Applied Mathematics and Mechanics (English Edition) Reiterated

Appl. Math. Mech. -Engl. Ed., 39(8), 1119–1146 (2018)

Applied Mathematics and Mechanics (English Edition) https://doi.org/10.1007/s10483-018-2352-6

Reiterated homogenization of a laminate with imperfect contact: gain-enhancement of effective properties∗ 1 ´ F. E. ALVAREZ-BORGES , J. BRAVO-CASTILLERO1,2,† , M. E. CRUZ3 , 4 ´ ´ R. GUINOVART-DÍAZ1 , L. D. PEREZ-FERN ANDEZ , 1 2 R. RODRÍGUEZ-RAMOS , F. J. SABINA

1. Facultad de Matem´ atica y Computaci´ on, Departamento de Matem´ atica, Universidad de La Habana, San L´ azaro y L, Habana 4, La Habana, CP 10400, Cuba; 2. Departamento de Matem´ aticas y Mec´ anica, Universidad Nacional Aut´ onoma de México, Instituto de Investigaciones en Matem´ aticas Aplicadas y en Sistemas, 01000 CDMX, AP 20-126, México; 3. Departamento de Engenharia Mecˆ anica, Universidade Federal do Rio de Janeiro, Politécnica/COPPE, Caixa Postal 68503, Rio de Janeiro, RJ, CEP 21941-972, Brasil; 4. Departamento de Matem´ atica e Estat´ıstica, Universidade Federal de Pelotas, Caixa Postal 354, Pelotas, Rio Grande do Sul, CEP 96010-900, Brasil (Received Jul. 8, 2017 / Revised Mar. 14, 2018)

Abstract A family of one-dimensional (1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method (RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed ´ ∗ Citation: ALVAREZ-BORGES, F. E., BRAVO-CASTILLERO, J., CRUZ, M. E., GUINOVART´ ´ DÍAZ, R., PEREZ-FERN ANDEZ, L. D., RODRÍGUEZ-RAMOS, R., and SABINA, F. J. Reiterated homogenization of a laminate with imperfect contact: gain-enhancement of effective properties. Applied Mathematics and Mechanics (English Edition), 39(8), 1119–1146 (2018) https://doi.org/10.1007/s10483-018-2352-6 † Corresponding author, E-mail: [email protected] Project supported by the Desenvolvimento e Aplica¸co ˜es de Métodos Matem´ aticos de Homogeneiza¸ca õ (CAPES) (No. 88881.030424/2013-01), the Homogeneiza¸ca õ Reiterada Aplicada a Meios Dependentes de M´ ultiplas Escalas con Contato Imperfeito Entre as Fases (CNPq) (Nos. 450892/2016-6 and 303208/2014-7), and the Caracterizaci´ on de Propiedades Efectivas de Tejidos Biol´ ogicos Sanos y Cancerosos (CONACYT) (No. 2016–01–3212) c Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

´ F. E. ALVAREZ-BORGES et al.

1120

here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces. Key words reiterated homogenization method (RHM), imperfect contact, variational formulation, effective coefficient gain Chinese Library Classification O241.82 2010 Mathematics Subject Classification

