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OPTIMALITY OF THE TRANSLATION BOUNDS
FOR LINEAR CONDUCTING COMPOSITES IN
TWO AND THREE DIMENSIONS
by
Nathan Lee Albin
A dissertation submitted to the faculty ofThe University of Utah
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
The University of Utah
May 2006
Copyright c© Nathan Lee Albin 2006
All Rights Reserved
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
SUPERVISORY COMMITTEE APPROVAL
of a dissertation submitted by
Nathan Lee Albin
This dissertation has been read by each member of the following supervisory committeeand by majority vote has been found to be satisfactory.
Chair: Andrej Cherkaev
Aaron Fogelson
Graeme Milton
Klaus Schmitt
Andrejs Treibergs
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
FINAL READING APPROVAL
To the Graduate Council of the University of Utah:
I have read the dissertation of Nathan Lee Albin in its final formand have found that (1) its format, citations, and bibliographic style are consistent andacceptable; (2) its illustrative materials including figures, tables, and charts are in place;and (3) the final manuscript is satisfactory to the Supervisory Committee and is readyfor submission to The Graduate School.
Date Andrej CherkaevChair, Supervisory Committee
Approved for the Major Department
Aaron BertramChair/Dean
Approved for the Graduate Council
David S. ChapmanDean of The Graduate School
ABSTRACT
We consider the long-standing problem of characterizing the set of all possible
effective tensors of two- and three-dimensional composites made of several isotropic
linearly conducting phases in prescribed volume fractions. For more than two
phases, a complete characterization of this set, the G-closure, is not known. The
results presented in this dissertation follow from and expand upon the work of a
number of people including Hashin and Shtrikman; Lurie and Cherkaev; Tartar;
Murat and Tartar; Milton; Milton and Kohn; and Gibiansky and Sigmund.
The “translation bound,” a generalization of the Hashin-Shtrikman bound,
consists of several inequalities that the effective tensors of any composite must
satisfy. The bound depends only on the G-closure parameters — the conductivities
of the phases and the relative volume fractions. It is independent of the layout of
the phases in the composite. The inequalities are known to be optimal for certain
parameters. It is also known that there exist parameters for which they are not
optimal. However, a complete characterization of the parameter set for which the
bound is optimal is still missing.
We use a systematic approach based on “field optimality conditions”. Using this
approach, we consider the general, anisotropic bound and prove that it is optimal
for a much larger range of parameters than was previously known. We do this
for three or more phases and in two or three dimensions by constructing laminate
composites that saturate one of the inequality bounds.
We illustrate a number of applications of this approach, including finite-rank
iterated laminates, special “block structures” resembling those of Gibiansky and
Sigmund, and infinite-rank laminates produced via a differential scheme. Finally,
we discuss a numerical algorithm that gives an approximation of the G-closure in
regions where the translation bounds may not be optimal.
In loving memory of my mother,
Jana Dee Albin
CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTERS
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Why multimaterial composites? . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Why a systematic approach? . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Why field optimality conditions? . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Multimaterial conducting mixtures: problem and notations . . . . . . . 7
1.3.1 The G-closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2. SEQUENTIAL LAMINATES IN TWO DIMENSIONS . . . . . . . 10
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Known bounds on the G-closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Outer bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Inner bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2.1 Two-material optimal structures . . . . . . . . . . . . . . . . . . . . 172.2.2.1.1 The lamination formula. . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2.2 Multimaterial optimal structures . . . . . . . . . . . . . . . . . . . . 182.2.2.2.1 Milton’s structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2.2.2 Lurie-Cherkaev multicoated spheres. . . . . . . . . . . . . . 192.2.2.2.3 Optimal structures for the Wiener bounds. . . . . . . . . 202.2.2.2.4 Milton-Kohn Matrix laminates. . . . . . . . . . . . . . . . . . 202.2.2.2.5 Gibiansky-Sigmund isotropic structures. . . . . . . . . . . 21
2.3 New optimal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 A convenient change of variable . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 T-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.3 Coating preserves optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Example: the optimality of two-material matrix laminates . . . . 272.3.5 Coated T-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.6 T2-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.7 The set of optimal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.8 The problem with fixed volume fractions . . . . . . . . . . . . . . . . . . 34
2.3.9 Applicability: volume fractions . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.10 An inner bound of the G-closure . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Fields in optimal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.1 Local fields required by the translation bound . . . . . . . . . . . . . 392.4.2 Rank-one connection and the fields in the optimal structures . . 432.4.3 Constructing optimal laminates . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.4 The general structure revisited . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.4.1 The optimal T-structures . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.4.2 The coating principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.4.3 The optimal T2-structures . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.5 Four and more materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.5 A supplementary bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5.1 The bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Optimality versus attainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3. BLOCK STRUCTURES IN TWO AND THREE DIMENSIONS 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Background: the structures of Gibiansky and Sigmund . . . . . . . . . . . 62
3.2.1 The structures introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.2 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2.3 The structures revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 New optimal structures in two dimensions . . . . . . . . . . . . . . . . . . . . . 663.4 New optimal structures in three dimensions . . . . . . . . . . . . . . . . . . . 66
3.4.1 The lower translation bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.2 Optimal structures for the lower bound . . . . . . . . . . . . . . . . . . . 69
3.4.2.1 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4.2.2 Applicability: volume fractions . . . . . . . . . . . . . . . . . . . . . 75
3.4.3 The upper translation bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.4.4 Optimal structures for the upper bound . . . . . . . . . . . . . . . . . . 77
4. A DIFFERENTIAL SCHEME: INFINITE-RANK LAMINATES 78
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 The differential scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1 The equations for the volume fractions . . . . . . . . . . . . . . . . . . . 814.2.2 The equation for the effective tensor . . . . . . . . . . . . . . . . . . . . . 814.2.3 An optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 The modified differential scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.1 Matrix laminates in replaced strips . . . . . . . . . . . . . . . . . . . . . . 844.3.2 The optimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 The optimal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
vii
5. NUMERICALLY ESTIMATING THE G-CLOSURE . . . . . . . . 90
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2 A finite scheme for laminate structures . . . . . . . . . . . . . . . . . . . . . . . 905.3 The optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4.1 Numerical estimates of G-closures . . . . . . . . . . . . . . . . . . . . . . . 935.4.2 Numerical estimates of the isotropic bound . . . . . . . . . . . . . . . . 955.4.3 Fields in numerically optimized structures . . . . . . . . . . . . . . . . 95
6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
viii
LIST OF FIGURES
2.1 Bounds and optimal points: a summary of previous results. . . . . . . . . 12
2.2 Bounds and optimal points: a summary of new results. . . . . . . . . . . . . 12
2.3 Two-material structures optimal for the translation bound. . . . . . . . . . 17
2.4 Three-material structures optimal for the translation bound. . . . . . . . 19
2.5 Extremal structures of Gibiansky and Sigmund. . . . . . . . . . . . . . . . . . 22
2.6 The general three-material laminate which we optimize. . . . . . . . . . . . 24
2.7 A selection of new optimal structures. . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 The set L(KT ) of coated T-structures. . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Optimal points for the lower translation bound for m2
m2+m3= 1
60. . . . . . 34
2.10 Optimal points for the upper translation bound for m2
m1+m2= 1
41. . . . . . 35
2.11 Domains of applicability of Theorem 2.1 and Theorem 2.6 in terms ofvolume fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.12 Inner and outer bounds on the G-closure for two different sets ofadmissible parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.13 Fields in optimal laminate structures. . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.14 The generalized T-structure for four materials . . . . . . . . . . . . . . . . . . . 51
2.15 The structures from Theorem 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1 The two-dimensional anisotropic optimal block structure. . . . . . . . . . . 67
3.2 The optimal three-dimensional block structure. . . . . . . . . . . . . . . . . . . 73
4.1 The differential scheme for isotropic composites in three dimensions. . 80
5.1 A numerical estimation of the G-closure for M1 = 0.4, M2 = 0.01. . . . 94
5.2 A comparison of numerical results to known inner bounds. . . . . . . . . . 94
5.3 The effective conductivity of numerically optimized isotropic resistorsfor m2 = 0.5(1 − m1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 The effective conductivity of numerically optimized isotropic resistorsfor m2 = 0.1(1 − m1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.5 The local fields in the numerically optimized structures used in Fig-ure 5.4 in the case when the bounds are known to be optimal . . . . . . . 97
5.6 The local fields in the numerically optimized structures used in Fig-ure 5.4 in the case when the bounds are not known to be optimal . . . . 97
x
ACKNOWLEDGEMENTS
This research was funded by the National Science Foundation through a VIGRE
grant to the University of Utah Department of Mathematics. I thank the NSF and
the Mathematics Department for this support.
It is also a great pleasure to thank the many people who have helped make this
dissertation possible.
I am forever indebted to my advisor, Prof. Andrej Cherkaev. His enthusiasm
for the subject captured me on our first meeting and his phenomenal intuition has
inspired me ever since. Without his frequent direction and advice, without the hours
and hours he spent sharing his knowledge of mathematics with me, without his
encouragement and gentle prodding, this dissertation would not have been possible.
I am likewise indebted to my “Italian mentor,” Prof. Vincenzo Nesi. His passion
for mathematics and his profound generosity are an inspiration to me. His advice,
his knowledge, and his uncanny ability to ask the right questions initiated a major
breakthrough in my research and have had an immeasurable effect in shaping this
dissertation ever since.
I would also like to thank my committee members for their support: Prof.
Aaron Fogelson, for the stimulating classes on fluids and numerical methods; Prof.
Graeme Milton, for sharing his knowledge of composite materials and numerical
algorithms; Prof. Klaus Schmitt, for the encouragement and advice he gave so
many times during my graduate career; and Prof. Andrejs Treibergs, for guiding
me through my first experience with graduate-level differential equations. I could
not have asked for a better or more supportive committee.
I owe special thanks to Prof. Arthur Sagle, for teaching me undergraduate
mathematics, for pointing me toward graduate school and for much advice and
encouragement throughout.
I am also grateful for the help and encouragement of numerous friends and family
members. In particular my grandpa Gene for the use of a certain Chevy Lumina
during my years as a graduate student, and my grandpa Marvin for some very wise
advice when I needed it badly. I also thank my siblings, Heather, Seth, Joshua,
Hannah and Caleb, for frequent phone calls and many smiles. I am fortunate to
have such a loving and supportive family.
Special thanks go to my parents-in-law and good friends, Paul and Cynthia.
Their constant encouragement helped make this dissertation possible. To my
parents, Rick and Carol, I am eternally grateful. Their loving support and charity
have made my life better in countless ways.
Most importantly, I wish to thank my wife, Courtney, for her love, reassurance
and patience throughout this adventure.
xii
CHAPTER 1
INTRODUCTION
In this dissertation we introduce a systematic approach for the construction
of optimal multimaterial conducting composites. The approach is based on the
fields optimality conditions associated with the “translation bounds.” Using this
approach we produce new laminate composites with properties beyond what was
previously known possible.
1.1 Motivation
1.1.1 Why multimaterial composites?
One of the key problems in multimaterial design is the characterization of
composite materials — fine-scale mixtures of two or more different materials. Such
mixtures are useful in a variety of physical applications because they tend to
exhibit properties that none of the constituent materials posses individually. While
the theory of homogenization has enjoyed great success in describing the effective
behavior of a given mixture, many very basic questions remain unanswered.
One such question is the characterization of composites formed from several
linear, isotropic, conducting materials with prescribed volume fractions. In both
the cases of two- and three-dimensional composites, a complete characterization
of composites in terms of their effective (homogenized) tensors exists only in the
case of two materials. Much remains unknown for composites of three and more
materials.
In a sense, the two-material problem always yields intuitively expected results.
To make a composite that conducts very well, one wraps the best conductor around
the worst. To make a composite that resists very well, one wraps the best resistor
around the worst. For some intermediate composite, one mixes these two extremes.
2
Moreover, in the physical world it is unlikely that any composite can truly be
classified as a “two-material” composite. Due to impurities or gaps in the mixture,
one can typically identify at least three materials.
1.1.2 Why a systematic approach?
When a third material is added, the problem becomes much more complicated.
It is still not known how to construct the best conducting isotopic composite of
three conducting materials for all volume fractions of the materials. We show in
this dissertation that, when there are several materials in a composite, the materials
take on more complex roles than “best” and “worst.” A seemingly bad material
may not necessarily “hide” in the center of the composite. Instead it can take a
more active role: helping direct current into an intermediate material, for example.
The added complexity makes finding optimal structures much more difficult.
Thus, the need for a systematic approach becomes clear. Without it, we can
never be sure we are not “missing something.” The goal of this dissertation is to
expand the work toward a general systematic approach to classifying multimaterial
composites. The work presented here is an extension of previous results of Hashin
and Shtrikman (1962); Tartar (1979, 1985); Lurie and Cherkaev (1981, 1982);
Milton (1981); Milton and Kohn (1988); Gibiansky and Sigmund (2000). Moreover,
in some cases this approach poses new interesting and challenging mathematical
questions. We do not pursue this direction in this dissertation. However, our work
makes a more transparent mathematical connection with work in the field referred
to as “partial differential inclusions” (see for example Muller (1999); Dolzmann
(2003); Conti et al. (2005a,b)). While we focus on the problem of linear conductivity
here, we believe that the methods can be extended to a wide range of physical
phenomena.
1.1.3 Why field optimality conditions?
In this dissertation, we discuss several inequality bounds which must be satisfied
by the effective tensor of a multimaterial composite. Each of these bounds has
an associated set of conditions which must be satisfied by the local fields in any
3
structure attaining the bound. These field optimality conditions are integral to the
derivation of the bounds. Such conditions can be seen, for example, in the works of
Voigt (1928); Reuss (1929) for the elasticity version of the arithmetic and harmonic
mean bounds. In this case, the optimality conditions require respectively that the
stresses and strains are constant throughout a composite that attains the bound.
Similar optimality conditions arise in the construction of the Hashin-Shtrikman
bounds (Hashin and Shtrikman (1962)) and in their anisotropic generalization, the
translation bounds (Lurie and Cherkaev (1982); Tartar (1979, 1985)). Furthermore,
the translation bounds are a specific case of a general class of bounds. Each of
these bounds has associated field optimality conditions which are satisfied by a
composite which is optimal for the bound. We invite the reader to refer to Milton
(2002) Section 25.3 for a discussion of these conditions and their importance in the
general case and to Section 25.5 for a discussion of the conditions for the translation
bound in particular. Field optimality conditions for various bounds have motivated
a number of optimal composites for the multimaterial and related problems. We
refer the reader to Milton (1981, 1986); Avellaneda et al. (1988); Avellaneda and
Milton (1989); Vigdergauz (1989); Nesi and Milton (1991); Avellaneda et al. (1996)
for some examples. These results provide an excellent argument for methods based
on field analysis.
1.2 Overview
In this dissertation, we consider the long-standing problem of bounding the
effective properties of composites made from several isotropic, linearly conduct-
ing materials given in prescribed proportions. The primary contributions of this
dissertation to the field of materials mathematics are as follows.
1. We prove that the so-called “translation bounds” are optimal for a much
larger range of parameters than previously known. We do this for two- and
three-dimensional composites of three or more materials.
2. We introduce a systematic approach to forming multimaterial conducting
composites that realize the translation bound. The approach is driven by
4
the known “field optimality conditions” associated with the bounds. During
the construction we temporarily remove volume fraction constraints. We
exploit rank-one connection arguments to methodically generate new optimal
composites from old allowing the volume fractions to change as necessary
to keep optimality. Though the volume fractions of the final composite are
unconstrained in this process, the final composite is guaranteed to be optimal
for the bound (which depends on the volume fractions of the composite).
3. We introduce the “coating principle,” which allows us to generate a large
family of optimal composites from a given optimal composite by lamination.
This is a key ingredient in the construction of the optimal composites in this
dissertation.
4. Using (2), we methodically construct a large class of optimal hierarchical
laminates. The algorithm is straightforward and at each step produces an
optimal composite. The final volume fractions of the composite are left free
until the end of the procedure. These optimal laminates prove (1) in two
dimensions and for three or more materials.
5. Using (2), we analyze and then generalize the two-dimensional isotropic “block
structures” of Gibiansky and Sigmund. These structures are characterized by
the fact that the periodicity cell is partitioned into a number of rectangular
blocks and each of these blocks is filled with a composite (with several well-
separated infinitesimal length scales). After the “coating principle” is applied
the resulting anisotropic composites generate an equivalent set of effective
tensors to (4) in the three-material case.
6. Using (2), we construct the three-dimensional analogue of the “block struc-
tures” in (5). These structures along with the “coating principle” prove (1)
in the three-dimensional case for three materials.
7. We introduce a new controllable differential scheme for constructing com-
posites. The scheme differs from previous schemes in that it tracks fields in
5
each material. We apply (2) to find optimal controls. The scheme produces
a family of composites parameterized by a real parameter µ ≥ 0. With
increasing µ, the family traces the transformation via infinitesimal inclusions
from an initial “seed” material to a final composite. Each of these composites
is optimal for the translation bound (for the appropriate volume fractions).
The scheme is applicable in two and three dimensions and for three or more
materials. The optimal composites it produces are equivalent to the compos-
ites produced in (4)–(6) when applied to the corresponding dimension and
number of materials.
8. In the case of three materials in two dimensions, we introduce a new sup-
plementary bound on the anisotropy of a periodic composite with smooth
interfaces which attains the translation bound. A subfamily of the structures
introduced in (4)–(7) satisfy this bound as equality.
9. We introduce a finite scheme that is used to numerically produce an inner
bound on the G-closure of three-material structures in two dimensions. The
results of the scheme are consistent with the analytical results in this disser-
tation and give an idea of the shape of the G-closure when the bounds are
not optimal.
10. We apply the scheme in (9) to the problem of finding the best resisting
isotropic, three-material composite in two dimensions. The results give an
indication of what the field optimality conditions may be for a bound that
improves the translation bound in this problem.
In the rest of Chapter 1, we introduce the problem and notations of effective
(homogenized) tensors and the G-closure set. This set characterizes all possible
effective tensors of composites made from the constituent materials in prescribed
proportions.
In Chapter 2, we discuss the G-closure problem for conducting materials in two
dimensions. For this problem, we consider the concepts of inner and outer bounds
6
on the G-closure and the concept of optimal structures for an outer bound. The
“translation bound” is of particular interest in this paper and we give it special
treatment, reviewing what is known about it. In this chapter we introduce new
optimal structures for the translation bound and describe a general algorithm for
producing these structures. The set of effective tensors of the structures produced
by this algorithm is strictly larger than the corresponding set for all previously
known optimal structures.
In Chapter 3, we consider the three-dimensional version of the translation
bounds. We find new optimal structures with a “block” layout analogous to the
two-dimensional structures of Gibiansky and Sigmund (2000). Once again, the
new types of optimal structures produce a larger set of effective tensors than the
previously known optimal structures.
In Chapter 4, we use a modified differential scheme to find new optimal struc-
tures in a different setting. The differential scheme approach allows us to explore
a structure parameterized by a real number µ ≥ 0. The effective tensor of this
parameterized structure is given by a controllable differential equation which de-
scribes how the structure changes infinitesimally with µ. By choosing the controls
correctly, we use the scheme to produce optimal structures equivalent to those of
the previous two chapters.
Finally, in Chapter 5, we use a finite-dimensional analogue of the differential
scheme to numerically optimize a class of high-rank laminates. We consider the
case of three conducting materials in two dimensions. The generalization to more
materials and higher dimensions is obvious. This method gives a numerically
estimated inner bound on the G-closure. We compare the numerical results to
the outer bounds and known inner bounds. Furthermore, we consider the fields in
the numerically optimal isotropic structures and discuss their interesting properties
when the translation bounds are not realized.
7
1.3 Multimaterial conducting mixtures:problem and notations
In this section, we recall the problems of homogenization and the G-closure,
fixing the notation. References for this section can be found, for example, in Dal
Maso (1993); Braides and Defranceschi (1998); Cherkaev (2000); Allaire (2002);
Milton (2002).
Consider a d-dimensional, periodic, multimaterial structure for d ≥ 2. The unit
periodicity cell Ω = [0, 1]d is partitioned into N disjoint sets Ω1, . . . , ΩN such that
N⋃
i=1
Ωi = Ω.
The relative volume fractions of each part, mi = |Ωi| (where |Ωi| represents the
area of the set Ωi), satisfy
mi ≥ 0 ∀i = 1, . . . , N,N∑
i=1
mi = |Ω| = 1. (1.1)
The Ωi are assumed to be filled by materials with isotropic conductivity tensors
Ki = kiI for i = 1, . . . , N (1.2)
where I is the d-by-d identity matrix. We assume the conductivities are ordered so
that
0 < k1 < · · · < kN . (1.3)
The conductivity equations applied to the periodicity cell are written as
div(K(x)∇u(x)) = 0 in Ω,
∫
Ω
∇u(x) dx = e, x = (x1, . . . , xd) ∈ Rd (1.4)
where K : Ω → K1, . . . , KN is the conductivity tensor defined by
K(x) =
K1 if x ∈ Ω1,...
KN if x ∈ ΩN ,
(1.5)
K1, . . . , KN are given by (1.2), and where e is the prescribed average field induced
by distant external sources.
