Optimal design through the sub relaxation method

But each situationcorresponds to a certain design problem in Engineering: to find optimal mixtures ofgiven materials providing the best performance under given working conditions.But eve

SEMA SIMAI Springer Series

Volume 11

Pablo Pedregal

Optimal Design through

the Sub-Relaxation Method

Understanding the Basic Principles

123

Universidad de Castilla-La Mancha

Ciudad Real, Spain

SEMA SIMAI Springer Series

DOI 10.1007/978-3-319-41159-0

Library of Congress Control Number: 2016948436

Mathematics Subject Classification (2010): 49J45, 74P05, 35Q74

© Springer International Publishing Switzerland 2016

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer Nature

The registered company is Springer International Publishing AG Switzerland

To my beloved sister Conchi, in memoriam

to be self-contained in such a way that, in addition to covering the aforementionedaspects, the book will serve as a sound basis for a masters or other postgraduatecourses in the subject.

Application to real problems in Engineering would almost demand a rate book On the one hand, many specific situations may have an interestingmechanical background (e.g., compliant mechanisms or vibrating structures), anelectric/electronic flavor (e.g., optimal design with piezoelectric materials), orrelevance in other fields On the other hand, there are many delicate issues associatedwith computational aspects which are well beyond the scope of this work and woulddemand a separate contribution written by somebody with extensive expertise inthose topics We simply illustrate analytical results with some simple, academicexamples and provide well-known references to cover all relevant aspects of optimaldesign

sepa-The book also aims to persuade young researchers, on both the analytical and thecomputational side, to further pursue the development of the sub-relaxation method

I firmly believe that there is still much room for improvement Although some newdirections may be very hard to examine (e.g., the analysis for the elasticity settingand the implementation of point-wise stress constraints, to name just two importantones), others may lie within reach In particular, applying the sub-relaxation method,appropriately adapted for numerical simulations, to realistic problems and situationsmay result in quite interesting approximation techniques

Some further training in Analysis is assumed, including basic Measure Theory,Sobolev spaces, basic theory of weak solutions for equations and systems of

vii

equilibrium, weak convergence, etc Moreover, it is desirable that the reader hassome previous experience with the basic techniques of the calculus of variations,the role of convexity in weak lower semicontinuity, and how the failure of thisfundamental structural property may result in special oscillatory behavior Again,some simple discussions and examples may serve to fill this gap, and so provide thereader with a basic, well-founded intuition on these important issues.

The book is intended for masters or graduate students in Analysis, Applied Math,

or Mechanics, as well as for more senior researchers who are new to the subject Atany rate, readers are expected to have sufficient analytical maturity to understandissues not fully covered here in order to appreciate the ideas and techniques that arethe basis for the sub-relaxation approach to optimal design

I would like to express my sincere gratitude to an anonymous reviewer whosepositive criticism led to various significant improvements in the presentation ofthis text Several colleagues from the editorial board of the Springer SEMA-SIMAISeries also helped a lot in making this project a reality Particular thanks go to L.Formaggia and C Pares for their specific interest in this book F Bonadei fromSpringer played an important role, too, in leading this project to final completion

