can be obtained by this approach with only slightly smaller improvements of the objective function. Optimization results for the MLS minimization of the rectangular plate with the coarse FE mesh using Hermite spline functions are introduced in Sec. 7.3.1. Section 7.3.2 presents the results for the same optimization problem, but for the refined FE discretization instead of the coarse mesh. Finally, Sec. 7.3.3 describes the results of the mass minimization of the rectangular plate with the coarse FE mesh using Hermite spline functions. All results are compared with their respective counterparts in Sec. 7.2 where the design variables are modified directly without using spline functions. 7.3.1

Minimization of MLS (Coarse FE Mesh)

The first example of an optimization calculation employing spline functions is the MLS minimization of the simply supported rectangular plate with the coarse FE mesh as described in Sec. 5.2.2 and depicted in Fig. 5.3 on p. 64. It is the same optimization problem as investigated in Sec. 7.2.2 without spline functions. This time, however, the nodal positions of the 77 movable surface nodes shown in Fig. 5.4 on p. 64 are not modified directly by COBYLA but indirectly by means of a Hermite spline function. The shape of the spline surface and therefore the thickness variations of the plate are now controlled by only 35 design variables, i.e., by the 7×5 key points of the spline surface as described in Secs. 6.1 and 6.3 and visualized in Fig. 6.3 on p. 76. The number of constraints stays the same as in Sec. 7.2.2, namely, 155.

100

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Optimization Results: Rectangular Plate

mean level of structure borne sound MLS [dB]

The iteration history in Fig. 7.25 looks quite different from the one in Fig. 7.7 on p. 85 because another start vector of design variables x as well as another initial trust-region radius ρbeg had to be used due to the splines. With the Hermite spline surface it takes 1850 iterations to meet the convergence criterion with a feasible design, 2287 or 55.3% less than that required for the direct manipulation of the movable surface nodes in Sec. 7.2.2. The consumed CPU time is 5 h, which is 5 h or 50% less than that needed without splines. The FE analyses by ABAQUS take about 4.4 h or 88% of the total CPU time. Upon convergence, the MLS is reduced by 7.4 dB (−81.8%) from 83.6 dB to 76.2 dB. This improvement is only 0.3 dB smaller than that without splines, but it is achieved after only 50% of the computation time. After about 830 iterations or 2.2 h of CPU time (−56%) an acceptable design is available with an MLS of 76.5 dB, a reduction of 7.1 dB or 80.5% compared to the original structure. 86 84 82 80 78 76 MLS value during optimization no constraint violation MLS of original structure

74 72 70

0

500

1000 number of iterations

1500

2000

Figure 7.25: Iteration history for the rectangular plate (COBYLA, Hermite splines, MLS minimization, coarse FE mesh, circles denote iterations without any mass or thickness constraint violation). The geometry of the optimized plate in Fig. 7.26 and the thickness distribution in Fig. 7.27 are remarkably similar to the ones for the direct thickness modification in Fig. 7.10 on p. 87 and Fig. 7.11 on p. 88, respectively. The general shape and thickness distribution are the same, but some small details to the left and right of the stiffening rib across the plate’s center are omitted due to the approximative character of the spline surface. For further interpretation of the optimization results see Sec. 7.2.2. The LS spectra of the optimized structure in Fig. 7.28 on p. 102 and Fig. 7.12 on p. 88 are also very similar up to a frequency of about 1500 Hz, above which they deviate, probably due to the influence of the small geometry differences mentioned above. The fundamental frequency rises by 144.8 Hz (+56.3%) from 257.4 Hz to 402.2 Hz, whereas the maximum LS decreases by 7.7 dB (−83.0%) from 106.8 dB to 99.1 dB. In general, the LS of the optimized plate is lower than that of the original structure, thus minimizing the MLS. The number of natural frequencies within the frequency band of interest (0–3000 Hz) does not change as a result of the optimization.

7.3

101

Rectangular Plate Optimized with COBYLA Using Splines

Z Y X

Figure 7.26: Optimized design of the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh, arrow indicates excitation point).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

X

1.8-03

1.0-03

Figure 7.27: Thickness distribution of the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh, values in m).

102

7

Optimization Results: Rectangular Plate

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.28: LS spectra of the original and the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh). Table 7.8 on p. 102 provides the usual summary of the optimization results for the MLS minimization of the simply supported rectangular plate with the coarse discretization using Hermite spline functions as introduced in this section. The results are compared with the ones obtained without spline functions given in Tab. 7.4 on p. 89. Table 7.8: Optimization results for the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh). Property MLS Compared to direct modification Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to direct modification CPU time until convergence Compared to direct modification Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

Initial design

Optimized design

83.6 dB

76.2 dB (−7.4 dB, −81.8%) +0.3 dB (0.3 dB less reduction) 2.564 kg (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 402.2 Hz (+144.8 Hz, +56.3%) 99.1 dB (−7.7 dB, −83.0%) 35 key points (−54.6%) 155 1850 −2287 (−55.3%) 5h −5 h (−50.0%) ≈ 830 76.5 dB (−7.1 dB, −80.5%) ≈ 2.2 h (−56%)

2.564 kg 4.0 mm 4.0 mm 257.4 Hz 106.8 dB 77 155

7.3

103

Rectangular Plate Optimized with COBYLA Using Splines

7.3.2

Minimization of MLS (Refined FE Mesh)

mean level of structure borne sound MLS [dB]

Next, Hermite spline functions are used to reduce the number of design variables for the same optimization problem as described in Sec. 7.2.1, namely, the MLS minimization of the simply supported rectangular plate with the refined FE discretization as introduced in Sec. 5.2.1 and shown in Fig. 5.1 on p. 62. This is done to investigate if substantially greater reductions of the number of design variables can decrease computation times considerably while still yielding acceptable optimized designs. As in Sec. 7.3.1, the thickness distribution is manipulated by means of a Hermite spline function whose shape is controlled by 7×5 key points, i.e., 35 design variables, as shown in Secs. 6.1 and 6.3 and depicted in Fig. 6.3 on p. 76. Thus, the number of design variables is reduced by 89.9% from 345 to 35, but there are still 691 mass and thickness constraints as in Sec. 7.2.1. The iterations converge after 18.35 h of CPU time on the usual Intel Pentium 4 PC, i.e., after 1645 iterations (Fig. 7.29). This is just 12% of the CPU time and 11% of the number of iterations required by the optimization calculation without splines in Sec. 7.2.1. The final optimized design is feasible, so none of the 691 constraints is violated. ABAQUS needs 87.2% of the total CPU time (16 h) for the FE analyses. The MLS decreases by 7.4 dB (−81.8%) from 83.9 dB to 76.5 dB. This reduction is 1.2 dB smaller than the one without splines (see Tab. 7.3 on p. 85). An acceptable result of 76.8 dB (−7.1 dB, −80.5%) is available after only 520 iterations, which take about 5.8 h of CPU time, a decrease of 68% compared to the fully converged solution. 86 84 82 80 78 76 MLS value during optimization no constraint violation MLS of original structure

74 72 70

0

500

1000 number of iterations

1500

Figure 7.29: Iteration history for the rectangular plate (COBYLA, Hermite splines, MLS minimization, refined FE mesh, circles denote iterations without any mass or thickness constraint violation). As in the previous section, the geometry and the thickness distribution of the optimized structure in Figs. 7.30 and 7.31, respectively, closely resemble the ones obtained without spline functions shown in Figs. 7.4 and 7.5 on p. 83. Again, some minor details are lost due to the rough approximation of the thickness distribution by the spline surface, but the general

104

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Optimization Results: Rectangular Plate

shape and thickness distribution are preserved. An interpretation of the results can be found in Sec. 7.2.1.

Z Y X

Figure 7.30: Optimized design of the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh, arrow indicates excitation point).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

X

1.8-03

1.0-03

Figure 7.31: Thickness distribution of the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh, values in m).

7.3

105

Rectangular Plate Optimized with COBYLA Using Splines

The LS spectra of the optimized plates in Fig. 7.32 and in Fig. 7.6 on p. 84 look alike for frequencies up to about 1300 Hz. Beyond that they more or less deviate, presumably due to the slightly different optimized shapes. In general, the LS of the optimized structure is below that of the original plate, which is the reason for the significantly reduced MLS. The fundamental frequency is increased by 119.9 Hz (46.8%) from 256.4 Hz to 376.3 Hz, and the maximum LS peak is decreased by 8.1 dB (−84.5%) from 107.2 dB to 99.1 dB. The greatest reduction of an LS peak can be observed for the third natural frequency: 24.5 dB (−99.7%) from 100.4 dB to 75.9 dB. The number of natural frequencies in the frequency range of interest goes up from 15 to 18 as a result of the optimization process.

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.32: LS spectra of the original and the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh). A summary of the optimization results for the MLS minimization of the rectangular plate with the refined FE mesh using Hermite spline functions as presented in this section is given in Tab. 7.9. Some of these results are compared with the ones provided in Tab. 7.3 on p. 85 for optimization calculations without using spline functions. 7.3.3

Minimization of Mass

The last example for the reduction of the number of design variables by applying Hermite spline functions to the rectangular plate is the mass minimization of the plate with the coarse FE discretization, whose outcome is compared with the one for the same optimization without splines shown in Sec. 7.2.3. As in the previous two sections, 7×5 key points control the shape of the Hermite spline surface that modifies the local thickness of the plate, resulting in 35 design variables instead of 77 to be controlled by COBYLA. There is a total of 155 MLS and thickness constraints as in Secs. 7.2.3 and 7.3.1. The iteration history in Fig. 7.33 shows that COBYLA’s convergence criterion is met after 1251 iterations, which is 1837 iterations or 59.5% less than required without splines. These 1251 iterations take about 4.1 h of CPU time, which is 6 h (59.4%) less than in Sec. 7.2.3. Since none of the 155 constraints is violated the optimized design is feasible. The FE analyses using

106

7

Optimization Results: Rectangular Plate

Table 7.9: Optimization results for the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh). Property MLS Compared to direct modification Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to direct modification CPU time until convergence Compared to direct modification Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

Initial design

Optimized design

83.9 dB

76.5 dB (−7.4 dB, −81.8%) +1.2 dB (1.2 dB less reduction) 2.564 kg (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 376.3 Hz (+119.9 Hz, +46.8%) 99.1 dB (−8.1 dB, −84.5%) 35 key points (−89.9%) 691 1645 −13 392 (−89.1%) 18 h 21 min (18.35 h) −134.4 h (−88.0%) ≈ 520 76.8 dB (−7.1 dB, −80.5%) ≈ 5.8 h (−68%)

2.564 kg 4.0 mm 4.0 mm 256.4 Hz 107.2 dB 345 691

ABAQUS require about 3.6 h or 87.8% of the total computation time. Upon convergence, the structural mass is brought down from 2.564 kg to 1.635 kg, a decrease of 0.929 kg or 36.2%. This improvement is only 0.061 kg or 6.2% smaller than the one in the optimization without splines, but the computation time is reduced by almost 60%. It is possible to obtain an acceptable optimization result ignoring COBYLA’s formal convergence criterion after only about 510 iterations and 1.7 h of CPU time, which is 59% less than that needed to reach convergence. At that point, the structural mass is reduced by 0.921 kg (35.9%) from 2.564 kg to 1.643 kg. This time, the optimized geometry in Fig. 7.34 and the thickness distribution in Fig. 7.35 on p. 108 look quite different from the ones in Fig. 7.14 on p. 90 and Fig. 7.15 on p. 91. This is due to the Hermite spline surface approximation, which is not able to model the fine details of the upside-down Y-shaped stiffening rib visible in Figs. 7.14 and 7.15. Instead, the optimization algorithm opts for an L-shaped rib with maximum allowable thickness at the excitation point. Interestingly, the little bulge in the lower left-hand corner is reproduced by the spline surface. Despite the differences in the optimized design, the LS spectra of the optimized structure in Fig. 7.36 on p. 108 and Fig. 7.16 on p. 91 are surprisingly similar. The fundamental frequency hardly changes due to the optimization (+0.9 Hz, +0.4%), the maximum LS peak is increased by 1.3 dB (34.9%) from 106.8 dB to 108.1 dB. Though the structural mass of the optimized plate is reduced by about 36% compared to the initial design, the overall LS does not go up, thus preventing the MLS from exceeding its upper limit. As for the plate optimized without using spline functions in Sec. 7.2.3, the number of natural frequencies in the frequency band from 0 to 3000 Hz increases from 14 to 19 due to the reduced thickness and the decreased mass of the plate.

7.3

107

Rectangular Plate Optimized with COBYLA Using Splines

3.0

mass [kg]

2.5

mass during optimization no constraint violation mass of original structure

2.0

1.5

0

500 1000 number of iterations

1500

Figure 7.33: Iteration history for the rectangular plate (COBYLA, Hermite splines, mass minimization, coarse FE mesh, circles denote iterations without any MLS or thickness constraint violation).

Z Y X

Figure 7.34: Optimized design of the rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, coarse FE mesh, arrow indicates excitation point). Table 7.10 on p. 109 gives an overview of the optimization results for the mass minimization of the simply supported rectangular steel plate with the coarse FE discretization employing Hermite spline functions, which are described in this section. Some of the results in Tab. 7.10 are compared with their respective counterparts in Tab. 7.5 on p. 92 for the same optimization without splines.

108

7

Optimization Results: Rectangular Plate

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

1.8-03

X

1.0-03

Figure 7.35: Thickness distribution of the optimized rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, coarse FE mesh, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.36: LS spectra of the original and the optimized rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, coarse FE mesh).

7.4

109

Rectangular Plate Optimized with PIKAIA

Table 7.10: Optimization results for the rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, coarse FE mesh). Property Mass m Compared to direct modification MLS Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to direct modification CPU time until convergence Compared to direct modification Iterations until acceptable result Mass of acceptable design CPU time until acceptable result

7.4

Initial design

Optimized design

2.564 kg

1.635 kg (−0.929 kg, −36.2%) +0.061 kg (0.061 kg less reduction) 83.6 dB (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 258.3 Hz (+0.9 Hz, +0.4%) 108.1 dB (+1.3 dB, +34.9%) 35 key points (−54.6%) 155 1251 −1837 (−59.5%) 4 h 7 min (≈ 4.1 h) −6 h (−59.4%) ≈ 510 1.643 kg (−0.921 kg, −35.9%) ≈ 1.7 h (−59%)

83.6 dB 4.0 mm 4.0 mm 257.4 Hz 106.8 dB 77 155

Rectangular Plate Optimized with PIKAIA: Minimization of Mass

This is the first example of an optimization using the genetic algorithm PIKAIA, which is explained in detail in Sec. 3.3. The objective of this optimization is to minimize the mass of the simply supported rectangular steel plate model with the coarse FE discretization, which is introduced in Sec. 5.2.2 and depicted in Fig. 5.3 on p. 64. As in previous mass minimizations of the same structure (see Secs. 7.2.3 and 7.3.3), there are 77 design variables and a total of 155 MLS and thickness constraints. There is no convergence criterion for PIKAIA as mentioned already in Sec. 4.3.2. However, experience shows that the objective function finally exhibits some sort of asymptotic behavior, so the optimization procedure is terminated manually after 50 000 iterations. Figure 7.37 visualizes how the fitness of the designs randomly created by PIKAIA, which is evaluated by means of Eq. (4.5), varies from iteration to iteration (gray line) and how the highest fitness value achieved by the fittest member of the population up to a particular iteration eventually increases as the optimization proceeds (black line). The greatest fitness prior to termination is achieved after 49 093 iterations. The fitness history in Fig. 7.37 directly corresponds to the actual objective function history in Fig. 7.38 due to the inversely proportional relation between fitness and objective function in Eq. (4.5). As can be seen from Fig. 7.38, most of the randomly created designs have a relatively great mass (gray line), which sometimes even exceeds that of the original structure (black dashed line). The solid black line represents the mass of those designs that have the highest fitness value up to that iteration, i.e., the lowest mass of a feasible design. There are

110

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Optimization Results: Rectangular Plate

600 500

fitness [−]

400

fitness value at individual iteration highest fitness value so far

300 200 100 0

0

5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 number of iterations

Figure 7.37: Iteration history for the rectangular plate (fitness, PIKAIA, mass minimization, MLS and thickness constraints). 3.5 mass at individual iteration lowest feasible mass so far mass of original structure

mass [kg]

3.0

2.5

2.0

1.5

0

5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 number of iterations

Figure 7.38: Iteration history for the rectangular plate (objective function, PIKAIA, mass minimization, MLS and thickness constraints).

iterations with a structural mass even below the solid black line. However, these are designs with a small mass but with some sort of MLS constraint violation, which results in a lower fitness value (see Sec. 4.3.1). There can be no thickness constraint violations since the design variables are scaled such that the local thicknesses remain within the allowable interval (see Sec. 4.3 and, in particular, Eq. (4.4) on p. 56). It seems worth noting that the MLS of the final design stays slightly below its upper limit, i.e., it is only 83.5 dB instead of the maximum allowable 83.6 dB.

7.4

Rectangular Plate Optimized with PIKAIA

111

After 49 093 iterations the plate’s mass is reduced by 0.883 kg or 34.4% from 2.564 kg to 1.681 kg. This improvement is 0.107 kg or 10.8% smaller than the one obtained by COBYLA in Sec. 7.2.3. Since the optimization calculations using PIKAIA were distributed to five different computers (see App. A, Sec. A.2) it would have been too complicated to record the pure CPU time. Instead, the so-called wall clock time, i.e., the real time elapsed during the optimization, was registered. This should give a rough estimate of the computation time since the computers were not used for other calculations when the PIKAIA computations were performed. PIKAIA’s 50 000 iterations took 13 days (312 h), which is about 31 times as long as the CPU time required by COBYLA to reach convergence (see Sec. 7.2.3). However, despite the longer time needed until termination, the mass reduction achieved by PIKAIA is about 11% smaller than the one obtained with COBYLA. Acceptable results can be obtained significantly earlier in the process but depend on the user’s assessment and decision. After only 7690 iterations or about 48 h the mass is reduced by 0.745 kg or 29.1% to 1.819 kg. If the user allows 12 364 iterations or 77.2 h (3 d 5 h 12 min) one can achieve a mass reduction of 0.776 kg or 30.3% to 1.788 kg. After 18 230 iterations or about 113.8 h (4 d 17 h 48 min) the objective function goes down by 0.812 kg (−31.7%) to 1.752 kg. And if the user is patient enough to wait until iteration 25 845 or for 161.3 h (6 d 17 h 18 min) one is rewarded with a mass decrease of 0.834 kg (−32.5 %) to 1.730 kg (see Fig. 7.38). Figures 7.39 and 7.40 depict the optimized geometry and the thickness distribution of the structure, respectively. The design of the mass minimized plate is completely different from the ones shown in Figs. 7.14 and 7.15 (using COBYLA) or Figs. 7.34 and 7.35 (employing COBYLA and Hermite splines). This time the stiffening rib is T-shaped instead of Y-shaped or L-shaped, but the input impedance is again increased by concentrating the remaining material at and near the excitation location up to the maximum allowable thickness of 10 mm, thus reducing the structural mass to a minimum.

Z Y X

Figure 7.39: Optimized design of the rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints, arrow indicates excitation point). The LS spectra in Fig. 7.41 indicate that the levels of the original and the optimized structure are similar, which yields even a slightly lower MLS. The fundamental frequency hardly changes from 257.4 Hz to 253.1 Hz (−4.3 Hz, −1.7%). As in the previous mass minimizations of the plate with the coarse FE mesh (cf. Fig. 7.16 on p. 91 or Fig. 7.36 on p. 108), the maximum LS at the fundamental frequency increases by 1.3 dB (34.9%) from 106.8 dB to 108.1 dB. The number of natural frequencies within the frequency band from 0 to 3000 Hz increases significantly from 14 to 20 because of the minimized mass and the reduced thickness of the plate.

112

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Optimization Results: Rectangular Plate

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Y Z

1.8-03

X

1.0-03

Figure 7.40: Thickness distribution of the optimized rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 7.41: LS spectra of the original and the optimized rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints).