8
Assume that the periodicity cell with material layout defined by K(x) is subject
to the average field e. The energy stored in the material is defined as
W (K, e) = infu∈H1
#(Ω)+e·x
∫
Ω
∇u(x) · K(x)∇u(x) dx
where H1#(Ω) is the space of locally H1 functions on R
d which are Ω-periodic and
have zero mean. The infimum is taken over functions that can be split into an affine
part, e · x, plus a periodic oscillating part:
u(x) = e · x + osc(x),
∫
Ω
∇u(x) dx = e.
Notice that the affine part, e · x, is prescribed by the loading. The minimization is
taken over the variable, oscillating part, osc(x).
The structure defined by the partition Ωi is associated with its effective tensor
Keff, the conductivity tensor of homogeneous material that stores the same energy
as the mixture under the same homogeneous loading. That is,
e · Keff e = infu∈H1
#(Ω)+e·x
∫
Ω
∇u(x) · K(x)∇u(x) dx ∀e ∈ Rd.
In order to completely determine Keff, it suffices to consider the response of the
same structure to d orthogonal loadings
e = rjej, j = 1, . . . , d (1.6)
where r1, . . . , rd ∈ R are the magnitudes of the loadings and e1, . . . , ed is the
canonical basis of Rd. If we assume that Keff is oriented so that its eigenvectors
are e1, . . . , ed, then the response refers to the sum of the energies of the separate
loadings:d∑
j=1
W (K, rjej) =
d∑
j=1
λjr2j , ∀r1, . . . , rd ∈ R (1.7)
where λ1, . . . , λd are the eigenvalues of Keff. This functional can be conveniently
rewritten in terms of two-by-two matrices. We write E = diag(r1, . . . , rd). Given
9
any d potentials U = (u1, . . . , ud), we define the gradient matrix as the matrix
whose rows are the gradients of the uj:
DU = (DUij), DUij =∂ui
∂xj∀i, j = 1, . . . , d.
The sum of energies (1.7) can then be written as
W(K, E) = infU∈H1
#(Ω)d+Ex
∫
Ω
〈DU(x) K(x), DU(x)〉 dx (1.8)
where 〈·, ·〉 is the inner product defined on two-by-two matrices by
〈A, B〉 = tr(ABT ).
The effective tensor Keff is the unique (symmetric) tensor satisfying the relation
〈E Keff, E〉 = infU∈H1
#(Ω)d+Ex
∫
Ω
〈DU(x) K(x), DU(x)〉 dx ∀E ∈ Rd×d.
1.3.1 The G-closure
The G-closure was introduced by Lurie and Cherkaev (1981). We think of the
volume fractions, mi = |Ωi|, and the material properties ki as parameters of the
problem. If m = (m1, . . . , mN ) and k = (k1, . . . , kN) satisfy (1.1) and (1.3), then we
say m and k are admissible parameters for the G-closure problem. The closure of
the set of all possible Keff available for parameters m and k is called the G-closure.
Specifically, the G-closure G(m; k) is defined as
G = G(m; k) = Keff : K as in (1.5); |Ωi| = mi, i = 1, . . . , N.
Observe that the variable in this problem is the partition of Ω into the Ωi, since
K and thus Keff are defined through the Ωi. Each partition defines a material
structure through (1.5). The question is: what is the set of possible Keff which can
be obtained by some partition into Ωi subject to the volume constraints |Ωi| = mi?
Note that for a given K∗ ∈ G(m; k), there need not exist a partition of Ω into Ωi
and associated K as in (1.5) such that Keff = K∗. Indeed, one often shows that a
particular K∗ lies in the G-closure by finding a sequence of structures Kǫ such
that Kǫeff → K∗. We use the notation Keff versus K∗ to distinguish these concepts.
CHAPTER 2
SEQUENTIAL LAMINATES IN TWO
DIMENSIONS
2.1 Introduction
In this chapter, we discuss new results for the long-standing problem of the
optimality of the so-called “translation bounds”1 on the effective properties of
two-dimensional composites of several isotropic conducting materials in prescribed
proportions. The bounds are known not to be optimal for all choices of parameters
— the constituent conductivities and prescribed proportions. However, it is still
not clear what the conditions of optimality are. In this chapter, we show that
the bounds are optimal for a much larger range of parameters than was previously
known.
The primary thrust of the chapter is two-fold. First, we introduce new optimal
structures which prove that the translation bounds are optimal for anisotropic
composites outside the region of parameters previously known. Second, we in-
troduce a general algorithm for constructing these structures and understanding
their optimality by looking at sufficient conditions on the local electrical fields.
In Section 2.2, we discuss known bounds on the G-closure, G, prior to the
results of this chapter. The set G can be associated with a compact subset of R2
by mapping tensors in G to their eigenvalue pairs. The bounds are divided into
two categories: inner bounds and outer bounds. An outer bound is a subset of the
plane B ⊇ G. The outer bound we discuss in this chapter is the intersection of the
Wiener bound (also known as the harmonic and arithmetic mean bound) and the
1The bounds are also frequently referred to as the Hashin-Shtrikman bounds, as they generalizethe pioneering work of Hashin and Shtrikman (1962).
11
translation bound discussed in Section 2.2.1. An inner bound on the other hand, is
a subset of the plane L ⊆ G. Inner bounds are constructed by exhibiting a set of
structures whose effective tensors form the set L. In this chapter, we do not attempt
to improve the outer bound in is full generality (though we do prove a limit on the
optimality of the bound for a special class of structures in Section 2.5). Instead,
we produce a new inner bound which includes previously known inner bounds on
the G-closure.
In Section 2.3, we construct this improved inner bound, using a class of finite-
but high-rank laminate structures. In particular, we extend the known region of
optimality of the translation bound. For definiteness, we primarily discuss the
lower bound defined by (2.4) for three-material mixtures. The upper bound is
dealt with analogously. The generalization to four and more materials is discussed
in Section 2.4.5. Generalizations to other problems such as two-dimensional linear
elasticity and three-dimensional linear conductivity are possible. In particular, we
discuss the problem of three-dimensional conductivity in Chapter 3.
The result of the section is that the “translation bound” is now known to be
optimal in the anisotropic case for a wider range of volume fractions m1, m2, m3
(m1 + m2 + m3 = 1), where mi > 0 is the fixed relative amount of the ith material
in the three-material G-closure problem. We assume the materials have isotropic
conductivity tensors Ki = kiI for i = 1, 2, 3 where I is the identity matrix. We label
the materials so that 0 < k1 < k2 < k3. For the previous results to be applicable
(see Section 2.2.2.2 and Theorem 2.1 in particular), the condition m1 ≥ 2Θ(1−m2)
must be satisfied. The constant Θ is defined in (2.9) and depends only on the
material properties Ki. In this section, we introduce new anisotropic structures
which attain the translation bound (2.4) with an improved applicability condition:
2Θ(√
m2 − m2) < m1 < 1. (2.1)
(Note that√
m2 − m2 < 1 − m2 if m2 ∈ (0, 1).)
As a first illustration of the results described in Section 2.3, consider Fig-
ures 2.1–2.2, which represent the outer bounds and their optimality in the plane of
12
Figure 2.1. The bounds (2.2)-(2.5) with previously known optimal points (left).A magnification of the upper-left corner (right).
Figure 2.2. The bounds (2.2)-(2.5) including new optimal points (left). Amagnification of the upper-left corner (right).
eigenvalues of an effective tensor K∗. The figures represent the case
k1 = 1, k2 = 2, k3 = 5
and
m1 = 0.4, m2 = 0.01, m3 = 0.59.
We have chosen an example where m2 is small to illustrate the extremeness of the
improvement in this case. While the amount of improvement depends on the values
of the parameters, our results always improve on previously known results if the
volume fractions satisfy (2.1).
13
First consider Figure 2.1. The thin dashed lines represents the Wiener bounds
(2.2) and (2.3) while the thin curved lines represent the translation bounds (2.4)
and (2.5). The thick portion of the upper bound represents known optimal points
on the bound as proved in Milton and Kohn (1988) (see Theorem 2.1). The single
isotropic point marked on the lower bound is the point proved optimal by Gibiansky
and Sigmund (2000) (see Theorem 2.2). The thick portions of the Wiener bounds
represent the optimal structures described by Cherkaev and Gibiansky (1996) (see
Theorem 2.9). The right side of the figure shows a magnification of the top-left
corner in order to show that the Wiener bounds are tighter than the translation
bounds near this corner and that the Cherkaev-Gibiansky structures do not quite
reach the translation bounds. Figure 2.2 shows the status after the results of the
present chapter are added.
In Section 2.4, we illustrate an algorithm for constructing the structures of
the previous section by examining the local electrical fields in each constituent
material. This approach is quite different from the traditional methods, which
rely on physical intuition and/or numerical optimization to suggest structures for
a given external loading and fixed volume fractions. Instead, the approach offered
in this section leaves the external loading and volume fractions free and instead
uses standard techniques for solving differential inclusions to produce structures
consistent with the pointwise field requirements. In this sense, our approach is
more systematic than traditional methods. After a composite has been produced
in this fashion, a straightforward calculation finds the external loading and volume
fraction constraints for which the structure is optimal. The algorithm resembles
the methods used to solve a number of problems involving gradients supported on
certain sets of matrices. We refer the reader to Muller (1999); Dolzmann (2003);
Conti et al. (2005a,b) for some examples.
Section 2.5 addresses the optimality of the translation bounds from a different
direction. It is known that the bounds cannot be optimal for all values of the
volume fractions. However, the exact conditions for optimality are still unknown.
In this section, we consider the lower bound (2.4) for three materials. We derive a
14
supplementary inequality bound that a special type of structure (periodic structures
with smooth interfaces) must satisfy if it attains the bound. The supplementary
bound limits the degree of anisotropy of such a structure. In particular, we show
that no periodic structure with smooth interfaces can attain the bound if
m1 < 2Θ(√
m2 − m2)
(compare to (2.1)). While this does not answer the optimality question in general,
we include this section for two reasons. First, the T 2-structures — which we describe
in Section 2.3 — satisfy the bound as equality. Second, the bound is a useful
application of the field analysis discussed in Section 2.4. Indeed, the bound is
obtained exactly by finding an inequality that the fields in the special structures
must satisfy pointwise. The bound can also be applied to finite-rank laminates
using properties of correctors as described in Briane (1994). However, no proof is
currently known for general structures.
2.2 Known bounds on the G-closure
The set G(m; k) is known to be a closed and bounded subset of the two-by-
two symmetric matrices, R2×2sym. It is also rotationally invariant, so it suffices to
consider the projection of the set into the two-dimensional plane of eigenvalues. As
mentioned in the introduction, there is a two-fold strategy to characterize G(m; k):
inner and outer bounds. By an outer bound, we mean a set B = B(m; k) such
that B(m; k) ⊇ G(m; k). On the other hand, by an inner bound, we mean a set
L = L(m; k) such that L(m; k) ⊆ G(m; k). Of course, if the sets L and B can
be constructed so that L = B, the we have characterized the entire G-closure. In
this section, we discuss known inner and outer bounds on the G-closure. As we
mentioned in the introduction, this chapter does not improve the outer bounds in
general, but instead produces a larger inner bound for admissible parameters m
and k satisfying certain conditions.
15
2.2.1 Outer bounds
In this chapter, we consider the outer bound B defined in the eigenvalue plane by
a set of inequalities. In particular, any tensor K∗ ∈ G(m; k) satisfies the following
inequality bounds which depend only on the parameters m and k. The set B(m; k)
of two-by-two symmetric tensors that satisfy these inequalities forms an outer bound
on the G-closure.
1. The Wiener bounds are
λmin(K∗) ≥
(N∑
i=1
mi
ki
)−1
, (2.2)
λmax(K∗) ≤
N∑
i=1
miki. (2.3)
where λmin(K∗) and λmax(K
∗) are the minimum and maximum eigenvalues of
K∗ respectively.
2. The translation bounds are
tr K∗ − 2k1
det K∗ − k21
≤ 2
N∑
i=1
mi
ki + k1, (2.4)
tr K∗ − 2kN
det K∗ − k2N
≥ 2
N∑
i=1
mi
kN + ki. (2.5)
The translation bounds are not optimal for all values of the parameters m and k.
Intuitively, we see this from the fact that the formulas for the bounds still depend
on k1 (respectively kN) when m1 = 0 (respectively mN = 0). Indeed, for m1 or
mN near 0, there are better bounds (see Talbot et al. (1995); Nesi (1995)). Despite
several results in the area, it is still not known for which parameters the translation
bounds are optimal. In the rest of this chapter, we address this issue and extend
the known range of parameters for which they are.
16
2.2.2 Inner bounds
In this section we summarize the known inner bounds on the G-closure prior
to this chapter. These bounds are found by proving that certain tensors K∗ lie
in G(m; k). One way to construct such a tensor is to exhibit a partition of Ω
into subsets Ωi with |Ωi| = mi so that with K defined as in (1.5), one has Keff =
K∗. However, such “exact structures” are often difficult to construct. A simpler
method is to construct a sequence of partitions Ωǫiǫ>0 so that |Ωǫ
i | = mi and the
corresponding effective tensors converge: Kǫeff → K∗ as ǫ → 0. We know from
homogenization theory that K∗ is then in G(m; k). This technique is simpler than
that of finding an exact structure because one can use simple formulas for computing
K∗ without explicitly computing any of the Kǫeff. Laminates are a common example
of such structures and are the structures we exploit in this chapter.
We focus on optimal structures — that is, structures (such as laminates) which
produce an effective tensor K∗ ∈ ∂G(m; k). We call the sequence of partitions an
optimal structure and the associated K∗ an optimal point on the G-closure. This
notion is based on the fact that the G-closure can be completely characterized
by its boundary. Indeed, G(m; k) is closed, bounded and simply connected (see
for example Cherkaev (2000)). The trick to finding optimal structures is to use
both inner and outer bounds. If we have L ⊆ G(m; k) ⊆ B and if there exists
K∗ ∈ L∩∂B then K∗ is optimal. Notice that for B defined by (2.2)-(2.4), K∗ ∈ ∂Bif and only if at least one of the inequalities (2.2)-(2.4) is satisfied as equality. To
differentiate these cases, we will say that K∗ is optimal for whichever inequality is
satisfied as equality.
Furthermore, we will sometimes need to distinguish the case when there exists an
effective tensor Keff = K∗ ∈ ∂B from the case when there exists a limit of effective
tensors Kǫeff → K∗ ∈ ∂B. In the former case, we will say the bound B is attainable
at the point K∗. In the latter case, we will say the bound is optimal at the point
K∗. Clearly, B is optimal at any point for which it is attainable. However, there
are examples of bounds which are optimal but not attainable. Section 2.6 discusses
such an example. This distinction is essentially only important in Section 2.5.
17
2.2.2.1 Two-material optimal structures
The case of two-material optimal structures (N = 2) is completely known due
to the work of Hashin and Shtrikman (1962); Lurie and Cherkaev (1982); Tartar
(1979, 1985). The optimal isotropic structures were constructed by Hashin and
Shtrikman (1962). They used the coated spheres construction (see Figure 2.3a):
a circle filled with K2 surrounded by a concentric annulus filled with K1. When
this construction is placed into an infinite plane with the conductivity K∗, K1 ≤K∗ ≤ K2 and a constant field is applied at infinity, the volume fraction of the circle
in the structure can be chosen to keep the outside field homogeneous. Thus, the
effective conductivity of the structure is K∗. The periodicity cell Ω can then be filled
with infinitely many homeothetic coated circles (on infinitely many length scales).
Hashin and Shtrikman showed that the effective conductivity of such structure is
optimal for the isotropic version of (2.4) (which they discovered as well).
The optimal anisotropic two-material structures were found by Lurie and Cherkaev
(1982) together with the translation bound (2.4) for N = 2 (see Figure 2.3b). The
structures are iterated laminates.
Remark 2.1 The figures in this chapter should be thought of as schematic rep-
resentation of the actual structures. In reality, the laminate structures must have
well-separated scales for the results to apply. For example, in Figure 2.3b, one
should imagine that the vertical strips of K1 and K2 are interleaved at a scale ǫ2,
while the horizontal strips of this laminate with K1 are interleaved at a scale ǫ. The
effective tensor K∗ is obtained by sending ǫ → 0.
= K2
= K1
(a) (b)
Figure 2.3. Two-material structures optimal for the translation bound.
18
2.2.2.1.1 The lamination formula. Recall the lamination formula of two
materials, KA and KB, with volume fraction of KA equal to m and normal of
lamination given by n. The effective tensor of the laminate is given by
K∗ = L(KA, KB, n, m) = mKA + (1 − m)KB −N (2.6)
where
N = m(1 − m)(KB − KA)n[nT (mKB + (1 − m)KA)n]−1
nT (KB − KA)
The optimal two-material structures use an iteration of this formula (see Tar-
tar (1985)). First K1 and K2 are laminated in some proportion, then this new
“auxiliary” material is laminated with K1 in the direction orthogonal to the first
lamination. The effective properties can be computed via
K∗ = L(K1, L(K1, K2, n1, c1), n2, c2) (2.7)
where n1 and n2 are orthogonal and c1, c2 ∈ [0, 1]. The fractions c1 and c2 are related
by (1 − c1)(1 − c2) = m2. These structures and their dual version (interchanging
K1 with K2 and m1 with m2) describe all optimal structures (they are optimal
for either (2.4) or (2.5)). Both classes of structures degenerate into laminates (for
example, when c1 = 0, c2 = 1 − m2).
Remark 2.2 While the optimal structures described above are sequences whose
effective conductivities converge to a point on the bound, there do exist exact
geometries whose effective conductivities attain each point of the two-dimensional
translation bound. We refer the reader to Vigdergauz (1989, 1999); Grabovsky and
Kohn (1995); Astala and Nesi (2003) for more details.
2.2.2.2 Multimaterial optimal structures
We now recall several optimal multimaterial structures. These structures are
illustrated in Figure 2.4.
2.2.2.2.1 Milton’s structure. The first type of isotropic, multimaterial
(N ≥ 3) structures which were proved optimal for the translation bound were
19
described by Milton (1981) (see Figure 2.4a). His construction is as follows. The
amount m1 is split into two parts m′1 and m′′
1 so that the coated circles structures
from K1 and K2 (in the proportionsm′
1
m′
1+m2and m2
m′
1+m2, respectively) and from K1
and K3 (in the proportionsm′′
1
m′′
1+m3and m3
m′′
1+m3, respectively) have the same effective
conductivity. Obviously, any mixture of these structures has the conductivity of
each of them. This mixture is optimal for the bound (2.4) (K∗ is isotropic). All
mixtures of this form clearly satisfy
K1 ≤ K∗ ≤ K2
since the mixture of K1 and K2 must lie in this range. Such a construction is
possible as long as there is enough of material k1. Specifically, this construction
requires that
m1 ≥ 2Θ(1 − m2) (2.8)
where Θ is a constant defined below in (2.9). Similar structures are optimal for the
opposite bound (2.5) with K3 taking the role of the “coating,” and K1 and K2 the
inclusions.
2.2.2.2.2 Lurie-Cherkaev multicoated spheres. An inner bound for the
G-closure problem was found in Lurie and Cherkaev (1985) by posing an additional
assumption that the structure is of the type of multicoated circles (see Figure 2.4b)
and then solving the corresponding optimal control problem. Their construction
is geometrically different from Milton’s, but the effective conductivities of both
structures coincide in the range of parameters where (2.8) holds. The structures
are not optimal for the translation bound if (2.8) is violated.
= K3
= K2
= K1
(a) (b) (c)
Figure 2.4. Three-material structures optimal for the translation bound.
20
2.2.2.2.3 Optimal structures for the Wiener bounds. Cherkaev and
Gibiansky (1996) introduced a class of three-material anisotropic structures with
the property that they are optimal for one of the Wiener bounds (2.2) or (2.3), but
not the other.2 In Section 2.6, we use this construction to illustrate the difference
between the attainability and optimality of an outer bound.
2.2.2.2.4 Milton-Kohn Matrix laminates. The matrix laminates intro-
duced by Milton and Kohn (1988) combine the idea of Milton (1981) with the
two-material anisotropic structures (2.7) (see Figure 2.4c). The amount m1 of K1 is
divided into two parts, which are used to form two different mixtures of materials,
one of K1 and K2 and the other of K1 and K3. The mixtures are chosen to be
optimal for the corresponding two-material G-closure problem and have effective
tensors given by
K∗ = L(K1, L(K1, K2, n1, c1), n2, c2)
K∗′ = L(K1, L(K1, K3, n3, c3), n4, c4)
where
nT1 n2 = 0, nT
3 n4 = 0, c1, c2, c3, c4 ∈ [0, 1].
From the results of two-material structures, we know both K∗ and K∗′ are optimal
structures. Furthermore, if the parameters can be chosen so that K∗ = K∗′, then
the linearity of the bounds (2.4) with respect to the volume fractions allows us
to mix the two constructions together in any way we wish to obtain an opti-
mal three-material structures. These constructions also require that (2.8) hold.
These structures are more general than those previously discussed: they include
anisotropic structures as well as isotropic. The results can be summarized as follows.
