On hierarchical structures and reiterated homogenization∗

On hierarchical structures and reiterated
homogenization ∗
Dag Lukkassen
Narvik University College, P.O. Box 385, N-8505 Narvik, Norway
Graeme W. Milton
Department of Mathematics, The University of Utah,
Salt Lake City, U10 841112-0090, USA
Abstract
In this paper we consider the concept of reiterated homogenization,
introduced on the physical level by Bruggeman in the 30’s and justified
mathematically by Bensoussan, Lions and Papanicolaou in 1978. We
present and discuss some recent developments of this theory and also
give some applications to linear and nonlinear problems.
Keywords: Reiterated homogenization, integral functionals, nonlinear partial
differential operators, bounds
MSC 2000: 35-99, 49-99
1 Introduction
Strongly non-homogeneous structures have fascinated people for a very long
time. Archaeological observations in Finland show that fibre-reinforced ceramics
were made about 4000 years ago, and that people already at this
∗ This paper is published in: Function Spaces, Interpolation Theory and Related Topics.
Proceedings of the International Conference in Honour of Jaak Peetre on his 65th Birthday,
Lund, Sweden, August 17-22, 2000, 355-368, Walter de Gruyter, Berlin 2002.
1

time had ideas and theories for intelligent combinations of materials and
structures. Analysis of the macroscopic properties of composites was investigated
by the physicists Maxwell, Rayleigh, and Einstein, among many others.
Around 1970 the problem of determining the physical properties of material
structures and composites was reformulated in such a way that this field became
interesting from a purely mathematical point of view. This formulation
initiated a new mathematical discipline called homogenization theory.
Reiterated homogenization was introduced on the physical level by Bruggeman
already in the 30’s. The mathematical justification of this theory was
given by Bensoussan, Lions and Papanicolaou in 1978. In this paper we
present and discuss some recent developments of this theory and also give
some applications to linear and nonlinear problems.
2 The periodic case
The conductivity problem on a periodic material structure with period equal
to 1/h can be formulated by the following minimum principle:
µ Z
Eh =min
u
Fh(u) − u(x)g(x) dx
Ω
, (1)
where
Z
Fh(u) =
¡ 2
λ(hx) |Du(x)| ¢ dx.
Ω
Here, λ(·) (the conductivity) is periodic relative to the unit-cube I of Rn and bounded between two strictly positive constants, Ω is a bounded open
subset of Rn and g is the source-field. The minimization is taken over some
suitable (subset of a) Sobolev-space which takes care of the given boundary
conditions. It is possible to prove that the energy Eh converges to a ”homogenized”
energy Ehom, as h →∞, of the form
Ehom =min
u
µ
Fhom(u) −
where Fhom is of the form
Z
Fhom(u) =
Ω
Z
Ω
u(x)g(x) dx , (2)
fhom(Du(x))dx
2

and
fhom(ξ) = min
W 1,2
per (I)
Z
λ(x) |ξ + Du(x)|
I
2 dx. (3)
Here, W 1,2
per (I) is the space of I-periodic functions of the Sobolev space W 1,2 (I).
Therefore, since the actual energy Eh (which is interesting for us to determine)
is difficult to find when h is large, and since fhom(ξ) and Ehom can
be found by many numerical methods, we can use Ehom as an approximation
of Eh. The convergence of energies is usually seen as a consequence of
Γ-convergence of the corresponding Lagrangians (see the definition below).
As an alternative to (1) we can formulate the conductivity problem via
the corresponding Euler equation
− div λ(hx)Duh(x) =g,
together with the given boundary conditions. It is possible to prove that the
solution uh (in this case the minimizer of (1)) converges weakly in the above
mentioned Sobolev space to the solution u (in this case also the minimizer
of (2)) of the ”homogenized” equation
− div AhomDu(x) =g.
Here, Ahom is defined by
Z
Ahomξ = λ(x) ¡ Du ξ (x)+ξ ¢ dx,
I
where uξ (in this case also the minimizer of (3)) is the I-periodic solution of
div ¡ λ(x)(Du ξ (x)+ξ) ¢ =0.
The convergence results above were first proved by De Giorgi and Spagnolo
in the late 60’s. Various kinds of simplifications of this proof were done
in the 70’s by Murat, Tartar, Bakhvalov, Bensoussan, Lions and Papanicolaou
(for more information, see e.g. the book [16]).
3 Reiterated homogenization of integral functionals
Let us consider the class of Lagrangians g such that g(x, ξ) is measurable in
x, convexinξand satisfying the standard growth condition
−c0 + c1 |ξ| p ≤ g(x, ξ) ≤ c0 + c2 |ξ| p , (4)
3