7.4

113

Rectangular Plate Optimized with PIKAIA

As usual, Tab. 7.11 presents the optimization results for the mass minimization of the rectangular plate with the coarse FE mesh using PIKAIA in a clear and compact form. The efficiency of the optimization strategy employing PIKAIA is assessed by means of comparing the results with those obtained by COBYLA without using splines given in Tab. 7.5 on p. 92. Table 7.11: Optimization results for the rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints). Property Mass m Compared to COBYLA result MLS Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Termination after Real time until termination Highest fitness value after Compared to COBYLA result Iterations until acceptable result Mass of acceptable design Real time until acceptable result

Initial design

Optimized design

2.564 kg

1.681 kg (−0.883 kg, −34.4%) +0.107 kg (0.107 kg less reduction) 83.5 dB (no constraint violation) 1.0003 mm (no constraint violation) 9.8439 mm (no constraint violation) 253.1 Hz (−4.3 Hz, −1.7%) 108.1 dB (+1.3 dB, +34.9%) 77 155 50 000 iterations 13 d 0 h 2 min (≈ 312 h) 49 093 iterations +46 005 (+1489.8%) 7690 1.819 kg (−0.745 kg, −29.1%) ≈ 48 h (−85%)

83.6 dB 4.0 mm 4.0 mm 257.4 Hz 106.8 dB 77 155

114

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Optimization Results: Rectangular Plate

8

Optimization Results: Two Plates Joined at 90◦

In this chapter the optimization results for the two plates joined at 90◦ are introduced. The FE model of this structure is described in detail in Sec. 5.3. Section 8.1 presents the analysis results for the initial model prior to optimization. Optimization results obtained with COBYLA for various objective functions are shown in Sec. 8.2, whereas those achieved for the maximization of the difference between two natural frequencies by means of the PIKAIA algorithm can be seen in Sec. 8.3.

8.1

Original Two Plates Joined at 90◦

This section provides the results of the FE analysis for the original FE model of the 90◦ plate joint prior to optimization as introduced in Sec. 5.3 and depicted in Fig. 5.6 on p. 67. This enables the reader to compare the properties of the optimized designs shown in Secs. 8.2 and 8.3 with those of the initial design and to assess the effectiveness of the design modifications. The initial FE model of the two-plate joint has an MLS of 81.5 dB in the frequency range of interest (0–3000 Hz) and a mass of 3.195 kg. Its maximum LS is 100.4 dB at the fundamental frequency 369.2 Hz as can be seen from the solid line representing the original structure in various LS spectrum plots such as, e.g., Fig. 8.4 on p. 118. A summary of these results for the original structure is given in Tab. 8.1. MLS, mass, and fundamental frequency can serve either as objective function or constraint for the optimization calculations in the following sections. Table 8.1: Properties of the original two-plate joint. Property

Value

MLS 81.5 dB Mass m 3.195 kg Fundamental frequency f1 369.2 Hz Max. LS (at f1 = 369.2 Hz) 100.4 dB

116

8.2

8

Optimization Results: Two Plates Joined at 90◦

Two Plates Joined at 90◦ Optimized with COBYLA

In the following sections, results of several optimization calculations using COBYLA are reported. These include an MLS minimization (Sec. 8.2.1), a mass minimization (Sec. 8.2.2), a mass minimization with an additional constraint on the fundamental frequency (Sec. 8.2.3), and a maximization of the difference between two natural frequencies (Sec. 8.2.4). 8.2.1

Minimization of MLS

mean level of structure borne sound MLS [dB]

First, the MLS of the two plates joined at 90◦ is minimized by optimizing their thickness distribution. The mass of the structure is not allowed to increase, and for each of the 220 movable surface nodes shown in Fig. 5.7 on p. 68 there is an upper and a lower limit on the thickness, namely, 10 mm and 1 mm, respectively. Thus, this optimization problem is defined by 220 design variables and 441 constraints. The optimization converges after 7665 iterations, which take about 59.6 h of CPU time on the Intel Pentium 4 PC and produce a feasible design (Fig. 8.1). The FE analyses by ABAQUS require about 52 h or 87.3% of the total computation time. The final design has an MLS of 71.3 dB, which is 10.2 dB or 90.5% less than that of the original structure. A satisfactory result with a negligibly small constraint violation of less than 0.01% is achieved after about 2000 iterations, which take only 15.6 h, a CPU time reduction of 74%. This acceptable, improved design exhibits an MLS of 71.4 dB, which is 10.1 dB or 90.2% lower than the initial one. 82 80 78

MLS value during optimization no constraint violation MLS of original structure

76 74 72 70 68

0

1000

2000

3000 4000 5000 number of iterations

6000

7000

Figure 8.1: Iteration history for the two-plate joint (COBYLA, MLS minimization, circles denote iterations without any mass or thickness constraint violation). The optimized design in Fig. 8.2 and the thickness distribution in Fig. 8.3 clearly indicate an increase of the input impedance due to a material concentration at the excitation point and in its vicinity as well as the formation of a stiffening rib across the upper plate. Both measures effectively serve the purpose of reducing the vibrational level. Since the excitation force is normal to the upper plate, it causes hardly any vibrations normal to the surface of the front plate. This explains why almost all of the material can be removed from the front

8.2

Two Plates Joined at 90◦ Optimized with COBYLA

117

plate, thereby reducing its thickness to the absolute minimum. This material is transferred to the upper plate to help increase the input impedance, shaping the stiffening rib, and, thus, reducing the MLS of the structure without exceeding the upper limit on the total mass of the structure, i.e., mmax = m(0) = 3.195 kg.

Z Y X

Figure 8.2: Optimized design of the two-plate joint (COBYLA, MLS minimization, mass and thickness constraints, arrow indicates excitation point). The LS spectra in Fig. 8.4 reflect the dramatic MLS reduction. The peak level at the fundamental frequency is reduced by 18.7 dB (−98.7%) from 100.4 dB to 81.7 dB, the one at the third natural frequency even by 35 dB (−99.97%) from 98.1 dB to 63.1 dB. The fundamental frequency increases by 19.9 Hz or 5.4% from 369.2 Hz to 389.1 Hz. The maximum LS at the initial fundamental frequency moves on to the second natural frequency of the optimized structure and drops from 100.4 dB to 92.9 dB (−7.5 dB or −82.2%). The optimization process does not change the number of natural frequencies in the frequency range of interest, which is 16. Table 8.3 contains a summary of the optimization results for the MLS minimization of the two joined plates presented in this section.

118

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Optimization Results: Two Plates Joined at 90◦

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Z Y

1.8-03 X 1.0-03

Figure 8.3: Thickness distribution of the optimized two-plate joint (COBYLA, MLS minimization, mass and thickness constraints, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 8.4: LS spectra of the original and the optimized two-plate joint (COBYLA, MLS minimization, mass and thickness constraints).

8.2

Two Plates Joined at 90◦ Optimized with COBYLA

119

Table 8.2: Optimization results for the two-plate joint (COBYLA, MLS minimization, mass and thickness constraints). Property MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

8.2.2

Initial design

Optimized design

81.5 dB 3.195 kg 4.0 mm 4.0 mm 369.2 Hz 100.4 dB 220 441

71.3 dB (−10.2 dB, −90.5%) 3.195 kg (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 389.1 Hz (+19.9 Hz, +5.4%) 92.9 dB (−7.5 dB, −82.2%) 220 441 7665 2 d 11 h 37 min (≈ 59.6 h) ≈ 2000 71.4 dB (−10.1 dB, −90.2%) ≈ 15.6 h (−74%)

Minimization of Mass

The inverse optimization problem to that described in the previous section is the mass minimization subject to MLS and thickness constraints. The thickness distribution of the two plates joined at 90◦ is to be optimized such that the mass is minimized while the MLS is not allowed to exceed its initial value of 81.5 dB and the usual wall thickness constraints apply. This again results in 220 design variables and 441 constraints. For this optimization there is no feasible design available, i.e., there is not a single iteration where all the constraints are satisfied in the strict mathematical sense. The lowest thickness value, for instance, never exceeds 0.99999 mm, which is only a microscopically small but yet existing constraint violation. Thus, Fig. 8.5 shows the iteration history where all iterations with a tolerably small constraint violation of less than 0.01% are marked with circles. About 50.4 h of CPU time and 3720 iterations are necessary to meet COBYLA’s convergence criterion. Approximately 44.2 h (87.7%) of the total computation time are devoted to the FE analyses of the modified designs. Upon convergence, the structural mass of the two-plate joint is reduced by 1.434 kg (−44.9%) from 3.195 kg to 1.761 kg. After only 1100 iterations or 14.9 h of computation time, which is equivalent to a CPU time reduction of about 78%, an acceptable design with a structural mass of 1.765 kg (−1.430 kg, −44.8%) is available, even though COBYLA’s formal convergence criterion is not yet met. Figures 8.6 and 8.7 depict the geometry of the optimized design and the thickness distribution, respectively. In order to minimize the mass of the structure, material is removed down to the absolute minimum thickness almost everywhere on the structure except for the excitation location where the thickness is at its allowable maximum in order to increase the input impedance and, thus, to keep the vibrational level from exceeding its upper limit of 81.5 dB. Particularly, the thickness at the relatively uninvolved front plate is reduced to the minimum of 1 mm almost everywhere. This design can be considered a case of a straightforward and “intuitive” solution to the optimization problem (see p. 24).

120

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Optimization Results: Two Plates Joined at 90◦

3.5

mass [kg]

3.0 mass during optimization max. constraint violation < 0.01% mass of original structure

2.5

2.0

1.5

0

500

1000

1500 2000 2500 number of iterations

3000

3500

Figure 8.5: Iteration history for the two-plate joint (COBYLA, mass minimization, circles denote iterations with a maximum constraint violation of less than 0.01%).

Z Y X

Figure 8.6: Optimized design of the two-plate joint (COBYLA, mass minimization, MLS and thickness constraints, arrow indicates excitation point). The LS spectra of the original and the optimized structure can be seen in Fig. 8.8. The fundamental frequency drops by 90.4 Hz from 369.2 Hz to 278.8 Hz (−24.5%). The maximum LS rises by 2.9 dB (+95.0%) from 100.4 dB to 103.3 dB, but this is compensated by levels that are lower than before in other regions of the spectrum so that the maximum MLS is not

8.2

Two Plates Joined at 90◦ Optimized with COBYLA

121

exceeded. The reduced thickness and minimized mass roughly lead to a doubling of the number of natural frequencies within the frequency range of interest (0–3000 Hz) from 16 to 33. 1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Z Y

1.8-03 X 1.0-03

Figure 8.7: Thickness distribution of the optimized two-plate joint (COBYLA, mass minimization, MLS and thickness constraints, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 8.8: LS spectra of the original and the optimized two-plate joint (COBYLA, mass minimization, MLS and thickness constraints).

122

8

Optimization Results: Two Plates Joined at 90◦

A summary of the optimization results of the mass minimization of the two joined plates subject to MLS and thickness constraints using COBYLA as reported in this section can be found in Tab. 8.3. Table 8.3: Optimization results for the two-plate joint (COBYLA, mass minimization, MLS and thickness constraints). Property Mass m MLS Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result Mass of acceptable design CPU time until acceptable result

8.2.3

Initial design

Optimized design

3.195 kg 81.5 dB 4.0 mm 4.0 mm 369.2 Hz 100.4 dB 220 441

1.761 kg (−1.434 kg, −44.9%) 81.5 dB (no constraint violation) 0.99999 mm (0.001% constr. viol.) 10.0 mm (no constraint violation) 278.8 Hz (−90.4 Hz, −24.5%) 103.3 dB (+2.9 dB, +95.0%) 220 441 3720 2 d 2 h 26 min (≈ 50.4 h) ≈ 1100 1.765 kg (−1.430 kg, −44.8%) ≈ 14.9 h (−78%)

Minimization of Mass with Fundamental Frequency Constraint

For the mass minimization in the previous section only the MLS and the thickness were constrained. As can be seen from the LS spectra in Fig. 8.8, the fundamental frequency decreases by about 90 Hz, which is only natural because of the reduced thickness and mass of the structure. Therefore, it seemed interesting to challenge the optimization algorithm with a more difficult optimization problem, namely, the same mass minimization but with an additional constraint on the fundamental frequency, i.e., the fundamental frequency is not allowed to fall below its initial value 369.2 Hz. Thus, the number of design variables stays the same (220) but the number of constraints is increased by one to 442. As in the previous section there is no iteration at all with a feasible design in the strict mathematical sense. The minimum thickness never exceeds 0.99999 mm, and the maximum thickness never falls below 10.00001 mm, yielding formal thickness constraint violations of 0.001% and 0.0001%, respectively. This is why the iteration history in Fig. 8.9 shows iterations with a tiny constraint violation of less than 0.01% marked with circles. The MLS and the fundamental frequency constraint are not violated. The optimization runs for 10 136 iterations or 135.8 h of CPU time on the usual Intel Pentium 4 PC (see App. A, Sec. A.2) before meeting COBYLA’s convergence criterion. These are 6416 iterations (172.5%) more than for the same optimization problem without the additional fundamental frequency constraint in Sec. 8.2.2. The computation time increases by 85.4 h or 169.4% due to the additional constraint. The mass of the optimized structure is reduced by 1.331 kg or 41.7% from 3.195 kg to 1.864 kg. This is 0.103 kg or 7.2% less improvement than for the optimization without the fundamental frequency constraint. ABAQUS takes about 118.7 h or 87.4% of the total computation time

8.2

Two Plates Joined at 90◦ Optimized with COBYLA

123

for the FE analyses. Satisfactory designs are available after about 3700 iterations or 49.6 h of CPU time, which is a CPU time reduction of 64%. The mass of these improved designs is 1.914 kg, a decrease of 1.281 kg or 40.1%. 3.5

mass [kg]

3.0 mass during optimization max. constraint violation < 0.01% mass of original structure

2.5

2.0

1.5

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 number of iterations

Figure 8.9: Iteration history for the two-plate joint (COBYLA, mass minimization, f1 constraint, circles denote iterations with a maximum constraint violation of less than 0.01%). The optimized geometry and the thickness distribution in Figs. 8.10 and 8.11, respectively, are very similar to the ones in Figs. 8.6 and 8.7 in the previous section. Because of the fundamental frequency constraint, however, the material is not concentrated at but only near the excitation point. In fact, directly at the excitation location the wall thickness is only 3.9 mm instead of the allowable 10 mm. In the vicinity of the excitation point, however, there are nodes at which the thickness is at its upper limit. Obviously, this configuration is able to minimize the structural mass while simultaneously keeping the MLS from exceeding its upper bound 81.5 dB (the MLS of the original structure) and preventing the fundamental frequency from falling below its lower limit of 369.2 Hz. This can be considered a case where any “intuitive” solution approach (see p. 24) will most likely fail due to the contradictory requirements of minimizing the mass without decreasing the fundamental frequency. The LS spectra in Fig. 8.12 prove the success of this optimization strategy. The fundamental frequency of the optimized structure is exactly at the same location as it was prior to optimization. The overall LS spectrum does not change much due to the optimization, thus keeping a constant MLS despite the mass reduction. The maximum LS peak moves from the fundamental frequency of the initial structure to the fifth natural frequency of the optimized structure and increases by 2.2 dB (66.0%) from 100.4 dB to 102.6 dB. As in the previous section, the number of natural frequencies in the frequency band from 0 to 3000 Hz is more than doubled from 16 to 33. An overview of the optimization results for the mass minimization of the two plates joined at 90◦ subject to MLS and thickness constraints as well as to an additional fundamental frequency constraint employing the COBYLA algorithm, which are introduced in this section, is given in Tab. 8.4.

124

8

Optimization Results: Two Plates Joined at 90◦

Z Y X

Figure 8.10: Optimized design of the two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints, arrow indicates excitation point).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Z Y

1.8-03 X 1.0-03

Figure 8.11: Thickness distribution of the optimized two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints, values in m).

8.2

Two Plates Joined at 90◦ Optimized with COBYLA

125

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 8.12: LS spectra of the original and the optimized two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints).

Table 8.4: Optimization results for the two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints). Property Mass m Compared to without f1 constr. MLS Fundamental frequency f1 Minimum thickness tmin Maximum thickness tmax Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to without f1 constr. CPU time until convergence Compared to without f1 constr. Iterations until acceptable result Mass of acceptable design CPU time until acceptable result

Initial design

Optimized design

3.195 kg

1.864 kg (−1.331 kg, −41.7%) +0.103 kg (0.103 kg less reduction) 81.5 dB (no constraint violation) 369.2 Hz (no constraint violation) 0.99999 mm (0.001% constr. viol.) 10.00001 mm (0.0001% constr. viol.) 102.6 dB (+2.2 dB, +66.0%) 220 442 10 136 +6416 (+172.5%) 5 d 15 h 50 min (≈ 135.8 h) +85.4 h (+169.4%) ≈ 3700 1.914 kg (−1.281 kg, −40.1%) ≈ 49.6 h (−64%)

81.5 dB 369.2 Hz 4.0 mm 4.0 mm 100.4 dB 220 442

126

8.2.4

8

Optimization Results: Two Plates Joined at 90◦

Maximization of the Difference between Two Natural Frequencies

The next example for an optimization of the 90◦ joint of two plates is the maximization of the difference between two natural frequencies ∆f . As explained in detail in Sec. 7.2.5, this original optimization problem is replaced with an equivalent one, namely, the maximization of the minimum difference between some specified frequency f ∗ and its two neighboring natural frequencies ∆fmin subject to MLS, mass, and thickness constraints. The frequency f ∗ is chosen to be 974 Hz, since this is approximately halfway between the fourth and the fifth natural frequency of the original structure, namely, 964.3 Hz and 983.4 Hz, respectively, which are close together (see the LS spectrum of the original structure in Fig. 8.16). Thus, the initial (0) minimum frequency difference is ∆fmin = 9.4 Hz. For this optimization problem there are 220 design variables and 442 constraints. As the iteration history plot in Fig. 8.13 indicates, the optimized design at convergence after 10 884 iterations and 115.6 h of CPU time is feasible, i.e., there is no constraint violation. The FE analyses contribute to the total computation time with about 101 h or 87.4%. The minimum frequency difference ∆fmin increases by 233.2 Hz or 2480.9% from 9.4 Hz to 242.6 Hz. If the optimization procedure is manually terminated although COBYLA’s convergence criterion is not yet met, acceptable results can be obtained after about 3500 iterations or 37.2 h of CPU time, a reduction of 68%. At this point, the minimum frequency difference ∆fmin is 239.5 Hz, an increase of 230.1 Hz or 2447.9%.

minimum frequency difference ∆fmin [Hz]

250

200

150 min. freq. diff. ∆fmin during optimization no constraint violation min. freq. diff. ∆fmin of original structure

100

50

0

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000011000 number of iterations

Figure 8.13: Iteration history for the two-plate joint (COBYLA, ∆f maximization, circles denote iterations without any MLS, mass, or thickness constraint violation). The optimized design and the thickness distribution in Figs. 8.14 and 8.15, respectively, can hardly be interpreted. As usual, the MLS is kept below its upper limit of 81.5 dB by increasing the thickness at the excitation point to the allowable maximum of 10 mm. However, it remains unclear why the formation of the other bulges brings any advantages. Interestingly, as a side effect the mass of the structure is reduced by 1.012 kg (31.7%) from 3.195 kg to 2.183 kg although it is not an objective function but rather a constraint.

8.2

Two Plates Joined at 90◦ Optimized with COBYLA

127

Z Y X

Figure 8.14: Optimized design of the two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, arrow indicates excitation point).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Z Y

1.8-03 X 1.0-03

Figure 8.15: Thickness distribution of the optimized two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, values in m).

128

8

Optimization Results: Two Plates Joined at 90◦

The maximized frequency difference ∆f is visualized in Fig. 8.16. The frequency f ∗ = 974 Hz is denoted by a dot-dashed line from which the neighboring natural frequencies are shifted away as far as possible. For the original configuration the difference between these two natural frequencies 964.3 Hz and 983.4 Hz is ∆f (0) = 19.1 Hz, for the optimized design the new natural frequencies are 731.4 Hz and 1216.6 Hz, yielding ∆f = 485.2 Hz, an improvement of 466.1 Hz or 2440.3%. The fundamental frequency drops by 51.2 Hz (−13.9%) from 369.2 Hz to 318.0 Hz. Though the LS spectra look quite different, the MLS of the original and the optimized structure are the same. The maximum LS peak at the fundamental frequency increases by 2.2 dB or 66.0% from 100.4 dB to 102.6 dB. The number of natural frequencies in the frequency range of interest (0–3000 Hz) goes up from 16 to 24.

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 8.16: LS spectra of the original and the optimized two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints, dot-dashed line marks f ∗ = 974 Hz). Table 8.5 presents a summary of the optimization results for the maximization of the difference between two natural frequencies of the two joined plates subject to MLS, mass, and thickness constraints utilizing the optimization algorithm COBYLA as shown in this section.