May 2016

1 Motivation and Framework 1

1.1 The Model Problem 1

1.2 Variations on the Same Theme 4

1.3 Why It Is an Interesting Problem 5

1.4 Why It Is a Difficult Problem 5

1.5 General Procedure 8

1.6 Subrelaxation 10

1.7 What is Known 11

1.7.1 Homogenization 11

1.7.2 Engineering 12

1.7.3 Some Brief, Additional Information 13

1.8 Structure of the Book 13

1.9 Bibliographical Comments 14

References 16

2 Our Approach 23

2.1 The Strategy 23

2.2 Young Measures 27

2.2.1 Some Practice with Young Measures 28

2.3 Relaxation 30

2.4 Basic Differential Information: The Div-curl Lemma 31

2.5 Subrelaxation 32

2.6 Bibliographical Comments 34

References 35

3 Relaxation Through Moments 37

3.1 The Relaxation Revisited: Constraints 37

3.2 The Moment Problem 39

3.3 Characterization of Limit Pairs 41

3.4 Laminates 43

3.5 Characterization of Limit Pairs II 49

3.6 Final Form of the Relaxation 52

ix

3.7 The Compliance Situation 58

3.8 Bibliographical Comments 60

References 61

4 Optimality 63

4.1 Descent Method 64

4.2 Optimality Conditions 67

4.3 Final Remarks 69

4.4 Bibliographical Comments 69

References 70

5 Simulation 71

5.1 A Direct Approximation Scheme 71

5.2 Some Selected Simulations 74

5.2.1 Dependence on Initialization 74

5.2.2 Dependence on Volume Fraction 74

5.2.3 Dependence on Contrast 75

5.3 Some Additional Simulations 76

5.4 Bibliographical Comments 77

References 78

6 Some Extensions 79

6.1 A Non-linear Cost Functional 80

6.1.1 Young Measures 81

6.1.2 Solution of the Mathematical Program 83

6.1.3 Subrelaxation 85

6.1.4 Optimality 88

6.1.5 Some Numerical Simulations 90

6.2 A Non-linear State Law 92

6.2.1 Reformulation and Young Measures for the Non-linear Situation 93

6.2.2 Moments 95

6.2.3 Necessary Conditions 95

6.2.4 Sufficient Conditions 98

6.3 A General Heuristic Approximation Method 102

6.3.1 Conductivity 104

6.3.2 Elasticity 107

6.4 Bibliographical Comments 110

References 111

7 Some Technical Proofs 113

7.1 Div-Curl Lemma 113

7.2 Riemann-Lebesgue Lemma 114

7.3 Young Measures 115

7.3.1 The Existence Theorem 115

7.3.2 Some Results to Identify Young Measures 119

7.3.3 Second-Order Laminares 120

Contents xi

7.4 A Non-linear Elliptic Equation 122

7.5 Covering Lemma 123

7.6 Bibliographical Comments 129

References 129

Motivation and Framework

It is not difficult to motivate, from a practical point of view, the kind of situations

we would like to deal with and analyze We have selected a typical example in heatconduction, but many other examples are as valid as this one Suppose we have twovery different materials at our disposal: the first, with conductivity˛1 D 1, is agood and expensive conductor; the other is a cheap material, almost an insulatorwith conductivity coefficient˛0D 0:001 These two materials are to be used to fill

up a given design domain Q, which we assume to be a unit square for simplicity

(Fig.1.1), in given proportions t1, t0, with t1C t0 D 1 Typically, t1 < t0 giventhat the first material is much more expensive than the second We will take, for

definiteness, t1 D 0:4, t0 D 0:6 The thermal device is isolated all over @Q, except

for a small sink0at the middle of the left side where we normalize temperature to

vanish, and there is a uniform source of heat all over Q of size unity The mixture

of the two materials is to be decided so that the dissipated energy is as small aspossible

If we designate u.x; y/ as temperature, and use a characteristic function  to indicate where to place the good conductor in Q, then we would like to find the

optimal such distribution minimizing the cost functional

Z

Q

u x; y/ dx dy

that measures dissipated energy, among all those mixtures complying with

divŒ.˛1.x; y/ C ˛0.1  .x; y///ru.x; y/ D 1 in Q;

uD 0 on 0; ˛1.x; y/ C ˛0.1  .x; y///ru.x; y/  n D 0 on @Q n 0;

Z

Q

.x; y/ dx dy D 0:4:

© Springer International Publishing Switzerland 2016

P Pedregal, Optimal Design through the Sub-Relaxation Method, SEMA SIMAI

Springer Series 11, DOI 10.1007/978-3-319-41159-0_1

1

2 1 Motivation and Framework

Fig 1.1 Design domain for a

thermal device

Distributed Heating

Fig 1.2 Optimal distribution

of the two materials

do not pay much attention to precise assumptions

1 The design domain˝  R2is a bounded, Lipschitz domain.

2 There are two conducting, homogeneous, isotropic materials, with conductivityconstants˛1> ˛0> 0, at our disposal in amounts t i , respectively, so that t1Ct0 Dj˝j

3 A source term f W˝ ! R

4 A boundary datum u0W @˝ ! R

There are many ways in which those two materials can be assembled to fill up thedesign domain˝ We can describe all of them by means of a characteristic function

.x/ which indicates where the first material (the one with conductivity constant ˛1

is being placed In this way, the non-homogeneous coefficient

Note that u depends (in a non-local way) upon: any time we change our design

, its corresponding potential will also be different Among the infinitely manypossibilities we have for these mixtures described by and complying with (1.1),

we would like to choose those that are optimal according to some reasonable andrelevant optimization criterium This can be formulated in terms of itself, and also

in terms of the associated potential u, unique solution of (1.2) There is a wholevariety of possibilities A typical one we will pay special attention to is

I./ DZ

which is identified as the compliance functional Let us stick for the moment to thisone to clarify what our aim is

Any time we have one complying with (1.1), we need to compute the number

I./ to be able to compare it with other admissible possibilities for  To do so,

as expressed in (1.3), we need to solve for u in (1.2) first, and then perform theintegration in (1.3) Our problem is to find the design, the, for which the value of

I, computed in this way, is as small as possible.

In compact, explicit form, we would write:

4 1 Motivation and Framework

Z

˝.x/ dx D t1;

divŒ˛.x/ru.x/ D f x/ in ˝; u D u0on@˝:

A specific, characteristic function is said to be optimal for the problem if it

is feasible, i.e it respects the volume constraint condition (1.1), and it realizes thisminimum:

I /  I./ for all admissible such designs :

There are many ingredients of our basic model problem which can be changed toproduce new interesting situations We list here a few

• The dimension N can be3 as well, or even higher

• Boundary conditions around@˝

• We could have more than just two materials to mix

• We might be asked to accommodate at the same time various different mental conditions (multi-load situations)

environ-• The materials do not have to be homogeneous and/or isotropic

• The cost functional can be quite general; in particular, it could depend also uponthe design itself, the derivatives ru of the potential u, etc.