1

35B27, 74Q05, 74Q15

Introduction

A typical feature of problems in nanofluids[1] , bone mechanics[2–3] , soil physics[5] , and porous media[6] , etc. is that the studied objects are heterogeneous media exhibiting several structural scales. In general, the mathematical modeling of such situations involves differential equations with rapidly-oscillating piecewise-smooth coefficients depending on several local or fast variables corresponding to such structural scales. The application of homogenization techniques to determine the macroscopic properties of such media plays an important role in the knowledge of the interest phenomena. Some basic elements from the homogenization theory may be found in Ref. [7]. A rigorous mathematical tool for this purpose is the reiterated homogenization method (RHM) introduced in Ref. [8]. Mathematically, the RHM is an asymptotic method. It allows to transform problems, which involve differential equations with periodic and rapidly oscillating coefficients dependent on several local variables, into the problems where the coefficients are not rapidly oscillating. The latter problems are known as homogenized problems, and their coefficients are the so-called effective coefficients of the original heterogeneous media. A crucial step is to show the proximity between the solutions of both types of the problems. The existence of several scales directly affects the effective coefficients, and it makes other intermediate coefficients appear, which are related to each structural scale. In addition, the presence of the structural imperfections related to the piecewise-continuous character of the coefficients must be considered by adding the contact conditions relevant to each physical situation under the analysis. The RHM has been applied to multidimensional elliptic problems by various authors. In the pioneering work of Ref. [8], the RHM was first introduced for scalar elliptic problems depending on three scales by using the asymptotic expansion. These results were generalized in Ref. [9] for several microscopic scales by introducing the multiscale convergence method. In Ref. [3], the Γ-convergence method was applied to an elastic problem with the coefficients depending on four scales. A detailed description of the periodic unfolding method and how it was applied to periodic homogenization multiscale elliptic problems could be found in Ref. [10]. However, in the classical books[11–12] , the reiterated homogenization was not studied. An interesting review of the different mathematical techniques for conventional two-scale homogenization was reported in Ref. [13]. The goal of this work is to illustrate, from the beginning to the ending, the main stages of the RHM by applying some basic elements of the theory of differential equations and functional analysis so as to provide an accessible material for a wide audience for the applied mathematicians, scientists, and engineers interested in this field. To this aim, the RHM is applied to a family of one-dimensional (1D) elliptic boundary-value problems with periodic and rapidly oscillating coefficients depending on two microscopic scales and a finite number of discontinuities. A mathematical justification of the asymptotic process is given based on an estimate. Although, the investigated family of problems is just as simple as to admit the analytical solutions, it is suitable to unfold the new aspects of the behavior of the RHM solution with respect to the conventional homogenization solution. This study could provide a relevant extension of previous perfect-contact RHM analyses[6,8,13]. The following content of this paper is organized as follows. In Section 2, a formal asymp-

Reiterated homogenization of a laminate with imperfect contact

1121

totic solution for a family of 1D two-point boundary-value problems with imperfect contact conditions is constructed. The expressions for the solutions of the local problems and the effective coefficients are given, and so is the statement of the related homogenized problem. A necessary and sufficient condition for the existence of periodic solutions is stated. In Section 3, some elements of the Sobolev space theory are extended to open non-connected sets. The variational formulation of the problem is considered. Finally, with the Lax-Milgram lemma, the convergence of the solution of the original problem to the solution of the homogenized one is proven. In Section 4, an example with piecewise-differentiable coefficients is developed mainly to illustrate the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts.

2

Reiterated homogenization for the 1D elliptic equation with piecewise differentiable coefficients and discontinuity conditions

In this section, the problem is stated. Following Refs. [8], [11], and [14], a formal asymptotic solution is constructed in order to obtain the local problems, the effective coefficients, and the homogenized problem. 2.1 Problem settings Let 1 2 Ω∗y = [0, 1]\{yi }li=1 , Ω∗z = [0, 1]\{zi}li=1 . The periodic extension to R2+ of the real function a(y, z) is differentiable, positive, and bounded for all (y, z) ∈ Ω∗y × Ω∗z . Then, a(y, z) can be regarded as 1-periodic with respect to y and z, and there exist α− , α+ ∈ R∗+ such that 0 < α− a(y, z) α+ for all (y, z). Let F (x) be a piecewise continuous real function in [0, 1]. Define the contrast operator as follows: [[F (ξ)ξi ]] = lim+ F (ξ) − lim− F (ξ) = F (ξ + ) − F (ξ − ). ξ→ξi

ξ→ξi

Let ε = 1/n (n ∈ N). The problem is to find uε (x) such that duε d ε a (x) = f (x), x ∈ Ω∗ε = (0, 1) \ xkij , − dx dx ε du ε aε (x) = βki [[uε (x)]]xkij , dx xkij

(1) (2)

duε aε (x) = 0, dx xkij

(3)

uε (0) = t0 ,

(4)

uε (1) = t1 ,

where aε (x) = a

x = a(y, z), ε ε2

x

,

t0 , t1 ∈ R.

The numbers xkij represent the discontinuity points of aε (x) in the interval (0, 1), i.e., x1ij = ε(yi + j), x2ij = ε2 (zi + j),

i = 1, 2, · · · , l1 , i = 1, 2, · · · , l2 ,

j = 0, 1, · · · , n − 1, j = 0, 3, · · · , n2 − 1.

ε That is, there are l1 n + l2 n2 discontinuity points, and βki = ε−k βki , where βki ∈ R+ (i = 1, 2, · · · , lk , and k = 1, 2) indicate the microscales where the related discontinuities take place.