8.3

Two Plates Joined at 90◦ Optimized with PIKAIA: Maximization of the Difference between Two Natural Frequencies

In this section the same optimization problem as in the previous section is investigated, namely, the maximization of the difference between two natural frequencies ∆f or, equivalently, the maximization of the minimum difference between some specified frequency f ∗ and its two neighboring natural frequencies ∆fmin . This time, however, the genetic optimization algorithm PIKAIA is used instead of the deterministic algorithm COBYLA in order to compare the effectivity and efficiency of the two algorithms. The specified frequency f ∗ , the initial objective function value, and the number of design variables and constraints are the same as in the (0) previous section, namely, f ∗ = 974 Hz, ∆fmin = 9.4 Hz, 220, and 442, respectively. For a detailed description of some of the special features of the results of an optimization using

8.3

Two Plates Joined at 90◦ Optimized with PIKAIA

129

Table 8.5: Optimization results for the two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints). Property Minimum frequency difference ∆fmin MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result ∆fmin of acceptable design CPU time until acceptable result

Initial design

Optimized design

9.4 Hz 81.5 dB 3.195 kg 4.0 mm 4.0 mm 369.2 Hz 100.4 dB 220 442

242.6 Hz (+233.2 Hz, +2480.9%) 81.5 dB (no constraint violation) 2.183 kg (−1.012 kg, −31.7%) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 318.0 Hz (−51.2 Hz, −13.9%) 102.6 dB (+2.2 dB, +66.0%) 220 442 10 884 4 d 19 h 37 min (≈ 115.6 h) ≈ 3500 239.5 Hz (+230.1 Hz, +2447.9%) ≈ 37.2 h (−68%)

PIKAIA such as the recording of real time instead of CPU time or the difference between objective function value and fitness, the reader is referred to Sec. 7.4, which describes the results of the mass minimization of the rectangular plate with PIKAIA. Figure 8.17 depicts the iteration history in terms of the objective function value ∆fmin , which is equal to the fitness function f (x) for maximization problems if the design is feasible (see Sec. 4.3.1 and, in particular, Eqs. (4.6) and (4.7)). If there is a constraint violation, the fitness is reduced to a tiny, yet positive number, which is the case during the first 2000 iterations. The gray line in Fig. 8.17 represents the objective function value, i.e., the minimum frequency difference ∆fmin , as it varies from iteration to iteration due to the random character of the design variable generation by PIKAIA. The solid black line visualizes the objective function value with the best fitness up to a particular iteration, i.e., the highest objective function value the corresponding design of which is feasible. There are gray lines that indicate an objective function value higher than that represented by the solid black line, but the respective designs turn out to be infeasible due to constraint violations and therefore have a low fitness. The (0) dashed line marks the objective function value of the original structure, i.e., ∆fmin = 9.4 Hz. The optimization procedure is manually terminated after 50 000 iterations due to the lack of a more suitable stop criterion. The highest objective function value of a feasible design, i.e., the greatest fitness, prior to termination is obtained after 48 940 iterations. The 50 000 iterations on 5 computers in parallel (see App. A, Sec. A.2) took about 27.5 days (≈ 658.9 h real time). This time is about 5.7 times as long as the CPU time required by COBYLA to reach convergence after only 10 884 iterations (see Sec. 8.2.4). After the aforementioned 48 940 iterations the minimum frequency difference ∆fmin is increased by 263.4 Hz (2802.1%) from 9.4 Hz to 272.8 Hz, which is 30.2 Hz or 12.5% more than the gain achieved by COBYLA in Sec. 8.2.4. This is the only case in the present study where PIKAIA finds a better solution than COBYLA, which indicates that COBYLA may have got trapped in a local optimum. It is also worthwhile noting that the MLS of the optimized design stays 0.2 dB (4.5%) below its

130

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Optimization Results: Two Plates Joined at 90◦

minimum frequency difference ∆fmin [Hz]

300 250 200 150 min. freq. diff. ∆fmin at individual iteration highest feasible min. freq. diff. ∆fmin so far min. freq. diff. ∆fmin of original structure

100 50 0

0

5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 number of iterations

Figure 8.17: Iteration history for the two-plate joint (objective function, PIKAIA, ∆f maximization, MLS, mass, and thickness constraints). upper limit. This is also true for the structural mass, which is 0.023 kg (0.7%) below its upper bound. Just as described in Sec. 7.4 and visible in Fig. 8.17, there are a number of iterations at which the optimization procedure could have been aborted earlier with acceptable results. An example is iteration 29 276 with a minimum frequency difference ∆fmin of 242.4 Hz (+233.0 Hz, +2478.7%) and a real time consumption of about 385.5 h (−42% compared to termination after 50 000 iterations).

Z Y X

Figure 8.18: Optimized design of the two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints, arrow indicates excitation point).

8.3

Two Plates Joined at 90◦ Optimized with PIKAIA

131

Figures 8.18 and 8.19 show the optimized geometry and the thickness distribution of the optimized design, respectively, which is almost impossible to interpret this time. It is completely different from the one created by COBYLA, depicted in Figs. 8.14 and 8.15. The primary reason for the difference is the fact that PIKAIA keeps the total mass of the structure almost constant. It thus has the potential to increase the thickness to almost 10 mm at a number of places on the structure. COBYLA, however, for some unknown reason reduced the mass by almost 32%, leaving little material left to shape stiffening ribs or to distribute “point masses” across the structure.

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

2.6-03

Z Y

1.8-03 X 1.0-03

Figure 8.19: Thickness distribution of the optimized two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints, values in m). Figure 8.20 illustrates the maximized frequency difference ∆f , which increases from ∆f (0) = 19.1 Hz to ∆f = 546.9 Hz (+527.8 Hz, +2763.4%) due to the optimization. This is an improvement of 61.7 Hz or 12.7% compared to COBYLA’s solution in Sec. 8.2.4. The dot-dashed line denotes the frequency f ∗ = 974 Hz. The fundamental frequency stays approximately the same (373.7 Hz compared to the initial 369.2 Hz, a slight increase of 4.5 Hz or 1.2%), whereas the maximum LS slightly decreases from 100.4 dB to 99.7 dB (−0.7 dB, −14.9%). The MLS of the optimized design hardly differs from that of the initial structure. The number of fundamental frequencies within the frequency range of interest from 0 to 3000 Hz decreases from 16 to 13. The summarizing Tab. 8.6 compares some of the optimization results for the maximization of the difference between two natural frequencies of the two joined plates subject to MLS, mass, and thickness constraints using PIKAIA with those obtained by COBYLA (cf. Tab. 8.5 on p. 129).

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Optimization Results: Two Plates Joined at 90◦

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 8.20: LS spectra of the original and the optimized two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints, dot-dashed line marks f ∗ = 974 Hz).

Table 8.6: Optimization results for the two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints). Property Minimum frequency difference ∆fmin Compared to COBYLA result MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Termination after Real time until termination Highest fitness value after Compared to COBYLA result Iterations until acceptable result ∆fmin of acceptable design Real time until acceptable result

Initial design

Optimized design

9.4 Hz

272.8 Hz (+263.4 Hz, +2802.1%) +30.2 Hz (30.2 Hz more increase) 81.3 dB (−0.2 dB, −4.5%) 3.172 kg (−0.023 kg, −0.7%) 1.0011 mm (no constraint violation) 9.9842 mm (no constraint violation) 373.7 Hz (+4.5 Hz, +1.2%) 99.7 dB (−0.7 dB, −14.9%) 220 442 50 000 iterations 27 d 10 h 51 min (≈ 658.9 h) 48 940 iterations +38 056 (+349.7%) 29 276 242.4 Hz (+233.0 Hz, +2478.7%) ≈ 385.5 h (−42%)

81.5 dB 3.195 kg 4.0 mm 4.0 mm 369.2 Hz 100.4 dB 220 442

9

Optimization Results: Half-Cylinder

This chapter provides the optimization results for the simply supported half-cylinder that is introduced in Sec. 5.4. First, Sec. 9.1 reports analysis results for the original half-cylinder prior to optimization. Then, results of various optimization calculations using COBYLA are presented in Sec. 9.2. This is followed by Sec. 9.3, which includes results of two optimization calculations using COBYLA and Hermite spline functions to reduce the number of design variables and, thus, the CPU time. Finally, the results of an MLS minimization employing PIKAIA are shown in Sec. 9.4.

9.1

Original Half-Cylinder

The original FE model of the half-cylinder (see Fig. 5.8 on p. 69) exhibits an MLS of 78.8 dB within the frequency band from 0 to 3000 Hz. Its structural mass is 1.973 kg. The fundamental frequency is 228.9 Hz, and the maximum LS within the frequency range of interest is 100.1 dB at the fundamental frequency. These numbers allow an easy comparison of the optimization results presented in the following sections with the analysis results of the initial configuration given above and facilitate an assessment of the effectivity of the optimizations. Table 9.1 provides a summary of the analysis results for the original half-cylinder model prior to optimization. Again, MLS, mass, and fundamental frequency can be either objective function or constraint in the optimization calculations in the following sections. Table 9.1: Properties of the original half-cylinder. Property

Value

MLS 78.8 dB Mass m 1.973 kg Fundamental frequency f1 228.9 Hz Max. LS (at f1 = 228.9 Hz) 100.1 dB

134

9.2

9

Optimization Results: Half-Cylinder

Half-Cylinder Optimized with COBYLA

This section comprises various optimization results for the half-cylinder, which were obtained by employing the COBYLA algorithm. These results include an MLS minimization presented in Sec. 9.2.1, a mass minimization shown in Sec. 9.2.2, and a fundamental frequency maximization described in Sec. 9.2.3. 9.2.1

Minimization of MLS

mean level of structure borne sound MLS [dB]

Figure 9.1 depicts the iteration history for the MLS minimization of the half-cylinder. This optimization problem is defined by 165 design parameters as described in Sec. 5.4 and 331 constraints, namely, a mass constraint (mmax = m(0) = 1.973 kg) as well as a lower (1 mm) and an upper limit (10 mm) on the thickness at each of the 165 movable surface nodes. COBYLA takes 5137 iterations and 17.5 h of CPU time on the Intel Pentium 4 PC to meet its convergence criterion. The final design is not feasible in the strict mathematical sense since somewhere on the structure the minimum thickness is 0.99999 mm and, thus, below the allowable lower limit. At other locations the thickness is 10.00001 mm, which is beyond the upper thickness bound of 10 mm. From the engineering point of view, however, these formal constraint violations are negligible. The FE software ABAQUS consumes about 15.1 h or 86.4% of the total computation time. The optimized design has an MLS of 72.6 dB, which is 6.2 dB or 76.0% below the initial value of 78.8 dB. Acceptable results with a tolerably small constraint violation are available after only 2000 iterations and 6.8 h of computation time, a reduction of 61%. At this point, the MLS is reduced by 6.0 dB (74.9%) to 72.8 dB. 90 88 MLS value during optimization max. constraint violation < 0.01% MLS of original structure

86 84 82 80 78 76 74 72 70

0

1000

2000 3000 number of iterations

4000

5000

Figure 9.1: Iteration history for the half-cylinder (COBYLA, MLS minimization, circles denote iterations with a maximum constraint violation of less than 0.01%). The optimized thickness distribution in Fig. 9.2 shows that the input impedance is increased by means of adding material at and near the excitation point (cf. Fig. 5.8 on p. 69), thus reducing the vibrational level. In addition to this “intuitive solution” (see p. 24), a stiffening rib in the circumferential direction is formed, which passes through the excitation point. Besides

9.2

135

Half-Cylinder Optimized with COBYLA

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

Z Y

2.6-03

Z Y

X

1.8-03 X 1.0-03

Figure 9.2: Thickness distribution of the optimized half-cylinder (COBYLA, MLS minimization, mass and thickness constraints, values in m). that, the wall thickness is increased at some other places as well, e.g., along the front edge of the half-cylinder and at the top back end. To compensate for this thickness increase, material is removed at other locations down to the minimum allowable thickness of 1 mm such that the structural mass does not increase. The LS spectrum in Fig. 9.3 illustrates the reduction of the LS and therefore of the MLS. The fundamental frequency increases from 228.9 Hz to 356.2 Hz (+127.3 Hz, +55.6%), thus enlarging the quasi-static frequency range. The maximum LS peak is reduced by 4.9 dB or 67.6% from 100.1 dB to 95.2 dB. The number of natural frequencies within the frequency range from 0 to 3000 Hz goes up from seven to nine. level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 9.3: LS spectra of the original and the optimized half-cylinder (COBYLA, MLS minimization, mass and thickness constraints).

136

9

Optimization Results: Half-Cylinder

A summary of the optimization results for the MLS minimization of the half-cylinder using COBYLA is given in Tab. 9.2. Table 9.2: Optimization results for the half-cylinder (COBYLA, MLS minimization, mass and thickness constraints). Property MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

9.2.2

Initial design

Optimized design

78.8 dB 1.973 kg 4.0 mm 4.0 mm 228.9 Hz 100.1 dB 165 331

72.6 dB (−6.2 dB, −76.0%) 1.973 kg (no constraint violation) 0.99999 mm (0.001% constr. viol.) 10.00001 mm (0.0001% constr. viol.) 356.2 Hz (+127.3 Hz, +55.6%) 95.2 dB (−4.9 dB, −67.6%) 165 331 5137 17 h 27 min (≈ 17.5 h) ≈ 2000 72.8 dB (−6.0 dB, −74.9%) ≈ 6.8 h (−61%)

Minimization of Mass

The second example of an optimization of the half-cylinder’s thickness distribution is the mass minimization subject to MLS and thickness constraints, which can be regarded as the inverse optimization problem with respect to the MLS minimization subject to a mass constraint presented in the previous section. Therefore, the number of design variables and constraints stays constant, namely, 165 and 331, respectively. The iteration history can be seen in Fig. 9.4 where iterations with a slight constraint violation of less than 0.01% are denoted by circles. The final design is feasible, i.e., neither the MLS constraint nor any of the thickness constraints are violated. COBYLA’s convergence criterion is met after 7905 iterations, which take about 29.1 h of computation time on the Intel Pentium 4 PC (see App. A, Sec. A.2). The FE analyses take up 86.5% of the total CPU time, i.e., 25.1 h. The optimized design has a structural mass of 1.344 kg, which is 0.629 kg or 31.9% less than the one of the original structure (1.973 kg). A satisfactory result is obtainable after only about 4500 iterations, which consume 16.6 h of CPU time, a reduction of 43% compared to the time it takes to reach convergence. Despite the significant CPU time reduction, a remarkable mass decrease of 0.626 kg (31.7%) from 1.973 kg to 1.347 kg is achieved. Figure 9.5 depicts the optimized thickness distribution. As can be seen, material is removed almost everywhere on the half-cylinder except for the excitation point where the thickness is increased to the maximum permissible 10 mm. This again represents the “intuitive solution” (see p. 24), i.e., the increase of the input impedance to keep the overall vibrational level below its upper limit of 78.8 dB despite the material removal at other locations. Likewise, the formation of a T-shaped rib at the excitation location and the isolated “point masses” at the left-hand side of the structure help to keep the MLS from exceeding its upper bound. These additional measures, however, cannot be predicted by the “intuitive solution”.

9.2

137

Half-Cylinder Optimized with COBYLA

2.0

mass [kg]

1.5

1.0

0.5

mass during optimization max. constraint violation < 0.01% mass of original structure

0

1000

2000

3000 4000 5000 number of iterations

6000

7000

8000

Figure 9.4: Iteration history for the half-cylinder (COBYLA, mass minimization, circles denote iterations with a maximum constraint violation of less than 0.01%).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

Z Y

2.6-03

Z X

Y

1.8-03 X 1.0-03

Figure 9.5: Thickness distribution of the optimized half-cylinder (COBYLA, mass minimization, MLS and thickness constraints, values in m). The LS spectra of the original and the optimized structure in Fig. 9.6 exhibit the same MLS. The fundamental frequency drops by 26.6 Hz (−11.6%) from 228.9 Hz to 202.3 Hz, whereas the maximum LS at the fundamental frequency goes up by 2.9 dB or 95.0% from 100.1 dB to 103.0 dB. The number of natural frequencies in the frequency band of interest increases from 7 to 11.

138

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Optimization Results: Half-Cylinder

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 9.6: LS spectra of the original and the optimized half-cylinder (COBYLA, mass minimization, MLS and thickness constraints). Table 9.3 provides an overview of the optimization results for the mass minimization of the simply supported half-cylinder subject to MLS and thickness constraints using COBYLA, which are presented in this section. Table 9.3: Optimization results for the half-cylinder (COBYLA, mass minimization, MLS and thickness constraints). Property Mass m MLS Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result Mass of acceptable design CPU time until acceptable result

Initial design

Optimized design

1.973 kg 78.8 dB 4.0 mm 4.0 mm 228.9 Hz 100.1 dB 165 331

1.344 kg (−0.629 kg, −31.9%) 78.8 dB (no constraint violation) 1.0 mm (no constraint violation) 9.99998 mm (no constraint violation) 202.3 Hz (−26.6 Hz, −11.6%) 103.0 dB (+2.9 dB, +95.0%) 165 331 7905 1 d 5 h 5 min (≈ 29.1 h) ≈ 4500 1.347 kg (−0.626 kg, −31.7%) ≈ 16.6 h (−43%)

9.2

139

Half-Cylinder Optimized with COBYLA

9.2.3

Maximization of the Fundamental Frequency

Next, the fundamental frequency f1 of the half-cylindrical structure is maximized by optimizing its thickness distribution subject to MLS, mass, and thickness constraints. Therefore, the number of constraints increases to 332, whereas the number of design variables remains at 165. Figure 9.7 shows the iteration history with circles indicating iterations with a tolerable constraint violation of less than 0.01%. The optimized design is not only feasible but even exhibits an MLS reduction of 2.7 dB (−46.3%) from 78.8 dB to 76.1 dB although the MLS is a constraint and not the objective function. The final design has a fundamental frequency of 375.0 Hz, which is 146.1 Hz or 63.8% higher than that of the initial design (228.9 Hz). This result is obtained after 6013 iterations and 20.7 h CPU time, 85% of which (17.6 h) are used by the FE analyses. If COBYLA’s formal convergence criterion is ignored, acceptable results are achieved after about 2400 iterations or 8.3 h, a reduction of 60%. At this point, the fundamental frequency is brought up by 146.0 Hz (63.8%) from 228.9 Hz to 374.9 Hz.

fundamental frequency f1 [Hz]

400 350 300 250 200 fund. freq. f1 during optimization max. constraint violation < 0.01% fund. freq. f1 of original structure

150 100

0

1000

2000 3000 4000 number of iterations

5000

6000

Figure 9.7: Iteration history for the half-cylinder (COBYLA, fundamental frequency maximization, circles denote iterations with a maximum constraint violation of less than 0.01%). The optimized thickness distribution is shown in Fig. 9.8. Material is removed from the back part of the half-cylinder and concentrated at two spots at the left-hand and right-hand side of the structure’s front part. This reinforcement simultaneously prevents an MLS increase (it even causes the MLS to decrease by 2.7 dB, see above) and results in an increase of the fundamental frequency f1 , whereas the structural mass is kept constant. The LS spectra in Fig. 9.9 clearly reflect the fundamental frequency rise from 228.9 Hz to 375.0 Hz. As mentioned above, the MLS of the optimized structure is even 2.7 dB below that of the original half-cylinder. The maximum LS at the fundamental frequency is reduced by 3.6 dB (56.4%) from 100.1 dB to 96.5 dB. The optimized design exhibits 10 natural frequencies within the frequency range of interest (0-3000 Hz), whereas the initial design has only 7. The usual summary of the optimization results, this time for the maximization of the fundamental frequency of the simply supported half-cylinder subject to MLS, mass, and thickness constraints using COBYLA as introduced above, is presented in Tab. 9.4.

140

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Optimization Results: Half-Cylinder

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

Z Y

2.6-03

Z Y

X

1.8-03 X 1.0-03

Figure 9.8: Thickness distribution of the optimized half-cylinder (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 9.9: LS spectra of the original and the optimized half-cylinder (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints).

