Original article Biomath 1 (2012), 1209021, 1–5

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 ISSN 1314-684X

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h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum

Multiscale Analysis of Composite Structures

Claudia Timofte
Faculty of Physics

University of Bucharest, Bucharest, Romania
Email: claudiatimofte@yahoo.com

Received: 15 July 2012, accepted: 2 September 2012, published: 9 October 2012

Abstract—The goal of this paper is to present some
homogenization results for diffusion problems in composite
structures, formed by two media with different features.
Our setting is relevant for modeling heat diffusion in
composite materials with imperfect interfaces or electrical
conduction in biological tissues. The approach we follow
is based on the periodic unfolding method, which allows
us to deal with general media.

Keywords-homogenization; the periodic unfolding me-
thod; dynamical boundary condition

I. INTRODUCTION AND SETTING OF THE PROBLEM

The analysis of diffusion phenomena in highly hetero-
geneous materials has been a subject of huge interest in
the last decades. The purpose of this paper is to analyze
the effective behavior of the solution of some nonlinear
problems arising in the modeling of diffusion in a
periodic structure formed by two media with different
properties, separated by an active interface. Our setting
is relevant for modeling heat conduction in composite
materials with imperfect interfaces or electrical conduc-
tion in biological tissues. We assume first that both
components are connected. Using the periodic unfolding
method, which allows us to deal with general heteroge-
neous media, we can describe the evolution in time of
the homogenized solution. The model we obtain at the
macroscale is a bidomain model, which conceives the
composite material, despite of its discrete structure, as
the coupling of two continuous superimposed domains.
Our model is a generalization of the so-called Barenblatt
model, arising in the context of diffusion in partially
fissured media. A similar model appears also in the study
of the bioelectrical activity of the heart at a macroscopic

level. In this case, at the microscopic scale, we deal with
a medium composed of two different conductive phases
(the intracellular and extracellular spaces), separated by
a dielectric interface (the cellular membranes), which
has a capacitive and a nonlinear conductive behavior.
The electric potential verifies elliptic equations in the
two conductive regions, coupled by a suitable evolutive
boundary condition involving the potential jump at the
interfaces between the two phases. The evolution in
time of the homogenized potential is governed exactly
by a bidomain model. We shall also briefly discuss a
different geometric situation, in which only one phase is
connected, while the other one is disconnected. In this
case, we are led at a different macroscopic model.

Let Ω be a bounded domain in Rn (n ≥ 3), with a
Lipschitz boundary ∂Ω consisting of a finite number of
connected components. We consider the case in which
Ω is a periodic structure formed by two components,
Ωε and Πε, representing two materials with different
features, separated by an interface Sε. We assume that
both Ωε and Πε = Ω \ Ωε are connected, but only
Ωε reaches the external fixed boundary of the domain
Ω. Here, ε represents a small parameter related to the
characteristic size of the our two regions. More precisely,
let Y1 be a Lipschitz open connected subset of the unit
cell Y = (0, 1)n. Let Y2 = Y \ Y1. We suppose that Y2
has a locally Lipschitz boundary Γ and we assume that
the intersections of the boundary of Y2 with the boundary
of Y are identically reproduced on opposite faces of the
cell, which are denoted, for any 1 ≤ i ≤ n, by

Σi = {y ∈ ∂Y | yi = 1},

Σ−i = {y ∈ ∂Y | yi = 0}.

Citation: C. Timofte, Multiscale Analysis of Composite Structures, Biomath 1 (2012), 1209021,
http://dx.doi.org/10.11145/j.biomath.2012.09.021

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C. Timofte, Multiscale Analysis of Composite Structures

We suppose that repeating Y by periodicity, the union of
all the sets Y1 is connected and has a locally C

2 boun-
dary. Also, we assume that the origin of the coordinate
system is set in a ball contained in this union (see [8]).

Let
Zε = {k ∈ Zn | εk + εY ⊆ Ω},

Kε = {k ∈ Zε | εk ± εei + εY ⊆ Ω, ∀i = 1, n},

where ei are the elements of the canonical basis of Rn.
We define

Πε = int(
⋃

k∈Kε

(εk + εY2))

and
Ωε = Ω \ Πε

and we set θ =
∣∣Y \ Y2∣∣.