Theorem 2.1 (Milton-Kohn) Let m = (m1, m2, m3) and k = (k1, k2, k3) be
admissible parameters for the G-closure problem such that
m1 ≥ 2Θ(1 − m2),
2The analogous two-material structures are geometrically impossible — that is, if one of thebounds is satisfied as equality for some K∗ then so is the other.
21
where
Θ =k1(k3 − k2)
(k2 + k1)(k3 − k1)≤ 1
2. (2.9)
Then there exists a family of structures with the given volume fractions which are
optimal for (2.4). The effective tensors of this family cover a connected subset of the
translation bound curve which includes the isotropic point. The most anisotropic
structure of this family has an effective tensor with eigenvalues given by
λ1 = νk1 + (1 − ν)k2, λ2 =k1k2
(1 − ν)k1 + νk2
where ν is defined to be
ν = m1 −2Θ
1 − 2Θm3.
Remark 2.3 Consult the thick, solid line in Figure 2.1 for an illustration of the
family structures for the other bound, (2.5). The family in the theorem covers a
segment of the bound near the isotropic point.
Remark 2.4 As in the two-material case, there also exist exact geometries which
attain certain points of the translation bound. In this case, however, it is known
that there exist G-closure parameters for which the bounds are not even optimal.
Once again, we refer the reader to Astala and Nesi (2003) for more details on these
exact structures.
After the paper by Milton and Kohn, the development slowed down. The
combination of two facts: the known nonoptimality of the translation bound as
m1 → 0 and the “natural” limit, K∗ ≤ k2I, of the known optimal structures
provocatively suggested that there were no other structures optimal for the bound.
However, in a surprising development 12 years later, Gibiansky and Sigmund (2000)
discovered new isotropic structures that are optimal for the translation bound for
smaller values of m1 than given in Theorem 2.1.
2.2.2.2.5 Gibiansky-Sigmund isotropic structures. Gibiansky and Sig-
mund (2000) announced a new construction that significantly increased the set
of optimal points of the translation bounds, (2.4) and (2.5). The paper focused
22
mainly on the problem of bulk moduli, but the results easily apply to the con-
ductivity problem as they describe in their Section 5.3. Their structures were
the surprising result of a numerical simulation. Using a “topology optimization”
algorithm developed earlier by Sigmund, the authors searched for optimal structures
by computer. We refer the reader to Figures 4–9 in their paper for examples of the
fascinating structures selected by the procedure. In particular, Figure 6 in their
paper illustrates a structure which violates the condition (2.8) of Theorem 2.1 but
which numerically appears to satisfy the translation bound. When the authors
attempted to replace the computer output with a similar, but simpler structure
for which the effective properties could be analytically computed, the simplified
structure was optimal for the translation bounds. In Figure 2.5, we illustrate a
special case of their structure.
Instead of iterated laminates or coated spheres, they used a construction (also
used earlier by Sigmund (2000)) which resembles the work of Marino and Spagnolo
(1969). In the latter paper, the authors consider a completely different issue.
However, they introduce (among other things), conductivities b(x) in d dimensions
which take the form
b(x) = b1(x1)b2(x2) · · · bd(xd) (2.10)
and study some special cases. Roughly speaking, Gibiansky and Sigmund consider
conductivities which have this form but on several different scales and only in an
approximate sense.
Reinterpreting their results slightly, we divide the cell of periodicity into four
rectangular subdomains. Two opposite squares are occupied by K2 and K3, and
= K3
= K2
= K1
Figure 2.5. Extremal structures of Gibiansky and Sigmund.
23
the remaining rectangles are filled with laminates from K1 and K3. The effec-
tive conductivity of the laminate depends on the volume fraction of materials in
it. This conductivity (or, equivalently, the volume fractions in the laminate) is
chosen in such a way that the conductivity equation (1.4) permits a separation
of the variables if the average fields are homogeneous. Because of this feature,
the solution is analytic, and so are the effective properties. Using Maple, the
authors then found that the structures are optimal for the translation bound (2.4).
The result is amazing because the structure is a mathematical approximation of a
numerical approximation of the optimization problem. The authors also described
more complicated structures that were optimal for larger values of m1 and which
coincided with the previously known structures at the point K∗ = K2. Their results
are summarized by the following theorem, which lowers the minimum value of m1
for which the bound (2.4) is optimal.
Theorem 2.2 (Gibiansky-Sigmund) Let m = (m1, m2, m3) and k = (k1, k2, k3)
be admissible parameters for the G-closure problem such that
2Θ(√
m2 − m2) ≤ m≤2Θ(1 − m2),
where Θ is given in (2.9). Then there exists an isotropic structure with the given
volume fractions and optimal for the bound (2.4).
Remark 2.5 The results of Gibiansky and Sigmund raise an interesting question.
If the volume fractions satisfy the inequalities
2Θ(√
m2 − m2) ≤ m1 < 2Θ(1 − m2),
is the isotropic point on the translation bound attainable?
In the next section, we introduce a class of anisotropic laminate structures
which contains structures with the same properties as the those of Gibiansky and
Sigmund. In Section 2.4, we analyze the fields in optimal structures and obtain a
clear picture of their features.
24
2.3 New optimal structures
In this section, we construct a family of optimal laminate structures. These
structures are all particular cases of the structure illustrated in Figure 2.6, an
orthogonal laminate of high rank with six design parameters and five well-separated
scales (see Remark 2.1). In this section, we choose the structural parameters so
that the structure satisfies the translation bound (2.4) as equality. We begin with
degenerate cases and work toward the structure in full generality. We delay until
Section 2.4 the discussion of why we should expect such a structure to be optimal.
The reader more interested in the reasons the structures are optimal than in the
structures themselves is invited to skip directly to Section 2.4.
2.3.1 A convenient change of variable
The material K1 and its volume fraction m1 play a special role in the bound
(2.4) and in the associated optimal structures. For this reason, it is convenient
to introduce (and fix) the relative fractions of the other two materials. Given
m1, m2, m3 > 0 with m1 + m2 + m3 = 1, define p ∈ (0, 1) by
p =m2
m2 + m3. (2.11)
Note that it follows that
1 − p =m3
m2 + m3
andm2
m3
=p
1 − p.
Using p-notation, the translation bound (2.4) for three material mixtures is rewrit-
ten as
= K3
= K2
= K1
Figure 2.6. The general three-material laminate which we optimize.
25
1
2· tr K∗ − 2k1
det K∗ − k21
≤ m1
2k1+ (1 − m1)
(p
k2 + k1+
1 − p
k3 + k1
)
.
We think of p ∈ (0, 1) as a parameter of the problem. With p fixed, we write the
requirement that a structure is optimal for (2.4) as
m1 =
12· tr K∗−2k1
det K∗−k21
−(
pk2+k1
+ 1−pk3+k1
)
12k1
−(
pk2+k1
+ 1−pk3+k1
) . (2.12)
2.3.2 T-structures
Figure 2.7a, b and c illustrate several special cases of the general structure
illustrated in Figure 2.6. The simplest optimal three-material structure in this
section, shown in Figure 2.7a, is the T-structure. It is assembled as a sequence of
laminates which depends upon two parameters. First, K1 and K3 are laminated
with normal in the x1-direction. Then, the resulting structure is laminated with K2
with the normal in the x2-direction. The effective properties are found by iterating
the formula (2.6)
KT = L
(
K2, L
(
K1, K3, n1,m1
m1 + m3
)
, n2, m2
)
where n1 = (1, 0)T and n2 = (0, 1)T .
Theorem 2.3 Let t ∈ (0, 1) and 0 < k1 < k2 < k3. Then there exist volume
fractions m1, m2, m3 > 0 so that p = t with p given by (2.11) and such that the
= K3
= K2
= K1
(a) (b) (c)
Figure 2.7. A selection of structures optimal for (2.4). The T-structure (a), theT-structure with one layer of “coating”(b), and the T 2-structure (c)
26
T-structure with these volume fractions is optimal for the translation bound (2.4).
The values of the volume fractions are
m1 =Θ(1 − p)
1 − pΘ, m2 =
p(1 − Θ)
1 − pΘ, m3 =
(1 − p)(1 − Θ)
1 − pΘ
where Θ is defined in (2.9). The eigenvalues λ1, λ2 of the optimal T-structure are
computed to be
λ1 =(1 − Θ)pk2 + (1 − p)k3β
(1 − Θ)p + (1 − p)
λ2 =(1 − Θ)pk2 + (1 − p)k2
(1 − Θ)p + (1 − p)β
where
β =k2 + k1
k3 + k1
. (2.13)
It may seem surprising that we have found that there is always an optimal T-
structure for any p. This happens because we consider structures with fixed relative
volume fractions of K2 and K3 but with arbitrary fraction of K1.
2.3.3 Coating preserves optimality
In order to describe the variety of the optimal structures, we make the following
observation.
Theorem 2.4 (The Coating Principle) If a structure with effective conductiv-
ity K∗ is optimal for the translation bound (2.4), then all structures obtained by
laminating it with material K1 are also optimal for (2.4), though with different
volume fractions. The laminating can be iterated several times with various normals
so that the original structure is “coated” by K1.
Proof : It is enough to apply the lamination formula (2.6) to K∗ (with volume
fractions m1, m2 and m3) and K1, specifying the normal of lamination, n, and the
volume fraction, c, of K1. This lamination produces a new material with effective
tensor
K∗′ = L(K∗, K1, n, c).
27
Substituting K∗′ into the bound (2.4) along with the updated volume fractions
m′1 = 1 − c + cm1, m′
2 = cm2, m′3 = cm3
and using the fact that K∗ satisfies the bound as equality, one can verify that K∗′
also satisfies the bound as equality, with the new volume fractions.
The theorem states that laminating optimal structures for the translation bound
with K1 preserves the optimality, though it changes the volume fractions. This
observation allows us to restrict ourselves to the description of only extremal struc-
tures that attain the bound (2.4). By extremal structures, we mean structures that
contain the minimal amount of K1.
2.3.4 Example: the optimality oftwo-material matrix laminates
As a particular example, the coating principle can be used to prove the opti-
mality of the two-material second-rank laminates discussed in Section 2.2.2.1. This
case can be considered a special case of the three-material problem with
p =m2
m2 + m3= 1
(that is, m3 = 0). Begin with Ω filled with pure K2 so that m1 = 0 and m2 = 1.
These volume fractions clearly satisfy the requirement on the ratio p. Furthermore,
it is easy to check that (2.12) holds for this structure since K∗ = K2 so
12· tr K∗−2k1
det K∗−k21
−(
pk2+k1
+ 1−pk3+k1
)
12k1
−(
pk2+k1
+ 1−pk3+k1
) =
12· 2k2−2k1
k22−k2
1
−(
1k2+k1
+ 0k3+k1
)
12k1
−(
1k2+k1
+ 0k3+k1
)
=1
k2+k1− 1
k2+k1
12k1
− 1k2+k1
= 0 = m1.
Therefore, the block of pure K2 is optimal for (2.4). Applying the coating principle
once, we then find that any lamination of K2 and K1 is also optimal. Finally,
applying the principle a second time, we find that the structures illustrated in
Figure 2.3b are optimal.
The coating principle also plays an important role in the analysis of multima-
terial mixtures. Notice that the coating changes the volume fractions, mi, but
28
it preserves the value of p. Since coating increases the value of m1, the principle
allows to look for the optimal structures with the lowest value of m1. Every optimal
structure generates a set of optimal coated structures. The set L(K∗) of optimal
structures obtained by coating is a domain in the plane of eigenvalues of the effective
tensor K∗. The two boundary components of this set correspond to the laminates
of the anisotropic, generating material K∗ and K1 with normal parallel to one of
the eigenvectors of K∗. To derive the equations for the boundaries, let K∗ be given
with volume fractions described by m1 and p. Let λ1 and λ2 be the eigenvalues of
K∗.
The boundary of L(K∗) is found from the lamination formula, (2.6), by lami-
nating K∗ with K1 with volume fractions c and 1 − c, respectively:
K∗′ = L(K∗, K1, n, c)
where n is parallel to an eigenvector of K∗. The laminate contains K1 in the fraction
m′1 = 1 − c + cm1, and the value of p is preserved, p′ = p. If n is chosen parallel
to the eigenvector associated with λ1, then the eigenvalues λ′1 and λ′
2 of K∗′ are
parameterized by c as
B1(K∗) =
((c
λ1+
1 − c
k1
)−1
, cλ2 + (1 − c)k1
)
: c ∈ [0, 1]
, (2.14)
while if n is parallel to the other eigenvector, the new eigenvalues are parameterized
by c as
B2(K∗) =
(
cλ1 + (1 − c)k1,
(c
λ2
+1 − c
k1
)−1)
: c ∈ [0, 1]
. (2.15)
We define Λ(K∗) ⊂ R2 to be the closed set bounded by B1(K
∗) ∪ B2(K∗) defined
in (2.14) and (2.15) where λ1 and λ2 are the eigenvalues of K∗. We define
L(K∗) = K∗′ ∈ R2×2sym : K∗′ has eigenvalues in Λ(K∗) (2.16)
where R2×2sym denotes the two-by-two symmetric matrices. The following is an im-
mediate corollary of Theorem 2.4.
29
Corollary 2.1 Let t ∈ (0, 1). Let k = (k1, k2, k3) and m = (m1, m2, m3) be
admissible parameters of the G-closure problem such that m2 = t(m2+m3). Suppose
K∗ ∈ G(m; k). Then, for any effective tensor K∗′ ∈ L(K∗), there exist admissible
volume fractions m′ = (m′1, m
′2, m
′3) such that m′
2 = t(m′2 + m′
3) and such that
K∗′ ∈ G(m′; k).
2.3.5 Coated T-structures
From the optimal T-structure, we obtain a set of optimal structures by coating
with K1. This set, L(KT ), is shaded in the eigenvalue plane in Figure 2.8 for the
parameters
k1 = 1, k2 = 2, k3 = 5, p =1
60.
It is convenient to represent an anisotropic material by two symmetric points
(λ1, λ2) and (λ2, λ1) in the plane of eigenvalues to avoid ordering them. Particularly,
the optimal T-structure is represented by two points, both labeled KT . The domain
L(KT ) of optimal structures as defined in (2.16) is the union of two lens-shaped
regions in the plane. The boundaries of this set are the laminate curves. Recall
that rather than fixing volume fractions, we fix the value p, which in turn fixes the
ratio of m2 to m3. The figure also includes some dotted curves of constant volume
fraction. Those closer to K1 indicate larger values of m1 than those farther away.
Any point where one of these curves intersects the region L(KT ) is an optimal point
for the translation bound (2.4) with the volume fractions given through m1 and p.
Remark 2.6 Observe the change in topology of the intersection of constant vol-
ume fraction curves with L(KT ). For large values of m1, the intersection is a
connected portion of the curve. As m1 decreases, the intersection suddenly becomes
disconnected (specifically, when the constant volume fraction curve passes through
the point K2). Letting m1 continue to decrease, one sees that the two connected
components of the intersection shrink to points and then vanish (when the curve
passes through the points KT ). For m1 lower than this, the intersection is empty.
30
λ1
λ2
b
b
b
b
b
b
K1
K3
KT
KT
KGS
K2
m1 = 0.8
m1 = 0.2469
m1 = 0.05488
Figure 2.8. The shaded region denotes the set L(KT ) of optimal structures formedby coating the optimal T-structure.
The two outer curves from KT to K1 represent the most anisotropic structures
of the class of coated T-structures. The pair of eigenvalues λ1, λ2 along these curves
are parameterized by
λ1 =
(
ν1
k1
+ (1 − ν)(1 − Θ)p + (1 − p)
(1 − Θ)pk2 + (1 − p)k3β
)−1
(2.17)
λ2 = νk1 + (1 − ν)(1 − Θ)pk2 + (1 − p)k2
(1 − Θ)p + (1 − p)β(2.18)
where the constants Θ and β are defined in (2.9) and (2.13). We can parameterize
the inner curves analogously. The parameter ν ∈ [0, 1] along the outer curve
controls the amount of material K1 added to the T-structure. The volume fractions
of the final structure depending on ν and p are given by
m1 = ν + (1 − ν)Θ(1 − p)
1 − pΘ, m2 = (1 − ν)
p(1 − Θ)
1 − pΘ, m3 = (1 − ν)
(1 − p)(1 − Θ)
1 − pΘ.
The more isotropic curves cross the line of isotropy at exactly λ1 = λ2 = k2 and do
so when
31
ν =(1 − p)(k3β − k2)
(1 − Θ)p(k2 − k1) + (1 − p)(k3β − k1).
At this point, we find that m1 = 2Θ(1 − m2) (see Theorem 2.1!) The region of
intersection (shaded darker) of the two lenses was proved to be optimal by Milton
and Kohn (1988) using the laminated structures described in Section 2.2.2.2. The
remaining portion of the region of L(KT ) represents the first new optimal structures
of this chapter. We will improve this region later. Observe that the only optimal
isotropic structures found in L(KT ) were already known to be optimal. However,
this construction has introduced a relatively large set of new optimal anisotropic
structures. Furthermore, this construction proves the optimality of the bound (2.4)
in a region of anisotropic points for a smaller value of m1 than was previously known
possible: m1 ≥ Θ(1−m2) rather than m1 ≥ 2Θ(1−m2). The coated T-structures
are a generalization of the Milton-Kohn structures in the sense that the latter always
have effective tensors in L(KT ). For reference, we have also indicated by the dashed
line from K2 to KGS the optimal isotropic structures introduced by Gibiansky and
Sigmund. This line intersects the set L(KT ) only at K2.
2.3.6 T2-structures
We now enlarge the class of optimal structures with a set of structures which
connects the points KT and KGS in Figure 2.8. We laminate the T-structure with a
laminate from K1 and K3 in the orthogonal direction as illustrated in Figure 2.7c.
The effective tensors of such T 2-structures are found from the iterative procedure
KT2 = L(KT , K ′13, n1, ω2),
KT = L(K2, K13, n2, ω1),
K ′13 = L (K1, K3, n2, ν
′) ,
K13 = L (K1, K3, n1, ν) .
(2.19)
Here, ν and ω1 are the parameters of the original T-structure. ν ′ is the relative
fraction of K1 in the additional K1-K3 laminate, and ω2 is the relative amount of
the T-structure compared to the additional laminate in the final T 2-structure.
The properties depend on four structural parameters: ν, ν ′, ω1, ω2 that all vary
in [0, 1] and are subject to the constraint that fixes p. (Recall that m1 is treated as a
32
variable.) Observe that the T 2-structures are a generalization of the T-structures.
(If ω2 = 0 in the equations above, then KT2 = KT .) Moreover, we show that
they describe a curve of anisotropic structures between the T-structures and the
isotropic structures of Gibiansky and Sigmund.
Theorem 2.5 Let t ∈ (0, 1) and let admissible parameters m = (m1, m2, m3) and
k = (k1, k2, k3) be given such that
p =m2
m2 + m3
= t
and
2Θ−p + 2p2Θ − pΘ +
√
p2Θ2 − 2p2Θ + p
(1 − 2pΘ)2 ≤ m1 ≤Θ(1 − p)
1 − pΘ. (2.20)
Then there exists a T 2-structure with the given volume fractions and optimal for
the bound (2.4). These structures vary between the anisotropic T-structure and the
isotropic point of Gibiansky and Sigmund (2000). The optimal volume fractions
(see (2.19)) in the structure satisfy
ν = Θ, ν ′ = ω1Θ
where Θ is defined in (2.9) and ω1, ω2 satisfy
ω1 + ω2 =1
Θ(m1 + 2Θm2), ω1ω2 = m2. (2.21)
The effective tensors have eigenvalues λ1, λ2, written in terms of ω1 and ω2 as
λ1 =ω1k2(k3 + k1) + (1 − ω1)k3(k2 + k1)
ω2(k3 + k1) + (1 − ω2)(k2 + k1), (2.22)
λ2 =ω2k2(k3 + k1) + (1 − ω2)k3(k2 + k1)
ω1(k3 + k1) + (1 − ω1)(k2 + k1). (2.23)
Notice that the relation between the effective properties of optimal mixtures is
symmetric to the interchanging of ω1 with ω2 in spite of the nonsymmetric iterative
procedure.
33
When m1 equals its upper bound in (2.20), then one of ω1 or ω2 must be equal to
1. Thus, the T 2-structure degenerate into the T-structure for this volume fraction.
On the other hand, when m1 equals its lower bound, then we find
ω1 = ω2 =1
2−√
1
4− m1
2Θ
and we obtain an isotropic structure whose volume fractions satisfy the equality
m1 = 2Θ(√
m2 − m2) (compare to Theorem 2.2!). The set of T 2-structures is a
generalization of both the T-structures and the Gibiansky-Sigmund structure with
minimal amount of m1 (see Figure 2.9).
Remark 2.7 We show in Section 2.4 how to find these structures by analysis
of the fields. In particular, we show how the parameters ω1, ω2, ν and ν ′ are
naturally determined and explain the remarkable properties of the local fields in
optimal structures that lead us to the construction.
