Applied Mathematical Sciences Volume 146 Editors S.S. Antman J.E. Marsden L. Sirovich Advisors J.K. Haie P. Holmes J. Keener J. Keller BJ. Matkowsky A. Mielke C.S. Peskin K.R.S. Sreenivasan Springer NewYork Berlin Heidelberg Barcelona HongKong London Milan Paris Singapore Tokyo Applied Mathernatical Sciences I. John: Partial Differential Equations, 4th ed. 34. KevorkianlCole: Perturbation Methods in Applied 2. Sirovich: Techniques of Asymptotic Analysis. Mathematics. 3. Haie: Theory of Functional Differential 35. Carr: Applications of Centre Manifold Theory. Equations, 2nd ed. 36. BengtssoniGhillKällen: Dynamic Meteorology: 4. Percus: Combinatorial Methods. Data Assimilation Methods. 5. von MiseslFriedrichs: Fluid Dynamics. 37. Saperstone: Semidynamical Systems in Infinite 6. FreibergerlGrenander: A Short Course in Dimensional Spaces. Computational Probability and Statistics. 38. LichtenberglLieberman: Regular and Chaotic 7. Pipkin: Lectures on Viscoelasticity Theory. 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Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA Mathematies Subjeet Classifieation (2000): 35B27, 49J45, 49K20, 65K10, 74Q20 Library of Congress Cataloging-in-Publieation Data Allaire, Gregoire. Shape optimization by the homogenization methodlGregoire Allaire. p. em. - (Applied mathematieal seienees; v. 146) Includes bibliographical referenees and index. ISBN 978-1-4419-2942-6 ISBN 978-1-4684-9286-6 (eBook) DOI 10.1007/978-1-4684-9286-6 I. Struetural optimization. 2. Homogenization (Differential equations) I. Tide. II. Applied mathematical seienees (Springer-Verlag New York, Inc.); v. 146. QA1.A647 [TA658.8] 624.l-de21 2001032845 Printed on acid-free paper. © 2002 Springer-Verlag New York, Inc. Softeover reprint of the hardcover 1s t edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in eonneetion with reviews or scholarly analysis. Use in eonneetion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general deseriptive names, trade names, trademarks, ete., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Produetion managed by Frank MCGuckin; manufacturing supervised by Jeffrey Taub. Camera-ready eopy provided by the author's TEX files using the svsing.sty macro. 9 8 765 432 1 SPIN 10838667 Springer-Verlag New York Berlin Heidelberg A member 0/ BertelsmannSpringer Science+Business Media GmbH Preface The topic of this book is homogenization theory and its applications to optimal design in the conductivity and elasticity settings. Its purpose is to give a self-contained account of homogenization theory and explain how it applies to solving optimal design problems, from both a theoretical and a numerical point of view. The application of greatest practical interest tar geted by this book is shape and topology optimization in structural design, where this approach is known as the homogenization method. Shape optimization amounts to finding the optimal shape of a domain that, for example, would be of maximal conductivity or rigidity under some specified loading conditions (possibly with a volume or weight constraint). Such a criterion is embodied by an objective function and is computed through the solution of astate equation that is a partial differential equa tion (modeling the conductivity or the elasticity of the structure). Apart from those areas where the loads are applied, the shape boundary is al ways assumed to support Neumann boundary conditions (i.e., isolating or traction-free conditions). In such a setting, shape optimization has a long history and has been studied by many different methods. There is, therefore, a vast literat ure in this field, and we refer the reader to the following short list of books, and references therein [39], [42], [130], [135], [149], [203], [220], [225], [237], [245], [258]. We shall not consider the case of Dirichlet boundary conditions (for which the homogenization theory is somehow very different), referring the reader only to [62], [63], [64], [267], and references therein. Nor shall we discuss any other types of optimal design problems (like material properties optimization), even in the cases where homogenization could be relevant. We note that optimal design may be seen as a special branch of optimal control, where the control is the domain itself or the coefficients in the domain (the material properties). There are also many relevant refer ences in optimal control, and more specifically in controllability, of so-called distributed systems (i.e., for which the state equation is a partial differential equation). In addition to the pioneering work of Pontryagin [223], we simply mention the classical books of Lions [164], [165]. In order to shed light on the originality of the homogenization method, we first briefly recall what can be called the classical method for shape op- VI Preface timization. The main idea goes, at least, as far back as Hadamard [129], and has been furt her developed by many authors, e.g., [39], [72], [203], [220], [255], [258]. Given an initial guess of the shape, one allows the boundary to vary along its normal (or, more generally, along any smooth velo city field). From both a theoretical and numerical pont of view, this approach gives a convenient framework for applying the calculus of variations. Neverthe less, the method suffers from two main drawbacks: It requires a smooth parametrization of the boundary, and it never changes the topology of the original domain (i.e., its number of holes or connected components). In struc tural design, this is often a serious drawback, since it is widely acknowledged that a porous structure of the same weight as a bulky one may be drastically more efficient. There have been some recent attempts to change the