Optimal design through the sub relaxation method

11
Pablo Pedregal

Optimal Design
through the
Sub-Relaxation Method
Understanding the Basic Principles

Se MA


SEMA SIMAI Springer Series
Series Editors: Luca Formaggia (Editor-in-Chief) • Pablo Pedregal (Editor-in-Chief)
Jean-Frédéric Gerbeau • Tere Martinez-Seara Alonso • Carlos Parés • Lorenzo Pareschi •
Andrea Tosin • Elena Vazquez • Jorge P. Zubelli • Paolo Zunino
Volume 11


More information about this series at />

Pablo Pedregal

Optimal Design through
the Sub-Relaxation Method
Understanding the Basic Principles

123


Pablo Pedregal
Universidad de Castilla-La Mancha

Ciudad Real, Spain

ISSN 2199-3041
SEMA SIMAI Springer Series
ISBN 978-3-319-41158-3
DOI 10.1007/978-3-319-41159-0

ISSN 2199-305X

(electronic)

ISBN 978-3-319-41159-0

(eBook)

Library of Congress Control Number: 2016948436
Mathematics Subject Classification (2010): 49J45, 74P05, 35Q74
© Springer International Publishing Switzerland 2016
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To my beloved sister Conchi, in memoriam



Preface

This book aims to introduce an alternative analytical method for the solution of
optimal design problems in continuous media. As such, it is not meant to serve as
an introductory text on optimal design. In fact, a certain degree of familiarity with
more classical approaches, especially the homogenization method, is required in
order to appreciate fully the comments and results. It is also assumed that the reader
will have had some exposure to the significance and relevance of these problems in
Engineering, as well as to the various numerical procedures developed to simulate
optimal designs in practical problems. The material and the treatment are intended
to be self-contained in such a way that, in addition to covering the aforementioned
aspects, the book will serve as a sound basis for a masters or other postgraduate
courses in the subject.
Application to real problems in Engineering would almost demand a separate book. On the one hand, many specific situations may have an interesting
mechanical background (e.g., compliant mechanisms or vibrating structures), an
electric/electronic flavor (e.g., optimal design with piezoelectric materials), or
relevance in other fields. On the other hand, there are many delicate issues associated
with computational aspects which are well beyond the scope of this work and would
demand a separate contribution written by somebody with extensive expertise in
those topics. We simply illustrate analytical results with some simple, academic
examples and provide well-known references to cover all relevant aspects of optimal

design.
The book also aims to persuade young researchers, on both the analytical and the
computational side, to further pursue the development of the sub-relaxation method.
I firmly believe that there is still much room for improvement. Although some new
directions may be very hard to examine (e.g., the analysis for the elasticity setting
and the implementation of point-wise stress constraints, to name just two important
ones), others may lie within reach. In particular, applying the sub-relaxation method,
appropriately adapted for numerical simulations, to realistic problems and situations
may 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


viii

Preface

equilibrium, weak convergence, etc. Moreover, it is desirable that the reader has
some previous experience with the basic techniques of the calculus of variations,
the role of convexity in weak lower semicontinuity, and how the failure of this
fundamental structural property may result in special oscillatory behavior. Again,
some simple discussions and examples may serve to fill this gap, and so provide the
reader 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. At
any rate, readers are expected to have sufficient analytical maturity to understand
issues not fully covered here in order to appreciate the ideas and techniques that are
the basis for the sub-relaxation approach to optimal design.
I would like to express my sincere gratitude to an anonymous reviewer whose

positive criticism led to various significant improvements in the presentation of
this text. Several colleagues from the editorial board of the Springer SEMA-SIMAI
Series 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 from
Springer played an important role, too, in leading this project to final completion.
Ciudad Real, Spain
May 2016

Pablo Pedregal


Contents

1

Motivation and Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 The Model Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2 Variations on the Same Theme . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3 Why It Is an Interesting Problem .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.4 Why It Is a Difficult Problem .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.5 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.6 Subrelaxation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7 What is Known .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7.1 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7.2 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7.3 Some Brief, Additional Information . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.9 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1
1
4
5
5
8
10
11
11
12
13
13
14
16

2 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1 The Strategy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2 Young Measures.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2.1 Some Practice with Young Measures . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Basic Differential Information: The Div-curl Lemma.. . . . . . . . . . . . . . . .
2.5 Subrelaxation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.6 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

23
23
27
28
30

31
32
34
35

3 Relaxation Through Moments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1 The Relaxation Revisited: Constraints .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 The Moment Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3 Characterization of Limit Pairs . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4 Laminates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5 Characterization of Limit Pairs II . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6 Final Form of the Relaxation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