where c1,c2 > 0 and p>1. If gh and g belong to this class we recall that g
is the Γ-limit of the sequence gh, denoted g = Γ−lim gh, if for any bounded
open set Ω with Lipschitz boundary the following two conditions hold:
(i) for any uh ∈ W 1,p (Ω), uh uweakly in W 1,p (Ω) it holds that
Z
Ω
g(x, Du)dx ≤ lim inf
h→∞
Z
Ω
gh(x, Du)dx,
(ii) for every u ∈ W 1,p (Ω) there is a sequence uh such that uh uweakly
in W 1,p (Ω) uh − u ∈ W 1,p
0 (Ω),
Z
Ω
g(x, Du)dx = lim
h→∞
Z
Ω
gh(x, Du)dx.
Let f(y, z, ξ) be I-periodic and measurable in the first and second variable,
respectively. Moreover, assume that f is piecewise continuous in the
first variable, i.e. of the form f(y, z, ξ) = PN i=1 χΩi (y)fi(y, z, ξ), where fi
satisfies
|fi(y, z, ξ) − fi(y 0 ,z,ξ)| ≤ ω(|y − y 0 |)(a(z)+fi(y, z, ξ))
for all y, y 0 ,z,ξ ∈ R n ,whereω and a are continuous positive real functions
with ω(0) = 0. Inthethirdvariableweletf be convex and satisfying the
growth condition (4).
Theorem 1 We have the existence of the Γ-limit fhom = Γ−lim fh where
fh(x, ξ) =f(hx, h 2 x, ξ). Moreover,
fhom(ξ) =f [2] (ξ),
where f [2] is found iteratively according to the following scheme:
f [2] (ξ) = min
u∈W 1,p
Z
f
per (I) I
[1] (y, ξ + Du(y))dy
Z
f [0] (x,y, ξ + Du(y))dy
f [1] (x, ξ) = min
u∈W 1,p
per (I)
f [0] (x, y, ξ) = f(x, y, ξ).
4
I

Figure 1: The laminate structure of rank 2. Here, m = 2 (h is a fixed
number).
The above theorem is easily extended to the case fh(x, ξ) =f(hx, ..., h m x, ξ)
in which
fhom(ξ) =f [m] (ξ),
where f [m−j] is found iteratively according to the following scheme:
Z
f [m−j−1] (x1, ..., xj,y, ξ + Du(y))dy
f [m−j] (x1, ..., xj,ξ) = min
u∈W 1,p
per (I)
f [0] (x1, ..., xm,ξ) = f(x1, ..., xm,ξ).
I
In Figure 1 and Figure 2 we have illustrated examples where fh(x, ξ) =
λ(hx, h 2 x) |ξ| 2 .InFigure1
λ(x, y) =2− k(x1)k(y2)
and in Figure 2
λ(x, y) =2− k(x1)k(x2)k(y1)k(y2),
where
½
1
0 t ∈ [0 + n, + n>
k(t) =
3
1 t ∈ [ 1
,nis an integer.
+ n, 1+n> 3
The conductivity λ(hx, h2x) takes the value 2 when x is in the black part
and the value 1 when x is in the white part, respectively.
5

Figure 2: The reiterated cube structure for m =2(h is a fixed number).
Remark 1 Theorem 1 of Braides and Lukkassen [9, 22] is a generalization
of the case of quadratic forms given by Bensoussan, Lions and Papanicolaou
in the book [7] where the concept of reiterated homogenization was introduced.
Later on Müller, Braides and Defranceschi generalized this result to standard
non-convex Lagrangians (see [8]). The corresponding proofs are quite different.
Remark 2 In its form Theorem 1 (and likewise its non-convex cousin) is
a natural generalization of the periodic case, and agrees with the physical
intuition that the effective properties should be obtained by first homogenizing
the medium on the finest microlevel and next on the second one. However,
in the case of non-standard Lagrangians, satisfying the growth condition
−c0 + c1 |ξ| q ≤ f(x, y, ξ) ≤ c0 + c2 |ξ| p , (5)
an unexpected phenomenon occurs: In contrast to the periodic case the limit
Γ−lim fh may not exist (see [25])! However, by compactness we always have
the existence of Γ-converging subsequences. It is possible to show that all such
limits are sharply bounded between two Lagrangians whose representation can
be found iteratively by a more general scheme than that above. These bounds
are strongly dependent on the power q and p of the Sobolev-spaces involved
and they are (certainly) equal when q = p. Non-trivial examples can be found
where these bounds are attained.
6