2.3.7 The set of optimal structures
Applying the coating principle to the extremal T 2-structures, we obtain a variety
of optimal structures because each T 2-structure can be coated, increasing the
amount m1 but keeping relative fraction p. The union of all the L(KT2) (see
(2.16)), for each KT2 with the given value of p, forms a set of structures optimal
for the translation bound (2.4). This set is illustrated in the eigenvalue plane in
Figure 2.9 for parameters
k1 = 1, k2 = 2, k3 = 5, p =m2
m2 + m3
=1
60.
The set of optimal structures is bounded by the solid boundary, which is the union
of coated T-structures (the curves between K1 and KT ) and the T 2-structures
(the curve passing through KGS). The closed region bounded by the dashed
lines represent the previously known optimal structures of Milton and Kohn, and
Gibiansky and Sigmund.
Curves of constant volume fraction are indicated by the dotted lines. In partic-
ular the curve passing through KT represents the case m1 = Θ(1 − m2) while the
34
λ1
λ2
b
b
b
b
b
b
K1
K2
K3
KGSKT
KT
m1 = 0.8
m1 = 0.4
m1 = 0.2469
m1 = 0.15
m1 = 0.05488
Figure 2.9. Optimal points for the lower translation bound for m2
m2+m3= 1
60.
The curve between KT and KGS corresponds to the optimal T 2-structures. Thecurve between KT and K1 corresponds to the coated T-structures illustrated inFigure 2.7b. The dashed line from K2 to KGS corresponds to the structures ofGibiansky and Sigmund. The dashed curve passing through K1 and K2 forms theboundary for the structures of Milton and Kohn.
curve passing through KGS represents the case m1 = 2Θ(√
m2 − m2). Recalling
Remark 2.6, we see that the topology of the intersection of the curves m1 = const
and the new region of optimal structures remains connected for all volume frac-
tions satisfying m1 > 2Θ(√
m2 − m2). As m1 decreases below this amount, the
intersection shrinks to a single point and then vanishes for all smaller m1.
2.3.8 The problem with fixed volume fractions
Until now, we have considered a problem of fixed p = m2/(m2 + m3). The clas-
sical G-closure problem, however, asks that we fix the volume fractions. Obtaining
this information from our results is quite straightforward because the pair (m1, p)
uniquely determines all volume fractions.
As an example, we find optimal structures in G(m; k) for the parameters
k1 = 1, k2 = 2, k3 = 5
and
35
m1 = 0.4, m2 = 0.01, m3 = 0.59
as in Figures 2.1–2.2. To accomplish this, we need only examine Figure 2.9 (notice
that m2/(m2 + m3) = 1/60) and the corresponding figure for the upper bound
with m2/(m1 + m2) = 1/41, Figure 2.10. The optimal points of the lower bound
marked by the thick portion of the curve in Figure 2.2 are the intersection of the
curve of constant m1 = 0.4 with the optimal region shown in Figure 2.9. The dot
in Figure 2.1 marks the point where this curve intersects the dashed Gibiansky-
Sigmund line. Similarly, the optimal points marked by the thick curve on the
upper bound in Figure 2.2 are where the line of constant volume fraction m3 = 0.59
intersects the set of optimal points in Figure 2.10. The thick portion of the upper
bound in Figure 2.1 marks the intersection with the Milton-Kohn region.
Look again at Figure 2.9. As long as m1 ≥ 2Θ(√
m2 − m2) (For the given
parameters, m1 ≥ 0.05488.), the intersection of the curve m1 = const and the
region of attainable points is a connected subset of the curve which includes the
isotropic point. Thus, the intersection is uniquely defined by the most anisotropic
λ1
λ2b
b
b
b b
b
K1
K2
K3
KGS
KTKT
m3 = 0.9
m3 = 0.59
m3 = 0.1749
m3 = 0.04616
Figure 2.10. Optimal points for the upper translation bound for m2
m1+m2= 1
41. The
curve between KT and KGS corresponds to the optimal T 2-structures. The curvebetween KT and K3 corresponds to coated T-structures. The dashed line from K2
to KGS corresponds to the structures of Gibiansky and Sigmund. The dashed curvepassing through K3 and K2 forms the boundary for the structures of Milton andKohn.
36
point of this subset. For m1 ≥ Θ(1−m2) (For the given parameters, m1 ≥ 0.2469.)
this most anisotropic point is a coated T-structure. For m1 ≤ Θ(1 − m2), it is a
T 2-structure. We summarize this in the following theorem.
Theorem 2.6 Let the volume fractions m1, m2, m3 > 0, and material properties
0 < k1 < k2 < k3 be given admissible parameters. Define p as in (2.11) and Θ as
in (2.9). Then the following hold.
(i) If m1 > Θ(1 − m2), then (2.4) is optimal. There exists a set of opti-
mal points on the bound which includes the isotropic point and whose most
anisotropic member is that given by the coated T-structure with eigenvalues
(2.17) and (2.18) where ν is chosen to satisfy the volume fraction constraints:
ν =1 − pΘ
1 − Θ
(
m1 −Θ(1 − p)
1 − pΘ
)
.
(ii) If 2Θ(√
m2−m2) ≤ m1 ≤ Θ(1−m2), then (2.4) is optimal. There exists a
set of optimal points on the bound which includes the isotropic point and whose
most anisotropic member is that given by the optimal T 2-structure satisfying
the volume fraction constraints. In particular, the parameters ω1 and ω2 for
this most anisotropic structure can be found by solving simultaneously the
equations (2.21).
2.3.9 Applicability: volume fractions
In Figure 2.11, we illustrate the difference between the applicability of Theo-
rem 2.1 and Theorem 2.6 in the case of isotropic structures. In both figures we take
k1 = 1, k2 = 2, k3 = 5. By the definition of admissible volume fractions, we know
that in the m1m3-plane (note that m2 = 1 − m1 − m3), the volume fractions are
constrained to the region bounded by the axes m1 = 0 and m3 = 0 and by the line
m1 + m3 = 1.
Consider Theorem 2.1 for the lower bound (2.4) which we illustrate on the left
of Figure 2.11. It implies that there is an isotropic structure that attains the bound
if m1 ≥ 2Θ(1 − m2) where Θ is defined in (2.9). We have indicated the line m1 =
37
m3
m1
m1 + m3 = 1
m3
m1
m1 + m3 = 1
Figure 2.11. Domain of applicability of Theorem 2.1 (left) and Theorem 2.6(right) in terms of volume fractions. (Here we consider the isotropic case fork1 = 1, k2 = 2, k3 = 5.) The volume fractions are physically constrained to theregion bounded by the axes m1 = 0 and m3 = 0 and the line m1 + m3 = 1. Thethick solid curves depict m1 = 2Θ(1−m2) (left) and m1 = 2Θ(
√m2 −m2) (right).
The thick dashed curves are similar but for the upper bound (2.5). The shadedregions indicate the region of parameters for which each theorem provides structureswhich attain both bounds.
2Θ(1−m2) by the thick solid curve passing from the origin (m2 = 1, m1 = m3 = 0)
to the line m1 + m3 = 1. (The point of intersection is m2 = 0, m1 = 2Θ, m3 =
1 − 2Θ.) For any admissible volume fractions which lie to the right of this curve,
Theorem 2.1 implies that there exists an isotropic structure that is optimal for the
lower bound (2.4). The thick dashed line gives similar information for the upper
bound (2.5). For any admissible volume fractions lying above this curve, there exists
an isotropic structure that satisfies the upper bound as equality. In particular, the
shaded region between the two curves indicates volume fractions for which both
bounds are optimal.
On the right of Figure 2.11, similar information is depicted for Theorem 2.6.
In particular, the thick solid curve represents m1 = 2Θ(√
m2 − m2). (Note the
endpoints m2 = 1, m1 = m3 = 0 and m2 = m1 = 0, m3 = 1.) The dashed
line represents the similar curve for the upper bound. The figure illustrates quite
clearly how powerful Theorem 2.6 is for small m2 (that is, for points near the line
m1 + m3 = 1). In this case, both bounds can be proved optimal for a range of
volume fractions including those where m1 or m3 are very close to zero.
38
We should remark that the corresponding figure for Theorem 2.2 coincides with
the right side of Figure 2.11. This is because Theorem 2.2 and Theorem 2.6 are iden-
tical for isotropic structures. However, Theorem 2.2 does not apply to anisotropic
structures, while Theorem 2.6 certainly does. Similar figures to Figure 2.11 can
be produced for varying degrees of anisotropy. In particular, if the ratio of the
eigenvalues of K∗ is nearly one, Figure 2.11 changes only slightly. Theorem 2.6 can
be considered a generalization of Theorem 2.2 to anisotropic structures.
2.3.10 An inner bound of the G-closure
From Theorem 2.6 and the results of Cherkaev and Gibiansky (1996), we can
produce a naive inner bound of the G-closure by lamination. Consider Figure 2.12a.
Here we have plotted an inner and outer bound of the G-closure for the parameters
k1 = 1, k2 = 2, k3 = 5,
m1 = 0.104, m2 = 0.5, m3 = 0.396.
The union of the unshaded and shaded region is an outer bound for the G-closure
formed from the Wiener and translation bounds. In this case, m1 is very close
to 2Θ(√
m2 − m2) so that case (ii) of Theorem 2.6 applies. We mark the most
anisotropic effective tensor given by this theorem by T . We also mark the least
anisotropic effective tensor on the Wiener bound (Cherkaev and Gibiansky (1996))
by H . By laminating the structures with effective tensors T and H , we form a
family of structures which lie on the uppermost curve connecting the two points.
Since this curve necessarily lies in the G-closure for the given parameters, we find
that the unshaded region of the figure depicts an inner bound on the G-closure.
The shaded region shows what is still not known for these parameters. The bounds
do not determine whether these points belong to the G-closure or not. We remark
that there is a similar region of “unknown points” near the point marked U in the
Figure. However, for these parameters and at the scale of the figure, the region is
impossible to see.
Figure 2.12b is similar. We use the same conductivity parameters, but less
“extreme” values of the volume fractions:
39
b
b
b
b
b
H
T
U
T ′
H ′
λ1
λ2
b
b
bb
H
T
T ′
H ′
λ1
λ2
(a) (b)
Figure 2.12. Inner and outer bounds on the G-closure for two different sets ofadmissible parameters. The bounds do not determine whether the points lying inthe shaded regions are a part of the G-closure or not.
m1 = 0.25, m2 = 0.5, m3 = 0.25.
There are very small (compared to the area of the G-closure) regions of unknown
points near the “corners” where the bounds intersect. That is, T and H are very
close to each other. In this case, both the inner and outer bounds are very close to
the G-closure itself.
If we allow m1 to decrease far enough, eventually this construction will not work.
When m1 = 2Θ(√
m2 − m2), the points T and T ′ of Figure 2.12a coincide. For
m1 < 2Θ(√
m2 −m2), we know of no structures which attain the lower translation
bound. In this case, a simple bound can be obtained by laminating the points H
and H ′ (that is, create a polycrystal of the anisotropic material H). In all cases,
a better inner bound could be obtained by more carefully mixing known optimal
points. This is a difficult problem, however, and we will not discuss it further.
2.4 Fields in optimal structures
2.4.1 Local fields required by the translation bound
For the reader’s convenience, we derive the optimality conditions for the transla-
tion bound below (see Lurie and Cherkaev (1982); Tartar (1979, 1985)). We sketch
the ideas of the derivation, focusing on the conditions on the fields inside each
material of an optimal structure.
40
The lower bound uses the quasi-affineness of the determinant function
∫
Ω
det DU dx = det E, ∀E ∈ R2×2, ∀U ∈ H1
#(Ω)2 + Ex
which we can verify for smooth functions by writing
det DU = div
(
u1
(∂u2
∂y,−∂u2
∂x
))
and applying the Divergence Theorem. The general result follows by approxima-
tion. The construction of the lower bound is as follows. We begin by adding and
subtracting the constant 2t det E for some t ∈ R.
W(K, E) = infU∈H1
#(Ω)2+Ex
∫
Ω
(〈DU K, DU〉 + 2t det DU) dx − 2t detE.
Next, we relax the differential constraint on the field DU by replacing the set
H1#(Ω)2 + Ex with the set of F ∈ L2(Ω; R2×2) such that
∫
ΩF dx = E. Thus, we
have
W(K, E) ≥ infF
∫
Ω
(〈F K, F 〉 + 2t det F ) dx − 2t det E (2.24)
such that
∫
Ω
F dx = E. (2.25)
This equation gives a family of bounds on W(K, E) parameterized by t. Here we
consider the case t = ±k1. We will choose t = k1 for the rest of this section.
The other case is analogous. It is not guaranteed that a minimizer, F , of the
right-hand side of (2.24) will be a gradient, but if it is (or if it can be approximated
in the appropriate sense by a sequence of gradients) then the translation bound
is optimal. In the rest of the section, we analyze the conditions of a minimizer,
F (with t = k1). Then we construct gradient fields DU ∈ H1#(Ω)2 + Ex which
approximate the minimizer F , proving that the bound is optimal in the cases we
discussed in Section 2.3.
To simplify the calculation, we use a rotation-invariant decomposition of the
quadratic forms on two-by-two matrices defined by Q1(F ) = |F |2 = 〈F, F 〉 and
41
Q2(F ) = 2 detF (see Astala and Miettinen (1998)). Namely, we take the zero sets
H+ = (Q1 − Q2)−1(0) and H− = (Q1 + Q2)
−1(0). Then
R2×2 = H+ ⊕H−
where
H+ =
(a b
−b a
)
: a, b ∈ R
, H− =
(a bb −a
)
: a, b ∈ R
. (2.26)
In particular, we can write
F = F+ + F−, F+ ∈ H+, F− ∈ H−
where
F+ =1
2(F + cof F ), F− =
1
2(F − cof F ). (2.27)
Here cof is the linear operator on matrices which returns the cofactor matrix.
cof
(f11 f12
f21 f22
)
=
(f22 −f21
−f12 f11
)
.
It is easy to verify that
|F+|2 + |F−|2 = |F |2, |F+|2 − |F−|2 = 2 det F.
Differentiating the integrand in (2.24) with respect to F we find the conditions
for a minimizer:
F (x)K(x) + k1 cof F (x) = A, a.e. in Ω (2.28)
or, equivalently,
F+(x)(K(x) + K1) + F−(x)(K(x) − K1) = A, a.e. in Ω
where A is a constant matrix of Lagrange multipliers enforcing (2.25). In particular,
we find that in Ω1 (where K ≡ K1) we have
2k1F+ = A a.e. in Ω1 (2.29)
which implies that A− = 0 in Ω. For Ωi with i = 2, . . . , N , we then have
(ki + k1)F+ = A, F− = 0 a.e. in Ωi. (2.30)
42
Using (2.29) and (2.30) we solve for A by noting that
E =
∫
Ω
F dx = AN∑
i=1
mi
ki + k1
+
∫
Ω1
F− dx
so that
A =
(N∑
i=1
mi
ki + k1
)−1
E+,
∫
Ω1
F− dx = E−.
Thus, we have the following theorem.
Theorem 2.7 A vector field F ∈ L2(Ω; R2×2) is a minimizer of the right-hand-side
of (2.24) if and only if
(i) F+ = 1ki+k1
(N∑
j=1
mj
kj+k1
)−1
E+ a.e. in Ωi for i = 1, . . . , N .
(ii)∫
Ω1F− dx = E−.
(iii) F− = 0 a.e. in Ωi for i = 2, . . . , N .
In particular, we can explicitly write the lower bound as
W(K, E) ≥(
N∑
i=1
mi
ki + k1
)−1
|E+|2 − 2k1 det E, ∀E ∈ R2×2. (2.31)
The traditional form of the translation bound (2.4) is obtained by choosing E to
make the bound above as tight as possible, and thus eliminating the dependence of
the bound on the fields. However, we are more interested in Theorem 2.7 since it
tells us exactly what the fields in each material of an optimal structure are.
Specifically, notice that the value of F is constant with F− ≡ 0 in all but the
first material: Ω2, . . . , ΩN . Furthermore, F+ is fixed in the remaining material
Ω1. The only “freedom” we have in our choice of F is the values of F− in Ω1
which are arbitrary as long as we satisfy the constraint on the average, (ii). If the
translation bound is optimal, then the corresponding structures contain pointwise
fields which are (nearly) constant in Ω2, . . . , ΩN and constrained to belong to the
two-dimensional manifold
1
2k1
(N∑
j=1
mj
kj + k1
)−1
E+ + H−
in Ω1. Here H− is defined in (2.26).
43
Remark 2.8 The problem of finding optimal structures has thus been reduced to a
differential inclusion problem: find U with a given average such that
DU ∈ K ⊂ R2×2 a.e. in Ω.
2.4.2 Rank-one connection and thefields in the optimal structures
As in Section 2.3, we deal with the class of orthogonal laminates — that is, the
class of laminates with mutually orthogonal normals. The effective properties of
this structures are found by iterating the equation (2.6) assuming that the normal
is equal either to n1 = (1, 0) or to n2 = (0, 1). It is enough to consider only diagonal
average fields (1.6) (see for example Milton (2002)) so that
E+ = αI, and A =
(N∑
i=1
mi
ki + k1
)−1
αI (2.32)
for some α ∈ R.
Remark 2.9 We choose laminates because the fields can be taken to be constant in
each layer, making them easy to work with. In actuality, there can be fluctuations,
especially near the boundaries. But with well-separated scales (see Briane (1994))
tending toward length zero, the fields may be assumed constant for the purpose of
computing the energy W(K, E). Technically speaking, what we will continue to call
“fields” are called “correctors” in the language of homogenization. The most elegant
mathematical formulation is given through the use of the so-called gradient Young
measures. The interested reader is referred to Muller (1999) for an introduction to
the subject in the context of materials science.
The fields of neighboring layers in a laminate structure are rank-one connected.
In other words, two neighboring layers with respective fields FA and FB must satisfy
det(FA − FB) = 0. (2.33)
On the other hand, the fields in each material in an optimal structure are given by
Theorem 2.7 with F = DU . We now analyze these conditions, showing how they
guide us in creating optimal structures.
44
The magnitude of the average field does not affect the effective properties,
therefore it is convenient to rescale the fields in the theorem so that
α = k1
N∑
i=1
mi
ki + k1, E+ = αI, A+ = k1I (2.34)
in (2.32). Thus, in an optimal structure, the field DU satisfies
DU+ =k1
ki + k1I a.e. in Ωi for i = 1, . . . , N.
Since trF = tr F+, we can rephrase the conditions of Theorem 2.7 as follows.
(P1) DU = DUT and tr DU = 1 a.e. in Ω1.
(P2)∫
Ω1DU− dx = E−.
(P3) DU = k1
ki+k1I a.e. in Ωi for i = 2, . . . , N .
Remark 2.10 We refer the reader to Grabovsky (1996) for a detailed derivation
of similar conditions for composites of two generally anisotropic linear elastic ma-
terials. The isotropic case is covered in (3.26) and (3.27) of his paper which the
reader may wish to compare to (P1)–(P3) above with N = 2. The same paper
also makes the connection between optimal composites in linear conductivity and in
special cases of linear elasticity. We further remark that conditions (P1)–(P3) are
a special case of (25.31) in Milton (2002).
One immediately observes that the fields in all materials except K1 are not
in rank-one connection with each other and therefore are incompatible, because
(P1)–(P3) and (2.33) are contradictory: if FA = k1
k2+k1I and FB = k1
k3+k1I then
det(FA − FB) 6= 0. In particular, no optimal laminate can contain a layer of K2
laminated with K3.
In a sense, K1 must be used as a glue between layers to ensure compatibility.
The volume fraction of K1 therefore cannot be too small, which indicates that the
translation bound (2.4) cannot be optimal for laminates if m1 is smaller than a
critical value.
45
2.4.3 Constructing optimal laminates
We now describe an algorithm for constructing laminates which are optimal for
(2.4). We always assume that the fields in the layers of the laminate satisfy (P1)–
(P3) and we find ways to join the materials by rank-one connection into laminate
structures. Since we require that (P1)–(P3) hold at all times in the process, the
final structures are necessarily optimal.
We leave the average field, E, and the volume fractions free until the end of
the process, when they are computed from the construction. In other words, by
following the procedure described in this section, we are guaranteed to produce
optimal structures, but we do not know the parameters of the optimization problem
(the average field and volume fractions) until the structure is complete. However,
at the end of the process, we have an algorithm to find all G-closure parameters.
2.4.4 The general structure revisited
Consider again the general structure illustrated in Figure 2.6. Notice that there
are six design parameters (the volume fractions in the laminate layers). Since the
structure is an orthogonal laminate and since we assume the average field E is
diagonal, we have that the local fields are also diagonal. Notice that material K2
appears in lamination only once, material K3 twice, and material K1 four times.
Thus, we have 6 design variables and 14 field variables. To make the structure
optimal, we must fill in the fields in the pure material components satisfying a
number of conditions. Let us count these conditions.
First, we have rank-one connections between the fields and the currents in each
layer of the laminate. This gives two condition per layer, since the tangential
component of the field and the normal component of the current must be continuous
across the interfaces. There are six laminate interfaces, so we have 12 continuity
conditions.
For optimality, we also need to satisfy (P1)–(P3). These give two conditions for
each layer of K2 and K3 and one condition for each layer of K1. Thus, (P1)–(P3)
impose 10 more conditions.