Let α1, β1 ∈ R such that 0 < α1 < β1. We denote
by M(α1, β1, Y ) the set of all the square matrices
A ∈ (L∞(Y ))n×n such that, for any ξ ∈ Rn, we have
(A(y)ξ, ξ) ≥ α1 | ξ |2, | A(y)ξ |≤ β1 | ξ |, almost
everywhere in Y . We consider a family of matrices
Aε(x) = A(

x

ε
) defined on Ω, where A ∈ M(α1, β1, Y )

is a symmetric smooth Y -periodic matrix. We shall
denote the matrix A by A1 in Y1 and by A2, respectively,
in Y2.

If (0, T ) is the time interval, we shall analyze the
macroscopic behavior of the solutions of the following
system:


−div (Aε1∇u
ε) + β(uε) = f in Ωε × (0, T ),

−div (Aε2∇v
ε) = f in Πε × (0, T ),

Aε1∇u
ε · ν = Aε2∇v

ε · ν on Sε × (0, T ),

Aε1∇u
ε · ν + αε

∂

∂t
(uε − vε) =

aεg(vε − uε) on Sε × (0, T ),
uε = 0 on ∂Ω × (0, T ),
uε(0, x) − vε(0, x) = c0(x) on Sε.

(1)

Here, ν is the unit outward normal to Ωε, a > 0, f ∈
L2(0, T ; L2(Ω)), c0 ∈ H10 (Ω), α > 0 and β and g are
continuous functions, monotonously non-decreasing and
such that β(0) = 0 and g(0) = 0. We shall suppose that
there exist a positive constant C and an exponent q, with
0 ≤ q ≤ n/(n − 2), such that

|β(v)| ≤ C(1 + |v|q), |g(v)| ≤ C(1 + |v|q). (2)

As examples of such functions, we mention the case of
Langmuir or Freundlich kinetics. For the case of elec-
trical conduction in biological tissues, we may consider
that f = 0, β = 0 and g is, in R3, a cubic function, like
in the Fitzhugh-Nagumo model (see, for instance, [9]).

Results concerning the well posedness of problem (1)
in suitable function spaces and proper energy estimates
were obtained in [1], [3] and [9].

Using the periodic unfolding method recently intro-
duced by D. Cioranescu, A. Damlamian, G. Griso, P.
Donato and R. Zaki (see [4] and [5]), we can prove
that the asymptotic behavior of the solution of our
problem is governed by a new nonlinear system (see (3)).
At a macroscopic level, the composite material can be
represented by a continuous model, which describes it as
the superimposition of two interpenetrating continuous
media, coexisting at every point of the domain. Our
macroscopic model is a degenerate parabolic system, as
the time derivatives involve the unknown v − u.

If we deal with a different geometry, i.e. we consider
that only one phase is connected, while the other one
is disconnected, we are led to a different macroscopic
model (see Remark 2.).

Similar problems have been considered, using diffe-
rent techniques, in [1] and [9], for studying electrical
conduction in biological tissues.

The results presented in this paper constitute a gene-
ralization of those obtained in [2], [8] and [10].

As already mentioned, our approach is based on the
periodic unfolding method, which allows us to deal with
general media (see Remark 3.). For dealing with such
two-component domains, we use unfolding operators,
which map functions defined on oscillating domains into
functions defined on fixed domains. In such a way, we
can avoid the use of extension operators. Therefore,
using this general method, we can deal with media
with less regularity than those usually considered in the
literature (composite materials and biological tissues are
highly heterogeneous and their interfaces are not very
smooth, in general).

The plan of the paper is as follows: in the second
section, we give the main convergence result of this
paper. The last section is devoted to the proof of our
result.

II. THE MAIN RESULT

Using the periodic unfolding method, we can pass to
the limit in the variational formulation of problem (1)
and we obtain the effective behavior of the solution of
our microscopic model.

Theorem 1. The solution (uε, vε) of system (1)
converges, as ε → 0, to the unique solution (u, v), with
u, v ∈ L2(0, T ; H10 (Ω)),

∂

∂t
(u − v) ∈ L2(0, T ; L2(Ω))

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C. Timofte, Multiscale Analysis of Composite Structures

and u, v ∈ C0([0, T ]; H10 (Ω)), of the following macro-
scopic problem:



α | Γ |
∂

∂t
(u − v) − div (A1∇u) + θβ(u)−

a | Γ | g(v − u) = θf in Ω × (0, T ),

α | Γ |
∂

∂t
(v − u) − div (A2∇v)+

a | Γ | g(v − u) = (1 − θ)f in Ω × (0, T ),
u(0, x) − v(0, x) = c0(x) on Ω.