9.3

141

Half-Cylinder Optimized with COBYLA Using Splines

Table 9.4: Optimization results for the half-cylinder (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints). Property Fundamental frequency f1 MLS Mass m Minimum thickness tmin Maximum thickness tmax Maximum LS Number of design variables Number of constraints Iterations until convergence CPU time until convergence Iterations until acceptable result f1 of acceptable design CPU time until acceptable result

9.3

Initial design

Optimized design

228.9 Hz 78.8 dB 1.973 kg 4.0 mm 4.0 mm 100.1 dB 165 332

375.0 Hz (+146.1 Hz, +63.8%) 76.1 dB (−2.7 dB, −46.3%) 1.973 kg (no constraint violation) 1.0 mm (no constraint violation) 9.99999 mm (no constraint violation) 96.5 dB (−3.6 dB, −56.4%) 165 332 6013 20 h 44 min (≈ 20.7 h) ≈ 2400 374.9 Hz (+146.0 Hz, +63.8%) ≈ 8.3 h (−60%)

Half-Cylinder Optimized with COBYLA Using Splines

A Hermite spline surface as introduced in Sec. 6.3 and depicted in Fig. 6.3 on p. 76 is used to reduce the number of design parameters and, consequently, the number of iterations and the computation time required for convergence. This time, the spline surface is defined by 8×6 key points instead of the usual 7×5 key points. As in the previous sections, COBYLA is utilized as the optimization algorithm. Section 9.3.1 presents the optimization results of an MLS minimization. This is followed by a description of the results of a mass minimization in Sec. 9.3.2. 9.3.1

Minimization of MLS

The first example of an optimization of the half-cylinder employing Hermite spline functions is the MLS minimization subject to 331 mass and thickness constraints using COBYLA. The 8×6 key points of the spline surface result in 48 design variables instead of 165 for the direct manipulation of the thickness distribution without splines, a reduction of 117 or 71%. The iteration history plot in Fig. 9.10 reveals that COBYLA’s convergence criterion is met after 2000 iterations, which take 6.2 h of computation time. About 88.7% of the total CPU time (5.5 h) accounts for the FE calculations. The final design is not feasible in the strict mathematical sense, since the upper thickness constraint is violated by 0.0001%. For practical applications, however, this tiny constraint violation does not have any effect. The optimized design has an MLS of 74.2 dB, which is a reduction of 4.6 dB or 65.3% compared to the initial design’s 78.8 dB. If the positions of the movable surface nodes are modified directly without using splines as is done in Sec. 9.2.1, a significantly greater MLS reduction of 6.2 dB (−76.0%) to 72.6 dB is achieved at the expense of more iterations (the use of spline functions results in a reduction of 3137 iterations or 61.1% from 5137 to 2000 iterations) and longer computation times (by using spline functions the CPU time is decreased by 11.2 h or 64.2% from 17.5 h to

142

9

Optimization Results: Half-Cylinder

mean level of structure borne sound MLS [dB]

6.2 h). As can be seen from Fig. 9.10, the improved design after about 1000 iterations exhibits an acceptable MLS reduction of 4.5 dB or 64.5% from 78.8 dB to 74.3 dB. The termination of the optimization procedure after only 1000 iterations saves 3.1 h (50%) CPU time. 80 78 76 74 72 MLS value during optimization no constraint violation MLS of original structure

70 68

0

500

1000 number of iterations

1500

2000

Figure 9.10: Iteration history for the half-cylinder (COBYLA, Hermite splines, MLS minimization, circles denote iterations without any mass or thickness constraint violation). The optimized thickness distribution in Fig. 9.11 looks quite similar to the one without using splines shown in Fig. 9.2 on p. 135. In particular, the thickness increase up to the maximum allowable value (10 mm) at the excitation point and the formation of stiffening rib-like structures in circumferential direction at the front part of the half-cylinder can be observed. Interestingly, this time this kind of circumferential reinforcement does not pass through the excitation point but is shifted to the front by one element. The domains on the structure in which the thickness is reduced to the minimum of 1 mm are relatively small compared to the ones in Fig. 9.2. Figure 9.12 shows that the overall LS level of the optimized structure is only slightly decreased compared to that of the original one. The MLS reduction is mainly due to the LS decrease at the first 5 natural frequencies. The fundamental frequency of the optimized halfcylinder is 30.6 Hz or 13.4% higher than that of the initial structure, a shift from 228.9 Hz to 259.5 Hz. The maximum LS peak at the fundamental frequency drops by 3.2 dB (52.1%) from 100.1 dB to 96.9 dB. The number of natural frequencies in the frequency range of interest from 0 to 3000 Hz increased from seven to eight. Table 9.5 with a summary of the optimization results for the MLS minimization of the halfcylinder subject to mass and thickness constraints employing a Hermite spline surface includes a comparison with the results obtained without spline functions (cf. Tab. 9.2 on p. 136).

9.3

143

Half-Cylinder Optimized with COBYLA Using Splines

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

Z Y

2.6-03

Z Y

X

1.8-03 X 1.0-03

Figure 9.11: Thickness distribution of the optimized half-cylinder (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, values in m).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 9.12: LS spectra of the original and the optimized half-cylinder (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints).

144

9

Optimization Results: Half-Cylinder

Table 9.5: Optimization results for the half-cylinder (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints). Property MLS Compared to direct modification Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to direct modification CPU time until convergence Compared to direct modification Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

9.3.2

Initial design

Optimized design

78.8 dB

74.2 dB (−4.6 dB, −65.3%) +1.6 dB (1.6 dB less reduction) 1.973 kg (no constraint violation) 1.05503 mm (no constraint violation) 10.00001 mm (0.0001% constr. viol.) 259.5 Hz (+30.6 Hz, +13.4%) 96.9 dB (−3.2 dB, −52.1%) 48 key points (−70.9%) 331 2000 −3137 (−61.1%) 6 h 14 min (≈ 6.2 h) −11.2 h (−64.2%) ≈ 1000 74.3 dB (−4.5 dB, −64.5%) ≈ 3.1 h (−50%)

1.973 kg 4.0 mm 4.0 mm 228.9 Hz 100.1 dB 165 331

Minimization of Mass

This section presents the results for the same mass minimization of the half-cylinder subject to MLS and thickness constraints using COBYLA as described in Sec. 9.2.2, but this time the positions of the movable surface nodes are modified by means of a Hermite spline surface defined by 8×6 key points. Thus, there are 48 design variables (117 or 71% less than in Sec. 9.2.2) and 331 constraints just as in the previous section. As can be seen from the iteration history plot in Fig. 9.13, there are hardly any iterations without a formal constraint violation, but the final optimized design is feasible. Convergence is achieved after 2106 iterations and 7.7 h of CPU time on the usual Intel Pentium 4 PC (see Tab. A.1 on p. 178 for details). ABAQUS consumes about 6.8 h or 88.3% of the total computation time for the FE analyses. The optimized design has a structural mass of 1.322 kg, which is 0.651 kg or 33.0% less than the 1.973 kg of the initial structure. This reduction is even greater by 0.022 kg (1.6%) than that achieved by COBYLA in Sec. 9.2.2, namely, 1.344 kg (see Tab. 9.3 on p. 138), and the number of iterations and the CPU time are drastically reduced by 5799 iterations (from 7905 to only 2106, −73.4%) and by 21.4 h (from 29.1 h to 7.7 h, −73.6%), respectively. Improved designs with a satisfactory mass reduction of 0.642 kg (−32.5%) from 1.973 kg to 1.331 kg are available after only about 1000 iterations and 3.7 h of computation time, a decrease of 52%. The optimization algorithm removes material almost everywhere on the structure except for a T-shaped area near the excitation point and a narrow strip at the very front edge on the left-hand side of the half-cylinder (Fig. 9.14). This way, the structural mass is minimized without increasing the vibrational level.

9.3

145

Half-Cylinder Optimized with COBYLA Using Splines

mass [kg]

2.0 mass during optimization no constraint violation mass of original structure 1.5

1.0

0

500

1000 1500 number of iterations

2000

Figure 9.13: Iteration history for the half-cylinder (COBYLA, Hermite splines, mass minimization, circles denote iterations without any MLS or thickness constraint violation).

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

Z Y

2.6-03

Z X

Y

1.8-03 X 1.0-03

Figure 9.14: Thickness distribution of the optimized half-cylinder (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, values in m).

146

9

Optimization Results: Half-Cylinder

A look at the LS spectra in Fig. 9.15 confirms the fact that the overall vibrational level is not changed due to the mass minimization. However, the natural frequencies are shifted to some extend. Surprisingly, the fundamental frequency hardly changes from 228.9 Hz to 222.3 Hz, a slight decrease of 6.6 Hz or 2.9%, despite the considerable mass and thickness reduction. This is most likely due to the aforementioned reinforcements on the left-hand and right-hand front part of the half-cylinder, which somewhat resemble those observed for the fundamental frequency maximization depicted in Fig. 9.8 on p. 140. The maximum LS at the fundamental frequency increases by 2.9 dB or 95.0% from 100.1 dB to 103.0 dB, and the number of natural frequencies within the frequency range of interest goes up from 7 to 11 due to the decrease of mass and thickness.

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 9.15: LS spectra of the original and the optimized half-cylinder (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints). Table 9.6 provides the usual overview of the optimization results for the mass minimization of the half-cylinder subject to MLS and thickness constraints employing Hermite spline functions as presented in the previous paragraphs. These results are compared with those of the mass minimization without splines as introduced in Sec. 9.2.2 and summarized in Tab. 9.3 on p. 138.

9.4

147

Half-Cylinder Optimized with PIKAIA

Table 9.6: Optimization results for the half-cylinder (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints). Property Mass m Compared to direct modification MLS Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until convergence Compared to direct modification CPU time until convergence Compared to direct modification Iterations until acceptable result Mass of acceptable design CPU time until acceptable result

9.4

Initial design

Optimized design

1.973 kg

1.322 kg (−0.651 kg, −33.0%) −0.022 kg (0.022 kg more reduction) 78.8 dB (no constraint violation) 1.0 mm (no constraint violation) 10.0 mm (no constraint violation) 222.3 Hz (−6.6 Hz, −2.9%) 103.0 dB (+2.9 dB, +95.0%) 48 key points (−70.9%) 331 2106 −5799 (−73.4%) 7 h 42 min (7.7 h) −21.4 h (−73.6%) ≈ 1000 1.331 kg (−0.642 kg, −32.5%) ≈ 3.7 h (−52%)

78.8 dB 4.0 mm 4.0 mm 228.9 Hz 100.1 dB 165 331

Half-Cylinder Optimized with PIKAIA: Minimization of MLS

The last optimization example for the half-cylinder is the MLS minimization subject to mass and thickness constraints, this time employing the genetic algorithm PIKAIA instead of the deterministic optimization strategy COBYLA. Except for the optimization technique used, this optimization problem is identical to the one described in Sec. 9.2.1, i.e., it is defined by 165 design variables and 331 mass and thickness constraints. General information on the optimization results produced by PIKAIA are also available from Sec. 7.4, the first section in this study describing results generated by PIKAIA. The iteration history in terms of fitness, which is calculated from Eq. (4.5), is shown in Fig. 9.16. The gray line marks the fitness value of the respective design as it varies from iteration to iteration due to the random modification of the design variables. The solid black line represents the highest fitness achieved up to a particular iteration. The asymptotic behavior of the black line is clearly visible and is the reason for terminating the optimization procedure after 50 000 iterations, which take 277.1 h (about 11.5 d) of real time on five computers in parallel (see Sec. A.2 in App. A). This is 259.6 h or 1483% more than the CPU time consumed by COBYLA to reach convergence (cf. Tab. 9.2 on p. 136). Figure 9.17 shows that the lowest MLS, i.e., the design with the highest fitness and without constraint violations, is obtained after 47 355 iterations. At this point, the MLS is reduced by 5.0 dB (68.4%) from 78.8 dB to 73.8 dB. This reduction is 1.2 dB smaller than the one obtained with COBYLA although the computation takes about 16 times as long (cf. Tab. 9.2 on p. 136). The optimized design is not only feasible but even has a structural mass that is 0.004 kg (0.2%) smaller than that of the initial design. The gray line visualizes the objective function value, i.e., the MLS, as it varies from iteration to iteration. The solid black line denotes

148

9

Optimization Results: Half-Cylinder

the MLS that corresponds to the solid black line in Fig. 9.16, i.e., the MLS of the design with the highest fitness up to a particular iteration. The dashed black line marks the MLS of the original design. An acceptable MLS reduction of 4.4 dB (63.7%) from 78.8 dB to 74.4 dB is obtained after 13 284 iterations and about 73.6 h, a real time reduction of 73%. 14 12

fitness [−]

10

fitness value at individual iteration highest fitness value so far

8 6 4 2 0

0

5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 number of iterations

mean level of structure borne sound MLS [dB]

Figure 9.16: Iteration history for the half-cylinder (fitness, PIKAIA, MLS minimization, mass and thickness constraints).

80 79 MLS value at individual iteration lowest feasible MLS so far MLS of original structure

78 77 76 75 74 73 72

0

5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 number of iterations

Figure 9.17: Iteration history for the half-cylinder (objective function, PIKAIA, MLS minimization, mass and thickness constraints).

9.4

149

Half-Cylinder Optimized with PIKAIA

The thickness distribution optimized by PIKAIA (Fig. 9.18) is similar to the one optimized by COBYLA (Fig. 9.2 on p. 135). Although it looks different, it features comparable thickness modifications such as the thickness increase at and near the excitation point or the narrow stiffening ribs in circumferential direction in the front part of the structure. In the back part of the half-cylinder the thickness is reduced to the minimum of 1 mm in many places in order to keep the structural mass below its upper limit of 1.973 kg.

1.0-02

9.2-03

8.4-03

7.5-03

6.7-03

5.9-03

5.1-03

4.3-03

3.5-03

Z Y

2.6-03

Z X

Y

1.8-03 X 1.0-03

Figure 9.18: Thickness distribution of the optimized half-cylinder (PIKAIA, MLS minimization, mass and thickness constraints, values in m). Figure 9.19 depicts the LS spectra of the original and the optimized structure, which reflect the MLS decrease. The fundamental frequency increases by 49.8 Hz or 21.8% from 228.9 Hz to 278.7 Hz, whereas the maximum LS peak at the fundamental frequency is lessened by 3.7 dB (−57.3%) from 100.1 dB to 96.4 dB. The number of natural frequencies in the frequency band of interest (0–3000 Hz) is only slightly increased from seven to eight. The results of the MLS minimization of the simply supported half-cylindrical structure subject to mass and thickness constraints using the genetic algorithm PIKAIA summarized in Tab. 9.7 are compared with those of the MLS minimization employing COBYLA provided in Tab. 9.2 on p. 136.

150

9

Optimization Results: Half-Cylinder

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure 9.19: LS spectra of the original and the optimized half-cylinder (PIKAIA, MLS minimization, mass and thickness constraints).

Table 9.7: Optimization results for the half-cylinder (PIKAIA, MLS minimization, mass and thickness constraints). Property MLS Compared to COBYLA result Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Termination after Real time until termination Highest fitness value after Compared to COBYLA result Iterations until acceptable result MLS of acceptable design Real time until acceptable result

Initial design

Optimized design

78.8 dB

73.8 dB (−5.0 dB, −68.4%) +1.2 dB (1.2 dB less reduction) 1.969 kg (−0.004 kg, −0.2%) 1.0024 mm (no constraint violation) 9.99259 mm (no constraint violation) 278.7 Hz (+49.8 Hz, +21.8%) 96.4 dB (−3.7 dB, −57.3%) 165 331 50 000 iterations 11 d 13 h 6 min (277.1 h) 47 355 iterations +42 218 (+821.8%) 13 284 74.4 dB (−4.4 dB, −63.7%) ≈ 73.6 h (−73%)

1.973 kg 4.0 mm 4.0 mm 228.9 Hz 100.1 dB 165 331

10

Optimization Results: Gearbox

This chapter introduces the optimization results for the MLS minimization of the gearbox as described in Sec. 5.5 and depicted in Fig. 5.10 on p. 70 subject to mass and thickness constraints. The optimization of the gearbox is regarded as the toughest test case in this study due to the great number of DOFs in the FE model and the great number of design variables, which both lead to extremely long computation times. This is the reason why only one optimization result is shown in this chapter. Likewise, the frequency range of interest is reduced from the usual 0–3000 Hz to 0–1000 Hz to reduce the CPU time required for the FE analysis, i.e., the natural frequency extraction and the subsequent mode-based steady-state dynamic analysis using a modal superposition technique (see Sec. 2.2). First, the analysis results for the initial gearbox design are presented in Sec. 10.1. This is followed by the optimization results of the MLS minimization using the optimization algorithm COBYLA in Sec. 10.2.

10.1

Original Gearbox

In this section the analysis results for the initial design of the gearbox are introduced to allow a comparison with the optimization results provided in the following section. The mass of the initial gearbox design is 88.999 kg. It has an MLS of 49.4 dB and a fundamental frequency of 355.9 Hz. The maximum LS is 64.0 dB at the 13th natural frequency 888.6 Hz as can be seen from the LS spectrum of the original structure (black solid line) in Fig. 10.3 on p. 155. Table 10.1 summarizes the above findings. The MLS serves as the objective function in the following section, whereas the structural mass is an additional constraint besides the thickness constraints. Table 10.1: Properties of the original gearbox. Property

Value

MLS 49.4 dB Mass m 88.999 kg Fundamental frequency f1 355.9 Hz Max. LS (at f13 = 888.6 Hz) 64.0 dB

152

10.2

10

Optimization Results: Gearbox

Gearbox Optimized with COBYLA: Minimization of MLS

mean level of structure borne sound MLS [dB]

The MLS minimization of the gearbox subject to mass and thickness constraints using COBYLA is the most difficult and complex optimization problem dealt with in this thesis. Its FE model consists of 2444 elements, 17 066 nodes, and 51 198 DOFs, which causes CPU times of about four to five minutes on the usual Intel Pentium 4 PC (see App. A, Sec. A.2) for every single FE analysis. Furthermore, there are 821 design variables and 1643 mass and thickness constraints that are to be influenced and controlled by COBYLA, which requires many iterations for the optimization procedure to converge. Thus, it was decided to reduce the frequency band of interest to the range from 0 to 1000 Hz (instead of the usual 0–3000 Hz for the previous optimization problems) in order to reduce computation times. Note that the abovementioned four to five minutes of CPU time refer to FE analyses covering the reduced frequency range from 0 to 1000 Hz. Figure 10.1 shows the iteration history plot. The optimization procedure is manually terminated after 10 000 iterations, which nevertheless take more than 33.5 days of CPU time (804.5 h). Although the iterations are not fully converged at that time, the trust-region radius ρ (see Secs. 3.2.3, 7.2.1, and 7.2.2) is reduced eight times from its initial value ρbeg = 6.0 to 2.34 · 10−2 . Experience from previous optimization calculations shows that, once the iterations start to converge, i.e., once the trust-region radius ρ starts to be reduced, it is usually only a matter of time until ρ reaches its final value ρend = 10−6 , but the objective function value hardly changes. About 666.9 h or 82.9% of the total CPU time are consumed by the FE analyses. 56 54

MLS value during optimization max. constraint violation < 0.01% MLS of original structure

52 50 48 46 44 42

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 number of iterations

Figure 10.1: Iteration history for the gearbox (COBYLA, MLS minimization, circles denote iterations with a maximum constraint violation of less than 0.01%). The MLS of the optimized design after 10 000 iterations is 5.7 dB or 73.1% lower than that of the initial design (43.7 dB instead of 49.4 dB). The structural mass is reduced by 0.046 kg (−0.05%) from 88.999 kg to 88.953 kg although it is merely a constraint. The maximum thickness constraint is slightly violated by 0.0039%, so the final design is not feasible in the strict mathematical sense, but it is, of course, fully acceptable from the engineering point of view. Improved designs with basically the same (due to rounding) MLS reduction of 5.7 dB

10.2

Gearbox Optimized with COBYLA: Minimization of MLS

153

(−73.1%) from 49.4 dB to 43.7 dB are available after only about 4000 iterations or 321.8 h of computation time, a decrease by 60%. The optimized thickness distribution of the gearbox can be seen in Fig. 10.2. The thickness is only modified in the domains marked in dark gray in Fig. 5.11 on p. 71. Note that Fig. 10.2 visualizes relative thickness changes (i.e., differences between the thickness distribution of the original and the optimized design) as opposed to absolute thickness values shown in all previous thickness distribution contour plots. Medium shades of gray represent areas where the thickness is left unchanged, which includes zones that are not part of the modification domains such as the flanges or bearing supports (cf. Fig. 5.11). In the lower part of the gearbox, where the initial thickness is 6 mm, the relative thickness change can vary between −5 mm (very light shades of gray) and +4 mm (medium dark shades of gray), yielding an allowable absolute thickness range from 1 mm to 10 mm as usual. The upper part of the gearbox, however, is initially 4 mm thick, so the relative thickness change there can range from −3 mm (medium light shades of grey) to +6 mm (very dark shades of grey), again resulting in an absolute thickness ranging from 1 mm to 10 mm. This time there is no “intuitive solution”, because the excitation point is not part of the modification domain, so there is no way of increasing the input impedance by reinforcing the structure at the excitation location. It is rather difficult to interpret the optimized thickness distribution. The thickness is increased on the right-hand and left-hand side at the front of the lower part but reduced in the middle (Fig. 10.2 a), whereas it is increased in the lower half and decreased at the upper half of the lower part of the gearbox’s back (Fig. 10.2 b). The thickness distributions at the sides of the lower part are rather similar except for some kind of reinforcement near the excitation point (Fig. 10.2 a). The thickness modifications at the upper part of the gearbox are even more difficult to describe and interpret. The thickness is increased mainly at the front and at the left-hand side of the top of the gearbox (Fig. 10.2 a), but it is reduced almost everywhere else. This material transfer from some parts of the gearbox to other ones helps to reduce the MLS but, at the same time, prevents the structural mass from exceeding its upper limit 88.999 kg. When comparing the LS spectra of the original and the optimized gearbox in Fig. 10.3 with those of other structures as shown in the previous chapters (e.g., Fig. 8.4 on p. 118 or Fig. 9.3 on p. 135), one can notice immediately that the gearbox’s LS spectra look very different. The LS is in the range of 25 to 65 dB instead of 50 to 110 dB as for the other structures, and the quasi-static frequency range is rather extended up to about 350 Hz. Both effects are due to the massive, heavy bearing supports and flanges around the gearbox and to the fact that the excitation force acts on these massive parts rather than somewhere on the thin plate-like parts of the gearbox. Furthermore, the vibrational levels in the lower frequency range are relatively low, whereas the maximum LS peaks can be found in the upper quarter of the frequency band from 0 to 1000 Hz. The LS of the optimized structure is generally lower than that of the original one, particularly between 400 Hz and 600 Hz as well as beyond about 750 Hz. The fundamental frequency slightly increases by 6.3 Hz or 1.8% from 355.9 Hz to 362.2 Hz. The maximum LS at the 13th natural frequency, namely, 64.0 dB, is shifted to the 15th natural frequency and reduced by 4.9 dB (−67.6%) to 59.1 dB. The number of natural frequencies in the frequency band of interest from 0 to 1000 Hz goes up from 15 to 16. Table 10.2 at the end of this sections provides the usual overview of the optimization results for the MLS optimization of the gearbox subject to mass and thickness constraints using COBYLA, which are described in the above paragraphs.