37
37
39
41
43
49
52
ix


x

Contents

3.7 The Compliance Situation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.8 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

58
60
61

4 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 Descent Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3 Final Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.4 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63
64
67
69
69
70

5 Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 A Direct Approximation Scheme . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 Some Selected Simulations .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.1 Dependence on Initialization .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.2 Dependence on Volume Fraction . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.3 Dependence on Contrast. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 Some Additional Simulations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

71

71
74
74
74
75
76
77
78

6 Some Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 A Non-linear Cost Functional . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.1 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.2 Solution of the Mathematical Program . . . .. . . . . . . . . . . . . . . . . . . .
6.1.3 Subrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.4 Optimality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1.5 Some Numerical Simulations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 A Non-linear State Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.1 Reformulation and Young Measures
for the Non-linear Situation . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.3 Necessary Conditions.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.4 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 A General Heuristic Approximation Method .. . . . .. . . . . . . . . . . . . . . . . . . .
6.3.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3.2 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.4 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

79
80

81
83
85
88
90
92
93
95
95
98
102
104
107
110
111

7 Some Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1 Div-Curl Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2 Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3 Young Measures.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.1 The Existence Theorem . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.2 Some Results to Identify Young Measures . . . . . . . . . . . . . . . . . . . .
7.3.3 Second-Order Laminares .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

113
113
114
115
115
119

120


Contents

7.4 A Non-linear Elliptic Equation .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5 Covering Lemma .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.6 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

xi

122
123
129
129


Chapter 1

Motivation and Framework

1.1 The Model Problem
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 heat
conduction, but many other examples are as valid as this one. Suppose we have two
very different materials at our disposal: the first, with conductivity ˛1 D 1, is a
good and expensive conductor; the other is a cheap material, almost an insulator
with conductivity coefficient ˛0 D 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 t1 C t0 D 1. Typically, t1 < t0 given
that 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 sink 0 at 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 as
possible.
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
u.x; y/ dx dy
Q

that measures dissipated energy, among all those mixtures
divŒ.˛1 .x; y/ C ˛0 .1
u D 0 on

0;

complying with

.x; y///ru.x; y/ D 1 in Q;

.˛1 .x; y/ C ˛0 .1
.x; y///ru.x; y/ n D 0 on @Q n
Z
.x; y/ dx dy D 0:4:

0;

Q

© 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

uy = 0
Distributed
Heating

u=0

?

ux = 0

min: compliant

u =0

y

Fig. 1.2 Optimal distribution
of the two materials

uy = 0

u=0

u =0
x

uy = 0

Vector n stands for the outer unit normal to @Q. Figure 1.2 shows the optimal
design (mixture) of the two materials. Black indicates the good conductor. We notice
that there are specific spots where the two materials tend to be mixed in finer and
finer scales. By changing the various elements of this example (the cost functional,
the boundary conditions, the total amount of the two materials, the conductivity
constants, etc.), one can study many other situations. Understanding the structure of
these optimal mixtures of two materials is the main, global aim of this book.
Once we have a better perspective on the significance of these problems, we
explain, in as clear terms as possible, what our objective is with this booklet. The
following are the basic ingredients of our model problem. At this initial stage, we
do not pay much attention to precise assumptions.
1. The design domain ˝ R2 is a bounded, Lipschitz domain.
2. There are two conducting, homogeneous, isotropic materials, with conductivity
constants ˛1 > ˛0 > 0, at our disposal in amounts ti , respectively, so that t1 Ct0 D
j˝j.
3. A source term f W ˝ ! R.

4. A boundary datum u0 W @˝ ! R.


1.1 The Model Problem

3

There are many ways in which those two materials can be assembled to fill up the
design 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
˛.x/ D ˛1 .x/ C ˛0 .1

.x//

is the conductivity coefficient for that particular mixture determined by . In
addition, we must demand that
Z
.x/ dx D t1 ;
(1.1)
˝

to emphasize that we have to use exactly the amount of the first material that we
have. The potential corresponding to this mixture solves the diffusion equation
u D u0 on @˝:

divŒ˛.x/ru.x/ D f .x/ in ˝;

(1.2)

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 many
possibilities 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 and
relevant 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 whole
variety of possibilities. A typical one we will pay special attention to is
Z
I. / D

˝

f .x/u.x/ dx

(1.3)

which is identified as the compliance functional. Let us stick for the moment to this
one 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 the
integration 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:
Z
Minimize in

W

I. / D

˝

f .x/u.x/ dx

subject to
(continued)


4

1 Motivation and Framework

Z
˝

.x/ dx D t1 ;

divŒ˛.x/ru.x/ D f .x/ in ˝;

u D u0 on @˝:

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 this
minimum:
I. / Ä I. / for all admissible such designs :

1.2 Variations on the Same Theme
There are many ingredients of our basic model problem which can be changed to
produce new interesting situations. We list here a few.