(3)

In (3), A
1

and A
2

are the homogenized matrices, defined
by:

A
1
ij =

∫
Y1

(
aij + aik

∂χ1j
∂yk

)
dy,

A
2
ij =

∫
Y2

(
aij + aik

∂χ2j
∂yk

)
dy

and χ
1k

∈ H1per(Y1)/R, χ2k ∈ H
1
per(Y2)/R, k =

1, ..., n, are the weak solutions of the cell problems

−∇y · ((A1(y)∇yχ

1k
) = ∇yA1(y)ek, y ∈ Y1,

(A1(y)∇yχ
1k

) · ν = −A1(y)ek · ν, y ∈ Γ,

−∇y · ((A2(y)∇yχ

2k
) = ∇yA2(y)ek, y ∈ Y2,

(A2(y)∇yχ
2k

) · ν = −A2(y)ek · ν, y ∈ Γ.

So, at a macroscopic scale, we obtain a new system,
which is similar to the bidomain model, appearing in the
context of diffusion in partially fissured media or in the
context of electrical activity of the heart (for this case,
f = 0, β = 0).

Remark 2. If we consider the case of a different
geometry, i.e. if we assume that Ωε is still connected,
but Πε is disconnected, then the homogenized matrix
A

2
= 0 and system (3) consists in the coupling of a

partial differential equation and an ordinary differential
one.

III. PROOF OF THE MAIN RESULT

We shall only sketch the proof of our main conver-
gence result. For details, we refer to [11].

Let us consider the variational formulation of problem
(1):∫ T

0

∫
Ωε

Aε1∇u
ε ·∇ϕdxdt +

∫ T
0

∫
Πε

Aε2∇v
ε ·∇ϕdxdt+

∫ T
0

∫
Ωε

β(uε)ϕdxdt + αε
∫ T

0

∫
Sε

(uε − vε)
∂

∂t
[ϕ]dσdt+

αε

∫
Sε

(uε − vε)(0)[ϕ](0)dσ+

aε

∫ T
0

∫
Sε

g(vε − uε)[ϕ]dσdt =
∫ T

0

∫
Ω

f ϕdxdt, (4)

for any ϕ ∈ L2(Ω × (0, T )) such that

ϕ|Ωε ∈ L
2(0, T ; H1(Ωε)), ϕ|Πε ∈ L

2(0, T ; H1(Πε)),

[ϕ] ∈ H1(0, T ; L2(Sε)),

ϕ vanishes on ∂Ω×(0, T ) and ϕ vanishes at t = T . Here,
we have denoted by [ϕ] the difference of the traces of
ϕ|Ωε and ϕ|Πε on S

ε.
There exists a unique weak solution (uε, vε) of (4),

with uε ∈ L2(0, T ; H1∂Ω(Ω
ε)), vε ∈ L2(0, T ; H1(Πε)),

where

H1∂Ω(Ω
ε) = { u ∈ H1(Ωε) | u = 0 on ∂Ω ∩ ∂Ωε}.

Under the above hypotheses on the data, we can obtain
suitable a priori estimates, independent of ε, for our
solution (see [6], [9] and [10]):∫ t

0

∫
Ωε

Aε1∇u
ε · ∇ϕdxdt +

∫ t
0

∫
Πε

Aε2∇v
ε · ∇ϕdxdt+

ε

∫
Sε

(uε − vε)2dσ ≤ C,

where 0 < t < T and C is independent of ε.
For dealing with such two-component domains, we

use two unfolding operators, T ε1 and T
ε
2 , which map

functions defined on the oscillating domains into func-
tions defined on fixed domains. In such a way, we can
avoid the use of extension operators (see [4] and [7]).
Also, we shall make use of the boundary unfolding
operator, T εb , introduced in [5]. Therefore, using the
above mentioned unfolding operators, we can prove that
there exist u, v ∈ L2(0, T ; H10 (Ω)), û ∈ L

2((0, T ) ×
Ω; H1per(Y1)), v̂ ∈ L2((0, T ) × Ω; H1per(Y2)) such that,
up to a subsequence, for ε → 0, we have:

T ε1 (u
ε) → u strongly in L2((0, T ) × Ω, H1(Y1)),

T ε1 (∇ u
ε) ⇀ ∇ u+∇yû weakly in L2((0, T )×Ω×Y1),

T ε2 (v
ε) → v strongly in L2((0, T ) × Ω, H1(Y2)),

T ε2 (∇ v
ε) ⇀ ∇ v+∇yv̂ weakly in L2((0, T )×Ω×Y2).