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Optimization Results: Gearbox

6.0-03

5.0-03

4.0-03

3.0-03

Z

2.0-03 Y X

a)

1.0-03

0

-1.0-03

-2.0-03

-3.0-03

-4.0-03

-5.0-03

Z X Y

b)

Figure 10.2: Thickness distribution of the optimized gearbox: a) front, b) back (COBYLA, MLS minimization, mass and thickness constraints, relative thickness change, values in m).

10.2

155

Gearbox Optimized with COBYLA: Minimization of MLS

level of structure borne sound LS [dB]

70 original structure optimized structure

65 60 55 50 45 40 35 30 25 20 200

300

400

500 600 700 frequency [Hz]

800

900

1000

Figure 10.3: LS spectra of the original and the optimized gearbox (COBYLA, MLS minimization, mass and thickness constraints).

Table 10.2: Optimization results for the gearbox (COBYLA, MLS minimization, mass and thickness constraints). Property MLS Mass m Minimum thickness tmin Maximum thickness tmax Fundamental frequency f1 Maximum LS Number of design variables Number of constraints Iterations until termination CPU time until termination Iterations until acceptable result MLS of acceptable design CPU time until acceptable result

Initial design

Optimized design

49.4 dB 88.999 kg 4.0 mm 6.0 mm 355.9 Hz 64.0 dB 821 1643

43.7 dB (−5.7 dB, −73.1%) 88.953 kg (−0.046 kg, −0.05%) 1.00133 mm (no constraint violation) 10.00039 mm (0.0039% constr. viol.) 362.2 Hz (+6.3 Hz, +1.8%) 59.1 dB (−4.9 dB, −67.6%) 821 1643 10 000 33 d 12 h 28 min (≈ 804.5 h) ≈ 4000 43.7 dB (−5.7 dB, −73.1%) ≈ 321.8 h (−60%)

156

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Optimization Results: Gearbox

11

Discussion

In this last chapter the procedures and results presented in this dissertation are discussed. Section 11.1 provides a summary of the methods and results introduced in this work. Some conclusions from these results are drawn in Sec. 11.2. Some known problems and suggestions for future research work are listed in Sec. 11.3. A table summarizing all optimization results can be found on p. 162.

11.1

Summary

In this dissertation a numerical method to optimize the thickness distribution of three-dimensional structures with respect to various vibrational and structural properties and subject to suitable constraints is introduced. Possible objectives of the optimization include the minimization of the mean level of structure borne sound (MLS), the minimization of the structural mass m, the maximization of the fundamental frequency f1 , and the maximization of the difference between two arbitrarily chosen natural frequencies ∆f . Possible candidates for constraints are the structural mass m, the MLS, and the fundamental frequency f1 . In addition, the allowable range of the design variable values, i.e., of the structure’s local thickness t, is restricted by an upper and a lower limit tmax and tmin , respectively. Finite element (FE) models of the structures to be optimized are modified iteratively and automatically in such a way that the objective function significantly improves. This process continues until some predefined convergence criterion is satisfied or until it is manually terminated. Chapter 1 gives information on the background and the motivation for this research project. It also contains a detailed literature review covering numerous books and papers that are related to this work. Finally, it introduces the scope and objectives of this project, explains which issues are not considered, and delimits the techniques and approaches used in the present work from those employed in a similar previous study by another author. Some basic theory of structural acoustics is outlined in Chap. 2. The fundamental equation of machine acoustics is used to show why it is considered sufficient to focus on the vibrational level of structures instead of on the radiated sound power level. The equations for calculating the level of structure borne sound (LS) and the MLS are provided. Chapter 3 first discusses various general aspects of numerical optimization methods before it elaborates on the two optimization algorithms used in this study, namely, the deterministic algorithm COBYLA and the genetic (stochastic) algorithm PIKAIA. The iterative numerical optimization procedure is described in detail in Chap. 4. First, aspects and features that are common to both optimization algorithms are presented. Then, issues that are unique to either of the two optimization strategies are introduced.

158

11

Discussion

In Chap. 5 the FE models of the structures investigated and optimized in this dissertation are shown. These structures are two rectangular steel plates, one with a coarse and the other with a refined discretization, two rectangular steel plates joined at 90◦ , a half-cylinder made of steel, and a steel gearbox. This chapter also contains a discussion of discretization errors and numerical pollution. Chapter 6 introduces an approach to reduce the number of design variables by employing spline functions. The use of splines makes it possible to model the structure’s thickness distribution with just a few key points rather than with many FE nodes, which drastically reduces computation times. A bicubic spline surface, a Hermite spline surface, and a tensor product B´ezier surface are presented. Various optimization results for the simply supported rectangular steel plate are reported in Chap. 7. First, some properties of the initial structure are determined. The two different discretizations result in only minimal deviations of MLS and fundamental frequency. Then, the optimization algorithm COBYLA is used to minimize the MLS of the FE model with the refined mesh and of the model with the coarse mesh. The results show that the final optimized designs look almost identical for both discretizations, and that the objective function reductions are comparable (−8.6 dB and −7.7 dB). However, the computation time is reduced by almost 94% by using the coarse FE mesh. Therefore, relatively coarse FE meshes are used for the remaining optimization problems in order to avoid excessive computation times. The mass minimization of the rectangular plate using COBYLA yields a mass reduction of almost 39%, whereas the maximization of the fundamental frequency utilizing COBYLA results in a fundamental frequency increase of about 91%. As a result of the maximization of the difference between the third and the fourth natural frequency, COBYLA increases this difference by 259%. Next, spline functions are used to reduce the number of design variables for some of the optimization calculations using COBYLA. A reduction in design variables of 55% or 90% leads to CPU time reductions of up to 59% and 89%, respectively, although the final objective function values hardly change. Finally, the genetic algorithm PIKAIA is used to minimize the mass of the rectangular plate. Although PIKAIA takes almost 15 times more iterations than COBYLA, the mass reduction is about 11% smaller than that achieved by COBYLA. The optimization results for the two-plate joint are presented in Chap. 8. By using COBYLA, the MLS is reduced by about 10 dB, the structural mass is decreased by 45%, and the difference between the fourth and the fifth natural frequency is increased by 2440%. If an additional fundamental frequency constraint is specified for the mass minimization, the mass decreases by only 42%. In case PIKAIA instead of COBYLA is used for the maximization of the natural frequency difference, an improvement of about 2760% is achieved. Chapter 9 contains the optimization results for the half-cylinder. When COBYLA is used as the optimization algorithm, the MLS decreases by 6.2 dB, the structural mass is reduced by 32%, and the fundamental frequency increases by 64%. Using Hermite spline functions reduces the number of design variables by 71%, which leads to a CPU time reduction of 64% for the MLS minimization (−4.6 dB) and of 74% for the mass minimization (−33%). If PIKAIA instead of COBYLA minimizes the MLS, the number of iterations is increased by a factor of eight, but the MLS reduction is only 5 dB. Finally, Chapter 10 provides the optimization results for the gearbox. Since this FE model requires extremely long computation times, the frequency band of interest is restricted to the range from 0 to 1000 Hz, and only one MLS optimization using COBYLA is performed, which reduces the MLS by 5.7 dB. A summary of all optimization results can be found in Tab. 11.1 on p. 162.

11.2

Conclusions

159

Appendix A describes the computation of quantities such as the structural mass, the surface area of the sound radiating surface, the local thickness, and the thickness direction vectors. It also introduces an approach to perform the ABAQUS FE analyses simultaneously on several computers if the PIKAIA algorithm is used. Appendix A concludes with some information on a graphical user interface and on the visualization of the optimization results. In App. B the optimization results are validated by comparing them with results obtained in a previous study by another author who used similar structures. Finally, App. C visualizes the nonlinearity and the multimodality of the objective function. This is accomplished by systematically varying only two of the design variables between their lower and upper bounds and keeping all other variables constant, thus reducing the “optimization problem” to a two-dimensional one, which can be displayed by means of a three-dimensional surface plot.

11.2

Conclusions

It has been shown in the previous chapters that the optimization procedures introduced in this dissertation are able to produce significant improvements of the objective function. This holds for both optimization algorithms employed, namely, the deterministic COBYLA algorithm and the genetic PIKAIA algorithm. Both algorithms are robust and stable despite the high number of design variables and constraints. Depending on the structure to be optimized and on the objective function, COBYLA achieved objective function improvements of up to 10.2 dB (−90.5%) for the MLS minimization, up to 44.9% for the mass minimization, up to 91.4% for the fundamental frequency maximization, and up to 2440% for the maximization of the difference between two natural frequencies (see Tab. 11.1 on p. 162). Whereas COBYLA is a local optimization algorithm, PIKAIA is theoretically able to find a global optimum. However, both PIKAIA’s efficiency and its effectivity are, in general, considerably inferior to COBYLA’s, i.e., it takes significantly more iterations (and therefore computation time) to achieve a result that is worse than COBYLA’s. For instance, PIKAIA obtained an MLS reduction of 5.0 dB (−68.4%) where COBYLA achieved 6.2 dB (−76.0%), and a mass reduction of 0.883 kg (−34.4%) where COBYLA achieved 0.990 kg (−38.6%). The only exception is the maximization of the difference between two natural frequencies where PIKAIA achieved an improvement of 527.8 Hz (+2763%), whereas COBYLA only obtained 466.1 Hz (+2440%). It is obvious that COBYLA found only a local optimum, whereas PIKAIA possibly got a step closer to the global one (cf. Tab. 11.1). The iteration numbers and CPU times required to achieve convergence or to reach the maximum number of iterations are thoroughly reported together with the number of design variables and constraints (see Tab. 11.1). These records show that some of the optimization runs require an extremely high number of iterations (up to 15 037 for COBYLA, 50 000 for PIKAIA) and, therefore, excessively long computation times of up to 33.5 days (804.5 h). However, this is most often due to the strict mathematical formulation of the constraints. If tiny constraint violations of no more than 0.01% are accepted and the formal convergence criterion is ignored, then the optimization process can be terminated significantly earlier with still impressive objective function improvements. A means of reducing computation times is the use of relatively coarse discretizations of the FE models, which decreases both the CPU time required for each single FE analysis and the number of design variables. However, a too coarse FE mesh can cause numerical pollution

160

11

Discussion

effects and lead to inaccurate analysis results. The mesh sizes used in this study are considered acceptable compromises between these contradictory requirements. Another way to reduce computation times is the use of spline functions to model the thickness distribution with just a few key points, which does not alter the FE models and, particularly, the FE discretization, but only the number of design variables. The optimization results seem to indicate that an approximately proportional relation between the number of design variables and the number of iterations (and therefore CPU times) can be assumed. This means that a halving of the number of design variables by employing spline functions yields (approximately) a halving of the number of iterations and, thus, of the computation time. The use of spline functions leads to similar designs but slightly smaller objective function improvements. The only exception is the mass minimization of the half-cylinder where the use of splines yields a mass reduction of 33% compared to 31.9% without splines. In general, the effectivity is slightly decreased if spline functions are used, but the efficiency is drastically increased since significant improvements are achieved within a much shorter timespan (see Tab. 11.1 on p. 162). “Classical” optimization problems such as the MLS minimization or the mass minimization, which are considered the standard test cases for all structures investigated except for the gearbox, yield final designs that are plausible and interpretable as the “intuitive” solution. These “intuitive” solutions are designs to which most acousticians would agree: “Yes, I would have done this, too.”, e.g., increase the input impedance or apply stiffening ribs and point masses. Thus, the optimization procedure seems to be reliable. On the other hand, it is able to find solutions to nontrivial problems as well, for which an “intuitive” solution is not readily available. Examples include the mass minimization with an additional constraint on the fundamental frequency and the maximization of the difference between two natural frequencies. These interesting and more challenging test cases can be considered “optional” as opposed to the standard test cases mentioned above. Additional constraints that are difficult to satisfy can lead to longer computation times and to less improvement of the objective function, but the optimization procedure is able to find a feasible design with a substantial objective function improvement anyway. Although the structures investigated in this thesis are rather academic examples that are not very realistic, they nevertheless constitute more complex and realistic models than most others reported in the literature. For example, all structures introduced in this work are truly three-dimensional models that consist of three-dimensional solid hexahedral elements (“bricks”) instead of two-dimensional plate or shell elements. This creates additional difficulties when calculating the nodal normal vectors on the structure’s surface or the local thickness. Thus, the concept of thickness direction vectors to determine the local thickness and the direction along which the position of the movable surface nodes can be varied during the optimization was developed and successfully applied to all optimization problems described in this study. The optimization procedure is a combination of the commercially available FE software ABAQUS and user-written programs. Due to the modular structure of the procedure it should be relatively easy to replace this particular FE software with any other FE program. Likewise, it should be possible to implement an alternative optimization algorithm. The genetic algorithm PIKAIA has been modified and adapted in a number of ways. Its original version is unconstrained except for the design variables that must be from the interval [0.0, 1.0]. Hence, constraints have been implemented in the form of penalty functions that artificially reduce the fitness value of any infeasible design. Furthermore, PIKAIA has been (pseudo-)parallelized, i.e., the time-consuming objective function evaluations can be carried out simultaneously on several different computers. This approach has the potential to speed

11.3

Future Work

161

up the process considerably and to reduce overall computation times despite high numbers of iterations. In practice, however, the speed-up was not quite so significant due to the fact that the participating computers differ greatly in computing power.

11.3

Future Work

Although this dissertation addresses a lot of issues there is still a lot left to be done and many problems still need to be solved. One can even get the impression that there are more questions raised than answers given. Many of the open problems can be derived from the list of issues that are not objectives of this dissertation (see Sec. 1.3.2). It would be interesting to include active control techniques in the optimization process. This could be done either sequentially, i.e., by first numerically optimizing the passive structure and then applying active noise and vibration control to increase the improvements, or simultaneously, i.e., by numerically simulating both passive modifications and active control and their mutual effects at the same time. The numerical optimization results should be validated by experimental measurements. Though they are difficult to prepare and to perform, they are the only means of proving that the theoretically obtained designs really yield a significant improvement of the objective function or even some sort of optimum. It should be investigated if the optimized designs are able to withstand the occurring static and dynamic loads when subjected to real-life operating conditions. Likewise, the problem of manufacturability should be considered, i.e., to determine if the optimized structures can be produced at reasonable cost by some metal forming, forging, or casting process. In particular, it would be interesting to see if and how the optimized design proposals can be converted into realistic designs and transferred to real-life applications. Though it is assumed that an MLS reduction leads to a noise reduction, i.e., to a reduction of the sound pressure level at some specified location or to a reduction of the total radiated sound power, this assumption has not been validated in the present work. Conversely, other researchers took great effort to minimize the radiated sound power by means of complex and time-consuming coupled FEM/BEM computations but did not investigate whether simple and fast FEM calculations of the vibrational behavior suffice to yield similar results after only a fraction of the computation time required by the coupled FEM/BEM calculations. Therefore, the radiated sound power of the optimized designs should be compared with that of the initial design to see whether a significant MLS reduction results in a significant reduction of the radiated sound power as well. One can also experiment with various spline function types and various numbers of key points to check if some additional CPU time savings can be achieved. A procedure should be developed to apply the spline function approach not only to simple geometries such as the rectangular plate or the half-cylinder but to more complex FE models such as the gearbox. It will probably be necessary to parameterize different parts of the structure’s surface with separate spline functions and to assure a smooth and continuous transition between these separate spline surfaces. Finally, it should be checked if the optimization algorithms used in this study can either be modified and improved so as to achieve better results with shorter computation times or even replaced with other, more efficient and effective optimization techniques. Although the methods and procedures presented in this work are quite effective and relatively efficient, they are certainly not the essence of all wisdom.

Objective AlgoInitial Optimized function rithm Splines design design Rectangular plate, refined FE discretization MLS minimization COBYLA 83.9 dB −8.6 dB (−86.2%) MLS minimization COBYLA × 83.9 dB −7.4 dB (−81.8%) Rectangular plate, coarse FE discretization MLS minimization COBYLA 83.6 dB −7.7 dB (−83%) MLS minimization COBYLA × 83.6 dB −7.4 dB (−81.8%) Mass minimization COBYLA 2.564 kg −0.990 kg (−38.6%) Mass minimization COBYLA × 2.564 kg −0.929 kg (−36.2%) Mass minimization PIKAIA 2.564 kg −0.883 kg (−34.4%) f1 maximization COBYLA 257.4 Hz +235.2 Hz (+91.4%) ∆f maximization COBYLA 96.7 Hz +250.5 Hz (259.0%) Two plates joined at 90◦ MLS minimization COBYLA 81.5 dB −10.2 dB (−90.5%) Mass minimization COBYLA 3.195 kg −1.434 kg (−44.9%) Mass min. + f1 constr. COBYLA 3.195 kg −1.331 kg (−41.7%) ∆f maximization COBYLA 19.1 Hz +466.1 Hz (+2440%) ∆f maximization PIKAIA 19.1 Hz +527.8 Hz (+2763%) Half-cylinder MLS minimization COBYLA 78.8 dB −6.2 dB (−76.0%) MLS minimization COBYLA × 78.8 dB −4.6 dB (−65.3%) MLS minimization PIKAIA 78.8 dB −5.0 dB (−68.4%) Mass minimization COBYLA 1.973 kg −0.629 kg (−31.9%) Mass minimization COBYLA × 1.973 kg −0.651 kg (−33.0%) f1 maximization COBYLA 228.9 Hz +146.1 Hz (+63.8%) Gearbox MLS minimization COBYLA 49.4 dB −5.7 dB (−73.1%) 15 037 1645 4137 1850 3088 1251 50 000 2004 3747 7665 3720 10 136 10 884 50 000 5137 2000 50 000 7905 2106 6013 10 000

345 35 77 35 77 35 77 77 77 220 220 220 220 220 165 48 165 165 48 165 821

No. of des. No. of variables iterations

Table 11.1: Summary of the optimization results.

h h h h h h

h h h h h

h h h h h h h

152

134 141 147 136 144 139

116 119 122 126 128

84 99 89 105 109 92 95

80 103

11

804.5 h

17.5 6.2 277.1 29.1 7.7 20.7

59.6 50.4 135.8 115.6 658.9

10 5 10.1 4.1 312 5.2 9.4

152.8 h 18.35 h

Comput. time Page

162 Discussion

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Some Remarks on the Implementation

As already described in Chap. 4, the optimization procedure that constitutes the Numerical Structural Acoustic Optimization (NSAO) program consists of various user-written Fortran programs, C programs, and Unix shell scripts. They are all interwoven with each other and use the commercial FE program ABAQUS to analyze the modified designs generated by the optimization algorithm. Fortran programs and subroutines calculate the LS and MLS, the local thickness directions and thickness distribution, normal vectors, objective function values, fitness values, constraint violations, etc. Unix shell scripts control the procedure, handle system calls, copy, move, and rename files, and so forth. C programs create a graphical user interface (GUI) that facilitates the creation and modification of a control parameter file. These user-written programs and shell scripts comprise approximately 19 000 lines of program code. About 12 000 lines are proper, executable commands, whereas the remaining about 7000 lines are comment lines. Section A.1 describes the calculation of quantities such as the structural mass, the surface area of the sound radiating surface, the normal vectors, the local thickness, and the thickness direction vectors. An approach to (pseudo-)parallelize the genetic PIKAIA algorithm is introduced in Sec. A.2. This is followed by a short description of NSAO’s graphical user interface in Sec. A.3. Finally, Sec. A.4 explains how the optimization results are visualized.