topology in this framework of shape sensitivity (e.g., the buble method [103], or the topology gradient [68], [257]), but they are either ad hoc methods, or very close in spirit to the homogenization method. Let us mention in passing that topology optimization can also be studied by stochastic approaches, like, for example, genetic algorithms, simulated annealing, or neural networks (e.g., [114], [143]). The homogenization method was developed in order to remove smooth ness and topological constraints inherent in the classical method of shape optimization. Another major difference of the homogenization method is the class of numerical algorithms it yields. As classical numerical methods are shape-tracking algorithms (where the shape fits to the mesh, which is deformed during the iterations), the homogenization-based numerical meth ods are rat her shape-capturing algorithms (where the shape is captured on a fixed mesh). Before going to numerical issues, we now describe the main features of the homogenization approach. In a first step, we rephrase a shape optimization problem as a two-phase optimization problem. Thanks to the homogeneous Neumann boundary con ditions that they support, the holes in the domain may be filled by a very weak material (a poor conductor or a compliant elastic phase). In the limit where the properties of this ersatz material go to zero, we recover holes with Neumann boundary conditions (note that such a limit process is not always easy to justify rigorously). In other words, shape optimization is just the degenerate limit of two-phase optimal design, where the interface between the two phases replaces the boundary of the shape. A precise definition of two-phase optimization is as follows. In the conductivity setting, given two conducting materials, the goal is to find the best arrangement of the ma terials in a fixed domain that maximizes, for example, overall conductivity. Preface Vll Similarly, in the elasticity setting, the goal is to find the best way of mixing two elastic materials yielding the most rigid structure. In both cases, there is a volume constraint on the best component phase, which may be seen as the extra cost on the good conductor or the stiff material (which are expected to be more expensive). Two-phase optimization is also an optimal design problem of interest, and we study it in great detail. A key feature of two-phase optimal design problems is that they usually do not admit a solution in the absence of any smoothness or topological con straint on the interface or on the shape boundary. The root of this pathology is that it is often more advantageous having many tiny inclusions of one phase into the other than just a few big ones. Therefore, any proposed design can be improved by making small variations of the phases arrangement, and can not be optimal. This process of making increasingly fine inclusions (or holes) shows that optimality can be achieved only in the limit of infinitely fine scale mixt ure of the two phases. Such mixtures are called composite materials and allow one to define optimal generalized designs. Homogenization is precisely the right mathematical tool for studying composite materials and defining generalized optimal designs. Therefore, in a second step we introduce generalized designs for which one can prove existence of an optimum and derive necessary conditions of optimality. This process of enlarging the space of admissible designs in order to make the problem weIl-posed is called relaxation. A generalized design is often equivalently called a relaxed, or composite, design. Homogenization theary is the main ingredient for performing the relaxation of two-phase optimal design problems. As such, homogenization appears as just a trick far proving existence theorems, but its importance goes far beyond that, since it yields new numerical algorithms. EventuaIly, it remains to justify this relaxation process in the degenerate limit of shape optimization, and to post-process numerically these optimal composite designs to recover classical shapes. The homogenization method in optimal design was initiated by Murat and Tartar in the late 1970's. Many joined their efforts in the development of that theory, and the first relevant references are the works of Murat and Tartar [205], [206], [269], [271] in France; Cherkaev, Lurie and their cowork ers [115], [172], [178], [177], as weIl as Raitum [227] in Russia; and Kohn and Strang [152] in the United States. Surprisingly enough, the numerical al gorithms based on the homogenization method have matured slowly. After the early contributions of Gibiansky and Cherkaev [115], Glowinski [117], Goodman, Kohn, and Reyna [119], and Lavrov, Lurie, and Cherkaev [160], viii Preface which were mostly restricted to academic problems, it was the paper of Bend soe and Kikuchi [47] in the late 1980's, which was the first to demonstrate the efficiency of such methods in shape optimization. After that pioneer ing contribution, many other works appeared that contributed to make the homogenization method one of the most popular and efficient methods for shape and topology optimization (see Chapter 5 and the References). Although the main motivation of this book is shape optimization, while homogenization is, to some extent, just a technical tool for implementing op timization methods, we somehow proceed in reverse order. Indeed, the first chapter is concerned solely with homogenization theory. Mathematically, it can be defined as a theory for averaging partial differential equations. Homogenization has many potential applications, but we consider it only as a tool for deriving macroscopic or effective properties of microscopically heterogeneous media. As such