4 Reiterated homogenization of differential
operators
We consider the class of partial differential equations of the form
− div(aε(x, Duε)) = F on Ω, uε ∈ W 1,p
0 (Ω), (6)
where Ω is an open bounded subset of R n , 1

where v ξ is the unique solution of the cell-problem
R
Y (b1(y, ξ + Dv ξ (y)),Dφ) dy =0for every φ ∈ W 1,p
per (Y ),
v ξ ∈ W 1,p
per (Y ).
Moreover, the operator b1 : Y × Rn → Rn is given by
b1(y, ξ) = 1
Z
a(y, z, ξ + Dv
|Z|
ξ,y (z)) dz,
Z
where v ξ,y is the unique solution of the cell-problem
R
Z (a(y, z, ξ + Dvξ,y (z)),Dφ) dz =0for every φ ∈ W 1,p
per (Z),
v ξ ∈ W 1,p
per (Z).
Definition 3 We say that a sequence uε 3-scale converges to u0(x, y, z) ∈
Lp (Ω,Y,Z) if
Z ³
lim uε(x)ϕ x,
ε→∞
x x
,
ε ε2 ´
dx =
Ω
1
Z
u0(x, y, z)ϕ (x, y, z) dx dy dz
|Y ||Z| Ω×Y ×Z
for all ϕ ∈ D(Ω; C ∞ per(Y,Z)) (which means functions ϕ(x, y, z) being C ∞ with
compact support in x with values of functions which are C ∞ and periodic in
y and z).
It is possible to extend Theorem 2 to the case aε(x, ξ) =a(x, x/ε, x/ε 2 ,ξ).
In the following theorem we let aε be of this form.
Theorem 4 Let uε be a solution of (6). Then as ε → 0, it holds that uε 3scale
converges to u0 ∈ W 1,p
0 (Ω) and Duε 3-scale converges to Du0 + Dyu1 +
Dzu2, where{u0,u1,u2} is a solution of
Z
1
a(x, y, z, Du0 + Dyu1 + Dzu2)(Dv0 + Dyv1 + Dzv2)dx dy dz =
|Y ||Z| Ω×Y ×Z
Z
= Fv0dx
for all v0 ∈ W 1,p
0 (Ω), v1 ∈ L p (Ω; W 1,p
per (Y )), v2 ∈ L p (Ω; W 1,p
per (Y × Z)).
Ω
8

Remark 3 The iterated homogenization theorem for monotone operators,
Theorem 2, was proved by Lions, Lukkassen, Persson and Wall in [20, 21].
Remark 4 The concept of multiscale convergence was used by Allaire and
Briane [2] to study linear homogenization problems with several scales (which
generalizes the concept of 2-scale convergence of Allaire and Nguetseng [1,
37]). The proof of the nonlinear version given in Theorem 4 can be found in
[21].
Remark 5 Theorem 2 and Theorem 4 can easily be extended to the case
when aε is on the form aε(x, ξ) =a(x,x/ε,...,x/ε m ,ξ). Following an idea
of Bensoussan and Lions [6] the case of infinitely many scales for linear
problems was also studied by Allaire and Briane [2]. It seems to be possible
to generalize such results to monotone problems as well.
4.1 Differential effective medium theory
Since reiterated homogenization is quite natural from a physical point of
view (see Remark 2) it does not come as a surprise that reiterated techniques
were used long before the homogenization theory itself was developed. Such
techniques led to the so called differential effective medium theory (DEM)
introduced in the 30’s by Bruggeman [10] for materials with two phases and
generalized by Norris to multiphase materials [38]. For two-phase materials
the theory is roughly speaking as follows: Let C be the tensor of a ”matrix
material”, let C1 be the tensor of the ”inclusion material” and let c be the
volume fraction of the inclusion material. Moreover, suppose that for small
c and all C we have
Ceff = C + cQ(C)+O(c 2 ), (7)
where Q is a continuous function. Then, the system of ordinary differential
equations
dC
= Q(C(t))
dt
gives a representation of the realizable effective properties which can be obtained
from an initial material C0 by iterating (7) (an incremental procedure).
It was rigorously proved by Milton [30, 31] that Bruggeman’s differential
scheme with spherical inclusions corresponds to a differential microstructure.
Later on this result was generalized by Avellaneda [4].
9