A.1

Calculation of Mass, Surface Area, Normal Vectors, and Thickness

This section deals with the calculation of important quantities that are necessary for the optimization calculations. The computation of the structural mass is described in Sec. A.1.1. Section A.1.2 explains how the surface area of the sound radiating surface is determined. In Sec. A.1.3 the calculation of the normal vectors is shown. Finally, Sec. A.1.4 gives the details of the thickness calculation. A.1.1

Calculation of the Structural Mass

The calculation of the mass of the structure to be optimized is necessary because it serves as either a constraint (e.g., when minimizing the MLS or maximizing the fundamental frequency) or even as the objective function. If the mass would not be constrained by an upper bound for the MLS minimizations, it would steadily increase in order to reduce the MLS, since a

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heavy structure exhibits lower vibrational levels. However, in most cases a mass increase is undesirable or the user even wants to reduce the mass. Since the mass is not readily available as an output variable to the binary ABAQUS result file (∗.fil) for this kind of analyses, i.e., natural frequency extraction and mode-based steady-state dynamic analysis [ABAQUS 2001a], it must be determined in another way. The mass m is calculated from the structure’s volume Vstruc and the material’s density ρ by m = ρVstruc .

(A.1)

Initially, the volume Vel of each finite element was computed separately and then added up to obtain the structure’s total volume Vstruc . This was achieved by calculating each element’s volume Vel by means of a three-dimensional Gaussian integration scheme using the global coordinates of the FE nodes x, y, and z and the shape functions Na (ξ, η, ζ) given in local coordinates ξ, η, and ζ. The volume Vel of a single finite element can be calculated from [Zienkiewicz 2000] Vel =

+1+1+1 det J dξ dη dζ ,

(A.2)

−1 −1 −1

where J is the Jacobian matrix 

 ∂x/∂ξ ∂y/∂ξ ∂z/∂ξ J =  ∂x/∂η ∂y/∂η ∂z/∂η  , ∂x/∂ζ ∂y/∂ζ ∂z/∂ζ

(A.3)

which consists of the partial derivatives of the global coordinates x, y, and z with respect to the local coordinates ξ, η, and ζ. The following transformations express the global coordinates x, y, and z in terms of the local coordinates ξ, η, and ζ 20  x(ξ, η, ζ) = Na (ξ, η, ζ) xa , y(ξ, η, ζ) = z(ξ, η, ζ) =

a=1 20 

a=1 20  a=1

(A.4)

Na (ξ, η, ζ) ya , Na (ξ, η, ζ) za ,

where Na (ξ, η, ζ) are the so-called shape functions for a 20-node hexahedral solid finite element (cf. Eqs. (A.6a) to (A.6t)), and xa , ya , and za are the global coordinates of the ath FE node on a particular element. Figure A.1 shows the node and face numbering and the local coordinates of such an element (ABAQUS element type C3D20) in ABAQUS nomenclature [ABAQUS 2001b]. Analogously, the partial derivatives needed in Eq. (A.3) can be calculated from ∂x/∂ξ = ∂y/∂ξ = ∂z/∂ξ =

20  a=1 20  a=1 20  a=1

(∂Na /∂ξ) xa , ∂x/∂η = (∂Na /∂ξ) ya , ∂y/∂η = (∂Na /∂ξ) za , ∂z/∂η =

20  a=1 20  a=1 20  a=1

(∂Na /∂η) xa , ∂x/∂ζ = (∂Na /∂η) ya , ∂y/∂ζ = (∂Na /∂η) za , ∂z/∂ζ =

20  a=1 20  a=1 20  a=1

(∂Na /∂ζ) xa , (∂Na /∂ζ) ya , (∂Na /∂ζ) za ,

(A.5)

A.1

165

Calculation of Mass, Surface Area, Normal Vectors, and Thickness

ζ η

face 2 15

8

7

16

14 13

5 face 6

face 5

6

20

19 ξ

17

face 4

18 11

4

3

12 10 1 face 3

9

2 face 1

Figure A.1: Nodes and faces of a 20-node hexahedral solid element (ABAQUS element type C3D20) in local coordinates (after [ABAQUS 2001b]). where ∂Na /∂ξ, ∂Na /∂η, and ∂Na /∂ζ are the partial derivatives of the shape functions Na (ξ, η, ζ) with respect to the local coordinates ξ, η, and ζ (cf. Eqs. (A.7a) to (A.7t)). The 20-node hexahedral solid elements (ABAQUS element type C3D20) used in this study are second-order elements, i.e., they have quadratic shape functions. The following equations show some of the twenty shape functions of such an element according to [ABAQUS 2001b] 1 N1 (ξ, η, ζ) = − (1 − ξ) (1 − η) (1 − ζ) (2 + ξ + η + ζ) , 8

(A.6a)

1 N2 (ξ, η, ζ) = − (1 + ξ) (1 − η) (1 − ζ) (2 − ξ + η + ζ) , (A.6b) 8 .. . 1 N20 (ξ, η, ζ) = (1 − ζ) (1 + ζ) (1 − ξ) (1 + η) . (A.6t) 4 The sixty derivatives, i.e., the partial derivatives of each of the twenty shape functions in Eqs. (A.6a) through (A.6t) with respect to each of the three local coordinates ξ, η, and ζ, are N1,ξ =

∂N2 (ξ, η, ζ) ∂N2 (ξ, η, ζ) ∂N2 (ξ, η, ζ) , N2,η = , N2,ζ = , ∂ξ ∂η ∂ζ ... ∂N20 (ξ, η, ζ) ∂N20 (ξ, η, ζ) ∂N20 (ξ, η, ζ) = , N20,η = , N20,ζ = . ∂ξ ∂η ∂ζ N2,ξ =

N20,ξ

∂N1 (ξ, η, ζ) ∂N1 (ξ, η, ζ) ∂N1 (ξ, η, ζ) , N1,η = , N1,ζ = , ∂ξ ∂η ∂ζ

(A.7a)

(A.7b)

(A.7t)

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The partial derivatives of the shape functions (A.7a) through (A.7t) can be used to compute the partial derivatives of the global coordinates in Eq. (A.5). These are then inserted into Eq. (A.3) to calculate the Jacobian matrix, which in turn is needed to determine the volume of each finite element by means of Eq. (A.2). This procedure worked quite well but it has one major disadvantage: One needs to manually program all the shape functions (A.6a) to (A.6t) of the finite element type(s) used as well as their partial derivatives (A.7a) to (A.7t). Thus, whenever another FE element type is used (e.g., tetrahedral or wedge elements instead of hexahedral elements or elements with linear shape functions instead of quadratic shape functions), the appropriate shape functions have to be programmed since they are not directly available in ABAQUS. This means a lot of work since the shape functions have to be transcribed from the ABAQUS Theory Manual [ABAQUS 2001b], and the derivatives have to be determined “manually” and typed in as well. Furthermore, this procedure is quite error-prone, since the slightest typing error results in wrong values for the partial derivatives and therefore for the element volume Vel , the total volume of the structure Vstruc , and eventually for the mass of the structure m. Later, Robers [Robers 2003] realized that this rather complicated and quite error-prone procedure described in the previous paragraphs is not the only way to determine the structural mass. As mentioned at the beginning of this section, it is not possible in ABAQUS explicitly to request the output of the structural mass to the binary ∗.fil file (at least not for this kind of analyses used here, i.e., natural frequency extraction and mode-based steady-state dynamic analysis). Nonetheless, ABAQUS implicitly records the mass of the structure at the beginning of the ∗.dat file (ASCII format) even if there are no other output requests at all. Therefore, the Fortran subroutine that determines the structural mass was modified in such a way that it now simply reads the mass from the ∗.dat file provided by ABAQUS. This means that the mass is now calculated internally by the commercial, professional FE software ABAQUS instead of by user-written, error-prone Fortran routines. Although the details of this calculation are not known to the author, it is assumed that ABAQUS internally uses the same or similar methods as described above. This alternative method (i.e., letting ABAQUS do the calculation of the mass) has some major advantages: First, it is more flexible, because it works with all element types without any modification or adaption. Second, it is more accurate and reliable, since it is professionally programmed and tested. Third, it is faster, because the mass is calculated by ABAQUS anyway, whereas additional subroutines were required for the Gaussian integration scheme described above. Thorough numerical tests [Robers 2003] showed excellent agreement with the results of the method previously used (i.e., Gaussian integration using the partial derivatives of the shape functions according to Eqs. (A.7a) to (A.7t)). However, as mentioned already, the alternative method is much simpler to use, easier to modify, more flexible in practice, and more exact and reliable. A.1.2

Calculation of the Surface Area

The surface area S of the sound radiating surface of the structure to be optimized is needed to calculate the LS by means of Eq. (2.7) on p. 30. Since not only the mass of the structure but also the surface area is not readily available in ABAQUS as an output variable to the binary result file (∗.fil) for this kind of analyses, i.e., natural frequency extraction and mode-based

A.1

Calculation of Mass, Surface Area, Normal Vectors, and Thickness

167

steady-state dynamic analysis [ABAQUS 2001a], it must be calculated some other way as well. The element faces and nodes that constitute the sound radiating surface are identified and marked by the user while creating the FE model of the structure. Although the nodal coordinates of the FE nodes on the sound radiating surface are modified by the optimization algorithm during the optimization, the nodes themselves and the corresponding element faces remain on the sound radiating surface. Initially, a numerical integration procedure similar to the one used for determining the structural mass (see Sec. A.1.1) was used to calculate the surface area. The surface area A of each finite element face on the sound radiating surface is computed separately, then all the face areas are added up to obtain the structure’s total sound radiating surface area S. The calculation of each element’s sound radiating surface area A is accomplished by means of a two-dimensional Gaussian integration scheme analogously to the three-dimensional integration scheme for determining the mass in Sec. A.1.1. For a three-dimensional second-order 20-node hexahedral solid element (ABAQUS element type C3D20), each element face can be interpreted locally as a two-dimensional second-order 8-node quadrilateral element (e.g., ABAQUS element type CPE8 or CPS8), an example of which could be face 2 in Fig. A.1. Therefore, the three-dimensional problem of mass determination described in Sec. A.1.1 reduces to a two-dimensional problem with only two local coordinates ξ and η for the surface area calculation. Analogously to Eq. (A.2), the surface area A of an element face can be calculated from

A=

+1+1 det J dξ dη ,

(A.8)

−1 −1

where J is the Jacobian matrix  J=

∂x/∂ξ ∂y/∂ξ ∂x/∂η ∂y/∂η

.

(A.9)

The eight shape functions of ABAQUS’s two-dimensional second-order 8-node quadrilateral element CPE8 or CPS8 [ABAQUS 2001b] are quite different from the 20 shape functions of the three-dimensional second-order 20-node hexahedral solid element C3D20 given in Eqs. (A.6a) through (A.6t), namely, for instance, 1 N1 (ξ, η) = − (1 − ξ) (1 − η) (1 + ξ + η) , 4

(A.10a)

1 N2 (ξ, η) = − (1 + ξ) (1 − η) (1 − ξ + η) , (A.10b) 4 .. . 1 (A.10h) N8 (ξ, η) = (1 − η) (1 + η) (1 − ξ) . 2 They result in 16 partial derivatives in analogy to Eqs. (A.7a) through (A.7t), which are used to calculate the Jacobian matrix in Eq. (A.9), which in turn is needed to determine the surface area of a finite element face by means of Eq. (A.8). This procedure worked very well, too, but had the same disadvantage as the one for calculating the mass in Sec. A.1.1: It was not flexible enough to allow a quick and easy change to

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other element types. Instead, it required lengthy and error-prone programming work to adapt the method for each additional or alternative element type. Robers [Robers 2003] found an alternative method for calculating the surface area that overcomes the above mentioned drawbacks. The surface area is not directly available in ABAQUS, but one can make ABAQUS calculate it and write it to a file by using a trick. While creating the FE model of the structure to be optimized, the user applies an element-uniform “virtual” pressure distribution on the element faces that belong to the sound radiating surface. The magnitude of this virtual pressure can be chosen arbitrarily because it is not used at all later on in the analysis. The pressure distribution serves just the one purpose of identifying and to mark the element faces on the sound radiating surface. An example of such a virtual pressure distribution can be seen in Fig. A.2. The arrows indicate the virtual pressure applied to each element face on the sound radiating surface.

Z Y X

Figure A.2: Virtual pressure applied to the half-cylinder’s surface. After the FE model creation process by means of the commercial FE pre- and post-processing software MSC.Patran [Patran 2003] is completed, a so-called ABAQUS input file (∗.inp) in ASCII format is created and saved to the hard disk. This file contains a complete description of the FE model to be analyzed (i.e., nodal coordinates, element definitions, boundary and loading conditions, node and element sets, material properties, analysis steps and types, etc.) and serves as an input file to start and control the FE analysis using ABAQUS. In the input file, the ∗DLOAD (“distributed load”, i.e., pressure) option of ABAQUS specifies the elements and the element faces (see Fig. A.1) to which the virtual pressure is applied [ABAQUS 2001a]. These lines in the ∗.inp file are edited manually. After that, they do not define a pressure applied to a surface (∗DLOAD option) any longer, but are rather used to define element surfaces themselves by means of the ∗SURFACE option [ABAQUS 2001a]. This option specifies just the elements or element sets and the respective element faces that had previously been marked by applying said virtual pressure. Therefore, there is no additional surface load applied to the element faces by the virtual pressure because the pressure does not exist any longer.

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Calculation of Mass, Surface Area, Normal Vectors, and Thickness

169

Usually, element surfaces defined by the ∗SURFACE option are used to define contact areas for FE models dealing with friction problems. However, they can be used as well to output the total surface area of the sound radiating surface to the human-readable ∗.dat file or the binary ∗.fil file. This is achieved by using the ∗SECTION PRINT or ∗SECTION FILE option of ABAQUS along with the SURFACE parameter and specifying the output variable SOAREA [ABAQUS 2001a]. The total surface area S of the sound radiating surface can then be read from either of these two files and used to calculate the LS by means of Eq. (2.7). This alternative method used to obtain the surface area was compared with the two-dimensional Gaussian integration scheme and showed perfect agreement as well [Robers 2003]. It features the same advantages as the alternative method used to determine the mass, i.e., ease of use, ease of modifications, flexibility, exactness, reliability, and robustness. A.1.3

Calculation of the Normal Vectors

The normal vectors at the nodes at the sound radiating surface are used to calculate the nodal normal rms surface velocities v⊥rms (scalars) of the vibrating structure, which are needed to calculate the mean squared transmission admittance h2t (f ) by means of Eq. (2.4). The FE software ABAQUS can only compute the nodal rms velocities v rms (three-dimensional vectors) caused by some force excitation. It cannot, however, directly calculate the components of the nodal velocities v⊥rms that are normal (i.e., perpendicular) to the sound radiating surface. Only these normal velocity components can excite the surrounding air and cause it to vibrate as well, thus transferring sound. Structure borne sound can only be radiated as air borne sound by means of the normal velocity components, whereas other effects are basically negligible. The normal rms surface velocity v⊥rms i at the ith surface node can be determined from the nodal rms velocity vector v rmsi computed by ABAQUS and the normal unit vector ni at that ith surface node v⊥rms i = v rmsi · ni . (A.11) Broschart [Broschart 1994] calculates the normal vector at a node of an FE face by using the Jacobian matrix and the partial derivatives of the shape functions introduced in Sec. A.1.2. The normal unit vector n pointing “outward” of the structure can then be calculated by n=

(∂x/∂ξ) × (∂x/∂η) , |(∂x/∂ξ) × (∂x/∂η)|

(A.12)

where x = (x, y, z)T are the global coordinates, and ξ and η are the local coordinates. Since this method of computing the normal vectors is based on the element shape functions and their derivatives, too, the same drawbacks as for the original mass and surface area calculation apply (see Secs. A.1.1 and A.1.2). In order to become independent of the finite element type(s) used, an alternative finite difference method based on the distance between the nodes on an element face was developed [Wesp 2002]. If one uses finite elements with plane faces and a constant thickness throughout an element (like the combined shell/membrane elements used by Hibinger [Hibinger 1998]), the thickness and the normal vector can either be calculated easily or even provided by the FE program. If, however, one uses general second-order solid elements such as wedge elements, tetrahedra, or hexahedra the faces of which do not necessarily have to be plane but can be curved, there is no readily available normal vector that is constant across an element face. Instead, for each node on an element face a “local” normal vector with respect to that face can be calculated by some finite difference approximation [Wesp 2002] as follows.

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Figure A.3 shows face 2 of an ABAQUS element C3D20 (cf. Fig. A.1 on p. 165), which can be curved in the general case. The local normal vector n at node 6 with respect to face 2 of the element is to be determined. Since node 6 is a corner node, the edge nodes 13 and 14 are the closest nodes on that element face. The vectors a and b (which are the vectors pointing from node 6 to node 14 and to node 13, respectively) span a local plane. The vector normal to this plane can be calculated by the cross product         x14 x6 x13 x6 a×b a =  y14  −  y6  , b =  y13  −  y6  , n = , (A.13) |a × b| z14 z6 z13 z6 where the normal vector n has been normalized to unit length. This is as good an approximation of the local normal vector at node 6 as one can get from the global coordinates of the FE nodes, and it is quite accurate if the FE discretization is not too coarse and if the curvature of the face is only moderate. The local normal vectors at nodes 7, 8, and 9 can be calculated analogously. If the element face is not curved but plane, all local normal vectors at that element face are equal.

Figure A.3: Local normal vector n at a corner node of an element face. Figure A.4 illustrates the calculation of the local normal vector at edge node 13, which is somewhat more difficult. The nodes closest to edge node 13 on element face 2 are the two neighboring corner nodes 5 and 6 and the two neighboring edge nodes 14 and 16. This time, two local normalized normal vectors n1 and n2 are determined from the coordinates of the neighboring nodes         x13 x5 x13 x16 a×b a =  y16  −  y13  , b =  y5  −  y13  , n1 = , (A.14a) |a × b| z16 z13 z5 z13         x13 x14 x13 x6 c×d c =  y6  −  y13  , d =  y14  −  y13  , n2 = . (A.14b) |c × d| z6 z13 z14 z13 The cross products a × b and c × d are not only perpendicular to the planes spanned by the vectors a and b and the vectors c and d, respectively, but also their length is equal to the

A.1

Calculation of Mass, Surface Area, Normal Vectors, and Thickness

171

Figure A.4: Local normal vectors n1 and n2 at an edge node of an element face. surface area of the parallelograms formed by the vectors a and b and the vectors c and d. Therefore, the one of the two normal vectors n1 and n2 with the smallest cross product value a × b or c × d, respectively, is chosen to be the local normal vector n at node 13 because, due to the smaller surface area of its corresponding parallelogram, it is assumed that it gives the better approximation to the local curvature. The local normal vectors at nodes 14, 15, and 16 are calculated in the same manner.

Figure A.5: Mean normal vector n at a corner node common to element faces A through D. More often than not, a node on the surface of a structure belongs not only to one but to several element faces as depicted in Fig. A.5. The four element faces A, B, C, and D share one common corner node. Depending on the element face, this node has the local node number 6 (with respect to face A), 5 (with respect to face B), 7 (with respect to face C), or 8 (with respect to face D). Accordingly, this node has four local normal vectors labeled n6A , n5B , n7C , and n8D , i.e., it has one normal vector for each of the four adjoining element faces it belongs to. Since there can be only one normal vector at each surface node, a mean local normalized normal vector n is calculated

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Some Remarks on the Implementation

1 (n6A + n5B + n7C + n8D ) . 4

(A.15)

If a node belongs to only two or three adjoining element faces (e.g., node 13A /15C in Fig. A.5 that is common to element faces A and C), a mean normalized local normal vector n is determined analogously. Figures A.6 and A.7 visualize local nodal normal vectors for the original and an optimized half-cylinder, respectively. These figures show that the normal vectors of the original structure in Fig. A.6 are well-aligned and kind of “parallel”. On the other hand, the normal vectors of the optimized structure in Fig. A.7 are askew and crisscrossed because the surface is now quite irregular.