46
By our count, we have 20 free variables and 22 conditions they must satisfy.
In fact, we observe that the general structure is a coated T 2-structure. The
T 2-structure has two free design parameters (ω1 and ω2). The coating introduces
two more parameters, so we actually have four degrees of freedom in spite of the
seemingly over-determined system. As we will show in the following sections, some
of the constraints are satisfied “for free” if the others are satisfied. In particular,
when we choose parameters to satisfy the rank-one connections and (P1)–(P3)
for the fields in the general structure, the currents are automatically rank-one
connected. This removes six constraints from the list above and we are left with 20
variables and 16 constraints, providing the four degrees of freedom we observe.
This is not as surprising as it may at first seem. Indeed, assume we have a
partition of Ω into Ωi with associated conductivity tensor K defined through (1.5).
Furthermore, assume that we find U ∈ H1#(Ω)2 + Ex that satisfies the optimality
condition (2.28) with F = DU . Then, by taking the divergence of both sides
of (2.28) and using the fact that div(cof DU) = 0, we find that U satisfies the
PDE div(K DUT ) = 0 automatically. The analogous statement for laminates is
that it is enough to check the jump conditions only of the piecewise constant field
approximating DU . We obtain “for free” the corresponding conditions for the
piecewise constant field approximating DU K.
2.4.4.1 The optimal T-structures
The construction of optimal laminates preserves the fields in the layers according
to (P1)–(P3). By restricting ourselves to orthogonal laminates and diagonal average
fields, we guarantee that the fields in each layer will be diagonal, which allows us
to simplify notation. We represent the diagonal two-by-two matrices as
M(α, β) =
(α 00 β
)
and associate M(α, β) with the point (α, β) in the plane. Observe that M(α1, β1)
and M(α2, β2) are rank-one connected if and only if (α2 −α1)(β2 − β1) = 0. In the
plane, this means that (α1, β1) and (α2, β2) lie on the same horizontal or vertical
line.
47
We illustrate the construction of optimal structures in Figure 2.13a and b. We
begin with the T-structure. The discussion is accompanied by Figure 2.13a. In this
figure, the points E2 and E3 represent the fixed fields given by (P3). The line l
represents the line of constant trace given by (P1). We wish to construct a laminate
which has its internal fields lying on the set E2 ∪ E3 ∪ l.
First, we look for a rank-one connection between the materials K1 and K3. The
admissible fields for K1 lie on the line l while the admissible field for K3 is the point
E3. Let us laminate in the x1-direction, which means the field in K1 must lie on
the intersection of l and the horizontal line through the fixed field E3. The optimal
fields E1 and E3 are
E1 = M
(k3
k3 + k1
,k1
k3 + k1
)
, E3 = M
(k1
k3 + k1
,k1
k3 + k1
)
,
which ensures that
(E1 − E3) ·(
01
)
=k1
k3 + k1− k1
k3 + k1= 0.
As we mentioned in Section 2.4.4, the rank-one connection condition on the currents
is automatic. So far, we have only ensured the condition on the fields. However,
note that
(k1E1 − k3E3) ·(
10
)
=k1k3
k3 + k1
− k1k3
k3 + k1
= 0
so the currents are also rank-one connected.
x
yb
b
bbc
r
E3
E2
E1
l
E13
ET
x
yb
b
b
b
bc rsr r
E3
E2
E′
1
E′′
1 l
E′
13ETET2
ECT
(a) (b)
Figure 2.13. Fields in optimal laminate structures.
48
The average field upon laminating K1 with K3 is
E13 = νE1 + (1 − ν)E3 = M
(νk3 + (1 − ν)k1
k3 + k1,
k1
k3 + k1
)
while the average current is
J13 = νk1E1 + (1 − ν)k3E3 = M
(k1k3
k3 + k1,νk2
1 + (1 − ν)k1k3
k3 + k1
)
where ν is the relative fraction of K1 to K3. In Figure 2.13a, we indicate the point
E1. The dashed line connecting this point to E3 represents the path of the point
E13 as ν varies between 0 and 1. All of these fields are available to us in an optimal
laminate of K1 and K3 by appropriate choice of ν.
We choose ν so that we can laminate this new material with K2 in the x2-
direction (Refer to Figure 2.7a.), again satisfying (P1)–(P3). Thus, we need to
adjust ν so that E13 and E2 lie on the same vertical line in the plane, where
E2 = M(
k1
k2+k1, k1
k2+k1
)
. Solving for ν, we find
ν =k1(k3 − k2)
(k2 + k1)(k3 − k1)= Θ.
Note that Θ is defined in (2.9) and is a parameter of the optimal structures in the
previous section.
Once again, we can verify that the currents in the laminate layers are compatible.
When ν = Θ,
J13 = M
(k1k3
k3 + k1,
k1k2
k2 + k1
)
which is rank-one connected to the current k2E2. Similar calculations for all
other structures discussed in this chapter show that the condition on currents is
automatically satisfied each time we satisfy the condition on the fields, so for the
rest of this section, we will keep track only of the fields and not the currents in the
construction.
Setting ν = Θ brings the point E13 to the vertical line passing through E2
and we are free to laminate with as much K2 as we please. The average field in
the T-structure can lie anywhere on the dashed line connecting E13 and E2. The
only condition we have is the ratio of m1 to m3, leaving m2 free. In this way, the
49
average field, E, depends on m2. We then find the volume fractions of the optimal
T-structure depending on the amount K2 parameterized by ω1:
m1 = Θ(1 − ω1), m2 = ω1, m3 = (1 − Θ)(1 − ω1). (2.35)
To find the effective properties, we need to find the average field,
ET = m1E1 + m2E2 + m3E3
=k1
k2 + k1
M
(
1,ω1(k3 + k1) + (1 − ω1)(k2 + k1)
k3 + k1
)(2.36)
and the average current,
JT = m1k1E1 + m2k2E2 + m3k3E3
=k1
k2 + k1
M
(ω1k2(k3 + k1) + (1 − ω1)k3(k2 + k1)
k3 + k1
, k2
)
.
Then we have that K∗ = E−1T JT . Substituting and simplifying, we get the eigen-
values of K∗:
λ1 =m2k2(k3 + k1) + (1 − m2)k3(k2 + k1)
k3 + k1
(2.37)
λ2 =k2(k3 + k1)
m2(k3 + k1) + (1 − m2)(k2 + k1). (2.38)
These expressions coincide with (2.22) and (2.23) for ω1 = m2, ω2 = 1. We can
now prove Theorem 2.3 by simple algebra using m2 = p(m2 + m3), (2.35), (2.37)
and (2.38).
2.4.4.2 The coating principle
It is now easy to see why the coating principle (Theorem 2.4) is true. Consider
Figure 2.13b. Starting from the optimal T-structure, ET , we look for compatible
fields for lamination. Notice that since the field in K1 can lie anywhere on the
line l, there are always two compatible fields: the intersection of the horizontal or
vertical line through ET and the line l. For this illustration, we have chosen the
50
point E ′1 ∈ l which lies on the same horizontal line as ET . Specifically, from (2.36)
we find
E ′1 =
1
(k3 + k1)(k2 + k1)M(ω1k2(k3 + k1) + (1 − ω1)k3(k2 + k1),
ω1(k3 + k1) + (1 − ω1)(k2 + k1)).
By varying the amount of K1 which is added in this layer, we can obtain a new
optimal structure ECT which lies anywhere on the dashed line connecting ET and
E ′1. (Refer to Figure 2.7b.) Of course, this operation can be iterated. For example,
we could then laminate in the x2-direction with K1 by choosing the point on l which
intersects the vertical line through KCT and so on. In this way, we obtain a whole
family of optimal structures from a single optimal structure. Again, the coating
will change the average field, E, and volume fractions. However, the construction
must give an optimal structure for some values of these parameters which we can
compute.
2.4.4.3 The optimal T2-structures
Finally, we illustrate how to obtain the optimal T 2-structures. Begin with the
optimal T-structure, indicated by ET in Figure 2.13b. Notice that we can laminate
E3 with the point E ′′1 = M
(k1
k3+k1, k3
k3+k1
)
. By adjusting the volume fraction ν ′ of
K1 to K3 in this laminate, we can move the resulting average field
E ′13 = ν ′E ′
1 + (1 − ν ′)E3 = M
(k1
k3 + k1,ν ′k3 + (1 − ν ′)k1
k3 + k1
)
to any point on the dashed line connecting E3 and E ′′1 . In particular, if we choose
the volume fraction so that E ′13 is rank-one connected to ET — that is if we choose
ν ′ so thatν ′k3 + (1 − ν ′)k1
k3 + k1=
ω1(k3 + k1) + (1 − ω1)(k2 + k1)
k3 + k1
(see (2.36)) — then we can laminate the E ′13 and ET structures to obtain a T 2-
structure (see Figure 2.7c). It is easy to check that the correct value of ν ′ is ν ′ = ω1Θ
51
with Θ defined in (2.9). We are then free to laminate ET with E ′13 in any fraction
we wish to obtain an optimal T 2-structure:
ET2 = ω2ET + (1 − ω2)E′13.
By computing the volume fractions, average field, and average current as we did
with the T-structure, one can verify all the statements of Theorem 2.5. We can
now apply the coating principle as many times as we wish to obtain a family of
optimal structures as described in Section 2.3.
2.4.5 Four and more materials
The generalization to N ≥ 4 is straightforward. In fact, Theorem 2.7 was proved
in the general case. The method for constructing optimal laminates is exactly the
same: we are given a set K of N−1 points and a two-dimensional plane in R2×2 and
we construct laminate structures whose internal fields are lie in K. As an example,
Figure 2.14a shows the rank-one construction of a four-material generalization of the
T-structure and Figure 2.14b shows the associated laminate structure. Laminates
of K1 and K3 (and K1 and K4) are brought into rank-one connection with material
K2 respecting (P1)–(P3), allowing for second-rank lamination. The improvements
of the applicability conditions for the method discussed in this chapter over previous
results become more pronounced with larger N .
x
y b
b
b
b
b
bc
bc
r
E4
E3
E2
l
= K4
= K3
= K2
= K1
(a) (b)
Figure 2.14. The rank-one construction of a generalized T-structure (marked bythe square) (a) and the generalized T-structure (b)
52
2.5 A supplementary bound
In Section 2.4, we showed that a structure is optimal for the bound (2.4)
provided certain conditions (P1)–(P3) hold in an approximate sense. Furthermore,
we constructed optimal structures exactly by satisfying these conditions. The
present section addresses a new issue. Do there exist structures optimal for (2.4)
which are not equivalent to those we have found? (Here we will consider two
structures equivalent if they have the same volume fractions m = (m1, . . . , mN)
and generate the same K∗ ∈ G(m; k).) An answer is not yet known for general
K∗ ∈ G(m; k). In this section, we prove a supplementary bound that any effective
tensor, Keff, must satisfy if it attains the translation bound, provided the interfaces
between materials is smooth. The supplementary bound limits the anisotropy of
such a structure and has an interesting relationship to the T 2-structures discussed
in Section 2.3. For this section, we assume the average field E has been rescaled as
in (2.34).
2.5.1 The bound
Theorem 2.8 Let k = (k1, k2, k3) and m = (m1, m2, m3) be admissible parameters.
Define
γ =
3∑
i=1
mi
ki + k1, δ =
m1k3
k1(k3 + k1)2+
m2
(k2 + k1)2+
m3
(k3 + k1)2. (2.39)
Assume that there exists three open sets Ω1, Ω2, Ω3 with smooth boundaries such
that |Ωi| = mi and3⋃
i=1
Ωi = Ω = [0, 1]2 and assume that there exists a field U ∈
H1#(Ω)2 + Ex which satisfies conditions (i)–(iii) in Theorem 2.7 for F = DU and
for average field E ∈ R2×2 such that
E+ = k1
(3∑
i=1
mi
k1 + k1
)
I = k1γI.
Let λ1 and λ2 be the eigenvalues of the effective tensor Keff. Then one has
(λ1 − λ2
λ1λ2 − k21
)2
≤ 4(γ2 − δ). (2.40)
53
Remark 2.11 To better understand the nature of (2.40), suppose we have labeled
the eigenvalues so that 0 < λ1 ≤ λ2 and that the right-hand side of (2.40) is
nonnegative. Since we assume that Keff is optimal, the eigenvalues must also satisfy
(see (2.4))λ1 + λ2 − 2k1
λ1λ2 − k21
= 2γ.
Using this together with the inequality (2.40), one finds an upper bound on λ2 in
terms of λ1:
(
γ −√
γ2 − δ)
λ2 ≤(
γ +√
γ2 − δ)
λ1 − 2k1
√
γ2 − δ.
Before proving Theorem 2.8, we need a bound on the pointwise fields in the
structure
Proposition 2.1 Under the hypotheses of Theorem 2.8, det DU satisfies the in-
equality
det DU ≥ k1k3
(k3 + k1)2in Ω1. (2.41)
Proof : Observe that (i)–(iii) in Theorem 2.7 implies that
det DU =1
4− 1
2|DU−|2 in Ω1 , (2.42)
det DU =k2
1
(ki + k1)2in Ωi for i = 2, 3.
Write the vector function U in terms of its components U = (u, v). Then
|DU−|2 = 12(ux − vy)
2 + 12(uy + vx)
2. Let w = ux − vy and z = uy + vx. Then
since u and v are harmonic in Ω1, so are w and z. We now show that − det DU is
subharmonic in Ω1. Indeed,
∆(− det DU) =1
2∆|DU−|2
=1
2∆(w2 + z2)
= w∆w + |∇w|2 + z∆z + |∇z|2
= |∇w|2 + |∇z|2 ≥ 0.
Therefore, det DU is smooth in Ω1 and satisfies the strong minimum principle there.
54
On the other hand, the assumption of smooth boundaries implies that the vector
potential satisfies the transmission conditions. If t is the tangent to the boundary
at a point and n is the normal, we have DU · t and DU K · n are continuous
across an interface. As a consequence of the invariance under rotations of the
determinant function we get that the function K det DU must be continuous across
the interfaces.
Using the fact that DU is constant on Ω2 and Ω3 along with the continuity of
K det DU and the minimum principle, we find that
det DU ≥ min
k1k2
(k2 + k1)2,
k1k3
(k3 + k1)2
=k1k3
(k3 + k1)2in Ω1.
Proof of Theorem 2.8: Begin with (2.28), which implies
DU(x)K(x) + k1 cof DU(x) = A = k1I, in Ω.
Integrating this equation on Ω, we have (by the definition of Keff)
EKeff + k1 cof E = k1I
which we can solve for
E = k1cof Keff − k1I
det Keff − k21
.
Assume that the material has been oriented in such a way that Keff = diag(λ1, λ2)
for λ1, λ2 > 0. Then we find
E+ =1
2(E + cof E) =
k1
2
(λ1 + λ2 − 2k1
λ1λ2 − k21
)
I,
E− =1
2(E − cof E) =
k1
2
(λ2 − λ1
λ1λ2 − k21
)(1 00 −1
)
.
Note that E+ = k1γI and the above equation for E+ are consistent since we assume
that the translation bound is attained. That is,
E+ = k1γI =k1
2
(λ1 + λ2 − 2k1
λ1λ2 − k21
)
I ⇐⇒ λ1 + λ2 − 2k1
λ1λ2 − k21
= 23∑
i=1
mi
ki + k1
.
55
To obtain the desired inequality, consider
2 detE = |E+|2 − |E−|2 = 2k21γ
2 − k21
2
(λ2 − λ1
λ1λ2 − k21
)2
. (2.43)
Using the quasiaffineness of the determinant, we also have
2 detE =
∫
Ω
2 det DU(x) dx =
∫
Ω1
2 det DU(x) dx +
∫
Ω\Ω1
2 det DU(x) dx.
We know det DU in Ω\Ω1 and we use (2.41) in Ω1 to obtain
2 detE ≥ 2m1k1k3
(k3 + k1)2+
2m2k21
(2k2 + k1)2+
m3k21
(k3 + k1)2= 2k2
1δ.
Combining this with (2.43), we obtain (2.40).
2.5.2 Discussion
Theorem 2.8 requires that any “exact structure” with smooth interfaces satisfy
(2.40). One can check that the right-hand side of the inequality vanishes when
m1 = 2Θ(√
m2 − m2) and is negative for
0 ≤ m1 < 2Θ(√
m2 − m2).
Thus no structure with smooth interfaces can attain the bound for this range of
volume fractions. The validity of the supplementary bound for arbitrary structures
is not addressed here. However, observe that this critical relationship of m1 =
2Θ(√
m2 −m2) is exactly the relationship that holds between the volume fractions
of the most extreme version of the isotropic structures introduced in Gibiansky and
Sigmund (2000). (This structure is represented by the point KGS in Figure 2.9.)
It is also interesting to note that if we formally set Keff = diag(λ1, λ2) with λ1, λ2
given in (2.22),(2.23) and set the volume fractions accordingly, then (2.40) holds as
equality. Thus the T 2-structures — which have the smallest values of m1 among
all known optimal structures — while not exact, satisfy the supplementary bound
(2.40) as equality. For all other structures in this chapter which are optimal for
(2.4), the inequality is strict.
56
2.6 Optimality versus attainability
Here we show the difference between the concepts of optimality and attainability
of an outer bound on the G-closure, as defined in Section 2.2.2. In particular, we
exhibit points on the Wiener bounds (2.2) and (2.3) which are optimal but not
attainable. First we recall the results published by Cherkaev and Gibiansky (1996).
These results can be summarized by the following theorem.
Theorem 2.9 (Cherkaev-Gibiansky) Let
0 < k1 < k2 < k3 and m1, m2, m3 > 0
such that m1 + m2 + m3 = 1 be admissible parameters to the G-closure problem.
Define
k =
3∑
i=1
miki, k =
(3∑
i=1
mi
ki
)−1
.
Then there exist numbers α and β (given explicitly below) such that
α < k and k < β
and such that
(i) The points on the closed line segment joining the points (k, k) and (k, α) are
optimal for the Wiener bound (2.2).
(ii) The points on the closed line segment joining the points (k, k) and (β, k) are
optimal for the Wiener bound (2.3).
The values of α and β are given by the following equations.
α =
α1 if m1 ≥ k1(k3−k2)k3(k2−k1)
m3,
α2 if m1 < k1(k3−k2)k3(k2−k1)
m3,
β =
β1 if m1 ≥ k3−k2
k2−k1m3,
β2 if m1 < k3−k2
k2−k1m3,
57
where
α1 = µ1k1 +(1 − µ1)
2k2(k1k2 + k2k3 − k1k3)
(k3 − k2)(k2 − k1)m2 + (1 − µ1)k22
, µ1 = m1 −k1(k3 − k2)
k3(k2 − k1)m3,
α2 = µ2k3 +(1 − µ2)
2k2(k1k2 + k2k3 − k1k3)
(k3 − k2)(k2 − k1)m2 + (1 − µ2)k22
, µ2 = m3 −k3(k2 − k1)
k1(k3 − k2)m1,
β1 =
(ν1
k1+
(1 − ν1)2(k1 + k3 − k2)
(k3 − k2)(k2 − k1)m2 + (1 − ν1)k1k3
)−1
, ν1 = m1 −k3 − k2
k2 − k1m3,
β2 =
(ν2
k3
+(1 − ν2)
2(k1 + k3 − k2)
(k3 − k2)(k2 − k1)m2 + (1 − ν2)k1k3
)−1
, ν2 = m3 −k2 − k1
k3 − k2
m1.
Proof : The proof uses one of the iterated laminates illustrated in Figure 2.15.
The important parameter is the relative volume fraction, c, of K1 to K3 in the inner
laminate layer. To find α, one fixes c so that the laminate has the same conductivity
in the x1 direction as K2. This ensures that when this composite is laminated with
K2 in the x2 direction, the current does not jump across the interface. It is easy to
check that
c = cα =k1(k3 − k2)
k2(k3 − k1).
If
m1 ≥cα
1 − cα
m3,
then the structure in Figure 2.15a is used, otherwise the structure in Figure 2.15b
is used.
To find β, one instead chooses c so that the conductivity of the laminate in the
x2 direction is the same as K2. This ensures that when this composite is laminated
with K2 in the x2 direction, the field does not jump across the interface. In this
case, one can check that
c = cβ =k3 − k2
k3 − k1.
If
m1 ≥cβ
1 − cβ
m3,
then the structure in Figure 2.15a is used, otherwise the structure in Figure 2.15b
is used.
58
= K3
= K2
= K1
(a) (b)
Figure 2.15. The structures from Theorem 2.9.
Remark 2.12 As a particular example, for the parameters of Figure 2.12b, one
has
ν1 =3
8and µ1 =
3
4,
so
m1 > ν1(1 − m2) and m1 < µ1(1 − m2).
Thus, we can calculate α and β from the theorem as
α =79
34≈ 2.3235 < k =
5
2= 2.5,
β =170
89≈ 1.9101 > k =
20
11≈ 1.8182.
On the other hand, we will now give a proof that the only attainable point on
the segments described in (i) and (ii) of the theorem is the point (k, k).
Theorem 2.10 Let
0 < k1 < k2 < k3 and m1, m2, m3 > 0
such that m1 + m2 + m3 = 1 be admissible parameters to the G-closure problem.