For more detailed information on DEM and its application we refer to
Avellaneda [4]. Other interesting variants and applications can be found
in the works of Beliaev and Kozlov [5], Jikov and Kozlov [15] and Kozlov
[18]. For the case of DEM and reiterated homogenization of linear equations
with random coefficients, see Kozlov [17]. DEM can also be used in case of
nonlinear problems (see [26]).
5 Further applications
By combining some suitable bounds for nonlinear homogenized functionals
with the nonlinear versions of the iterated homogenization above it is possible
to give very sharp estimates of the homogenized functional also in cases
when it cannot be computed exactly. These estimates can be used to study
some two-component reiterated structures with rather surprising macroscopic
behavior. In [9] and [22] three types of structures were analyzed: the laminate
structure (Figure 1), the iterated cube structure (Figure 2) and a mixed
iterated structure (Figure 5), the latter being a mixture of the first two
structures and a structure of chess-board type. In particular, in [9] and [22]
we pointed out some cases where the macroscopic behaviors of the iterated
cube structure and the mixed reiterated structure possess a higher or lower
effective energy density than that of the best possible laminate structure of
rank n (where n isthedimensionofthespace). Thefactthatthishasproven
to be impossible for two-phase linear conductivity problems makes these
structures particularly interesting (it should be mentioned that infinite rank
laminates do as well as these structures in the non-linear case). Moreover,
the results in [9] imply that the effective properties corresponding to the
mixed reiterated structure are extremely sensitive to the growth on the local
energy density. Other examples of surprising phenomena have been reported
for nonlinear iterated checkerboard structures [25].
Reiterated homogenization of linear elliptic operators and effective differential
medium theory has proven to be an important tool in the construction
of iterated laminates and other iterated structures with optimal effective behavior.
The discoveries so far even include globally isotropic structures with
negative Poisson’s ratio (for more information see Milton [33] and Lakes [19]).
Concerning optimal structures and bounds on effective material properties
in general we refer to the collection of classical papers [12], where the introduction
gives a good selection of references, and particularly the recent book
10

Figure 3: Iterated hexagonal honeycombs (illustrated for m =3).
of Milton [34]. Here, we only mention a few examples of optimal iterated
structures. Concerning optimal iterated laminates for the conductivity problem
see Lurie and Cherkaev [27], Milton [29], Schulgasser [42] and Tartar
[45]. On optimal microstructures for the elasticity case in two-dimensions
(transversely isotropic case) there exist periodic (i.e. non-iterated) geometries
found by Vigdergauz [46, 47, 48, 49, 50] which are optimal for the effective
bulk modulus. A very readable treatment of his work can be found in
[14]. There is also a new class of optimal iterated structures found recently
by Sigmund [43]. For the effective shear modulus, there appear to be no
”simple microstructures”. All known ones have structure on at least 5 wellseparated
length scales (iteration levels). They simultaneously attain both
the shear and bulk modulus bounds. The result that the Hashin-Shtrikman
shear bound is attained is in essence in an article of Roscoe [41]. The only
missing part is a rigorous proof that the differential scheme (in the effective
differential medium theory) corresponds to some geometry. Such a proof can
be found in Avellaneda [4]. Independent of this, the attainability of the shear
modulus bound has been shown by Norris [38], Francfort and Murat [13] and
Milton [32]. There was also an appendix in a preprint of Lurie and Cherkaev,
written during the same time, that showed realizability. Unfortunately their
paper was published without the appendix. Another iterated structure which
yieldsthesameeffective behavior can be found in [24] (by similar methods
one can also verify that this even holds for iterated triangular honeycombs).
Moreover, in that paper (see also [22]) an iterated cube structure consisting
of m iteration levels was analyzed for the elasticity and the conductivity
problem. In particular it was proven that this structure turns optimal in the
class of square symmetric structures when m →∞for the elasticity case and
11

Figure 4: Globally isotropic structures with negative Poisson’s ratio (of Milton).
also optimal with respect to effective conductivity (recently, we have become
aware of the fact that in the conductivity case this was already proven in
[17] for a very similar structure. The derivation, though, is different). Similar
results were obtained for iterated hexagonal honeycombs in [23] for the
conductivity case (see Figure 3). It is also possible to prove optimality for
the effective bulk modulus for these structures. For numerical computations
on such structures we refer to [11].
For non-well ordered materials the same microstructures attain the bulk
modulus bounds. In three dimensions the Walpole bound on the shear moduluscanbeimprovedasshownin[28,35].
Intwodimensionsitisnotknown
if the Walpole bounds can be improved. Microgeometries found by Sigmund
(see above) come close to attaining them. There are also a number of papers
on optimal microgeometries for multiphase materials (see [34] and the
references given there). Besides these structures there is also another familyofmicrostructuresintroducedin[34]inthenameofpartial
differential
microstructures.
6 A final comment
The homogenized integrand fhom(ξ) can be seen as some kind of average of
the local function fh(·,ξ). Inthelaminatecaseweevenhaveadirectlink
to the well known arithmetic and harmonic mean. A similar example is the
12

Figure 5: A mixed iterated structure.
checkerboard structure for which fhom(ξ) =g(λ) |ξ| 2 , where g(λ) denotes the
geometric mean. Power means and compositions of such means often serve as
upper and lower bounds for fhom(ξ) (see e.g. [22], [26], [36] and [44]). Hence,
there is an obvious link between reiterated homogenization and iteration of
means. The latter subject has fascinated many (see e.g. Jaak Peetre [3], [39],
[40] and the references given there).
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17