Z Y X

Figure A.6: Local normal vectors of the original half-cylinder. Fig. A.8 shows a detail of the front edge of the two joined plates shown in Fig. 5.6 on p. 67. It can be seen that the normal vectors are perpendicular to the respective reference surface or adjoining element faces. Right at the joint edge the normal vectors are oblique at 45◦ because they belong to two adjacent element faces that are perpendicular to each other. Wesp [Wesp 2002] compared the two different methods of calculating the normal vectors, namely, the one using Eq. (A.12) and the one using Eqs. (A.13) through (A.15). He used the rectangular plate (see Fig. 5.3 on p. 64) and a half-cylinder similar to the one depicted in Fig. 5.8 on p. 69, only with fewer elements. The average angular difference between the corresponding normal vectors obtained from the two different methods was about 1◦ . This shows that the results from the alternative method agree very well with the results from the original method.

A.1

Calculation of Mass, Surface Area, Normal Vectors, and Thickness

Z Y X

Figure A.7: Local normal vectors of the optimized half-cylinder.

Z Y X

Figure A.8: Normal vectors at the edge of the two joined plates.

173

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Some Remarks on the Implementation

Calculation of the Local Thickness

The optimization procedures described in the present paper are based on variations of the local thickness of the structure to be optimized. The optimization algorithm tries to determine a thickness distribution for the structure that offers advantages with respect to the objective function and constraints and that is superior to other possible thickness distributions. Some authors (e.g., Marburg [Marburg 2002e, Marburg 2002f] or Fritze et al. [Fritze 2003]) do not vary the local thickness but the local curvature of a structure by employing beads. This way, they do not need to care about the thickness or the structural mass as a constraint during the optimization, since the mass does not change due to local curvature variations. However, they cannot use the mass as an objective function, i.e., they cannot minimize the mass of a structure subject to some other constraints. In contrast, the method presented here can be used to create design proposals for lightweight structures with nevertheless acceptable structural acoustical properties. “What is the thickness?” seems to be an almost philosophical question that is not as trivial as one may think at first glance. If one uses two-dimensional shell elements to model the structure with finite elements, the answer is simple: Each finite element has a constant thickness that is to be defined by the user while creating the FE model. However, if general three-dimensional solid continuum elements are used, the answer is not quite so simple: The thickness can vary throughout the element, and at each node there can be a different local thickness. How can these local thicknesses be determined? A mechanic would just take a caliper rule and measure the thickness of a real machine part by moving the caliper rule around until he cannot move its two jaws together any closer. In other words, he tries to determine the shortest distance between one point on one side of the machine part and a second point on the other side. He intuitively uses the mathematical definition of the term “distance”: the shortest possible route between two given points. The same approach is used by Wesp [Wesp 2002] to determine the local thickness for each FE node on the surface of a structure. While creating the original FE model, all FE nodes on the sound radiating surface are identified and collected in a so-called node set named “outer nodes”. Likewise, all FE nodes on the “inner side” or “underside” of the structure are grouped together in another node set called “inner nodes”. Prior to the start of the optimization iterations (iteration 0), the Euclidian distance between every outer node p and every inner node q is calculated from their nodal coordinates xp and xq by ! " 3 "  2 , p = 1, 2, . . . , n , q = 1, 2, . . . , n , (A.16) x −x d = |x − x | = # pq

p

q

pi

qi

out

in

i=1

where nout is the number of outer nodes and nin is the number of inner nodes. The combination of a specific outer node p and that inner node q  with the smallest value of dpq according to Eq. (A.16) defines the local thickness tp = dpq min for this particular outer node p. This procedure is repeated for every outer node p until the thickness distribution of the whole structure is determined. During iteration 0, the initial local thickness at each outer node p is saved in a file for reference (see steps (7) and (8) in Fig. 4.2 on p. 52). During every following iteration of the optimization procedure, the local thicknesses tp are calculated anew using Eq. (A.16). They can serve as constraints, i.e., in order to check if the local thickness falls below a predefined lower limit or exceeds a predefined upper limit anywhere on the structure. At the same time as the initial local thicknesses are calculated during iteration 0, something which can be called the local “thickness direction” is determined and saved to another file as

A.1

Calculation of Mass, Surface Area, Normal Vectors, and Thickness

175

well. This vector tp is of unit length and defines for each outer node p the direction in which the smallest value of dpq according to Eq. (A.16), i.e., the local thickness tp = dpq min , was detected. The corresponding inner partner-node with the smallest distance to the outer node p (i.e., defining the local thickness tp ) is marked q  . The local thickness direction vector tp for the outer node p can be calculated from the nodal coordinates xp and xq
and the local thickness tp by tp = (xp − xq
) /tp . (A.17) The vectors of the wall thickness direction marked with ① shown in Fig. A.9 represent this approach. Depending on the discretization of the structure, the “measured” or calculated thickness can deviate slightly from the thickness one would expect (see vector ②). In this special case it may have been better to use the normal vector on the surface as the thickness direction. If the discretization is sufficiently refined, though, the deviations are negligible. Figure A.9 also shows an example of what would happen if one would just use the direction normal to the surface as the thickness direction (vector ③). Obviously, the normal vector would probably be a rather poor representation of the thickness direction.

Figure A.9: Determining the local thickness and thickness direction. Admittedly, this does not seem to be the most elegant and convincing method to determine the local thickness and the local thickness direction since it can yield slightly inaccurate results (see vector ② in Fig. A.9). On the other hand, however, there does not seem to be a superior method that would always produce accurate results within a reasonable amount of computation time [Wesp 2002]. A sufficiently refined discretization limits the errors to an acceptable order of magnitude. Besides that, as mentioned above, it is always arguable anyway which is the “correct” thickness and thickness direction. Whereas the local thicknesses at the surface nodes have to be calculated anew during each iteration, the local thickness directions are calculated only once prior to the first iteration (iteration 0) because they do not change during the iterations. The nodes on the surface of the structure which are moved by the optimization algorithm in order to vary the wall thickness can only be moved along the previously calculated corresponding local thickness directions. Thus, the general shape of the structure is preserved because locally it “breathes in” or “breathes out” but the outer contour is not distorted. Figures A.10 and A.11 demonstrate the results of this procedure. Figure A.10 seems to be identical to Fig. A.6. However, it does not show the normal vectors but the local thickness directions of the original half-cylinder. The differences become obvious when one compares the optimized half-cylinder in Fig. A.11 with the one in Fig. A.7. The thickness direction vectors in Fig. A.11 are all still well-aligned and “parallel” because they do not change during the

176

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Z Y X

Figure A.10: Local thickness direction vectors of the original half-cylinder. optimization, whereas the normal vectors in Fig. A.7 are askew and crisscrossed, since they represent the irregular surface of the optimized structure.

Z Y X

Figure A.11: Local thickness direction vectors of the optimized half-cylinder.

A.2

(Pseudo-)Parallelizing PIKAIA

177

A comparison between Figs. A.8 and A.12 reveals the differences between normal vectors and thickness direction vectors. Whereas the normal vectors in Fig. A.8 are perpendicular to their corresponding element faces (except for the ones at the joint edge), the thickness direction vectors in Fig. A.12 are not. Some of them point to the closest nodes on the “underside” or backside of the structure, which are located right at the inner edge, thus resulting in an oblique angle between the vectors and the surface. When the nodes on the surface move along the thickness direction vectors due to the optimization process, the structure is locally scaled, i.e., it is blown up or it shrinks but it cannot get more distorted than it was before.

Z Y X

Figure A.12: Thickness direction vectors at the edge of the two joined plates.

A.2

(Pseudo-)Parallelizing PIKAIA

As mentioned earlier in Sec. 3.3, it is possible to carry out the computations on several computers in parallel when using the PIKAIA algorithm. This does not mean that PIKAIA’s program code itself is parallelized, but rather are the time-consuming FE computations distributed to several machines. Typically, the FE analyses take 80% to 90% of the total CPU time, whereas the calculation of the objective function value, of the constraints, and of the fitness value take only 10% to 20% of the computation time. When using the COBYLA algorithm, the FE computations and the calculation of the objective function and constraint values must be done sequentially because each new design proposal depends on the previous optimization history and success. In contrast, PIKAIA allows parallel FE computations because the new design parameters hardly depend on previous results. In order to take advantage of the parallel computation, the population size is chosen to be rather large, i.e., Np = 50, and in each loop there are not two but twenty new offspring

178

A

Some Remarks on the Implementation

Table A.1: List of computers available for parallel computation using PIKAIA. Hostname

Type

Amazone

PC

Isolde Haegar Elvis Arnold

CPU

Intel Pentium 4, 2.533 GHz IBM RS/6000 PowerPC, 43P-140 233 MHz IBM RS/6000 PowerPC, 43P-140 233 MHz IBM RS/6000 Power2, 3AT 59 MHz IBM RS/6000 Power2, 3AT 59 MHz

RAM

Operating system

1.5 GB

Linux (Debian, Kernel 2.4.20) Unix (IBM AIX 4.3.3) Unix (IBM AIX 5.1L) Unix (IBM AIX 4.3.3) Unix (IBM AIX 4.3.3)

768 MB 768 MB 256 MB 128 MB

individuals produced that replace some individuals of the parent generation if they have a higher fitness value than the parent individuals. Table A.1 lists all the computers that were available for parallel computation in the order of decreasing computational power. Amazone is by far the fastest of the available machines and therefore serves as the central computer (“master”) that controls the other ones and coordinates the parallel computations. The IBM Unix workstations serve as “slaves”. Unfortunately, they are relatively old and therefore significantly slower than Amazone: Isolde and Haegar are slower than Amazone by a factor of about four to five, Elvis and Arnold even by a factor of six to seven. Nonetheless, up to 32% of all FE computations are carried out on the slave computers, thus lessening the load on Amazone and speeding up the optimization procedure. As a first step, the algorithm randomly produces fifty different vectors of design variables on the master computer Amazone and converts them to fifty different ABAQUS input files. A central control routine, which runs on Amazone, distributes the various input files one after another to the master computer and to the slave computers that are available for parallel computation. Then the FE analysis of the respective individual input file is started using ABAQUS. When a particular ABAQUS job on one of the computers (master or slave) is finished, it creates a little file in a special directory on the master computer. This allows the control routine to determine which computation job is done, if that ABAQUS job exited without errors, and how much time that ABAQUS job consumed. If an ABAQUS job exited with an error (e.g., because some finite elements were too distorted or because there was temporarily no connection to the license server—but that rarely happens), a very low artificial fitness value is assigned to the particular individual that caused or was affected by this error. Thus, the chance that a “defective” individual produces any offspring individuals is reduced or eliminated, and the algorithm proceeds as usual. Based on the time an ABAQUS job took on a particular slave computer and on the time an ABAQUS job takes on Amazone, the algorithm estimates if it is better to start the next job on that now available slave computer or if it is better (i.e., faster) to let Amazone do the work. For example, if there are only three jobs left it is not sensible to start the next job on, say, Arnold, which takes six or seven times as long for an FE analysis as Amazone does. Amazone would have finished all three remaining jobs before Arnold had finished even half the first job.

A.3

Graphical User Interface

179

When all fifty FE analyses are completed and all fifty *.fil and *.dat files are available, the objective function, constraint and fitness values of each individual are calculated in rapid succession on the master computer Amazone. Since each single fitness evaluation takes just between one and three seconds it would not be sensible to include the slave computers into this process. As soon as the fitness values are calculated, the members of the population are ranked based on their fitness. Then, a new generation of twenty offspring individuals is bred, employing the crossover and mutation processes described in Sec. 3.3. Parents with a high fitness value have a higher chance of being selected for crossover. After that, a new loop starts, and twenty new FE analyses are carried out on several computers in parallel as described above. If some of the new offspring individuals turn out to have a greater fitness than some of the parent individuals, they replace the corresponding number of parents that have the lowest fitness values (steady-state-delete-worst reproduction plan, see p. 47), and again a new generation of offspring individuals is produced. This process is repeated iteratively until a user-defined preset maximum number of iterations is reached. A specific convergence criterion is not available in PIKAIA. The procedure described above works very well, and all the optimization results involving PIKAIA were produced this way. Unfortunately, there is quite some discrepancy between the computation speeds of the participating computers. The so-called speed-up due to the parallelization would be much greater if all the computers were as fast as Amazone. Furthermore, the binary ABAQUS result files (*.fil files) are encoded slightly differently on Linux systems (Amazone) than on Unix systems (IBM workstations). Thus, ABAQUS result files created on IBM workstations cannot be read and processed by Amazone. To circumvent this problem, binary result files on the slave computers are translated to ASCII files using an ABAQUS routine called ascfil (ASCII to *.fil and vice versa), moved to Amazone, and reconverted to binary result files again. This, of course, takes some additional time, particularly since the ASCII files are somewhat larger than the binary *.fil files so that it takes more time to transfer them to the master computer over the network, thereby creating an additional bottleneck that slows down the process. Unfortunately, the speed-up of the computations due to this kind of parallelization cannot really be quantified because it would take a lot of effort to keep track of the CPU time consumption of the various computation jobs on the various computers. Instead, only the real, “wall clock” time of the complete optimization procedure is available for the optimizations using PIKAIA. When some other user runs a job on the same computer, it does not affect the CPU time but it increases the real time of the PIKAIA procedure, thus distorting the real computation time of PIKAIA. As a positive side effect, it was very interesting and educational to implement some sort of parallelization, and it was much fun and quite fascinating to see it actually work.

A.3

Graphical User Interface

The optimization procedure is controlled by a parameter file, which is to be created or edited by the user. This parameter file contains the maximum number of iterations, the name of the ABAQUS input file of the original structure, an estimation of the model size (number of nodes, number of elements, etc.), the values of ρbegin and ρend for the COBYLA algorithm (see Sec. 3.2.3), statements concerning which physical quantities (MLS, mass, fundamental frequency, thickness, etc.) serve as objective function or constraints, and many other things. The entries are accessible to the Fortran routines and Unix shell scripts via unique key numbers.

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Some Remarks on the Implementation

The parameter file became longer and longer the more options and features were included in the optimization program. As one can imagine, it is somewhat awkward, confusing, and timeconsuming for the user to edit and adapt this parameter file. Therefore, a so-called widget toolkit named GTK+ (version 2.0.2) [GTK+ 2004] was used to create a simple graphical user interface (GUI) that allows the user to enter and manipulate values in the parameter file graphically without having to know the respective key numbers or variable names. Figures A.13 and A.14 show two examples of this GUI.

Figure A.13: Example of the graphical user interface: the “File” page. Elements of such a GUI are programmed in C including special libraries for the functions and layout of GTK+. Some of the features of GTK+ can be seen in Figs. A.13 and A.14. It is possible to group similar or related entries into pages of a so-called notebook. The list of available widgets includes various buttons (plain buttons, check buttons, toggle buttons, etc.), entry fields for numeric or text data entries, drop-down menus with entries to choose from, file selection dialogs, pop-up windows, progress bars, various elements for the design and layout of a GUI (e.g., frames, boxes, scrollbars, horizontal and vertical lines, tooltips), and many other things. It is worth noting that GTK+ [GTK+ 2004] is free software under the GNU Library General Public License (GNU LGPL) [GNU 2004].

A.4

Visualization of Optimization Results

181

Figure A.14: Another example of the graphical user interface: the “FEM/Algorithm” page.

A.4

Visualization of Optimization Results

It is also worth mentioning how the various optimization results were displayed and visualized. The three main ways of displaying results and additional information are line graphs, marker plots, and contour plots. Line graphs such as the ones showing iteration histories (e.g., Fig. 7.1 on p. 81), the history of the trust-region radius ρ (e.g., Fig. 7.3 on p. 82), or the spectra of the level of structure borne sound (e.g., Fig. 7.6 on p. 84) were created using the free XY plotting tool xmgr [xmgr 2004]. Features of xmgr include various layout options (such as user-defined scaling, tick marks, labels, symbols, line styles, colors, etc.) and mathematical operations for data processing such as polynomial regression, splines, running averages, DFT/FFT, and cross/auto-correlation. The line graph illustrating various types of extrema in Fig. 3.1 on p. 34 was also created with xmgr using splines. The lines in the LS spectra plots (e.g., Fig. 7.6 on p. 84) have sharp corners at the lower levels and are not as round and smooth as in, e.g., Hibinger’s [Hibinger 1998] spectra plots (compare Figs. B.1 and B.2 on p. 184). The reason for this is the two-step procedure for the dynamic FE analyses mentioned in Sec. 2.2, which consists of a natural frequency extraction and a subsequent mode-based steady-state dynamic analysis [ABAQUS 2001a,ABAQUS 2001b]. The latter allows to specify the number of frequency increments between natural frequencies (set to 10 for the present study) as well as a bias factor, which spaces the sampling points on the frequency scale closer together at natural frequencies. This bias factor is set to 5.0 for the FE analyses presented in this thesis, thus significantly increasing the frequency intervals in the middle between any two natural frequencies. This leads to the aforementioned sharp corners in the LS spectra due to the poor frequency resolution at the “antipeaks”.

182

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Some Remarks on the Implementation

The marker (arrow) plots used to illustrate the normal vectors (e.g., Fig. A.6 on p. 172) and thickness direction vectors (e.g., Fig. A.10 on p. 176) described in Secs. A.1.3 and A.1.4, respectively, were produced using the commercial pre- and post-processing software MSC.Patran [Patran 2003]. Since both the local normal vectors and the thickness direction vectors are computed quantities that are not readily available in ABAQUS or MSC.Patran, some tricks have to be used in order to display these quantities. A copy of the ABAQUS input file of the original structure to be optimized is edited in such a way that the FE program calculates static stresses instead of natural frequencies and dynamic vibrations. The geometry, material properties, boundary conditions, etc., stay the same, but the natural frequency calculation is canceled, the dynamic load is replaced by some arbitrary static load, and the requested output is changed from nodal velocities to nodal stresses. After an FE analysis using the modified input file, the previously mentioned ABAQUS routine ascfil [ABAQUS 2001a] converts the binary result file (*.fil) into a human readable ASCII file. A user-written Fortran program then replaces the three-dimensional stress components at the surface of the structure (which serve only as placeholders) with the components of the normal vectors or thickness direction vectors computed during the optimization procedure (steps (6) and (7) in Fig. 4.2 on p. 52). All stress components at nodes that are not located at the surface of the structure are set to zero. After that, the ABAQUS routine ascfil reconverts the modified ASCII file into a binary result file (*.fil) again. MSC.Patran can import the results of the FE analysis from that *.fil file and can display them in various ways. One option is to create marker (arrow) plots, leading MSC.Patran to believe that it displays static stresses. In reality, however, it displays the computed normal vectors or thickness direction vectors. Setting all nonsurface vector components to zero ensures that the local normal or thickness direction vectors are displayed at the surface only. The results of this procedure can be seen in, e.g., Figs. A.6 through A.8 or A.10 through A.12. The contour plots displaying the thickness distribution of the optimized structures (e.g., Fig. 8.3 on p. 118) were created in a very similar way as the marker plots described above. The only difference is that the local thickness at a surface node is just a scalar value, not a vector. Thus, in the converted *.fil file only the x components of the virtual placeholder stresses at the surface are replaced by the respective local thickness values, whereas all other numbers (y and z components at surface nodes, all stress components at nonsurface nodes) are replaced by zero. As before, MSC.Patran can read the reconverted binary *.fil file, interpreting and eventually displaying the local thickness values in a contour plot (or fringe plot, as it is called by MSC.Patran) as if they were stress components in x direction.

B

Verification of Optimization Results

The numerical optimization results introduced in the present work have not been verified experimentally due to various reasons given in Sec. 1.3.2. However, an attempt to validate the theoretical optimization results implicitly is presented in this appendix. In his Ph.D. dissertation, Hibinger [Hibinger 1998] numerically optimized the thickness distribution or rib heights of various three-dimensional structures with respect to various objective functions such as MLS, mass, or natural frequencies and subject to suitable constraints. For some of these structures he verified his numerical results with experimental measurements. One of the structures investigated by Hibinger is the same 90◦ joint of two plates as studied in the present paper (see Fig. 5.6 on p. 67 and the optimization results shown in Sec. 8). Unfortunately, Hibinger performed comparative measurements on this structure only for the optimized rib height arrangement but not for the optimized thickness distribution. Since all of his measurements agreed very well with the numerical predictions, it can be assumed that the measurements and simulations for the two-plate joint probably would have also exhibited a similarly good agreement. Thus, it can be argued that the numerical results of the present work can be validated by comparing them with Hibinger’s numerical results even though he did not experimentally verify his results for this particular optimization problem. Figure B.1 shows the LS spectra of the two joined plates as obtained numerically by Hibinger. The LS spectra obtained in the present thesis are depicted in Fig. 8.4 on p. 118 and are repeated in Fig. B.2 for convenience. Both graphs show the LS spectrum of the original and the optimized structure after an MLS minimization by means of thickness optimization (see Sec. 8.2.1). As can be seen, there is excellent agreement between Hibinger’s results and the new results for the original structure over the whole frequency range. This is remarkable since the following differences between Hibinger’s model and the present model can be observed: • Hibinger used combined shell/membrane elements, whereas solid hexahedral elements are used in this dissertation. • Hibinger utilized a coarser FE discretization than the FE mesh used in this paper, which is made exactly twice as fine as Hibinger’s in order to apply the excitation force at the same physical location. • Hibinger used linear elements, whereas the elements used in the present work have quadratic shape functions. • Hibinger employed a constant modal damping coefficient of 0.5%, whereas a damping coefficient of 0.4% is used in this dissertation. This accounts for the slightly higher peak levels in Fig. B.2.