• One of the materials can degenerate to void˛0 D 0 In this situation, we speakabout shape optimization problems

• The state equation can be non-linear, so that the materials have a non-linearbehavior, or could be a dynamical equation so that the structure of the mixture isallowed to change with time

• The equation of state can be a system rather than a single equation The mostimportant situation here corresponds to linear elasticity

• There could also be further important constraints either in integral-form, orperhaps more importantly, in a pointwise way

Each one of these new situations requires new ideas Though we will stick to ourmodel problem above to show how one can understand these situations and describethe basic tools of our approach, any variation of the previous list will require newtechniques for a full analysis We are not talking about more-or-less straightforwardgeneralizations

This text aims at being an introduction to the subject of optimal design problemstreated with tools from non-convex variational analysis Our goal is to use this modelproblem to describe the basic concepts and techniques But the treatment of more

sophisticated situations, at the level of analysis and/or numerical simulation, wouldrequire much more work to understand the distinctive features of those basic toolsfor each particular situation.

From the practical point of view, the general meaning and interest of all thoseproblems discussed in the previous section is pretty clear: we would like to findthe best mixture of several given materials according to a certain criterium One ofthe paradigmatic situation in Mechanical Engineering is to find the most resistant,non-collapsing structure under given environmental conditions This is probably themost difficult, and most important, problem in the list above But each situationcorresponds to a certain design problem in Engineering: to find optimal mixtures ofgiven materials providing the best performance under given working conditions.But even from a strict analytical viewpoint, optimal design problems arefascinating They pose to the applied analyst highly non-trivial problems that requirefine analytical tools They challenge constantly known ideas, and one is forced toinnovate for new problems and situations As a matter of fact, quite often, problemsbecome so difficult that they look rather unsolvable

We are going to spend some time with a simple variational problem, with the goal

in mind to convey the difficulties we expect to face in understanding our optimaldesign problems We hope that such an example may help in appreciating, at least

in a first round, the subtleties of non-convex variational problems

Consider the integrand.z/ W R2! R defined explicitly by

Note that the boundary condition is u0.x/ D 1=2/.a1C a0/  x Suppose Q D

.0; 1/2is the unit cube We realize that the integrand is non-negative, and it attains

6 1 Motivation and Framework

its vanishing global minimum at the two values ai , i D1; 2 The question is then if

one can arrange a function u 2 H1.Q/, complying with the given boundary datum,

in such a way that ru.x/ 2 fa1; a0g This is clearly impossible in a neat way for a

single function u But, indeed, it is possible to find a whole sequence fu jg of feasible

functions so that I u j/ ! 0 Set

v.x/ D

Z x a1

0 .s/ ds; rv.x/ D .x  a1/a1;where.s/ is the characteristic function of the interval 0; 1=2/ over the unit interval

.0; 1/ extended by periodicity to all of R We clearly see that rv.x/ takes on only the two values a0and a1in “proportions”1=2  1=2 However, v hardly takes on

the appropriate boundary values given by u0around@˝ This can be achieved by abit of “surgery” First, put

Choose, next, a sequence of cutoff functionk.x/ enjoying the properties

This full family of functions is admissible as its members comply with the boundarycondition Its gradient is

rvj ;k.x/ D k.x/rvj.x/ C 1  k .x//ru0.x/ C vj .x/  u0.x//rk.x/:

If, thanks to the uniform convergencevj ! u0, we take j D j.k/ so that the last term

in this gradient is arbitrarily small as k ! 1, we would have a sequencevk vj k/;k

of feasible functions, whose gradient takes the two values a0, a1except in a small

and negligible boundary layer This shows that indeed I.v k/ ! 0, and such sequence

is a minimizing sequence for our variational problem It is interesting to realize thatany minimizing sequence will have to be essentially like the one we have built:

gradients need to alternate between the two vectors a1, a0 in proportions that are

determined by the boundary datum u0 This persistent oscillatory behavior of the

gradients of minimizing sequences on alternate strips with normal n D a1 a0, andrelative proportion given, is what we will intuitively refer to as “microstructure”

It shows in clear terms the behavior one can expect when non-convexity is afundamental ingredient of our optimization problem Notice that the density.z/

for our integral functional I is indeed non-convex, and so the direct method of the Calculus of Variations could not be applied As a matter of fact, the functional I

is not weak lower semicontinuous because vk * u0 in H1.Q/, and yet I.u0/ >

lim I.v k/ D 0

The problem can be made more sophisticated if the two constant vectors a1, a0depend on the spatial variable x In this situation, those alternate layers will take place locally around each point x with normals, and relative proportions depending upon x If the integrand .z/ is only allowed to be finite when z is either a1, a0, wewould have a more rigid scenario for a binary variable that is only permitted to take

on two values This provides also an intuitive explanation of why the use of binaryvariables in optimization problems leads to non-convexity and persistent oscillatorybehavior