•
•
•
•
•
•
•
•

•
•

The dimension N can be 3 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 environmental conditions (multi-load situations).
The materials do not have to be homogeneous and/or isotropic.
The cost functional can be quite general; in particular, it could depend also upon
the 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 speak
about shape optimization problems.
The state equation can be non-linear, so that the materials have a non-linear
behavior, or could be a dynamical equation so that the structure of the mixture is
allowed to change with time.
The equation of state can be a system rather than a single equation. The most
important situation here corresponds to linear elasticity.
There could also be further important constraints either in integral-form, or
perhaps more importantly, in a pointwise way.

Each one of these new situations requires new ideas. Though we will stick to our

model problem above to show how one can understand these situations and describe
the basic tools of our approach, any variation of the previous list will require new
techniques for a full analysis. We are not talking about more-or-less straightforward
generalizations.
This text aims at being an introduction to the subject of optimal design problems
treated with tools from non-convex variational analysis. Our goal is to use this model
problem to describe the basic concepts and techniques. But the treatment of more


1.4 Why It Is a Difficult Problem

5

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

1.3 Why It Is an Interesting Problem
From the practical point of view, the general meaning and interest of all those
problems discussed in the previous section is pretty clear: we would like to find
the best mixture of several given materials according to a certain criterium. One of
the paradigmatic situation in Mechanical Engineering is to find the most resistant,
non-collapsing structure under given environmental conditions. This is probably the
most difficult, and most important, problem in the list above. But each situation
corresponds to a certain design problem in Engineering: to find optimal mixtures of
given materials providing the best performance under given working conditions.
But even from a strict analytical viewpoint, optimal design problems are
fascinating. They pose to the applied analyst highly non-trivial problems that require
fine analytical tools. They challenge constantly known ideas, and one is forced to
innovate for new problems and situations. As a matter of fact, quite often, problems

become so difficult that they look rather unsolvable.

1.4 Why It Is a Difficult Problem
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 optimal
design 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
.z/ D jz

a1 j jz

a0 j;

ai D .i; i/; i D 0; 1;

and let us examine the problem
1

Minimize in u 2 H .Q/ W

Z
I.u/ D

.ru.x// dx
Q

under the further constraint
u.x/ D 0:5.x1 C x2 / on @Q;

x D .x1 ; x2 /:

Note that the boundary condition is u0 .x/ D .1=2/.a1 C a0 / x. Suppose Q D
.0; 1/2 is 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 D 1; 2. The question is then if
one can arrange a function u 2 H 1 .Q/, complying with the given boundary datum,
in such a way that ru.x/ 2 fa1 ; a0 g. This is clearly impossible in a neat way for a
single function u. But, indeed, it is possible to find a whole sequence fuj g of feasible
functions so that I.uj / ! 0. Set
Z
v.x/ D

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 a0 and a1 in “proportions” 1=2 1=2. However, v hardly takes on
the appropriate boundary values given by u0 around @˝. This can be achieved by a
bit of “surgery”. First, put

vj .x/ D

1
v. jx/;
j

rvj .x/ D rv. jx/;

so that the two values a0 , a1 are taken on in the same proportion over Q but in a
smaller and smaller scale as j ! 1. This fact translates into vj * u0 in H 1 .Q/, but
vj ! u0 uniformly all over Q. See a one-dimensional version of this fact in Fig. 1.3.
The full two-dimensional version of this construction will be retaken later in Chap. 3
(see Figs. 3.1 and 3.2).
Choose, next, a sequence of cutoff function k .x/ enjoying the properties
0Ä

k .x/

Ä 1;

jr k .x/j Ä

k .x/

D 0 on @Q;

jQ n f

k

D 1gj ! 0 as k ! 1;

M
for all x 2 Q and all k, for some constant M:
k

Finally, define
vj;k .x/ D
Fig. 1.3 One dimensional
version of a paradigmatic
oscillatory sequence for the
derivatives

k .x/vj .x/

C .1

k .x//u0 .x/:


1.4 Why It Is a Difficult Problem

7

This full family of functions is admissible as its members comply with the boundary
condition. Its gradient is
rvj;k .x/ D

k .x/rvj .x/

C .1

k .x//ru0 .x/

C .vj .x/

u0 .x//r k .x/:

If, thanks to the uniform convergence vj ! 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 sequence vk Á vj.k/;k
of feasible functions, whose gradient takes the two values a0 , a1 except in a small
and negligible boundary layer. This shows that indeed I.vk / ! 0, and such sequence
is a minimizing sequence for our variational problem. It is interesting to realize that
any 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 , and
relative 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 a
fundamental 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 H 1 .Q/, and yet I.u0 / >
lim I.vk / D 0.
The problem can be made more sophisticated if the two constant vectors a1 , a0
depend 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 , we
would 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 binary

variables in optimization problems leads to non-convexity and persistent oscillatory
behavior.
Our optimal design problems are difficult to analyze and to simulate. Conceptually, it is not hard to understand the reason from the very beginning, as we have
tried to convey with the previous discussion. The design (or optimization) variable
is , a binary variable taking two possible values f0; 1g. As such, the problem
cannot 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 the
preceding paragraphs.
As one thinks more about our model problem, and how it works, one realizes
that the set of characteristic functions has a complicated structure for they cannot
have any regularity, and we have an overwhelming amount of possibilities. The
non-convexity is typically associated with lack of optimal solutions: the infimum
is not attained, so that minimizing sequences of characteristic functions do not
converge 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 mixture
has 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 knew
that the optimal mixture around this point should have 30 % of one material and
70 % of the other, this still leaves open the door to determining the optimal geometry
itself 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 functional
will generally depend not only on the underlying optimal density of material, but
also on the geometry with which the materials are arranged microscopically. This
issue about optimal micro-geometries is what makes these problems so hard. Even
so, we are still interested in understanding how optimal (minimizing) sequences of
characteristic functions look like. Even better, we would like to understand how to
build some of those minimizing sequences for our problem, and how to encode those
optimal micro structural features in analytical tools of some sort. This is the main
objective, and main reason for all that follows.
The way in which these optimization problems are tackled is by means of
relaxed formulations. A relaxed form of a given optimal design problem should
take into account all those microstructural features that may exhibit sequences of
characteristic 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 of
the problem at hand. Intuitively, a relaxation is like an enlargement of the original
problem 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 the
relaxation 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 cannot
forget that the relaxation is a means towards the goal of understanding optimal
designs for the initial structural problem, so that once one succeeds in having a
true relaxation for one of these problems, then their optimal solutions have to be
interpreted in terms of the original problem.

1.5 General Procedure
Let us pause further on the general procedure to establish a relaxation of a complex
optimization problem in continuous media. Though the discussion may sound a bit
abstract at some point, readers may benefit from such a general discussion in order to

have 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 be
better understood and appreciated in subsequent chapters.
Assume we have an interesting optimization problem like our model problem
above for a linear conductivity equation. We are very much interested in finding


1.5 General Procedure

9

optimal solutions, but after resorting to standard literature on the topic we realize
that 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) an
optimal 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 of
optimal 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 the
objective in mind of being able to build very precisely at least one such minimizing
sequence. The process of going from the original problem to a new one, yet to be
formulated, in which feasible objects are identified with sequences of the original
problem 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 new
generalized variable may be related at least to a sequence of the feasible set for
the original problem.

2. The other important ingredients of every optimization problem should be
reconsidered for the new scenario. In particular, a generalized cost functional
ought to be specified, and constraints to be respected must be explicitly written.
Both the new objective functional, and the constraints have to be derived taken
into account very carefully the same ingredients for the original problem, in
the sense that the limit of costs for a sequence in the original problem must be
the new cost of the new feasible object determined by that sequence. This limit
process ensures that we are not changing the nature of the problem in its relaxed
formulation.
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 whole
point of a relaxation is to find an optimal generalized object through which we
could understand the structure of at least one minimizing sequence of the initial
problem, which is, after all, the objective of our analysis.
In a compact formal way, we can write an optimization problem like
Minimize in u 2 A W

I.u/:

Minimize in U 2 A W

I.U/;

Its relaxation would read
(1.4)

where every feasible u 2 A must be somehow identified within A , and any time
fuj g 7! U, then I.uj / ! I.U/. The relaxation link that ties together these two

10

1 Motivation and Framework

problems can be expressed in writing
inf I.u/ D min I.U/:

u2A

U2A

Note the use of min in the relaxed version of the problem to stress the fact that a
relaxed version of an optimization problem seeks to find a complete, equivalent
version of the problem with optimal solutions which should encode optimal
information in some way. If this is not the case, we may not have quite succeeded
in understanding minimizing sequences though we might be closer than before.