Define k and k as in the previous theorem. Then a point Keff is attainable for (2.2)
or (2.3) if and only if Keff has eigenvalues k and k.
Proof : The fact that (k, k) is attainable is easy; we simply use a first-rank
59
lamination in the x1 direction. Choose the partition
Ω1 = (0, m1) × (0, 1),
Ω2 = (m1, m1 + m2) × (0, 1),
Ω3 = (m1 + m2, 1) × (0, 1).
Then Keff = diag(k, k).
Now assume that there exists a partition of Ω into Ωi such that for K defined
as in (1.5), the effective tensor Keff = diag(λ1, λ2) is such that λ2 = k. That is, we
have
infu∈H1
#(Ω)
∫
Ω
(∇u(x) + e2) · K(x)(∇u(x) + e2) dx = k
where we denote by e1, e2 the canonical basis in R2. Then by uniqueness of weak
solutions to the PDE (1.4), we have u ≡ 0. It follows that for the K defined through
the hypothetical partition, we have
div(K(x)e2) = 0 or∂K(x1, x2)
∂x2
= 0
in the sense of distribution. It is well known that this implies the existence of a
distribution K = K(x1) such that K(x1, x2) = K(x1) as distributions.
Let us now compute the other eigenvalue λ1 by considering the orthogonal
applied field e1. We solve the equation
div(K(x)(∇v(x) + e1)) = 0 in Ω.
Define v ∈ H1#(Ω) such that
v(x) = v(x1), ∇v(x) =k
K(x1)− e1 .
Then
div(K(x)(∇v(x) + e1)) = div
(
K(x1) ·k
K(x1)
)
= 0.
60
By uniqueness of the solution, we have that
infu∈H1
#(Ω)
∫
Ω
(∇u(x) + e1) · K(x)(∇u(x) + e1) dx
=
∫
Ω
(∇v(x) + e1) · K(x)(∇v(x) + e1) dx
=
∫
Ω
k2
K(x1)dx = k2
3∑
i=1
mi
ki= k,
which proves that λ1 = k. A similar argument shows that if λ1 = k then λ2 = k.
CHAPTER 3
BLOCK STRUCTURES IN TWO AND
THREE DIMENSIONS
3.1 Introduction
In this chapter, we construct optimal anisotropic two- and three-dimensional
structures that generalize the isotropic structures introduced by Gibiansky and
Sigmund (2000). The structures have a special “block” form in which the periodicity
cell, Ω = [0, 1]d with d = 2, 3 the dimension of the composite, is split into 2d
rectangular blocks with sides parallel to the sides of the cell. Each block contains
a composite of the constituent materials. By choosing special composites for each
block, we are able to easily compute the effective tensor of the overall composite
and ensure that the composite is optimal.
In Section 3.2, we recall the Gibiansky-Sigmund structures, which we use as
inspiration for the new structures introduced in this chapter. We have introduced
these structures already in the previous chapter, Section 2.2.2.2 and illustrated
a special case in Figure 2.5. In this section we discuss the structures in more
detail and explain the special characteristics that allow the conductivity PDE to
be solved by separation of variables. These concepts motivate the rest of the
chapter. In Section 3.3, we describe a simple modification of the Gibiansky-Sigmund
block structures that produces optimal anisotropic structures equivalent to the
T 2-structures introduced previously in Section 2.3.
In Section 3.4, we use the optimal two-dimensional block structures to motivate
a class of three-dimensional block structures optimal for the three-dimensional
version of the translation bounds. We refer the reader to Cherkaev (2000); Milton
(2002) for an overview of the history of the bounds optimal structures. The
62
structures introduced in this section improve the region of parameters for which
the translation bounds are known optimal.
3.2 Background: the structures ofGibiansky and Sigmund
3.2.1 The structures introduced
In Figure 2.5, we have illustrated the structures introduced by Gibiansky and
Sigmund (2000). The reader should imagine that the two laminate layers are mixed
at an arbitrarily fine scale. This structure is optimal for the lower two-dimensional
translation bound (2.4) if the relative volume fractions in each rectangular block
are chosen correctly. In particular, one should choose the relative volume fractions
of K1 and K3 in the laminate layers to be Θ and 1 − Θ, respectively, where Θ is
the constant defined in (2.9) which depends only on the materials’ properties.
Remark 3.1 In the paper by Gibiansky and Sigmund (2000), the lower-left block
need not contain pure K2 but can also contain inclusions of “coated spheres” of
K1 and K3 so long as the relative amounts of K1 and K3 are chosen so that the
effective conductivity of the coated sphere is exactly K2. The results of this section
apply to these more general structures as well. In fact, by coating the structure in
Figure 2.5 with layers of K1 so that the final structure is isotropic, one can always
generate a structure equivalent to those structures with coated spheres inside the
block of K2. For simplicity we restrict the discussion to the case where no coated
spheres are present.
Let us now assume that the mixture in each rectangular block has been homog-
enized so that the conductivity tensor, K, in Ω = [0, 1]2 is piecewise constant with
the form
K(x) =
k2I if x ∈ (0, ω)× (0, ω)
k3I if x ∈ (ω, 1)× (ω, 1)
diag(µ1, µ2) if x ∈ (0, ω)× (ω, 1)
diag(µ2, µ1) if x ∈ (ω, 1)× (0, ω).
(3.1)
Here ω ∈ (0, 1) is the parameter that defines the sizes of the rectangular blocks,
and µ1 and µ2 are the conductivities of the laminates of K1 and K3. Specifically,
63
µ1 =
(Θ
k1+
1 − Θ
k3
)−1
, µ2 = Θk1 + (1 − Θ)k3. (3.2)
We will soon see that we are able to solve the conductivity equation with this
conductivity tensor by separation of variables. First, we must take a short detour
and examine the implications of a conductivity tensor of special form related to
that of Marino and Spagnolo (1969) discussed in the previous chapter.
3.2.2 Separation of variables
Suppose that the conductivity, K, in d dimensions is diagonal on Ω = [0, 1]d
with
K(x) = diag(λ1(x), λ2(x), . . . , λd(x)) (3.3)
such that λ1, . . . , λd have the form
λi(x) = λ0i (xi)λ
′i(x1, . . . , xi−1, xi+1, . . . , xd), for i = 1, . . . , d. (3.4)
A particular case of such a λ resembles (2.10). However, we need the more general
case for the three-dimensional structures to follow.
Now we show that this hypothesis allows us to separate variables to solve the
PDE
div(DU(x) K(x)) = 0 in Ω,
U ∈ H1#(Ω)d + Ex
(3.5)
with E ∈ Rd×d and diagonal: E = diag(r1, . . . , rd). Begin with the ansatz U =
(u1(x1), . . . , ud(xd)) so that
DU = diag
(du1
dx1
, . . . ,dud
dxd
)
.
Then the ith row of the PDE has the form
0 =∂
∂xi
(
λi(x)dui
dxi(xi)
)
= λ′i(x1, . . . , xi−1, xi+1, . . . , xd)
d
dxi
(
λ0i (xi)
dui
dxi(xi)
)
.
Thus, the solution we seek is the U ∈ H1#(Ω)d +Ex that solves the system of ODEs
d
dxi
(
λ0i (xi)
dui
dxi(xi)
)
= 0 in (0, 1), for i = 1, . . . , d.
In particular, if K is piecewise constant, then U is piecewise affine. Now let us
return to the two-dimensional structures in question.
64
3.2.3 The structures revisited
To understand the relationship of the separation of variables approach to the
structures of Gibiansky and Sigmund, let us consider (3.5) with d = 2 and K given
by (3.1). Assume there is a piecewise affine solution U such that DU is diagonal
and
DU(x) =
A1 if x ∈ (0, ω) × (0, ω)
A3 if x ∈ (ω, 1) × (ω, 1)
A2 if x ∈ (0, ω) × (ω, 1)
A′2 if x ∈ (ω, 1) × (0, ω).
Since we can always rescale the average field, let us assign the field A1 as in the
previous chapter:
A1 =k1
k2 + k1
I.
Then the transmission conditions immediately imply that
A2 = diag
(k1
k2 + k1,
k1k2
µ2(k2 + k1)
)
,
A′2 = diag
(k1k2
µ2(k2 + k1),
k1
k2 + k1
)
.
Now we see the potential for trouble. Since it shares boundaries with both A2
and A′2, A3 has four transmission conditions to satisfy. Looking at the interfaces
separately, we obtain the following two forms for A3.
A3 = diag
(µ1k1
k3(k2 + k1),
k1k2
µ2(k2 + k1)
)
,
A3 = diag
(k1k2
µ2(k2 + k1),
µ1k1
k3(k2 + k1)
)
.
For these to be consistent, we must have µ1µ2 = k2k3. But by multiplying µ1 and
µ2 given in (3.2), we find that this condition is satisfied.
From this, we find the decomposition
K(x) = diag(λ11(x1)λ12(x2), λ21(x1)λ22(x2)).
65
Indeed, if µ1µ2 = k2k3, then (3.1) admits the decomposition
λ11(x1) =µ1√k3
χ(0,ω)(x1) +√
k3 χ(ω,1)(x1),
λ12(x2) =µ2√k3
χ(0,ω)(x2) +√
k3 χ(ω,1)(x2),
λ21(x1) =µ2√k3
χ(0,ω)(x1) +√
k3 χ(ω,1)(x1),
λ22(x2) =µ1√k3
χ(0,ω)(x2) +√
k3 χ(ω,1)(x2).
Now let us analyze the fields inside the structure. By our earlier assumption,
A1 =k1
k3 + k1
I.
Now consider A3. With µ1 and µ2 given by (3.2), we find
A3 =k1
k3 + k1I.
Turning our attention to the laminate layer A2, we find that
A2 = diag
(k1
k2 + k1
,k1
k3 + k1
)
.
In this laminate, material K1 has volume fraction Θ while K3 has volume fraction
1−Θ. Thus, the average field A2 is achieved through lamination as the average of
the two approximately constant fields of these materials, A21 and A23 respectively.
By writing
A2 = ΘA21 + (1 − Θ)A23
and satisfying the transmission conditions in this laminate, we find that the fields
inside K1 and K3 of the laminate layer are, respectively,
A21 = diag
(k3
k3 + k1
,k1
k3 + k1
)
,
A23 = diag
(k1
k3 + k1,
k1
k3 + k1
)
.
We see immediately that these fields satisfy (P1)–(P3) of the previous chapter,
which explains the optimality of these structures.
Here, then, is the connection between the construction of Gibiansky and Sig-
mund and the special form of a conductivity tensor that allows for the separation
66
of variables. The following two requirements on the laminate layers of K1 and K3
are equivalent!
(i) The relative volume fractions of K1 and K3 are chosen so that a separation
of variables is possible in the conductivity equation when each rectangular
block is considered homogenized.
(ii) The relative volume fractions of K1 and K3 are chosen so that the local fields
approximately satisfy the optimality conditions (P1)–(P3) of the previous
chapter.
3.3 New optimal structuresin two dimensions
It is not difficult to see that the same arguments can be used to prove the
optimality of a more general form of (3.1). Indeed, with the same laminate of K1
and K3 as before, consider the following modified version of K.
K(x) =
k2I if x ∈ (0, ω1) × (0, ω2)
k3I if x ∈ (ω1, 1) × (ω2, 1)
diag(µ1, µ2) if x ∈ (0, ω1) × (ω2, 1)
diag(µ2, µ1) if x ∈ (ω1, 1) × (0, ω2).
Now there are two parameters ω1, ω2 ∈ (0, 1). K still has the correct form to admit
the separation of variables solution. Furthermore, the calculation of the piecewise
constant approximate fields is unchanged. Thus, such a structure is optimal for any
ω1 and ω2. In fact, it can be shown that these structures are exactly equivalent to
the T 2-structures of the previous chapter. Figure 3.1 illustrates such a structure.
3.4 New optimal structuresin three dimensions
In the rest of this chapter, we will use the idea behind this generalization of the
Gibiansky-Sigmund structure as inspiration for optimal three-dimensional block
structures. These structures are optimal for the lower or upper translation bounds
in three dimensions. These bounds are similar to the two dimensional bounds in
their derivation. In both cases, we make use of a “translator”, φ : R3×3 → R, (the
67
ω2
ω1
= K3
= K2
= K1
Figure 3.1. The two-dimensional anisotropic optimal block structure.
analogue of det DU in the two-dimensional case) with a convexity property of the
form
φ
(∫
Ω
F (x) dx
)
≤∫
Ω
φ(F (x)) dx
for all F in a certain class of vector fields (curl-free for the lower bound, divergence-
free for the upper bound).
3.4.1 The lower translation bound
The lower translation bound in three dimensions is derived similarly to the
bound (2.4). In this case, the dimension d = 3, so the periodicity cell Ω is the
unit three-dimensional cube and the field matrix E lies in R3×3. In place of the
“translator” det DU from the two-dimensional case, we use a sum of two-by-two
minors:
(trE)2 − tr E2 = 2(e11e22 − e12e21) + 2(e11e33 − e13e31) + 2(e22e33 − e23e32).
This function has the property that
∫
Ω
((trDU)2 − trDU2
)dx = (trE)2 − tr E2, ∀E ∈ R
3×3, ∀U ∈ H1#(Ω)3 + Ex.
This can be seen by integration by parts or by the fact that (trE)2 − trE2 is
polyconvex and therefore quasiconvex. In this case, it is convenient to decompose
R3×3 into the direct sum
R3×3 = M1 ⊕M2 ⊕M3
68
where
M1 = αI : α ∈ R , M2 =E ∈ R
3×3 : E = ET , tr E = 0
,
M3 =E ∈ R
3×3 : E = −ET
The projections of a matrix E onto these orthogonal subspaces are
E1 :=1
3tr E I, E2 :=
1
2
(E + ET
)− E1 and E3 :=
1
2
(E − ET
)
respectively. It is easy to verify the following identities.
|E|2 = |E1|2 + |E2|2 + |E3|2,
(trE)2 = 3|E1|2,
trE2 =⟨E, ET
⟩= 〈E1 + E2 + E3, E1 + E2 − E3〉
= |E1|2 + |E2|2 − |E3|2.
Remark 3.2 This decomposition was used by Nesi and Rogora (2004) to show that
certain functions (“translators” in our language) are the most efficient for bounds
in homogenization. These translators are the ones used in this chapter to derive
the translation bounds.
As in the two-dimensional case, we fix the matrix E and add and subtract our
translator:
〈E Keff, E〉 = infU∈H1
#(Ω)3+Ex
∫
Ω
(〈DU K, DU〉 + t
((trDU)2 − tr DU2
))dx
−t((trE)2 − tr E2
).
We then relax the problem by replacing U with F ∈ L2(Ω; R3×3) such that∫
ΩF dx =
E so that
〈E Keff, E〉 ≥ infF :
R
ΩF dx=E
∫
Ω
(〈F K, F 〉 + t
((trF )2 − trF 2
))dx
−t((trE)2 − tr E2
).
(3.6)
Thus, using the identities calculated above, a minimizer F must satisfy
K(x) F (x) + 2tF1(x) − tF2(x) + tF3(x) = A a.e. in Ω (3.7)
69
for some constant A ∈ R3×3. Alternatively, we can use the specific form of K given
in (1.5) and the identities calculated above to rewrite the bound as
〈E Keff, E〉 ≥ infF :
R
ΩF dx=E
N∑
i=1
∫
Ωi
[(ki + 2t)|F1|2 + (ki − t)|F2|2 + (ki + t)|F3|2
]dx
−t((trE)2 − tr E2
).
This bound is not trivially −∞ only if t ∈ [−k1/2, k1]. The bound we consider is
for the choice t = k1. For this choice, we obtain a theorem similar to Theorem 2.7
Theorem 3.1 F is a minimizer of the right-hand-side of (3.6) with t = k1 if and
only if
(i) F1 = 1ki+2k1
(N∑
j=1
mj
kj+2k1
)−1
E1 a.e. in Ωi for i = 1, . . . , N .
(ii) F3 = 1ki+k1
(N∑
j=1
mj
kj+k1
)−1
E3 a.e. in Ωi for i = 1, . . . , N .
(iii)∫
Ω1F2 dx = E2.
(iv) F2 = 0 a.e. in Ωi for i = 2, . . . , N .
Thus, we can write the lower bound as
〈E Keff, E〉 ≥(
n∑
i=1
mi
ki + 2k1
)−1
|E1|2 +
(n∑
i=1
mi
ki + k1
)−1
|E3|2
−k1
((trE)2 − tr E2
), ∀E ∈ R
3×3.
Once again, however, we are more interested in the fields than in the optimal value
of the bound. As in the two-dimensional case, the only freedom in the fields in a
structure which attains this bound is in the value of DU2 in the first material.
3.4.2 Optimal structures for the lower bound
Now we are in a position to produce three-dimensional three-material structures.
Assuming that E is diagonal and properly scaled, the conditions of Theorem 3.1
become the following.
(P1) DU = DUT and tr DU = 1 a.e. in Ω1.
70
(P2)∫
Ω1DU2 dx = E2 = 1
2(E + ET ) − 1
3tr E I.
(P3) DU = k1
ki+2k1I a.e. in Ωi for i = 2, 3.
As in the two-dimensional case, we will see that the transmission conditions on
DU K are free if we satisfy those on DU . Indeed, assume that U ∈ H1#(Ω)3 + Ex
and that DU satisfies (3.7) with F = DU , so
DU(x) K(x) + 2tDU1(x) − tDU2(x) + tDU3(x) = A a.e. in Ω. (3.8)
Note that
2tDU1 − tDU2 + tDU3 = t trDU I − tDUT .
Taking the divergence of each terms on the right-hand side respectively, we find
(div(t tr DU I))i = t3∑
j=1
∂
∂xj
(trDU I)ij
= t∂
∂xi
(tr DU I)ii
= t
3∑
j=1
∂2uj
∂xi∂xj
and
(div(tDUT )
)
i= t
3∑
j=1
∂
∂xj
(DUT )ij
= t3∑
j=1
∂
∂xj
(DU)ji
= t
3∑
j=1
∂2uj
∂xi∂xj.
Thus, by taking divergence of both sides of (3.8), we find that U automatically
solves the PDE
div(DU(x) K(x)) = 0.
The corresponding statement for laminates is that the jump conditions on currents
are “free” if we satisfy the jump conditions on the fields.
71
In this chapter, we construct structures in which the unit cube is divided
into eight blocks. Each of the eight blocks will contain either a pure material
or a laminate. If we consider each block to be homogenized, then we will find
conductivity tensors K with the form (See Figure 3.2 for an illustration of the final
structure.)
K(x) =
k2I if x ∈ (0, ω1) × (0, ω2) × (0, ω3),
k3I if x ∈ (ω1, 1) × (ω2, 1) × (ω3, 1),
diag(µ1, µ1, µ2) if x ∈ (0, ω1) × (0, ω2) × (ω3, 1),
diag(µ1, µ2, µ1) if x ∈ (0, ω1) × (ω2, 1) × (0, ω3),
diag(µ2, µ1, µ1) if x ∈ (ω1, 1) × (0, ω2) × (0, ω3),
diag(η1, η2, η2) if x ∈ (0, ω1) × (ω2, 1) × (ω3, 1),
diag(η2, η1, η2) if x ∈ (ω1, 1) × (0, ω2) × (ω3, 1),
diag(η2, η2, η1) if x ∈ (ω1, 1) × (ω2, 1) × (0, ω3).
(3.9)
We perform this construction in the “easy” direction, considering the fields
and ensuring that they satisfy (P1)–(P3) above. There are three blocks which
share a border with pure material K2. For definiteness, let us consider the block
(0, ω1)×(0, ω2)×(ω3, 1). We construct a laminate of K1 and K3 in this block whose
local fields satisfy (P1)–(P3) and whose average field is in rank one connection with
the field in K2 given by (P3). This is a matrix laminate.
First we need a field for K1 which is compatible with the field in K3. We
laminate first in the direction e1 = (1, 0, 0)T , so the required field, E1, in K1 is
E1 = diag
(k3
k3 + 2k1,
k1
k3 + 2k1,
k1
k3 + 2k1
)
.
Laminating this with the field E3 = k1
k3+k1I in K3 and with relative volume fraction
ν1, we obtain a composite with average field
E13 = diag
(ν1k3 + (1 − ν1)k1
k3 + 2k1
,k1
k3 + 2k1
,k1
k3 + 2k1
)
.
We know we need this field to be rank-one connected to the field in K2, so we
choose ν1 so thatν1k3 + (1 − ν1)k1
k3 + 2k1=
k1
k2 + 2k1.
72
Thus, we find
ν1 =k1(k3 − k2)
(k2 + 2k1)(k3 − k1)≤ 1
3(3.10)
and
E13 = diag
(k1
k2 + 2k1,
k1
k3 + 2k1,
k1
k3 + 2k1
)
.
Now we will laminate this composite with K1 again in the e2 = (0, 1, 0)T
direction. The compatible field for K1 is
E ′1 = diag
(k1
k2 + 2k1
,k1k2 + k1k3 + k2k3
(k3 + 2k1)(k2 + 2k1),
k1
k3 + 2k1
)
.