Our optimal design problems are difficult to analyze and to simulate tually, it is not hard to understand the reason from the very beginning, as we havetried to convey with the previous discussion The design (or optimization) variable

Concep-is , a binary variable taking two possible values f0; 1g As such, the problemcannot be convex, for it is “not defined for intermediate values in Œ0; 1”, and

so it takes an infinite price for such intermediate values This is rather a naive,though essentially correct, reason, for even if would allow 2 Œ0; 1, and regard

.x/ D s.x/ as a density, the problem would still be equally difficult We have

made an attempt to provide some intuition on the nature of non-convexity with thepreceding paragraphs

As one thinks more about our model problem, and how it works, one realizesthat the set of characteristic functions has a complicated structure for they cannothave any regularity, and we have an overwhelming amount of possibilities Thenon-convexity is typically associated with lack of optimal solutions: the infimum

is not attained, so that minimizing sequences of characteristic functions do notconverge weakly to a characteristic function, but to a density This means that

8 1 Motivation and Framework

optimal mixtures will sometimes tend to have microstructural features as the mixturehas to be very fine spatially to be represented by a density (one can think in terms

of black and white, and grey levels) But there is much more

Suppose we focus on a certain neighborhood of a point x 2˝ Even if we knewthat the optimal mixture around this point should have30 % of one material and

70 % of the other, this still leaves open the door to determining the optimal geometryitself because many different micro-geometries can have in common the same value

of the density, and perhaps not all of those will be optimal The cost functionalwill generally depend not only on the underlying optimal density of material, butalso on the geometry with which the materials are arranged microscopically Thisissue about optimal micro-geometries is what makes these problems so hard Even

so, we are still interested in understanding how optimal (minimizing) sequences ofcharacteristic functions look like Even better, we would like to understand how tobuild some of those minimizing sequences for our problem, and how to encode thoseoptimal micro structural features in analytical tools of some sort This is the mainobjective, and main reason for all that follows

The way in which these optimization problems are tackled is by means ofrelaxed formulations A relaxed form of a given optimal design problem shouldtake into account all those microstructural features that may exhibit sequences ofcharacteristic functions Or in other words, a relaxed formulation must, somehow,

be a new, though intimately connected, optimization problem “defined for sequences

of characteristic functions” through their relevant features from the perspective ofthe problem at hand Intuitively, a relaxation is like an enlargement of the originalproblem without changing its nature It is like going from the rationals to the reals:

a completeness process But an overwhelmingly huge one

The main issue in finding a relaxation is to decide the variables in which therelaxation is going to be defined, how these variables relate to the original variables,and how they are going to encode information about micro-geometries We cannotforget that the relaxation is a means towards the goal of understanding optimaldesigns for the initial structural problem, so that once one succeeds in having atrue relaxation for one of these problems, then their optimal solutions have to beinterpreted in terms of the original problem

Let us pause further on the general procedure to establish a relaxation of a complexoptimization problem in continuous media Though the discussion may sound a bitabstract at some point, readers may benefit from such a general discussion in order tohave an overall picture of where we are heading with our optimal design problems

We ask for a bit of patience, as many of these steps will be made precise and will bebetter understood and appreciated in subsequent chapters

Assume we have an interesting optimization problem like our model problemabove for a linear conductivity equation We are very much interested in finding

optimal solutions, but after resorting to standard literature on the topic we realizethat there is no result to be applied to conclude the existence of optimal solutions.This does not mean that there is never (depending on the particular data set) anoptimal solution, but at least general theorems cannot be applied On the contrary,

we have learnt in the process that, or come across examples where non-convexity

in any form may very seriously interfere with the existence and approximation ofoptimal solutions Yet we insist in that we would like to understand the structure

of optimality, perhaps not reflected in a single feasible object of our problem but

in a full sequence going to the infimum: a minimizing sequence This is the issue

of understanding the nature of minimizing sequences for our problem with theobjective in mind of being able to build very precisely at least one such minimizingsequence The process of going from the original problem to a new one, yet to beformulated, in which feasible objects are identified with sequences of the originalproblem is what we refer to when we use the term relaxation

A complete understanding of this passage proceeds in various steps

1 New generalized variables need to be defined and analyzed Its connecting link

to the initial variables should be very clearly established, so that each newgeneralized variable may be related at least to a sequence of the feasible set forthe original problem

2 The other important ingredients of every optimization problem should bereconsidered for the new scenario In particular, a generalized cost functionalought to be specified, and constraints to be respected must be explicitly written.Both the new objective functional, and the constraints have to be derived takeninto account very carefully the same ingredients for the original problem, inthe sense that the limit of costs for a sequence in the original problem must bethe new cost of the new feasible object determined by that sequence This limitprocess ensures that we are not changing the nature of the problem in its relaxedformulation

3 The process of going back from a new (optimal) generalized object to a sequence

of the original problem has to be described without ambiguity The wholepoint of a relaxation is to find an optimal generalized object through which wecould understand the structure of at least one minimizing sequence of the initialproblem, which is, after all, the objective of our analysis