Moreover,

∂

∂t
(u − v) ∈ L2(0, T ; L2(Ω))

and
u, v ∈ C0([0, T ]; H10 (Ω)).

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C. Timofte, Multiscale Analysis of Composite Structures

In order to obtain the limit problem (3), we take,
in a first step, Φ1, Φ2 ∈ C∞0 (Ω) = D(Ω) and Ψ ∈
C∞0 ((0, T )) = D(0, T ). We have:∫ T

0

∫
Ωε

Aε1∇u
ε · ∇Φ1Ψdxdt+

∫ T
0

∫
Πε

Aε2∇v
ε · ∇Φ2Ψdxdt +

∫ T
0

∫
Ωε

β(uε)Φ1Ψdxdt

+αε
∫ T

0

∫
Sε

(uε − vε)(Φ2 − Φ1)
dΨ
dt

dσdt+

aε

∫ T
0

∫
Sε

g(uε − vε)(Φ2 − Φ1)Ψdσdt =

∫ T
0

∫
Ωε

f Φ1Ψdxdt +
∫ T

0

∫
Πε

f Φ2Ψdxdt. (5)

Applying the corresponding unfolding operators in (5)
and passing to the limit, with ε → 0, we get (see, for
details, [4], [6], [10] and [11]):∫ T

0

∫
Ω×Y1

A1(∇u + ∇yû) · ∇Φ1Ψdxdydt+

∫ T
0

∫
Ω×Y2

A2(∇v + ∇yv̂) · ∇Φ2Ψdxdydt+

∫ T
0

∫
Ω×Y1

β(u)Φ1Ψdxdydt+

α

∫ T
0

∫
Ω×Γ

(u − v)(Φ2 − Φ1)
dΨ
dt

dxdσdt+

a

∫ T
0

∫
Ω×Γ

g(u − v)(Φ2 − Φ1)Ψdxdσdt =

∫ T
0

∫
Ω×Y1

f Φ1Ψdxdydt+

∫ T
0

∫
Ω×Y2

f Φ2Ψdxdydt. (6)

In a second step, we take the test functions wεi =
εΦi(x)ϕi(

x

ε
)Ψ(t), with i = 1, 2, where Φ ∈ D(Ω), ϕi ∈

H1per(Yi), Ψ ∈ D((0, T )). Observing that T εi (w
ε
i ) → 0

strongly in L2((0, T )×Ω×Yi) and T εi (∇w
ε
i ) → Φi∇yϕi,

strongly in L2((0, T )×Ω×Yi), we can pass to the limit
and we get:∫ T

0

∫
Ω×Y1

A1(∇u + ∇yû) · ∇yϕ1Φ1Ψdxdydt+

∫ T
0

∫
Ω×Y2

A2(∇v + ∇yv̂) ·∇yϕ2Φ2Ψdxdydt = 0. (7)

Putting together (6) and (7) and using standard density
arguments, we obtain exactly the variational formulation
of the limit problem (3). We can easily pass to the
limit, with ε → 0, in the initial condition and we obtain
u(0, x) − v(0, x) = c0(x), ∀x ∈ Ω.

As u and v are uniquely determined (see [9]), the
whole sequences of microscopic solutions converge to a
solution of the unfolded limit problem and this completes
the proof of Theorem 1.

Remark. 3 The above results can be extended to
the case in which Aε is a sequence of matrices in
M(α1, β1, Ω) such that

T εi (A
ε) → A a.e. in Ω × Y,

with i = 1, 2 and A = A(x, y) ∈ M(α1, β1, Ω × Y ).
The only difference is that in this case the homogenized
matrices are no longer constant and depend on x.

IV. CONCLUSION

Using the periodic unfolding method, the effective
behavior of the solution of some problems arising in the
modeling of diffusion processes in a periodic structure
formed by two media with different properties, separated
by an active interface, was analyzed. Two interesting
geometric situations were discussed, leading to different
macroscopic models. Our setting is relevant for studying
the heat conduction in composite materials with imper-
fect interfaces or the electrical conduction in biological
tissues.

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http://dx.doi.org/10.11145/j.biomath.2012.09.021

	Introduction and Setting of the Problem
	The Main Result
	Proof of the Main Result
	Conclusion
	References