1.6 Subrelaxation
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 the
optimal 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 set A . This
is indeed the deepest issue we are facing, to the point that we cannot expect to be
able to find an efficient, complete, practical description of this relaxed set. All we
will aim at is to retain the most manageable of the fundamental constraints in A ,
and build with them a new set A , together with a new optimization problem
min I.U/:

(1.5)

U2A

Two main issues are:
1. The fact that we are using minimum instead of infimum in (1.5) indicates that
this new problem ought to admit optimal solutions, and so it is no longer in need
of relaxation.
2. Because in defining A we have ignored some constraints of A (but retain some
important ones too), the set A might be, in general, larger than A , and so
min I.U/ Ä min I.U/:
U2A

(1.6)

U2A

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 them
will be useless. We need to say something else about when a given subrelaxation
could 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


1.7 What is Known

11

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 procedure
will be:
1. Study the original problem, the one we are interested in, and decide which
constraints are going to be retained.
2. Define a subrelaxation by determining in clear terms A , 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 to A , the true
relaxation of the problem. Whenever this is so, the subrelaxation method will have
been successful. If not, we may have valuable information about our problem, but
may not fully understand optimal structures. On the other hand, it is also possible
that, in seeking a sub-relaxation, the structure of the design problem is such that
one ends up having a true relaxation. The sub-relaxation method is, above all, a
procedure of going about setting up a relaxation. If the process can be carried out to
the 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 out
for our model problem in conductivity, when generalized objects are identified with
Young measures associated with suitable sequences of a convenient reformulation of
the initial problem. A main goal is to understand the important piece of information
to be retained in setting a good, efficient subrelaxation.

1.7 What is Known
We will devote a short subsection to each of three important topics: analytical
viewpoint, engineering perspective, and additional information. Many important
references 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 this
kind of optimization problems, it has been very successfully used in optimal
design in continuous media as a main application. To relate the perspective of
homogenization 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 basic
knowledge 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 the
original problem is transformed through this passage. In this regard, we can talk
about a new feasible set A , 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 I.U/ D inf I.U/ D inf I.u/;
U2A

U2A

u2A

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 in A resulting

from optimality or simulation yield, quite often, a very good idea about the optimal
way in which the two materials are to be mixed. The term super-relaxation is used
here in a rather loose way to indicate that the feasible A is smaller than the true
admissible set for the relaxation, whereas A for the sub-relaxation is bigger. Note
that it is never true that
min I.U/ < inf I.U/;

U2A

U2A

inequality that might be suggested by the term in parallelism with sub-relaxation.
Our approach is intimately connected to homogenization. It is based on the same
fundamental facts, but used in a slightly different way. See the final section for
suitable and fundamental references on this field.

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 require
complicated new computations, or new insight into the problem, to the point that
a practical way to produce sensible quasi-optimal solutions for the original problem
turns 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 in
industry, these robust, direct methods of approximation are of great relevance. Most
definitely the ones that are used nowadays are well founded on solid ideas coming,
above all, from super-relaxations setup after homogenization techniques. Notice that
even 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, we
will be asked to provide a more-or-less macroscopic answer to the original problem

1.8 Structure of the Book

13

that could eventually be manufactured. This post-process is as important, from a
practical viewpoint, as the analysis itself.
We will mention just three main practical philosophies to deal with numerical
approximation. We refer to the abundant bibliography in the final section of the
chapter 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 force
extreme values f0; 1g, and recover in this way a true design. This is typically
accomplished through the introduction of an artificial material with rigidity
proportional to a power of the density.
2. Level-set methods. These have become more popular lately, and it amounts to the
clever application of the standard level-set method of Osher and Sethian (see the
final section for specific references) in the context of optimal design problems. It
suffers from some important disadvantages, but it apparently works very well in
some 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 to
add, to the compliance cost functional or whatever relevant functional we
are examining, a term that penalizes excessive perimeter for the characteristic
function determining the mixture. This perturbation introduces an additional
compactness 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 homogenization or sub-relaxations. See the final section of the chapter about bibliography.
On the one hand, there is the approach to optimal design based on the shape

derivative. This typically requires smoothness of shapes, and its applicability
is limited through this smoothness requirement. On the other, there are other
approaches that make use in various different ways of densities and measures.
Finally, it is interesting to mention some recent work to deal with optimal design
problems subjected to some randomness in the environmental conditions. This area
will most probably start attracting more and more the attention of researchers.

1.8 Structure of the Book
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 of
paramount 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