In a compact formal way, we can write an optimization problem like

Minimize in u 2A W I.u/:

Its relaxation would read

where every feasible u 2 A must be somehow identified within A , and any time

fu j g 7! U, then I.u j / ! I.U/ The relaxation link that ties together these two

10 1 Motivation and Framework

problems can be expressed in writing

in understanding minimizing sequences though we might be closer than before

The preceding discussion is rather neat in a very abstract, analytical sense Practice,however, is much more complicated than that The main difficulty, at least for theoptimal design problems we are interested in this text, is hidden in the fundamental

constraint that feasible objects u 2 A should respect, and, more specifically, on

how those constraints translate into the corresponding relaxed feasible setA This

is indeed the deepest issue we are facing, to the point that we cannot expect to beable to find an efficient, complete, practical description of this relaxed set All wewill aim at is to retain the most manageable of the fundamental constraints inA ,

and build with them a new setA , together with a new optimization problem

min

Two main issues are:

1 The fact that we are using minimum instead of infimum in (1.5) indicates thatthis new problem ought to admit optimal solutions, and so it is no longer in need

of relaxation

2 Because in definingA we have ignored some constraints of A (but retain some

important ones too), the setA might be, in general, larger than A , and so

min

I.U/  min

Inequality (1.6) clearly expresses the idea that problem (1.5) is a subrelaxation of

our initial problem

It is very easy to find subrelaxations of optimization problems Many of themwill be useless We need to say something else about when a given subrelaxationcould be a good one for a given problem By a good one we mean a successful one,one through which we can find at least one non-trivial solution of the true relaxation(1.4) In other words, if m, m, and m are the values of the infima/minima of the three

problems, then a good subrelaxation is one for which those three numbers turn out

to be equal How is this to be accomplished? In any given problem the procedurewill be:

1 Study the original problem, the one we are interested in, and decide whichconstraints are going to be retained

2 Define a subrelaxation by determining in clear termsA , and I.

3 Examine optimality and/or approximation for (1.5), and find an optimal object

U0 2 A

4 Conclude, if you can, that in fact U0 2 A , and interpret through it optimal

structures for the initial optimal design problem

The main advantage of a subrelaxation over a true relaxation, is that in setting up

A , constraints may be much more flexible, but at the same time sufficiently tight

so that at least one optimal solution of (1.5) may turn out to belong toA , the true

relaxation of the problem Whenever this is so, the subrelaxation method will havebeen successful If not, we may have valuable information about our problem, butmay not fully understand optimal structures On the other hand, it is also possiblethat, in seeking a sub-relaxation, the structure of the design problem is such thatone ends up having a true relaxation The sub-relaxation method is, above all, aprocedure of going about setting up a relaxation If the process can be carried out tothe end, we will have a true relaxation; if not, or if we are not interested in doing so

to the end, we will have just a sub-relaxation

In this book, we will describe in full detail how this process can be carried outfor our model problem in conductivity, when generalized objects are identified withYoung measures associated with suitable sequences of a convenient reformulation ofthe initial problem A main goal is to understand the important piece of information

to be retained in setting a good, efficient subrelaxation

We will devote a short subsection to each of three important topics: analyticalviewpoint, engineering perspective, and additional information Many importantreferences for each of these areas are given in the final section of the chapter

1.7.1 Homogenization

Although Homogenization Theory cannot be reduced to its relationship with thiskind of optimization problems, it has been very successfully used in optimaldesign in continuous media as a main application To relate the perspective ofhomogenization to our own here in this text, we could use the term “super-

12 1 Motivation and Framework

relaxation” to define it The short discussion that follows requires some basicknowledge of homogenization which is not provided here

Through a basic cell, homogenization theory aims at describing (micro)structures

of mixtures of the two materials modeled after that given unit cell, and how theoriginal problem is transformed through this passage In this regard, we can talkabout a new feasible setA , which, by construction, is always a (smaller) part of A

instead of a bigger one as in a sub-relaxation, and the extension I of I to this new

set All of this is done in a coherent way so that

min

U2A I.U/ D inf

U2A I.U/ D inf

but the infimum in the middle may not be a minimum In practice, the feasible set

A is parameterized in a very efficient manner, in such a way that the corresponding

infimum can be found or approximated The (quasi)optimal elements inA resulting

from optimality or simulation yield, quite often, a very good idea about the optimalway in which the two materials are to be mixed The term super-relaxation is usedhere in a rather loose way to indicate that the feasibleA is smaller than the true

admissible set for the relaxation, whereasA for the sub-relaxation is bigger Note

that it is never true that

1.7.2 Engineering

At any rate, either for the sub-relaxation or for the super-relaxation, the analysis

to be carried out is far from being straightforward New situations may requirecomplicated new computations, or new insight into the problem, to the point that

a practical way to produce sensible quasi-optimal solutions for the original problemturns out to be as important as the analysis of sub-relaxations or super-relaxations.Especially when one is talking about a realistic design problem of interest inindustry, these robust, direct methods of approximation are of great relevance Mostdefinitely the ones that are used nowadays are well founded on solid ideas coming,above all, from super-relaxations setup after homogenization techniques Notice thateven if we can formulate in a very precise way an exact relaxation through which

we can fully understand optimal mixtures, at the end, for realistic problems, wewill be asked to provide a more-or-less macroscopic answer to the original problem

that could eventually be manufactured This post-process is as important, from apractical viewpoint, as the analysis itself.