184

B

Verification of Optimization Results

Figure B.1: Hibinger’s optimization results (dashed line: original structure, solid line: optimized structure; reprinted by kind permission of the publisher from [Hibinger 1998]).

level of structure borne sound LS [dB]

110 original structure optimized structure

105 100 95 90 85 80 75 70 65 60 55 50

0

500

1000

1500 2000 frequency [Hz]

2500

3000

Figure B.2: Optimization results of the present study. • Hibinger modeled the boundary conditions by torsion spring elements, whereas the structure is assumed to be simply supported along its edges in the present study. • Hibinger used a noncommercial, somewhat experimental FE code called FEAP (Finite Element Analysis Program [FEAP 2004]). In this paper the well-known commercial FE program ABAQUS [ABAQUS 2001a] is used. The figures also show that the LS spectra for the optimized structures depicted in Figs. B.1 and B.2 deviate significantly. This is not surprising since the following differences in the optimization procedures exist in addition to the differences in the FE models shown above:

B

Verification of Optimization Results

185

• Hibinger used a completely different optimization algorithm, namely, the revised simplex method of linear programming instead of the COBYLA or PIKAIA algorithm used in this dissertation. • Hibinger calculated the MLS, which is the objective function for the optimization, by adding up the LS peak levels only. In this paper the MLS is computed by integrating the area beneath the LS spectrum. • The original plate thickness is 4 mm for both Hibinger’s and the present model. However, Hibinger set the lower and upper limit on the design variables to 2 mm and 20 mm, respectively, whereas in the present work the thickness is allowed to vary between 1 mm and 10 mm. • Hibinger permitted a maximum mass increase of 16%, whereas the mass is not allowed to increase at all in this study. In conclusion, it can be stated that the agreement between Hibinger’s results and the present results for the original structure is excellent. This means that at least the FE model is reliable. The agreement between Hibinger’s results and the present results for the optimized structure is inferior, which can easily be explained by the limitations listed above. Interestingly, the overall MLS attenuation in the frequency band 0–3000 Hz achieved by Hibinger was 11.6 dB, whereas in the present study the overall MLS reduction is 10.2 dB, which is very close to Hibinger’s results.

186

B

Verification of Optimization Results

C

Nonlinearity and Multimodality of the Objective Function

This appendix presents an attempt to visualize the nonlinear relationship between the objective function and the design variables and to illustrate that the objective function has many local minima and maxima. The nonlinearity prevents the use of efficient linear or quadratic optimization algorithms and is the cause for applying general nonlinear optimization techniques. The many local optima make it more difficult for an optimization algorithm to find the global optimum. Consider the rectangular plate with the coarse mesh as described in Sec. 5.2 and depicted in Fig. 5.3 on p. 64. This model has 77 design variables, i.e., the wall thickness at 77 surface nodes is to be optimized. However, it is impossible to visualize a 77-dimensional function on plane two-dimensional paper. Thus, two pairs of surface nodes, namely, surface nodes 317 and 459 (excitation point) on the one hand and surface nodes 398 and 449 on the other hand, were chosen more or less arbitrarily to produce two three-dimensional surface plots that can be visualized on paper. These nodes are shown in Fig. C.1.

398 317

459

449

Z Y X

Figure C.1: Position of the four surface nodes 317, 398, 449, and 459 (excitation point) used to visualize the nonlinearity and multimodality of the objective function. The wall thickness at each of the surface nodes 317 and 459 is varied from 1 mm to 10 mm in steps of 0.2 mm, whereas the thickness at all other surface nodes is held constant at the initial value of 4 mm. The mean level of structure borne sound MLS is calculated for each of these configurations as the objective function and plotted vs. the thickness variations at the two nodes. The same is done for the two other surface nodes 398 and 449. The results are the combined surface and contour plots in Figs. C.2 and C.3.

188

mean level of structure borne sound MLS [dB]

C

Nonlinearity and Multimodality of the Objective Function

85 84 83 82 81 80 79 78 1

2

3

4 5 6 thickness at 7 node 459 [mm] 8

9 10

10

9

8

4 5 6 thickness at node 317 [mm]

3

2

1

7

mean level of structure borne sound MLS [dB]

Figure C.2: Change of the objective function value due to thickness variations at surface nodes 317 and 459 from 1 mm to 10 mm (combined surface and contour line plot).

85 84 83 82 81 80 79 78 1

2

3

4 5 6 thickness at 7 node 449 [mm] 8

9 10

10

9

8

4 5 6 thickness at node 398 [mm]

3

2

1

7

Figure C.3: Change of the objective function value due to thickness variations at surface nodes 398 and 449 from 1 mm to 10 mm (combined surface and contour line plot).

C

Nonlinearity and Multimodality of the Objective Function

189

Though the nonlinearity of the objective function is clearly visible in both figures, it is obviously not dramatic. Also, in both figures there is only one local minimum (which therefore is the global minimum) and only one local maximum (which therefore is the global maximum) rather than several local optima. However, if one recalls that during the “real” optimization procedure the wall thickness can be varied simultaneously at 77 and not only at two surface nodes, and if one further recalls that this is done with arbitrary stepsize, one can easily imagine that the “true” 77-dimensional objective function exhibits many local minima and maxima. One should also recall that the coarsely discretized rectangular plate used here for illustrative purposes is the least complex structure of all structures investigated in this study. All other models have a more complex geometry and significantly more design variables (up to 821). Therefore, they surely exhibit an even higher degree of nonlinearity and multimodality than this simple example.

190

C

Nonlinearity and Multimodality of the Objective Function

List of Figures 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 7.6

Examples of local and global extrema of some arbitrary function f (x) . . . . . . Some hypothetical steps of the simplex algorithm . . . . . . . . . . . . . . . . . Linear approximation of the objective function F (x) . . . . . . . . . . . . . . . Generation of two new vertices x(p) and x(q) . . . . . . . . . . . . . . . . . . . . Flowchart of the general optimization procedure . . . . . . . . . . . . . . . . . . Flowchart of program module OFAC (objective function and constraints) . . . . Flowchart of design evaluation program module . . . . . . . . . . . . . . . . . . Flowchart of the PIKAIA algorithm . . . . . . . . . . . . . . . . . . . . . . . . . Initial FE model of the rectangular plate with 24×16 elements . . . . . . . . . . Movable nodes at the surface of the rectangular plate with 24×16 elements . . . Initial FE model of the rectangular plate with 12×8 elements . . . . . . . . . . . Movable nodes at the surface of the rectangular plate with 12×8 elements . . . . Comparison of the LS spectra of the rectangular plate for various discretizations Initial FE model of the two plates joined at 90◦ . . . . . . . . . . . . . . . . . . Movable nodes at the surface of the two plates joined at 90◦ . . . . . . . . . . . Initial FE model of the half-cylinder . . . . . . . . . . . . . . . . . . . . . . . . Movable nodes at the surface of the half-cylinder . . . . . . . . . . . . . . . . . . Initial FE model of the gearbox (dimensions, various views) . . . . . . . . . . . Initial FE model of the gearbox (excitation point and modification domains) . . Modification of surface node positions by means of a spline function . . . . . . . Bicubic spline surface defined by 7×5 key points . . . . . . . . . . . . . . . . . . Hermite spline surface defined by 7×5 key points . . . . . . . . . . . . . . . . . Tensor product B´ezier surface defined by 7×5 key points . . . . . . . . . . . . . Iteration history for the rectangular plate (COBYLA, MLS minimization, no mass or thickness constraint violation, refined FE mesh) . . . . . . . . . . . . . Iteration history for the rectangular plate (COBYLA, MLS minimization, maximum mass or thickness constraint violation < 0.01%, refined FE mesh) . . . . . History of the trust-region radius ρ for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . . Optimized design of the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . . . . . . . . . . . . . Thickness distribution of the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . . . . LS spectra of the original and the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . .

34 38 40 41 50 52 54 57 62 63 64 64 66 67 68 69 69 70 71 74 75 76 77 81 81 82 83 83 84

192

7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29

List of Figures

Iteration history for the rectangular plate (COBYLA, MLS minimization, no mass or thickness constraint violation, coarse FE mesh) . . . . . . . . . . . . . . 85 Iteration history for the rectangular plate (COBYLA, MLS minimization, maximum mass or thickness constraint violation < 0.01%, coarse FE mesh) . . . . . 86 History of the trust-region radius ρ for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . 86 Optimized design of the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . . . . . . . . . . . . 87 Thickness distribution of the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . . . 88 LS spectra of the original and the optimized rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . 88 Iteration history for the rectangular plate (COBYLA, mass minimization, maximum MLS or thickness constraint violation < 0.01%) . . . . . . . . . . . . . . . 90 Optimized design of the rectangular plate (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Thickness distribution of the optimized rectangular plate (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . 91 LS spectra of the original and the optimized rectangular plate (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . 91 Iteration history for the rectangular plate (COBYLA, fundamental frequency maximization, maximum MLS, mass, or thickness constraint violation < 0.01%) 93 Optimized design of the rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . 93 Thickness distribution of the optimized rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . . . . 94 LS spectra of the original and the optimized rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . 95 Iteration history for the rectangular plate (COBYLA, ∆f maximization, no MLS, mass, or thickness constraint violation) . . . . . . . . . . . . . . . . . . . . 96 Optimized design of the rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Thickness distribution of the optimized rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . 97 LS spectra of the original and the optimized rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . 98 Iteration history for the rectangular plate (COBYLA, Hermite splines, MLS minimization, no mass or thickness constraint violation, coarse FE mesh) . . . . 100 Optimized design of the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . 101 Thickness distribution of the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh) . . 101 LS spectra of the original and the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh)102 Iteration history for the rectangular plate (COBYLA, Hermite splines, MLS minimization, no mass or thickness constraint violation, refined FE mesh) . . . . 103

List of Figures

193

7.30 Optimized design of the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . . 104 7.31 Thickness distribution of the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh) . . 104 7.32 LS spectra of the original and the optimized rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh)105 7.33 Iteration history for the rectangular plate (COBYLA, Hermite splines, mass minimization, no MLS or thickness constraint violation, coarse FE mesh) . . . . 107 7.34 Optimized design of the rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . 107 7.35 Thickness distribution of the optimized rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, coarse FE mesh) . . 108 7.36 LS spectra of the original and the optimized rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints) . . . . . . . . . 108 7.37 Iteration history for the rectangular plate (fitness, PIKAIA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.38 Iteration history for the rectangular plate (objective function, PIKAIA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . 110 7.39 Optimized design of the rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.40 Thickness distribution of the optimized rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . 112 7.41 LS spectra of the original and the optimized rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . 112 8.1 Iteration history for the two-plate joint (COBYLA, MLS minimization, no mass or thickness constraint violation) . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.2 Optimized design of the two-plate joint (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.3 Thickness distribution of the optimized two-plate joint (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . 118 8.4 LS spectra of the original and the optimized two-plate joint (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . 118 8.5 Iteration history for the two-plate joint (COBYLA, mass minimization, maximum MLS or thickness constraint violation < 0.01%) . . . . . . . . . . . . . . . 120 8.6 Optimized design of the two-plate joint (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.7 Thickness distribution of the optimized two-plate joint (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . 121 8.8 LS spectra of the original and the optimized two-plate joint (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . 121 8.9 Iteration history for the two-plate joint (COBYLA, mass minimization, maximum MLS, f1 , or thickness constraint violation < 0.01%) . . . . . . . . . . . . . 123 8.10 Optimized design of the two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.11 Thickness distribution of the optimized two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints) . . . . . . . . . . . . . . . . . . . 124

194

List of Figures

8.12 LS spectra of the original and the optimized two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints) . . . . . . . . . . . . . . . . . 8.13 Iteration history for the two-plate joint (COBYLA, ∆f maximization, no MLS, mass, or thickness constraint violation) . . . . . . . . . . . . . . . . . . . . . . . 8.14 Optimized design of the two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.15 Thickness distribution of the optimized two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . 8.16 LS spectra of the original and the optimized two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . 8.17 Iteration history for the two-plate joint (objective function, PIKAIA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . 8.18 Optimized design of the two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.19 Thickness distribution of the optimized two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . 8.20 LS spectra of the original and the optimized two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . 9.1 Iteration history for the half-cylinder (COBYLA, MLS minimization, maximum mass or thickness constraint violation < 0.01%) . . . . . . . . . . . . . . . . . . 9.2 Thickness distribution of the optimized half-cylinder (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . 9.3 LS spectra of the original and the optimized half-cylinder (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . 9.4 Iteration history for the half-cylinder (COBYLA, mass minimization, maximum MLS or thickness constraint violation < 0.01%) . . . . . . . . . . . . . . . . . . 9.5 Thickness distribution of the optimized half-cylinder (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . 9.6 LS spectra of the original and the optimized half-cylinder (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . 9.7 Iteration history for the half-cylinder (COBYLA, fundamental frequency maximization, maximum MLS, mass, or thickness constraint violation < 0.01%) . . . 9.8 Thickness distribution of the optimized half-cylinder (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . . . . . . 9.9 LS spectra of the original and the optimized half-cylinder (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . . . . 9.10 Iteration history for the half-cylinder (COBYLA, Hermite splines, MLS minimization, no mass or thickness constraint violation) . . . . . . . . . . . . . . . . 9.11 Thickness distribution of the optimized half-cylinder (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . 9.12 LS spectra of the original and the optimized half-cylinder (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . 9.13 Iteration history for the half-cylinder (COBYLA, Hermite splines, mass minimization, no MLS or thickness constraint violation) . . . . . . . . . . . . . . . . 9.14 Thickness distribution of the optimized half-cylinder (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . .

125 126 127 127 128 130 130 131 132 134 135 135 137 137 138 139 140 140 142 143 143 145 145

List of Figures

9.15 LS spectra of the original and the optimized half-cylinder (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . 9.16 Iteration history for the half-cylinder (fitness, PIKAIA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.17 Iteration history for the half-cylinder (objective function, PIKAIA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . 9.18 Thickness distribution of the optimized half-cylinder (PIKAIA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . 9.19 LS spectra of the original and the optimized half-cylinder (PIKAIA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . 10.1 Iteration history for the gearbox (COBYLA, MLS minimization, maximum mass or thickness constraint violation < 0.01%) . . . . . . . . . . . . . . . . . . . . . 10.2 Thickness distribution of the optimized gearbox (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 LS spectra of the original and the optimized gearbox (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . A.1 Nodes and faces of a 20-node hexahedral solid element . . . . . . . . . . . . . . A.2 Virtual pressure applied to the half-cylinder’s surface . . . . . . . . . . . . . . . A.3 Local normal vector n at a corner node of an element face . . . . . . . . . . . . A.4 Local normal vectors n1 and n2 at an edge node of an element face . . . . . . . A.5 Mean normal vector n at a corner node common to several element faces . . . . A.6 Local normal vectors of the original half-cylinder . . . . . . . . . . . . . . . . . . A.7 Local normal vectors of the optimized half-cylinder . . . . . . . . . . . . . . . . A.8 Normal vectors at the edge of the two joined plates . . . . . . . . . . . . . . . . A.9 Determining the local thickness and thickness direction . . . . . . . . . . . . . . A.10 Local thickness direction vectors of the original half-cylinder . . . . . . . . . . . A.11 Local thickness direction vectors of the optimized half-cylinder . . . . . . . . . . A.12 Thickness direction vectors at the edge of the two joined plates . . . . . . . . . . A.13 Example of the graphical user interface: the “File” page . . . . . . . . . . . . . A.14 Another example of the graphical user interface: the “FEM/Algorithm” page . . B.1 Hibinger’s optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Optimization results of the present study . . . . . . . . . . . . . . . . . . . . . . C.1 Surface nodes for the visualization of nonlinearity and multimodality . . . . . . C.2 Change of the objective function value: thickness variations at surface nodes 317 and 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Change of the objective function value: thickness variations at surface nodes 398 and 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195

146 148 148 149 150 152 154 155 165 168 170 171 171 172 173 173 175 176 176 177 180 181 184 184 187 188 188

196

List of Figures

List of Tables 3.1 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 8.1 8.2 8.3 8.4 8.5 8.6 9.1 9.2

Some hypothetical steps of the simplex algorithm . . . . . . . . . . . . . . . . . Properties of the original rectangular plate (refined FE discretization) . . . . . . Properties of the original rectangular plate (coarse FE discretization) . . . . . . Optimization results for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . . . . . . . . . . Optimization results for the rectangular plate (COBYLA, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . . . . . . . . . Optimization results for the rectangular plate (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the rectangular plate (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . Optimization results for the rectangular plate (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, coarse FE mesh) . . . . . . . . . . Optimization results for the rectangular plate (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints, refined FE mesh) . . . . . . . . . Optimization results for the rectangular plate (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints, coarse FE mesh) . . . . . . . . . . Optimization results for the rectangular plate (PIKAIA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the original two-plate joint . . . . . . . . . . . . . . . . . . . . . . Optimization results for the two-plate joint (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the two-plate joint (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the two-plate joint (COBYLA, mass minimization, MLS, f1 , and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the two-plate joint (COBYLA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the two-plate joint (PIKAIA, ∆f maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the original half-cylinder . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the half-cylinder (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 80 80 85 89 92 95 99 102 106 109 113 115 119 122 125 129 132 133 136

198

9.3 9.4 9.5 9.6 9.7 10.1 10.2 11.1 A.1

List of Tables

Optimization results for the half-cylinder (COBYLA, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the half-cylinder (COBYLA, fundamental frequency maximization, MLS, mass, and thickness constraints) . . . . . . . . . . . . . . . Optimization results for the half-cylinder (COBYLA, Hermite splines, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . Optimization results for the half-cylinder (COBYLA, Hermite splines, mass minimization, MLS and thickness constraints) . . . . . . . . . . . . . . . . . . . . . Optimization results for the half-cylinder (PIKAIA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the original gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization results for the gearbox (COBYLA, MLS minimization, mass and thickness constraints) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the optimization results . . . . . . . . . . . . . . . . . . . . . . . . List of computers available for parallel computation using PIKAIA . . . . . . .

138 141 144 147 150 151 155 162 178

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Resume of the Author

Personal Data Name: Date of birth: Place of birth: Marital status: Children:

Joachim B¨os April 27, 1972 Fulda, Germany married to Sonja B¨os one daughter

School Education 8/1978 – 6/1982 8/1982 – 6/1991

elementary school “Florenberg” in K¨ unzell-Pilgerzell near Fulda high school “Rabanus-Maurus-Schule (Domgymnasium)” in Fulda focus on mathematics and physics degree: Abitur (German high school diploma)

Military Service 7/1991 – 6/1992

mandatory basic military service in Sontra, Germany

University Education 10/1992 – 7/1998 9/1994 8/1996 – 9/1997

7/1998 5/1998 – 4/2003 4/2004

studies in Mechanical Engineering, Darmstadt University of Technology, Darmstadt, Germany preliminary examination “Vordiplom” (comparable to a bachelor’s degree) studies at the Mechanical and Aerospace Engineering Department, State University of New York (SUNY) at Buffalo, Buffalo, NY, USA degree: Master of Science (M.S.) in Mechanical Engineering degree: Diplom-Ingenieur (Dipl.-Ing., German master’s degree) Ph.D. student at the Department of Mechatronics and Machine Acoustics, Darmstadt University of Technology degree: Doktor-Ingenieur (Dr.-Ing., German Ph.D. degree)

Employment 5/1998 – 4/2003

research assistant at the Department of Mechatronics and Machine Acoustics, Darmstadt University of Technology 5/2003 until present research scientist at Fraunhofer Institute for Structural Durability LBF, Competence Center of Mechatronics/Adaptronics, Darmstadt, Germany