Laminating this in volume fraction ν2 with E13, we obtain a new composite with
average field
E113 = diag
(k1
k2 + 2k1,ν2(k1k2 + k1k3 + k2k3) + (1 − ν2)k1(k2 + 2k1)
(k3 + 2k1)(k2 + 2k1),
k1
k3 + 2k1
)
.
We bring this into rank-one connection with the field in K2 by choosing ν2 so that
ν2(k1k2 + k1k3 + k2k3) + (1 − ν2)k1(k2 + 2k1)
(k3 + 2k1)(k2 + 2k1)=
k1
k2 + 2k1.
Thus,
ν2 =k1(k3 − k2)
k1k3 + k2k3 − 2k21
=ν1
1 − ν1≤ 1
2(3.11)
and
E113 = diag
(k1
k2 + 2k1,
k1
k2 + 2k1,
k1
k3 + 2k1
)
.
Now we have all the components of the structure and we can construct it by
placing the matrix laminate with average field E113 along with its two orthogonal
rotations into the three blocks neighboring K2. Furthermore, since E13 is rank one
connected to both E113 and to E3, we can fill the remaining three blocks with this
laminate and its two orthogonal rotations. In particular, the laminate with average
field E13 can be placed in the block (0, ω1)× (ω2, 1)× (ω3, 1). The final structure is
illustrated in Figure 3.2. For illustrative convenience, the matrix laminate has
been replaced by an equivalent “coated cylinders” construction. These should
be visualized as a three-dimensional extrusion of the two-dimensional structures
73
1 − ω2
ω3
ω2
ω1
ω1
ω3
= K3
= K2
= K1
Figure 3.2. The optimal three-dimensional structure (left) and the details of the“hidden” block (right).
illustrated in Figure 2.3a. The block on the right of the figure shows the details of
the “hidden” block behind the block of pure K2.
Now we can compute the volume fractions of the structure. Note that by (3.11),
we get that the volume fraction in the matrix laminate is
ν2 + (1 − ν2)ν1 =ν1
1 − ν1+
ν1(1 − 2ν1)
1 − ν1= 2ν1.
With a little simplification, the volume fractions can be shown to be
m1 = ν1(ω1 + ω2 + ω3 − 3ω1ω2ω3),
m2 = ω1ω2ω3,
m3 = 1 − m2 − m3.
Computing the effective tensor, K∗, is also straightforward, since we know the
approximate piecewise constant field in each piece of the structure. Keeping track
of the transmission conditions further helps simplify the calculation. For example,
consider the eigenvalue λ1 in the e1 direction. We see that
DU(x) · e1 =
k1
k2+2k1if x1 ∈ (0, ω1),
k1
k3+2k1if x1 ∈ (ω1, 1)
and
74
DU(x) K(x) · e1 =
k1k2
k2+2k1if (x2, x3) ∈ (0, ω2) × (0, ω3),
k1k3
k3+2k1if (x2, x3) ∈ (ω2, 1) × (ω3, 1),
k1(k1k2+k1k3+k2k3)(k2+2k1)(k3+2k1)
if (x2, x3) ∈ (0, ω2) × (ω3, 1),k1(k1k2+k1k3+k2k3)(k2+2k1)(k3+2k1)
if (x2, x3) ∈ (ω2, 1) × (0, ω3).
We define the average field, E, and average current, J , to be
E =
∫
Ω
DU(x) dx and J =
∫
Ω
DU(x) K(x) dx.
From the above calculation, we find that E and J satisfy
E · e1 =ω1k1(k3 + 2k1) + (1 − ω1)k1(k2 + 2k1)
(k2 + 2k1)(k3 + 2k1),
J · e1 =k1k3(k2 + 2k1) − k2
1(k3 − k2)(ω2 + ω3)
(k2 + 2k1)(k3 + 2k1).
Thus,
λ1 =J · e1
E · e1=
k3(k2 + 2k1) − k1(k3 − k2)(ω2 + ω3)
ω1(k3 + 2k1) + (1 − ω1)(k2 + 2k1). (3.12)
Similarly, we can find the other two eigenvalues in the e2 and e3 directions respec-
tively:
λ2 =k3(k2 + 2k1) − k1(k3 − k2)(ω1 + ω3)
ω2(k3 + 2k1) + (1 − ω2)(k2 + 2k1),
λ3 =k3(k2 + 2k1) − k1(k3 − k2)(ω1 + ω2)
ω3(k3 + 2k1) + (1 − ω3)(k2 + 2k1).
In particular, one can verify that when ω1 = ω2 = ω3, then the structure is
isotropic (λ1 = λ2 = λ3 = λ) and is optimal for the three-dimensional version of
the Hashin-Shtrikman bound:
1
λ + 2k1=
3∑
i=1
mi
ki + 2k1.
3.4.2.1 Separation of variables
Now let us return to the tensor K as in (3.9) with the composites inside each
block of Ω considered homogenized already. From the construction above, it is
straightforward to find
µ1 =k1k2 + k1k3 + k2k3
k3 + 2k1, µ2 =
k2(k3 + 2k1)
k2 + 2k1,
η1 =k3(k2 + 2k1)
k3 + 2k1
, η2 =k1k2 + k1k3 + k2k3
k2 + 2k1
.
75
We now show that this K admits a decomposition as in (3.3) and (3.4) so that we
can solve the PDE by separation of variables. Indeed, if we define
λ01(x1) = k2χ(0,ω1) + µ2χ(ω1,1)
and
λ′1(x2, x3) =
1 if (x2, x3) ∈ (0, ω2) × (0, ω3),µ1
k2if (x2, x3) ∈ (ω2, 1) × (0, ω3),
µ1
k2if (x2, x3) ∈ (0, ω2) × (ω3, 1),
η1
k2if (x2, x3) ∈ (ω2, 1) × (ω3, 1),
then λ1(x) = λ01(x1)λ
′1(x2, x3) exactly because the following two identities hold.
µ1µ2
k2= η2 and
η1µ2
k2= k3.
The other two eigenvalues are analogous. Once again, we see that the conditions for
a separation of variables are identical to those of optimality. Note that λ1 does not
have a decomposition as in (2.10). The more general decomposition is necessary.
3.4.2.2 Applicability: volume fractions
As we did with the two-dimensional structures, let us analyze the applicability
of this construction in terms of volume fraction. The optimal structures of Milton
(1981); Milton and Kohn (1988) applied to three materials in three dimensions
require that the volume fractions satisfy the inequality
3ν1(1 − m2) ≤ m1 ≤ 1
where ν1 is defined in (3.10). On the other hand, if ω1 = ω2 = ω3 in the above
structure, then we see that
m1 = 3ν1( 3√
m2 − m2) (3.13)
which lies outside this region. By using a generalization of the “coating principle”
of the previous chapter, we find a set of optimal anisotropic structures strictly larger
than was previously known. In particular, such structures are possible as long as
3ν1( 3√
m2 − m2) ≤ m1 ≤ 1.
Furthermore, similar three-dimensional constructions are also possible for any num-
ber of materials N ≥ 3. In all cases, one can obtain structures outside the realm
76
of previously known optimal structures. In fact, as N increases, the improvement
over previous results becomes more and more pronounced.
3.4.3 The upper translation bound
Though the derivation of the upper bound is somewhat different, the calcula-
tions are very similar. In fact, we will see that one can produce optimal structures
for this bound by reversing the roles of K1 and K3 in the previous structures. First
we recall the derivation of the bound, which uses the dual variational principle and
the “translator” φ(B) = tr BBT +trB2− (tr B)2. This functional is A-quasiconvex
on 3 × 3 divergence-free matrices in the sense that
φ
(∫
Ω
B(x) dx
)
≤∫
Ω
φ(B(x)) dx (3.14)
for any divergence-free vector field B : Ω → R3×3. We can use this to obtain bounds
in the dual variational principle as follows. Let Ωi be a partition of Ω and let
K be the conductivity defined through (1.5). Then Keff is characterized by the
following principle.
⟨J (Keff)
−1, J⟩
= infB∈B
∫
Ω
⟨B K−1, B
⟩dx ∀J ∈ R
3×3
where B is the set of divergence-free vector fields with mean J . Now we can add
and subtract φ(B) in the integral and use (3.14) to obtain
⟨J (Keff)
−1, J⟩≥ inf
B∈B
∫
Ω
(⟨B K−1, B
⟩− tφ(B)
)dx + tφ(J) ∀J ∈ R
3×3, ∀t ≥ 0.
Once again, we obtain the bound by relaxing the differential constraint. This time,
we drop the constraint that B is divergence-free, replacing it instead with a vector
field F with average J . The analog of (3.7) for the upper bound is
K−1(x)F (x) + tF1(x) − 2tF2(x) = A a.e. in Ω.
Alternatively, we can rewrite the energy using the same identities as for the lower
bound, which gives
⟨J (Keff)
−1, J⟩≥ inf
F :R
ΩF=J
N∑
i=1
∫
Ωi
[(1
ki+ t
)
|F1|2 +
(1
ki− 2t
)
|F2|2
+1
ki|F3|2
]
dx + tφ(J).
(3.15)
77
This bound is nontrivial for 0 ≤ t ≤ 12kN
and we will analyze the case t = 12kN
. It
is not difficult to verify the following theorem.
Theorem 3.2 F is a minimizer of the right-hand-side of (3.15) with t = k1 if and
only if
(i) F1 = kikN
ki+2kN
(N∑
j=1
mjkjkN
kj+2kN
)−1
J1 a.e. in Ωi for i = 1, . . . , N .
(ii) F3 = ki
(N∑
j=1
mjkj
)−1
J3 a.e. in Ωi for i = 1, . . . , N .
(iii)∫
ΩNF2 dx = J2.
(iv) F2 = 0 a.e. in Ωi for i = 1, . . . , N − 1.
3.4.4 Optimal structures for the upper bound
Using the constitutive relation DU K = B and rescaling the average field E
as usual, we obtain the following conditions for a the fields DU = B K−1 in a
three-material optimal structure.
(P1′) DU = DUT and tr DU = 1 a.e. in Ω3.
(P2′)∫
Ω3DU2 dx = E2 = 1
2(E + ET ) − 1
3trE I.
(P3′) DU = k3
ki+2k3I a.e. in Ωi for i = 1, 2.
Note that if U ∈ H1#(Ω) + Ex satisfies (P1′)–(P3′) then a similar identity to
(3.8) holds. Specifically,
DU(x) K(x) + 2k3DU1(x) − k3DU2(x) + k3DU3(x) = k3I a.e. in Ω.
Which implies that U is a solution of the PDE and so we once again do not need
to worry about the transmission conditions on the currents DU K. We can repeat
the construction of the structure in Figure 3.2 interchanging the roles of K1 and K3
to find a family of structures optimal for the upper translation bound. By coating
with K3, we obtain a class of structures that generalizes and improves previous
results.
CHAPTER 4
A DIFFERENTIAL SCHEME: INFINITE-
RANK LAMINATES
4.1 Introduction
In this chapter, we introduce a modified differential scheme and use it to produce
optimal infinite-rank laminates. The traditional differential scheme uses the strat-
egy of inserting infinitesimal inclusions into an existing composite and calculating
the increment of its effective properties. By repeatedly performing this inclusion,
one finds that the effective tensor of the structure changes according to an ordinary
differential equation. If the inclusions can be of several different types at each step,
the differential equation becomes controllable. We refer the reader to such papers
as Bruggemann (1935); Norris (1985); Lurie and Cherkaev (1985); Avellaneda et al.
(1988); Hashin (1988) for a more in-depth look at the general idea of differential
schemes.
In Section 4.2 we introduce a differential scheme for the three-material G-closure
problem using a special type of inclusions. The inclusions are composites of the
constituent materials and are always placed into infinitesimally thin strips parallel
to one of the sides of the periodicity cell. We derive the equations of the differential
scheme, which is controlled by the effective tensors and orientations of these strips.
Furthermore, we derive the equations for the volume fractions of the composite
which change as the scheme progresses. Finally, we introduce an optimal control
problem which is equivalent to finding effective tensors on the boundary of the
G-closure.
In Section 4.3, we modify the differential scheme to allow us to easily solve
the optimal control problem of the previous section in the case where the three-
dimensional translation bound is optimal. The strategy is to ensure that the
79
optimality conditions on the fields can be satisfied at each infinitesimal step of
the differential scheme. We leave the parameters of the optimal control problem
free until the end of the scheme. Once the structure is produced, it is guaranteed
optimal since the optimality conditions on the fields can be satisfied. By tracking
the effective tensor and volume fractions throughout the scheme, we can easily find
the parameters of the optimal control problem that was solved.
In Section 4.4, we compute the effective tensor of the optimal structures pro-
duced in the previous section. We show that in the isotropic case, these structures
are exactly equivalent to the three-dimensional optimal block structures constructed
in the previous chapter. This is also true of the anisotropic case. The basic strategy
is very similar to the isotropic case, but the calculations are more complex, so we
do not prove it here.
In Section 4.5, we discuss the applicability of the differential scheme method to
other problems. Specifically, similar methods can be applied to the problem with
two dimensions (d = 2) and/or more materials (N ≥ 4). Furthermore, we point
the reader to a discussion of the differential scheme applied to three-dimensional
isotropic polycrystals.
4.2 The differential scheme
The differential scheme described in this section is a method of generating
effective tensors for an inner bound in the G-closure problem. We imagine a
sequence of d-dimensional structures in Ω = [0, 1]d parameterized by µ ≥ 0 with
effective tensors K∗(µ) and with volume fractions mi(µ) for i = 1, . . . , N . We
describe the scheme in d dimensions here and then specialize to d = 2, 3 in the
following sections.
The idea is to describe the infinitesimal transition between the tensors K∗(µ)
and K∗(µ + dµ) with dµ ≪ 1. The basic strategy is as follows. For µ ≥ 0, begin
with the homogenized material with effective tensor K∗(µ) filling Ω. For each
j = 1, . . . , d, we replace the infinitesimal strip
[0, 1] × · · · × [0, 1]︸ ︷︷ ︸
j−1 times
×[0, ρj(µ)dµ] × [0, 1] × · · · × [0, 1]︸ ︷︷ ︸
d−j times
80
by some new material K∗j (µ). The tensors K∗
j (µ) and parameters ρj(µ) ≥ 0 are
controls of the construction. The ρj are subject to the constraint ρ1(µ) + · · · +
ρd(µ) = 1. Thus, after performing the d replacements, we have replaced a total
volume of dµ of the original material with new material. K∗(µ + dµ) is found by
homogenizing the new conductivity tensor in Ω. Figure 4.1 illustrates a special
example of this procedure for d = 3. Note that the order of replacement does not
matter since the pairwise intersections of the infinitesimal strips have volume of the
order o(dµ).
To simplify the calculations, we choose the K∗j to be diagonal for each µ and we
select the initial tensor K∗(0) to be diagonal. This ensures that K∗(µ) is diagonal
for each µ ≥ 0. We now specialize the scheme to the problem of multimaterial
composites.
Let 0 < k1 < · · · < kN be the constituent material properties and fix µ ≥ 0.
Let ρ1(µ), . . . , ρd(µ) ≥ 0 be such that ρ1(µ) + · · · + ρd(µ) = 1. Let νij(µ) ≥ 0 for
i = 1, . . . , N , j = 1, . . . , d be such that ν1j (µ) + · · · + νN
j (µ) = 1 for j = 1, . . . , d.
As described earlier, the ρi control the thickness of the layers replaced in each of
the d directions. The νij act as controls for the volume fractions in the replacement
layers where νij represents the relative amount of the material with conductivity Ki
present in the jth layer. Specifically, we choose
K∗j ∈ G
((ν1
j (µ), . . . , νNj (µ)) ; (k1, . . . , kN)
)∀j = 1, . . . , d.
1
3dµ
Kadd
Kcore
Figure 4.1. The differential scheme for isotropic composites in three dimensions.
81
4.2.1 The equations for the volume fractions
Suppose that for some fixed µ ≥ 0, we have successfully constructed a composite
with effective tensor K∗(µ) and with volume fractions m1(µ), . . . , mN (µ). Let us
analyze the change in volume fraction after an infinitesimal volume dµ has been
replaced by the differential scheme described above. Since the volume of the original
material, K∗(µ), in the new composite is 1 − dµ, we find that
mi(µ + dµ) = (1 − dµ)mi(µ) + dµ
d∑
j=1
ρj(µ)νij(µ)
= mi(µ) + dµ
(d∑
j=1
ρj(µ)νij(µ) − mi(µ)
)
.
Taking dµ → 0, we find the differential equation satisfied by the volume fractions:
dmi
dµ=∑d
j=1 ρjνij − mi,
mi(0) = m0i
∀i = 1, . . . , N
where m0i is the initial volume fraction of the ith material in the “seed” composite
with effective tensor K∗(0). Thus, we find the volume fractions parameterized by
µ are
mi(µ) = m0i e
−µ +
∫ µ
0
d∑
j=1
ρj(ξ)νij(ξ)e
ξ−µ dξ ∀i = 1, . . . , N. (4.1)
4.2.2 The equation for the effective tensor
Now we consider the differential equation of the effective tensor K∗. By our
assumptions, we have ensured that K∗ and the K∗i are diagonal for every µ. To
simplify notation, we write
K∗(µ) = diag(λ1(µ), . . . , λd(µ))
and
K∗j (µ) = diag(ηj
1(µ), . . . , ηjd(µ)) ∀j = 1, . . . , d.
First consider the case when ρ1(µ) = 1 and ρ2(µ) = · · · = ρd(µ) = 0 so that only
one infinitesimal strip is replaced. In this case, the lamination formula shows that
λ1(µ + dµ) =
(1 − dµ
λ1(µ)+
dµ
η11(µ)
)−1
= λ1(µ) + dµ λ1(µ)
(
1 − λ1(µ)
η11(µ)
)
+ o(dµ)
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and
λl(µ + dµ) = (1 − dµ)λl(µ) + dµ η1l (µ) = λl(µ) + dµ(η1
l (µ) − λl(µ))
for l = 2, . . . , d. From this, it is straightforward to find the differential equation
satisfied by λ1 for general ρj :
dλ1
dµ= ρ1λ1
(
1 − λ1
η11
)
+∑d
j=1 ρj(ηj1 − λ1),
λ1(0) = λ01
(4.2)
where λ01 is the first eigenvalue of K∗(0). The equations for the other eigenvalues
are similar.
4.2.3 An optimal control problem
One way to characterize the optimal structures in the two- and three-dimensional
problems discussed in the previous chapters (see for example Cherkaev (2000)) is
to solve the following optimal control problem (or the related maximum problem).
min λ1(µf)
subject to the constraints of the differential equations above and
λj(µf) = βj ∀j = 2, . . . , d,
mi(µf) = Mi ∀i = 1, . . . , N
where the βj and Mi are problem parameters. The controls for the optimization
problem are the measurable functions ρj, νij : [0, µf ] → [0, 1] that are subject to the
constraints
N∑
j=1
ρj = 1 a.e. µ,
N∑
i=1
νij = 1 a.e. µ ∀i = j, . . . , d.
Rather than attack the problem directly at this point, we instead make the
following observation. If the effective tensor K∗(µf) is optimal for one of the bounds
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we have discussed in the previous chapters (with the volume fractions given by the
Mi), then K∗(µf) is a solution to the optimal control problems with the parameters
chosen appropriately. The next section shows how simplify the control problem by
ensuring that the structure saturates the translation bound. Such a structure is
necessarily optimal. After we have completed the construction, we can easily find
the βj and Mi of the optimal control problem the structure solves.
4.3 The modified differential scheme
In this section, we describe a convenient method for finding optimal multimate-
rial structures in three dimensions, using a modification of the differential scheme
of the last section. The scheme is modified to incorporate the optimality conditions
of the fields at each step. To keep the calculations simple, we will construct only
isotropic composites by requiring
ρ1(µ) = ρ2(µ) = ρ3(µ) =1
3, (4.3)
νi1(µ) = νi
2(µ) = νi3(µ) = νi(µ) for i = 1, 2, 3, (4.4)
K∗1(µ) = diag(ηn(µ), ηt(µ), ηt(µ)), (4.5)
K∗2(µ) = diag(ηt(µ), ηn(µ), ηt(µ)), (4.6)
K∗3(µ) = diag(ηt(µ), ηt(µ), ηn(µ)). (4.7)
The scheme generalizes easily to anisotropic structures, however, and produces a
set of structures equivalent to those of the previous chapter.
Consider a three-material translation-optimal composite. After rescaling the
average field, the fields in the phases are described by (P1)–(P3) of the previous
chapter, which we repeat here for the reader’s convenience.
(P1) DU = DUT and tr DU = 1 a.e. in Ω1.
(P2)∫
Ω1DU2 dx = E2 = 1
2(E + ET ) − 1
3tr E I.
(P3) DU = k1
ki+2k1I a.e. in Ωi for i = 2, 3.
The construction of optimal isotropic structures proceeds as follows. We begin with
an initial core of the second material K2, which we assume is subject to an average
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isotropic field k1
k2+2k1I. Then at each step in the differential scheme process we add
three orthogonal layers of a transversely isotropic composite, K13, of materials K1
and K3 formed by placing a cylindrical inclusion of K3 into a matrix of K1.