it provides a firm basis to the notion of com posite material obtained by mixing, on a very fine scale, several phase com ponents. Although this quest ion of averaging and finding effective properties is very old in physics or mechanics, the mathematical theory of homogeniza tion is quite recent, going back to the 1970's. We expose the most general framework, known as the H-convergence, or G-convergence, introdueed by Spagnolo [260], [261], and generalized by Tartar and Murat [204], [270]. In the framework of homogenization theory, the second chapter focuses on the study of two-phase composite materials. These composite materials (obtained as fine mixt ure limits of the two original phases) playa key role in the homogenization method used for shape optimization: N amely, as gener alized designs. It is therefore erucial to find the range of all possible effective properties of such composites. This is called the G-closure problem, and it was solved in the conductivity setting for two isotropie phases by Murat and Tartar [205], [274], and Lurie and Cherkaev [175], [176]. Unfortunately, a similar answer is yet unknown in the elasticity setting. In this latter case, one can only obtain bounds that must be satisfied by the effective properties. In their most general form such bounds are called Hashin-Shtrikman energy bounds since they are derived by using the famous variational prineiple in troduced by Hashin and Shtrikman [133]. They turn out to be optimal, i.e., the values of these bounds are exactly attained by special choices of composites. In particular, optimal composites can be chosen in the dass of so-called sequential laminates, which will play an important part in shape optimization. The third ehapter is devoted to the application of homogenization to problems of two-phase optimization in conductivity. After introdueing a Preface IX precise mathematical framework, we give explicit examples of the nonex istence of optimal designs in the original space of "classical" designs. It is therefore necessary to relax the problem by enlarging the space of admissible designs, i.e., by allowing for generalized designs that are made up of com posite materials. The key tool of this relaxation process is homogenization theory. As an additional advantage, the relaxed formulation allows one to derive optimality conditions that are at the root of numerical algorithms. Our exposition follows the original work of Murat and Tartar [205J. The fourth chapter is also concerned with the homogenization method for optimal design, but in the elasticity setting. Since the G-closure set of two elastic isotropie phases is still unknown, a rigorous relaxation procedure is available only for special objective functions including compliance (i.e., the work done by the loads). In such a case, the optimization problem being self-adjoint, the relaxation requires only the knowledge of optimal energy bounds instead of the full G-closure. Those bounds (called Hashin Shtrikman bounds) are precisely studied in Chapter 2, where they are shown to be attained by sequentiallaminates. Therefore, in the relaxation process the unknown full G-closure set can be replaced by its explicit subset of sequentiallaminates. EventuaIly, a complete relaxation procedure for shape optimization (i.e., when one of the phases degenerates to holes) is rigorously established. Here we follow mainly our work [15], [21], that of Gibiansky and Cherkaev [115], [116], and that of Kohn and Strang [152J. FinaIly, the fifth chapter is devoted to numerical issues for the homoge nization method in optimal design. We mostly focus on shape optimization for elastic structures, since it is by far the most important application of the homogenization method. Building upon our knowledge of the relaxed, or homogenized, formulation, we discuss the two main types of algorithms, the optimality criteria and gradient methods, as weIl as several numerical technicalities. In particular, we explain how a numerical procedure, called penalization, allow one to post-process the optimal generalized designs in or der to recover classical shapes. This chapter is illustrated by several results of two-dimensional and three-dimensional computations. Our exposition is complementary to that of Bendsoe in his book [42], which is a very good introduction to the numerical aspects of the homogenization method for a more practically inclined reader. In the sequel, we do not point out open problems, but it is clear that the range of applications covered by this book, although very important, is somehow narrow. Of course, there are many other types of optimal design problems, apart from shape optimization, that have not yet been attacked x Preface with the homogenization method (or that cannot be treated by this method). It is our hope that this book can serve as a basis for furt her developments in new directions. The references are not exhaustive, by any means, and so I apologize to those inadvertently left off of the list. To conclude this introduction, I want to thank all my coworkers, with whom I learned so much about shape optimization: S. Aubry, Z. Belhachmi, E. Bonnetier, G. Francfort, F. Jouve, R. Kohn, V. Lods, and F. Murat. I benefited from many stimulating discussions with G. Milton and L. Tartar, that I acknowledge with much pleasure. Special thanks are due to F. Jouve, who was the corner stone of the development of a three-dimensional shape optimization code, and to G. Francfort for his comments on an earlier version of the manuscript. Last but not least, I owe a lot to F. Murat, who had so much influence on my understanding of homogenization, and to R. Kohn, who introduced me to optimal design. Paris, France G REGOIRE ALLAIRE May 2001