1122


Since the condition in Eq. (2) becomes [[uε (x)]]xkij = 0 as βki → ∞, uε (x) is continuous. In the context of composite media, this continuity condition together with the condition in Eq. (3) are known as perfect contact conditions. The expressions related to the homogenization of the problems with the perfect contact conditions are obtained by taking the limit as βki → ∞ in the more general expressions to be obtained for Eqs. (1)–(4). In the context of heat conduction, uε (x) is the temperature distribution in the heterogeneous medium, aε (x) is its thermal conductivity, and f is the heat source. In addition, Eq. (3) is the continuity condition on the heat flux, and Eq. (2) is a discontinuity condition for the temperature to state that the size of the heat flux across each discontinuity point is proportional to the temperature change at the point. The proportionality constants are the thermal contact ε conductances βki , at the discontinuity points xkij , related to the Biot number defined in Ref. [15]. 2.2 Homogenization, local problems, and effective coefficients The first step in this homogenization process is to construct a formal asymptotic solution for Eqs. (1)–(4) as follows: u(m) (x, ε) =

m

εi ui (x, y, z),

(5)

i=0

where ui (x, y, z) (i = 0, 1, · · · , m) are piecewise differentiable and 1-periodic with respect to the local or fast variables y = x/ε and z = x/ε2 . By substituting the expansion (5) into Eqs. (1)–(4), using the chain rule d ∂ ∂ ∂ = + ε−1 + ε−2 , dx ∂x ∂y ∂z and equating the terms corresponding to equal powers of ε, we can obtain the following recurrent family of partial differential equations: ε−4 : Lzz u0 = 0,

(6)

ε−3 : Lzz u1 = −Lzy u0 − Lyz u0 ,

(7)

ε−2 : Lzz u2 = −Lzy u1 − Lzx u0 − Lyz u1 − Lyy u0 − Lxz u0 ,

(8)

ε−1 : Lzz u3 = −Lzy u2 − Lzx u1 − Lyz u2 − Lyy u1 − Lyx u0 − Lxz u1 − Lxy u0 ,

(9)

ε0 : Lzz u4 = −Lzy u3 − Lzx u2 − Lyz u3 − Lyy u2 − Lyx u1 − Lxz u2 − Lxy u1 − Lxx u0 − f (x),

(10)

and εi : Lzz ui+4 = − Lzy ui+3 − Lzx ui+2 − Lyz ui+3 − Lyy ui+2 − Lyx ui+1 − Lxz ui+2 − Lxy ui+1 − Lxx ui

(11)

for 1 i m − 4, where, with p, q ∈ {x, y, z}, the differential operators Lpq are defined as follows: ∂ ∂ a(y, z) . Lpq = ∂p ∂q Analogously, the contact conditions can be expressed as follows:


1123

(i) For ε−2 , ∂u0 a(y, z) ∂z yi ∂u0 a(y, z) ∂z zj ∂u0 a(y, z) ∂z yi ∂u0 a(y, z) ∂z zj

= 0,

(12)

= β2j [[u0 ]]zj ,

(13)

= 0,

(14)

= 0.

(15)

(ii) For ε−1 , ∂u 1 a(y, z) ∂z ∂u 1 a(y, z) ∂z ∂u 1 a(y, z) ∂z ∂u 1 a(y, z) ∂z

∂u0 = β1i [[u0 ]]yi , ∂y yi ∂u0 + = β2j [[u1 ]]zj , ∂y zj ∂u0 + = 0, ∂y yi ∂u0 + = 0. ∂y zj +

(16) (17) (18) (19)

(iii) For ε0 ,

∂u 2 a(y, z) ∂z ∂u 2 a(y, z) ∂z ∂u 2 a(y, z) ∂z ∂u 2 a(y, z) ∂z

∂u1 ∂u0 + ∂y ∂x yi ∂u1 ∂u0 + + ∂y ∂x zj ∂u1 ∂u0 + + ∂y ∂x yi ∂u1 ∂u0 + + ∂y ∂x zj +

= β1i [[u1 ]]yi ,

(20)

= β2j [[u2 ]]zj ,

(21)

= 0,

(22)

= 0.

(23)

(iv) For εi (1 i m − 4),

∂u i+4 a(y, z) ∂z ∂u i+4 a(y, z) ∂z ∂u i+4 a(y, z) ∂z ∂u i+4 a(y, z) ∂z

∂ui+3 ∂ui+2 + ∂y ∂x yj ∂ui+3 ∂ui+2 + + ∂y ∂x zj ∂ui+3 ∂ui+2 + + ∂y ∂x yj ∂ui+3 ∂ui+2 + + ∂y ∂x zj +

= β1j [[ui+3 ]]yj ,

(24)

= β2j [[ui+4 ]]zj ,

(25)

= 0,

(26)

= 0.