At the same time, we alter to average field so that the field in the core does
not change. We further choose the composite K13 to ensure that (P1)–(P3) can be
satisfied. The infinitesimal layers are always oriented so that the axis of cylindrical
inclusions in K13 coincides with normal to the layer.
Remark 4.1 While it is impossible to describe the infinite-rank laminate in finite
length scales, the following coated spheres structure is useful for visualizing its main
features. A spherical core of K2 is surrounded by a spherical layer of K1 stuffed
with radially-oriented conical inclusions of K3. The cones become thicker with the
increase of the radius.
4.3.1 Matrix laminates in replaced strips
The three-dimensional transversely isotropic extremal structure can be assem-
bled either as coated cylinders, or as second-rank matrix laminates, or as Vigdergauz-
type structures (see for example Cherkaev (2000); Milton (2002)). In all cases, the
volume fractions can be chosen in such a way that it is possible to satisfy (P1)–(P3).
We show the calculation using the matrix laminate structure here. We do this only
for the composite that will fill the strip [0, dµ/3] × [0, 1] × [0, 1]. The composites
for the other strips are simply rigid rotations of this composite (see (4.3)-(4.7)).
We begin by laminating K1 and K3 in the x2-direction. In order to satisfy
(P1)–(P3) and the rank-one connectedness condition, the fields in K1 and K3 must
be
E1 = diag
(k1
k3 + 2k1,
k3
k3 + 2k1,
k1
k3 + 2k1
)
and E3 =k1
k3 + 2k1I
respectively. Laminating these two fields with relative proportions ν11 and 1− ν11,
we obtain a laminate composite with average field
E13 = diag
(k1
k3 + 2k1
,ν11k3 + (1 − ν11)k1
k3 + 2k1
,k1
k3 + 2k1
)
.
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Now we laminate this composite with K1 in the x3 direction. The field in
this new layer of K1, which satisfies the rank-one connectedness condition and the
optimality conditions on the fields is
E ′1 = diag
(k1
k3 + 2k1,ν11k3 + (1 − ν11)k1
k3 + 2k1,ν11k1 + (1 − ν11)k3
k3 + 2k1
)
.
Laminating this with the composite with average field E13 in relative proportions
ν12 and 1 − ν12, we obtain a matrix laminate with average field
E113 = diag
(k1
k3 + 2k1,ν11k3 + (1 − ν11)k1
k3 + 2k1,(1 − ν12(1 − ν11))k1 + ν12(1 − ν11)k3
k3 + 2k1
)
.
We will keep the average field in the core proportional to the identity, so for E113
to be rank-one connected to the core, we need first of all that the fields in the x2
and x3 direction are equal:
ν11k3 + (1 − ν11)k1
k3 + 2k1=
(1 − ν12(1 − ν11))k1 + ν12(1 − ν11)k3
k3 + 2k1
which requires
ν12 =ν11
1 − ν11.
The field in the infinitesimal strip is controlled by the single parameter ν11 and is
given by
E113 = diag
(k1
k3 + 2k1,ν11k3 + (1 − ν11)k1
k3 + 2k1,ν11k3 + (1 − ν11)k1
k3 + 2k1
)
.
The volume fraction ν1 of K1 in this composite is given by
ν1 = ν12 + (1 − ν12)ν11 =ν11
1 − ν11+
ν11(1 − 2ν11)
1 − ν11= 2ν11. (4.8)
Furthermore, since we know piecewise approximations to the fields, we can find
the effective tensor for this composite to be
K∗1 = (ηn, ηt, ηt), (4.9)
ηn = 2ν11k1 + (1 − 2ν11)k3, (4.10)
ηt =k1(ν11k1 + (1 − ν11)k3)
ν11k3 + (1 − ν11)k1
. (4.11)
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4.3.2 The optimal controls
Now we are in a position to produce optimal controls for the problem described
in Section 4.2.3. As mentioned previously, we initiate the process with a pure block
of material K2
K∗(0) = K2, m2(0) = 1, m1(0) = m3(0) = 0.
Recall that to simplify calculations, we have chosen to keep K∗ isotropic for all µ,
so we may write K∗(µ) = λ(µ)I for a scalar function λ. In order to check (P1)–(P3)
for each µ, we also track the average field, E(µ), in the homogenized composite,
K∗(µ). In fact, it will be enough to assume E(µ) = γ(µ)I for all µ. To satisfy
(P1)–(P3) at µ = 0, we set
γ(0) =k1
k2 + 2k1
.
Suppose for some µ ≥ 0, we have constructed a composite in which we can
satisfy the field conditions (P1)–(P3) and that the average field for which these
conditions are satisfied is E(µ) = γ(µ)I = γI. To find the composites for the
replacement strips for the next infinitesimal step, we need to choose ν11 so that the
field E(µ) and E113 are rank-one connected. Solving, we find
ν11 =γ(k3 + 2k1) − k1
k3 − k1. (4.12)
Of course, we must make sure that we satisfy the physical constraints 0 ≤ ν11 ≤ 1,
which requiresk1
k3 + 2k1≤ γ ≤ k3
k3 + 2k1. (4.13)
We will show later that this constraint is always satisfied. For now, note that γ(0)
satisfies it and so we expect to be able to proceed for at least some small range of
µ.
Using (4.12) together with (4.9)-(4.11), we find
ηn = (1 − 2γ)(k3 + 2k1), (4.14)
ηt = k1
(k3 + k1
γ(k3 + 2k1)− 1
)
. (4.15)
87
Provided we know γ(µ), then, we can find the optimal controls. By analyzing the
transmission conditions on the fields, we find that γ changes at each infinitesimal
step as
γ(µ + dµ) =
(
1 − dµ
3
)
γ(µ) +dµ
3
k1
k3 + 2k1
= γ(µ) + dµ
(1
3
(k1
k3 + 2k1
− γ(µ)
))
.
Thus, with the infinitesimal strips chosen in this way, γ satisfies the differential
equation
dγdµ
= 13
(k1
k3+2k1− γ)
,
γ(0) = k1
k2+2k1.
Solving, we find
γ(µ) = e−µ/3 k1
k2 + 2k1
+ (1 − e−µ/3)k1
k3 + 2k1
. (4.16)
This combined with (4.14) and (4.15) provides the optimal control for the problem
until the final structure is obtained at µ = µf provided (4.13) is satisfied for every
µ ∈ [0, µf ]. But these inequalities hold for any µ ≥ 0 since for such a µ, γ(µ) is a
convex sum of two quantities, each of which satisfies the inequalities.
4.4 The optimal structures
From (4.2), we know that K∗(µ) = λ(µ)I satisfies the differential equation
dλdµ
= 13λ(
1 − ληn
)
+ 23(ηt − λ),
λ(0) = k2
(4.17)
where ηn, ηt are functions of µ defined through (4.14) and (4.15) respectively which
depend on γ(µ) defined in (4.16). Suppose λ(µ) is a solution to this equation. Then
for any µf ≥ 0, the corresponding structure with effective tensor K∗(µf) = λ(µf)I
is optimal for the three dimensional translation bound.
Let us now make more precise the earlier statement that these structures are
equivalent to those discussed in the previous chapter. Consider the isotropic case
of (3.12) where ω1 = ω2 = ω3 = 3√
m2 and K∗ = λI. Then
λ =k3(k2 + 2k1) − 2 3
√m2 k1(k3 − k2)
3√
m2(k3 + 2k1) + (1 − 3√
m2)(k2 + 2k1). (4.18)
88
By (4.1), we find
m2(µ) = e−µ.
Substituting this into (4.18), we find that the function
λ(µ) =k3(k2 + 2k1) − 2e−µ/3 k1(k3 − k2)
e−µ/3(k3 + 2k1) + (1 − e−µ/3)(k2 + 2k1)
solves (4.17).
The final quantity to check is the volume fraction m1. From (4.1), we find
m1(µ) =3k1(k3 − k2)
(k2 + 2k1)(k3 − k1)
(e−µ/3 − e−µ
)
=3k1(k3 − k2)
(k2 + 2k1)(k3 − k1)
(3√
m2(µ) − m2(µ))
.
Comparing this to (3.13), we see that the modified differential scheme produces
isotropic structures with exactly the same effective tensors and volume fractions as
the three-dimensional block structures described in the previous chapter.
4.5 Discussion
This construction is easily extended to larger numbers of materials (N > 3).
Choosing the initial core to be K2 was convenient, but we may start with any core
we wish as long as (P1)–(P3) are satisfied there. Additionally, instead of adding
a matrix laminate at any given step, we can also add a layer of pure K1 because
the optimal field in this material can always be brought into rank-one connection
with whatever field is in the core. As we mentioned before, there is no need to
require that the final structure be isotropic. Instead, we can choose the ρj ≥ 0 with
ρ1 + ρ2 + ρ3 = 1 and materials in each infinitesimal strip however we like as long
as we can satisfy the field requirements. The general scheme is as follows.
At any step and in any direction, we may add one of up to N different types of
layers: N − 1 types of composites of some Ki in a matrix of K1 or a layer of pure
K1. Some bookkeeping is required, but the idea is straightforward. At any step we
can add either pure K1 or any matrix laminate for which we can choose the volume
fractions to satisfy (P1)–(P3). The general scheme produces structures equivalent
to all currently known structures optimal for the translation bound. Furthermore,
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this scheme is not limited to dimension d = 3. Indeed for d = 2, the scheme also
produces structures equivalent to all currently known structures optimal for the
two-dimensional translation bound.
Finally, we remark that the differential scheme can be used for other types of
problems. In particular, we point the reader to Albin and Cherkaev (2006) where a
similar scheme was applied to the example of generating optimal three-dimensional
polycrystals. The result was consistent with the earlier analysis of the fields in these
structures due to Nesi and Milton (1991) and shows that the modified differential
scheme may be useful in a larger class of optimization problems.
CHAPTER 5
NUMERICALLY ESTIMATING THE
G-CLOSURE
5.1 Introduction
In this chapter, we introduce a numerical method for estimating the G-closure
from inside. The method is a finite-dimensional version of the differential scheme
discussed in the last chapter.
In Section 5.2, we introduce the finite scheme, which produces hierarchical
laminates by alternately laminating the current composite to the side and to the
top with simple laminates and re-homogenizing. The evolution of the structures are
controlled at each step by six parameters. In Section 5.3, we rephrase the optimal
control problem from the previous chapter in terms of the finite scheme.
In Section 5.4, we illustrate some numerical results obtained by combining
the finite scheme with a nonlinear optimizer to seek a numerical solution to the
optimization problem. We apply the scheme to the G-closure problem of three ma-
terials in two dimensions and to the problem of finding the most resistive isotropic
structure for a range of volume fractions. In the latter case, we also plot the local
fields in the laminates and discuss some interesting properties they exhibit.
5.2 A finite scheme for laminate structures
In this section, we discuss an algorithm for obtaining a numerical inner bound
on the G-closure using a finite version of the differential scheme introduced in the
last chapter. We work with the two-dimensional problem and with three materials
whose isotropic conductivities are given by k1, k2, k3 > 0. The generalizations to
higher dimension and more materials are obvious.
91
The basic step for the scheme consists of two parts and depends on six param-
eters: µ1, µ2, ν11 , ν
12 , ν
21 , ν
22 ∈ [0, 1] with the additional constraint
ν1k + ν2
k ≤ 1 k = 1, 2.
The νik define two auxiliary laminates in a similar way to the differential scheme.
Specifically, νik gives the volume fraction of the ith material in the kth auxiliary
laminate for i, k = 1, 2.
For k = 1, the materials are laminated with vertical normal n = (0, 1)T forming
a material with eigenvalues
η11 = ν11k1 + ν2
1k2 + (1 − ν11 − ν2
1)k3, η12 =
(ν1
1
k1+
ν21
k2+
1 − ν11 − ν2
1
k3
)−1
directed in the horizontal and vertical directions respectively. A similar construction
for k = 2 is done with all normals rotated 90 degrees. This gives a laminate
materials with eigenvalues
η21 =
(ν1
2
k1+
ν22
k2+
1 − ν12 − ν2
2
k3
)−1
, η22 = ν12k1 + ν2
2k2 + (1 − ν12 − ν2
2)k3
directed in the horizontal and vertical directions respectively.
The two parts of the basic step, which resembles a step of the differential scheme
without the assumption of the smallness of the change, are as follows.
(I) Laminate the current composite with the first laminate (eigenvalues η11 and
η12) in the x1 direction with relative fractions 1 − µ1 and mu1 respectively.
Homogenize the new composite.
(II) Laminate the current composite with the second laminate (eigenvalues η21
and η22) in the x2 direction with relative fractions 1−µ2 and µ2 respectively.
Homogenize the new composite.
This basic step can be repeated any number of times to produce a composite.
We will classify the different schemes obtained as N-step schemes. The one-step
scheme begins with a seed material and produces a new composite by performing (I)
and then (II); the two-step scheme begins with the seed and performs (I), then (II),
92
then (I) and then (II); and so on. From a theoretical standpoint, the seed material
is irrelevant (unless it cannot be obtained by the N -step scheme to begin with)
since the choice of µ1 = 1 or µ2 = 1 are allowed in (I) and (II) which completely
replaces the existing material.
5.3 The optimization problem
Now, motivated by the differential scheme of the previous chapter, we fix num-
bers β > 0, M1, M2 ≥ 0 such that M1 + M2 ≤ 1 and consider the following
minimization problem (or the corresponding maximization problem)
min λ1
subject to the constraints
λ2 = β, m1 = M1, m2 = M2.
The controls of the problem are the values of the µk and νik at each step and λ1, λ2
are the eigenvalues of the effective tensor of the produced composite with volume
fractions given by m1, m2. λ1, λ2, m1, m2 are computed from the µk and νik through
the lamination formula applied by (I) and (II). For the constraints to be satisfied,
we must have that β lies inside the arithmetic-harmonic mean bounds:
(3∑
i=1
Mi
ki
)−1
≤ β ≤3∑
i=1
Miki
where M3 = 1 − M1 − M2.
5.4 Numerical results
In this section, we present numerical results for the optimization problem. The
finite scheme was implemented in the C programming language and the optimiza-
tion was performed by the nonlinear optimizer DONLP21. The initial guess was
chosen to be the T 2-structure (not necessarily optimal) with the correct volume
fractions. Experimentally, we found that the nonlinear optimizer worked best if we
1http://www.netlib.org/opt/donlp2/
93
slowly increased the number of degrees of freedom. For this reason, we began by
optimizing the one-step scheme then used the results to seed the two-step scheme
and used this to seed the three-step scheme. In passing from the (N − 1)-step
scheme to the N -step scheme, we retain the previously optimized structure as a
small core (about 1% of the new material). The rest of the composite is made of
two additional laminate layers, which maintain the volume fraction constraints.
5.4.1 Numerical estimates of G-closures
In Figures 5.1–5.2, we show the results after the three-step scheme has been
numerically optimized in this way. In all cases the conductivities for the problem
are
k1 = 1, k2 = 2, k3 = 5,
and β ranged through 50 values between the arithmetic and harmonic mean. Both
the minimization and maximization problems were performed.
Figure 5.1 displays the results corresponding to Figures 2.2–2.1. Recall that the
volume fractions for this problem are
M1 = 0.4, M2 = 0.01.
The numerically optimized conductivity tensors are indicated by dots. The arith-
metic/harmonic mean bounds and the translation bounds are indicated by dashed
lines. Note that the translation bounds are difficult to see since they are covered
by the dots.
Figure 5.2 displays the results corresponding to Figure 2.12. The volume frac-
tions are
M1 = 0.104, M2 = 0.5 (5.1)
and
M1 = 0.25, M2 = 0.5
respectively. In addition to the numerically optimized points indicated by solid
dots, we also show the inner and outer bounds on the G-closure discussed in
Section 2.3.10. Observe that the numerical scheme does better than the naive
analytic inner bound.
94
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Figure 5.1. A numerical estimation of the G-closure for M1 = 0.4, M2 = 0.01.
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Figure 5.2. A comparison of numerical results to known inner bounds.
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5.4.2 Numerical estimates of the isotropic bound
In Figures 5.3–5.4, we consider a similar problem. In this case, m1 is allowed to
vary and the constraints in the earlier optimization problem are replaced by
λ2 = λ1, m2 = p(1 − m1)
for some p ∈ [0, 1]. This problem characterizes the most resisting isotropic com-
posite for the various volume fractions given through m1. The figures illustrate the
results for p = 0.5 and p = 0.1 respectively. For reference, we have also included
the limits of construction (minimum m1) for the results of Milton (1981); Gibiansky
and Sigmund (2000), marked by the points M and GS respectively. For values of
m1 > 0 and below the GS point, it is not known whether the translation bound is
optimal. Observe that the numerical scheme tracks the translation bound when it
is known to be optimal and then leaves the bound (but remains very close!) beyond
this point.
5.4.3 Fields in numerically optimized structures
Finally, we remark that the fields in the numerically optimized structures can be
found automatically by satisfying the jump conditions in each layer. In Figures 5.5–
5.5 we illustrate the results corresponding to the second case above (m2 = 0.1(1 −m1)). We have split the data into two parts. Figure 5.5 displays the fields when the
translation bound is known to be optimal. Figure 5.6 displays the other cases. The
data in both figures come from numerically optimizing structures for 50 equally-
spaced values of m1 in the respective range. The fields are computed by assuming
an isotropic average field scaled so that the average field in the third material is
the same for each composite.
Figure 5.5 is much as we expect from the field optimality conditions. The fields
tend to cluster about two points and a line. Once we move from this region of m1,
however, the fields that were free to move along the line suddenly appear to cluster
about two points while the fields in the second material gain a degree of freedom.
This gives some indication of where to look for new bounds on the G-closure. We
expect the field optimality conditions to have the following properties.
96
u
u
M
GS
bbbbbbbbb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
b
bb
bb
b
b
b
b
b
b
m1
λ
Figure 5.3. The effective conductivity of numerically optimized isotropic resistorsfor m2 = 0.5(1 − m1). The points M and GS represent the limits of applicabilityof the results of Milton, and Gibiansky and Sigmund, respectively.
u
u
M
GS
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
m1
λ
Figure 5.4. The effective conductivity of numerically optimized isotropic resistorsfor m2 = 0.1(1 − m1). The points M and GS represent the limits of applicabilityof the results of Milton, and Gibiansky and Sigmund, respectively.
97
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
local diagonal fieldsconstant trace line
isotropic line
Figure 5.5. The local fields in the numerically optimized structures used inFigure 5.4 in the case when the bounds are known to be optimal
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
local diagonal fieldsconstant trace line
isotropic line
Figure 5.6. The local fields in the numerically optimized structures used inFigure 5.4 in the case when the bounds are not known to be optimal
98
1. The field in K3 is constant and isotropic.
2. The field in K1 is always rank-one connected to the constant isotropic field
of K3
3. The field in K2 exhibits a degree of “freedom,” which allows it to take values
along some manifold. (This manifold should degenerate to the line required
by the two-material translation bounds when m1 = 0.)
CHAPTER 6
CONCLUSION
We have seen that the use of the sufficient conditions on pointwise fields allows
us to methodically generate structures optimal for the translation bound from a very
general class of structures. We illustrated this in Chapter 2 for two-dimensional
iterated laminates, in Chapter 3 for a class of two- and three-dimensional block
structures and in Chapter 4 for a class of infinite rank-laminates generated by a
controllable differential scheme.
Each of these classes of structures is extremely rich and nonunique in the sense
that several different structures produce the same effective tensor. This makes it
difficult to guess the best members of the class. On the other hand, by requiring
that the sufficient conditions on the fields are satisfied, we easily generate such
members from each class. In fact, since we require the field optimality conditions
to hold at each step of the process, the approach cannot create a composite which
is not optimal. Furthermore, up to a “coating argument,” this method produces
equivalent optimal structures from each general class of structures. The method
not only reproduces previous results, but also shows that the G-closure for multi-
material conducting composites in two or three dimensions is strictly larger than
was previously known.
In Chapter 5, we introduced a simple numerical scheme for optimizing a class of
structures. This scheme proves very effective in several examples of estimating an
inner bound on the G-closure. Furthermore, when applied to the problem of best
resisting isotropic composites, it is consistent with known results and gives some
idea of field optimality conditions that would arise in a bound improving on the
translation bound.
100
In Section 2.5, we presented some evidence that the classification of optimality of
the translation bounds for three-phase composites in the plane is nearly known. We
introduced a supplementary bound on the anisotropy of a structure which attains
the translation bound and satisfies certain conditions. We conjecture that this
bound can also be shown to hold for general three-phase composites. If this is true,
then the question of attainability of the translation bound for this problem will be
completely closed for isotropic composites.
The achievement of this dissertation has been to show that a very careful
analysis of the field optimality conditions in several general classes of laminate
composites provides a pronounced improvement of the known region of optimality
of the translation bounds. We have shown the effectiveness of such an approach for
the translation bounds in linear conductivity with isotropic phases. We believe that
such methods are also possible for a much wider range of physical phenomena. In
particular, as we mentioned earlier (Remark 2.10), the work of Grabovsky (1996)
allows these results to immediately apply to certain cases of linear elasticity. Our
hope is that the work presented in this dissertation will give rise to new systematic
methods for classifying G-closure problems.
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