We will mention just three main practical philosophies to deal with numericalapproximation We refer to the abundant bibliography in the final section of thechapter for many sources where these (and some other) methods can be studied

1 SIMP method Capitals stand for Solid Isotropic Material with Penalization.The basic idea is to penalize intermediate values of the density so as to forceextreme values f0; 1g, and recover in this way a true design This is typicallyaccomplished through the introduction of an artificial material with rigidityproportional to a power of the density

2 Level-set methods These have become more popular lately, and it amounts to theclever application of the standard level-set method of Osher and Sethian (see thefinal section for specific references) in the context of optimal design problems Itsuffers from some important disadvantages, but it apparently works very well insome situations

3 Perimeter penalizations This is also a favorite alternative when one is willing

to put a limit to the fineness of the microstructure It basically amounts toadd, to the compliance cost functional or whatever relevant functional weare examining, a term that penalizes excessive perimeter for the characteristicfunction determining the mixture This perturbation introduces an additionalcompactness property that limits the intricacy of the mixture

1.7.3 Some Brief, Additional Information

There is much more work on optimal design than the material related to nization or sub-relaxations See the final section of the chapter about bibliography

homoge-On the one hand, there is the approach to optimal design based on the shapederivative This typically requires smoothness of shapes, and its applicability

is limited through this smoothness requirement On the other, there are otherapproaches that make use in various different ways of densities and measures.Finally, it is interesting to mention some recent work to deal with optimal designproblems subjected to some randomness in the environmental conditions This areawill most probably start attracting more and more the attention of researchers

A given optimal design problem may admit many various equivalent formulations

in terms of different sets of variables In finding a successful subrelaxation, it is ofparamount importance to decide on the best formulation of the problem This will be

a constant concern in this text The problem may come formulated in a natural way

14 1 Motivation and Framework

in a given set of variables But it may nevertheless be advantageous to reformulate

it in a different set of variables One must have this possibility in mind at whateverstage of the relaxation or sub-relaxation processes

As pointed out earlier in the chapter, we will pursue to cover all of the steps ofour analysis as clearly as possible for the compliance problem given explicitly inthe first section of this chapter Our objective is to show in a rigorous way how to

go from the original formulation of the problem to its relaxation, or sub-relaxation.This corresponds to Chap.2 The relaxation is setup in terms of underlying Youngmeasures associated with suitable fields Chapter 3 revolves around the issue offinding the more transparent and flexible form of the relaxation This will be aconsequence of the observation that Young measures occurring in the relaxationalways do so through integration against a finite and limited number of integrands,and so the relaxation can be formulated just in terms of those moments This is

a tremendous advantage from the viewpoint of having an explicit relaxation InChap.4, optimality conditions are examined Again for the compliance situation,these are quite explicit and lead to very important information, that is directly used

in producing some simulations in Chap.5 Once we have gained some familiaritywith the sub-relaxation method, and mastered its main tools, we treat two moredifficult situations where at least partial answers are known: a non-linear (in thederivatives of the potential) cost functional, and a non-linear conductivity stateequation This is the material in Chap.6 We also introduce in this chapter anotherdirect, computational method, argue why it might be an interesting alternative tobear mind, and test it with some easy situations in elasticity Finally, technicalproofs have been deferred to Chap.7to avoid cutting the thread of the discussion inpreceding chapters

Our references here to the abundant bibliography about this subject focuses mainly

on the areas mentioned in Sect.1.7, while references to the sub-relaxation methoditself will be deferred to subsequent chapters The introductory example in the firstsection is taken directly from [57]

There are thousands of articles, books, surveys, etc., about HomogenizationTheory We will mention here only a selection of those directly related to optimaldesign or shape optimization, not claiming in the least that our selection isexhaustive

Perhaps it is better to start with a number of different textbooks on the subject

of optimal design Some of the gaps and the more basic issues not explained herecan be found in them In particular, we will mention [3,19,29,48,52,105,124].They cover both formal rigorous analysis as well as computational aspects, andmechanical issues Some are elementary (like [52]), some others are rather advanced(like [48] or [124], or [105] for a more mechanical-oriented one), and yet others looklike intermediate as they try to pay attention to several perspectives

There are some very good, classical presentations of Homogenization Theoryfrom a general viewpoint We just mention [4,108,109,125] See [35] for a morerecent account.