(27)

1124


The boundary conditions (4) for the unknown functions ui (x, y, z) are u0 (0, 0, 0) = t0 ,

(28)

u0 (1, n, n2 ) = t1 ,

(29)

ui (0, 0, 0) = ui (1, n, n2 ) = 0,

i 1.

(30)

The following lemma guarantees the existence of the 1-periodic solutions of the recurrent chain of Eqs. (6)–(30), and, consequently, allows the construction of a formal asymptotic solution for Eqs. (1)–(4). Lemma 1 Let a(ξ), F0 (ξ), and F1 (ξ) be 1-periodic piecewise-differentiable functions with finite jump discontinuities in ξ ∈ {ξ1 , ξ2 , · · · , ξn } (0 < ξj < 1) and positive a(ξ) bounded over [0, 1]. Then, a necessary and sufficient condition for the existence of a 1-periodic solution N (ξ) of the problem dN d dF1 Lξξ N ≡ a(ξ) = F0 + , ξ ∈ {ξ1 , ξ2 , · · · , ξn }, (31) dξ dξ dξ dN − F1 (ξ) a(ξ) = βj [[N (ξ)]]ξj , βj > 0, (32) dξ ξj dN − F1 (ξ) a(ξ) =0 (33) dξ ξj is

1

F0 (ξ)ξ ≡

F0 (ξ)dξ = 0.

(34)

0

Remark 1 The lemma is stated for functions depending on a single variable. Functions depending on several variables can be considered as functions of a single variable by assuming that the other variables are parameters with fixed values. Remark 2 If N (ξ) is a 1-periodic solution of Eqs. (31)–(33), there exists a family of 1 (ξ) = N (ξ) + c, where c ∈ R is an arbitrary constant, i.e., the periodic solutions given by N solution is unique up to an additive constant. Thus, without loss of generality, the condition N (0) = 0 can be added to Eqs. (31)–(33) in order to guarantee the uniqueness of the 1-periodic solution. Now, Lemma 1 is applied to each of the problems formed by Eqs. (6)–(10) and the corresponding conditions (12)–(23). First, we consider the functions ui of the form ui (x, y, z) = Ni (x, y, z) + Mi (x, y) + vi (x),

(35)

where Ni (x, y, z), Mi (x, y), and vi (x) are infinitely piecewise-differentiable functions. The main results obtained after the application of Lemma 1 to Eqs. (6)–(10) are the first and second local problems along with the homogenized problem. The first local problem, for each fixed y, is enunciated as to find the 1-periodic solution N (y, z) for ∂ ∂N a(y, z) + a(y, z) = 0, z = zj , (36) ∂z ∂z ∂N + a(y, z) a(y, z) = β2j [[N (y, z)]]zj , (37) ∂z zj ∂N + a(y, z) a(y, z) = 0, (38) ∂z zj


N (y, 0) = 0.

1125

(39)

This problem also satisfies the hypotheses of Lemma 1 with ξ = z, F0 = 0, and F1 = −a(y, z) for every y. Therefore, it has a solution N (y, z) which is 1-periodic with respect to z. Besides, according to Remark 2, Eq. (39) could ensure the uniqueness of the solution. Since Eq. (36) states that a(y, z) ∂N ∂z + a(y, z) is independent of z, we can introduce the intermediate effective coefficient a(y) (the effective coefficient related to the first microscale of the original problem) as follows: a(y) = a(y, z)

∂N + a(y, z). ∂z

(40)

Some manipulations of this relation yield l2 −1 −1 a(y) = a−1 (y, z)z + β2j .

(41)

j=1

Note that a(y) is a positive and 1-periodic function of y. It is important to emphasize that the calculation of the effective coefficients depends on the solution of the respective local problems. Typically, these problems will demand numerical solutions for the physical situations of practical interest. The analytical solution of the first local problem (36)–(39) is N (y, z) = a(y)

z

0

ds −1 + − z, β2i a(y, s) i=0 j

(42)

where z ∈ (zj , zj+1 ), and j = 0, 1, · · · , l2 . The second local problem consists in finding the 1-periodic solution M (y) for the problem d dM a(y) + a(y) = 0, y = yi , (43) dy dy dM + a(y) a(y) = β1i [[M (y)]]yi , (44) dy yi dM + a(y) a(y) = 0, (45) dy yi M (0) = 0.