Various terms occur often in connection with the homogenization method applied

to optimal design and related areas They are all intermingled, and it is hard toseparate them from each other, and to set up limits about where one starts and theother ends We are talking about composite materials, effective properties and theirbounds, extremal properties and their realization, G-closure problems, etc Most ofthese areas have stirred a lot of work and interest Just as a sample we mention

• relationship with quasiconvexity and relaxation: [8,37,41,47,62,89,104];

• stress and other constraints: [45,71,72,79,91,92];

16 1 Motivation and Framework

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Chapter 2

Our Approach

To introduce our analytical strategy, let us focus on the particular situation described

in Sect.1.1, but changed in a way to avoid any distraction from our main objective:

The various ingredients are taken as indicated earlier, except that we have taken a

cost functional which is linear on ru, with a fixed given factor F, and a vanishing

© Springer International Publishing Switzerland 2016

P Pedregal, Optimal Design through the Sub-Relaxation Method, SEMA SIMAI

Springer Series 11, DOI 10.1007/978-3-319-41159-0_2

23

What do we know about this pair if it is coming from the state diffusion equation of

our problem? On the one hand, V must be a divergence-free field in˝; on the other,

V and ru must be colinear with a multiple that depends on the point x, but can only

be chosen between two possibilities˛1or˛0 To formalize this last condition, let us

put

iD f.y; z/ 2 R2 R2W z D ˛iyg; i D 1; 0;

a couple of linear manifolds inR2 R2 If we further put, to uniformize notation,

UD ru, then the pair U; V/ should comply with:

1 U is a gradient, and so curl-free in˝;

It is immediate to check that if a pair.U; V/ complies with all of these conditions,

then, except for the boundary restriction on@˝, there is a feasible characteristicfunction.x/ so that

VD ˛1.x/ C ˛0.1  .x///ru.x/;

and so, because V is div-free,

divŒ.˛1.x/ C ˛0.1  .x///ru.x/ D 0 in ˝:

What are the main reasons of using this reformulation of the problem? Why may

it be advantageous with respect to the primal formulation in terms of?

To provide some insight on this important issue, compare the same problemunder these two equivalent dependences

 7! I./; U; V/ 7! I.U; V/:

In the first case, there is an intermediate ingredient, the potential u, which is obtained

through a non-local operation from the design variable; while in the second there

is no such intermediate step since the cost is computed by integrating directly thedesign variables It is true, however, that there are important differential restrictionsplaced on feasible pairs.U; V/ as indicated earlier, but we do hope to gain something

in the overall picture This will become clear in the sequel

Notice that weak convergence is the main basic concept when one has todeal with integrals and sequences Yet, it does not behave well when non-linear

2.1 The Strategy 25

operations are involved, and when non-local processes are an important ingredient

We do have some tools to get reasonably well around the former (Young measures),but it is much more delicate to treat the latter Hence our insistence on trying toavoid design variables for which some non-local process is to be taken into account

to compute costs

After the proposed reformulation, we face the problem

Minimize in.u; V/ 2 H1.˝/  L2.˝I R2/ W I.u; V/ D

How are we to seek an optimal solution for this problem? As in every continuous

optimization problem, we should first check that there is a finite infimum m (it

cannot go all the way to 1), and then take a minimizing sequence.u j; Vj/ with

I u j; Vj / & m.

Suppose that for such a minimizing sequence, we have u j * u; u 2 H1.˝/,where, as usual,* stands for weak convergence Then it is clear that

mDZ

˝F.x/  ru.x/ dx:

This is also true even if the sequence of pairs.u j; Vj/ is not minimizing but simply

u j * u The important issue we face is to detect what weak limits u; V/ can be

generated or achieved through sequences of feasible vector fields.u j; Vj/ Amongthe constraints on feasible fields, there are some which are global in ˝, like the

Dirichlet-boundary condition u D u0on@˝, or the constraint involving the volume

fraction t1 The ones that truly, and jointly, require a non-trivial understanding are:

1 U is curl-free in˝;

2 V is div-free in˝;

3 .U; V/ 2 1[ 0

Therefore we want to provide an answer to the problem:

Characterize the set of pairs.U; V/ which can be weak limits (in L2.˝/) of asequence of pairs.Uj; Vj/ such that

curl UjD 0; divVjD 0 in ˝; Uj; Vj/ 2 1[ 0:

Once we have a precise answer for this problem, we will be able to set up anew variational problem whose optimal solutions will codify minimizers and/orminimizing sequences for our optimal design problem

Before proceeding any further, and to motivate our next step, let us reflect a bit

on the characterization above On the one hand, we have the differential informationthat pairs.Uj; Vj/ are curl-div-free On the other, we have a very precise pointwiseinformation about where those pairs are to take values on Suppose, for a minute, wepretend not to have any differential information on feasible pairs.Uj; Vj/ If U; V/

is a weak limit of (a subsequence of) such a sequence taking values on D 1[0,what kind of information do we have on.U; V/? All we can say, in this generality,

is that.U; V/ is to take values on the convex hull of  After all, averages of weak

limits are limits of averages over arbitrary sets This condition truly implies ourconclusion on the convex hull We therefore see that the information on where asequence takes values on is transferred to where the weak limit may take values on,and this passage involves some kind of convexity of sets How is the differentialinformation we have on pairs going to change this general picture?

There is a fundamental result, in the form of a structural restriction, that involves

a certain non-linear operation (the inner product) with weak limits when we haveadditional information, as we do in our situation It is called the div-curl lemma,and it is the central result in the theory of compensated compactness The proof hasbeen deferred until Chap.7

Fj



be a sequence of bounded fields in L2.˝I Mm N / converging

weakly to F, such that˚

Fj ru j/T * Fru T

in the sense of distributions.

What this fact ensures is that the non-linear operation of taking inner productbehaves as linear quantities do with respect to weak convergence, as long as wehave some crucial differential information It is, by no means, valid without thisinformation

2.2 Young Measures 27

Our task is then to understand how the div-curl lemma interferes and restrictsthe discussion above on where weak limits are supposed to take values To deal withthis sort of questions in general, and not just for the particular situation here, we willget into exploring an analytical tool that is really helpful in taming the relationshipbetween non-linear functionals, and weak convergence

We forget momentarily our problem until the end of the last section, to focus on ananalytical tool that can be helpful for our purposes It is called Young measure, and

it was originally introduced in the context of non-convex control problems

To motivate this new concept, consider the following situation Let fujg be

a sequence of fields uniformly bounded in L1.˝I Rm/, where ˝  RN is acertain domain Let W Rm 7! Rd be a continuous mapping The composition

f.uj /g will also be uniformly bounded in L1.˝I Rd/ After extracting appropriatesubsequences, we will have

uj * u in L1.˝I Rm/;

.uj / * u in L1.˝I Rd/:

The question is: what is the relationship among u, , and u? The answer is not,

in general, u D .u/ It is furnished by the basic existence theorem for Young

measures

p  1 There exists a subsequence (not relabeled), and a family of probability

measures D f xgx2˝, supported in Rm , such that for every continuous  W

Rm! Rd for whichf.uj /g converges weakly in some L q.˝I Rd /, we have

.uj/ * ; .x/ D

Z

Rm .z/ d x z/:

This fundamental result motivates the following definition

inRm, is called the (underlying) Young measure corresponding to a sequence of

functions fujW ˝ 7! Rmg if for every continuous  W Rm! Rdfor which f.uj/g

converges weakly in some L q.˝I Rd/, we have

.uj/ * ; .x/ D

Z

Rm .z/ d x z/:

We could therefore rephrase the fundamental existence theorem by simply statingthe following.

L p.˝I Rm /, p  1, there is always a subsequence (not relabeled) that admits a

Young measure D f xgx2˝in the sense of Definition 2.1 .

The formula for is typically referred to as the representation formula in terms ofthe underlying Young measure What is remarkable is that this family of probability

measures is determined by the sequence fujg, but does not depend on the quantity .The conclusion of the statement of this basic existence theorem for Young measuresmeans that

for all appropriate test fields

Carathéodory integrands ˚.x; z/ for which the sequence f˚.x; uj.x//g weakly

2.2.1 Some Practice with Young Measures

We include in this subsection a potpourri of basic information, basic questions,simple examples, some general facts, etc

• A simple, clarifying example that shows that composition with non-linearfunctions does not commute with weak convergence Take ˝ D 0; =2/,

u j x/ D sin jx/, v j x/ D u j x/2 By elementary trigonometry, we have

Z b a

sin.jx/ dx D cos.ja/  cos.jb/

j ;

Z b a

sin2.jx/ dx D b  a

2 C 14j .sin.2ja/  sin.2jb//:

The arbitrariness of a and b implies that u j * 0 while vj D u2

j * 1=2 But1=2 ¤ 02! Non-linear quantities are not conserved by weak convergence: u jweak

converges to some u, butvj D u2

j does not to u2but some otherv

2.2 Young Measures 29

• The situation in the previous item is a particular case of a more general fact asfollows

in L p QI R m /, where Q is the unit cube in R N Then the sequenceffj .x/ D f.jx/g

generates the homogeneous Young measure  xfor a.e x 2 Q, where

The weak limit u of a sequence ujis always represented by the first moment of

the Young measure generated by (a subsequence of) fujg

• How can one measure sets through each particular member x of a wholeYoung measure D f xgx2˝generated by a specific sequence fujg? This is notstraightforward, but there is a funny formula to measure an arbitrary measurable

strictly positive value to a set E, which is not a singleton, then the sequence

u j has to oscillate as j becomes larger and larger around the point x.

• How is strong convergence captured by the Young measure device? As a quence of the previous item, triviality of the Young measure is translated (undersome additional technical hypotheses depending on the particular framework)into strong convergence of the generating sequence: x D ıu x/ if and only if

conse-(under further restrictions) uj! u strongly.

As usual, we take the Dirac massıu , u 2 Rm, as the Radon measure definedthrough

hıu ; i D u/

for every continuous