(46)

a(y). This problem also satisfies the hypotheses of Lemma 1 with ξ = y, F0 = 0, and F1 = − Therefore, it has a 1-periodic solution M (y). Since Eq. (43) states that a(y) dM + a (y) is dy independent of y, we can introduce the global effective coefficient a related to the macroscale of the original problem as follows: a= a(y)

dM + a(y). dy

(47)

Some manipulations in this relation give l1 l2 −1 −1 −1 −1 a = a (y, z)z y + β1j + β2j . j=1

(48)

j=1

Note that a is a positive constant. Therefore, it is 1-periodic. In the case of a single microscale, in which a(y, z) = a(y) is a piecewise-constant function that takes only two different values,


1126

a given by Eq. (48) becomes the 1D realization of the elementary lower bound on the effective conductivity reported in Eq. (3.1) of Ref. [16]. Moreover, it is interesting to note that, in higher-dimension single-microscale problems, the remaining sum in the corresponding version of Eq. (48) becomes an integral over the discontinuity surfaces. In the present case, the analytical solution of the second local problem (43)–(46) is M (y) = a 0

y

ds −1 + − y, β1i a(s) i=0 j

(49)

where y ∈ (yj , yj+1 ), and j = 0, 1, · · · , l1 . The so-called homogenized problem is − a

d2 v0 = f (x), dx2

x ∈ (0, 1),

v0 (0) = t0 ,

v0 (1) = t1 .

(50)

The solutions from those three problems play an important role in the expressions for functions u0 , u1 , u2 , and u3 . With Eq. (35), we have that N0 (x, y, z) = M0 (x, y) = 0, and v0 (x) is the solution of Eq. (50). For u1 (x, y, z), we have N1 (x, y, z) = 0, and M1 (x, y) = M (y)

dv0 , dx

(51)

where M (y) is the solution of the second local problem. The function v1 will be determined later. For u2 (x, y, z), we have dv dM 0 +1 , N2 (x, y, z) = N (y, z) dy dx

(52)

where N (y, z) is the solution of the first local problem. The function M2 (x, y) will be considered 0 1 to have the form M2 (x, y) = M20 (x, y)+ M (y) dv dx , where M2 (x, y) is the solution of the problem ∂M20 d2 v0 ∂ a(y) a3 (x, y) = 0, + a(y)M (y) 2 + ∂y ∂y dx

y = yi ,

(53)

∂M20 d2 v0 a(y) + a(y)M (y) 2 + a3 (x, y) = β1i [[M20 (x, y)]]yi , ∂y dx yi

(54)

∂M20 d2 v0 a(y) + a(y)M (y) 2 + a3 (x, y) = 0, ∂y dx yi

(55)

M20 (x, 0) = 0.

(56)

The function a3 (x, y) will be introduced later. Note that this problem satisfies the hypotheses of Lemma 1 with ξ = y,

F0 = 0,

F1 = − a(y)M (y)

d2 v0 − a3 (x, y) dx2

for every x. Then, the solution M20 (x, y) exists, and is 1-periodic in y. In addition, the equation of the problem for M20 (x, y) states that a(y)

d2 v0 ∂M20 + a(y)M (y) 2 + a3 (x, y) ∂y dx


1127

does not depend on y. Therefore, it is possible to define a(y) a2 (x) =

d2 v0 ∂M20 + a(y)M (y) 2 + a3 (x, y). ∂y dx

(57)

The function v2 (x) will be determined later. For u3 (x, y, z), we have ∂M d2 v0 dv1 2 + M (y) 2 + , N3 (x, y, z) = N30 (x, y, z) + N (y, z) ∂y dx dx

(58)

where N30 (x, y, z) is the 1-periodic solution of the following problem: ∂ ∂N30 ∂N2 a(y, z) + a(y, z) = 0, ∂z ∂z ∂y

z = zj ,

(59)

∂N30 ∂N2 a(y, z) + a(y, z) = β2j [[N30 (x, y, z)]]zj , ∂z ∂y zj

a(y, z)

(60)

∂N30 ∂N2 + a(y, z) = 0, ∂z ∂y zj

(61)

N30 (x, y, 0) = 0.

(62)

The problem (59)–(62) satisfies the hypotheses of Lemma 1 with ξ = z,

F0 = 0,

F1 = −a(y, z)

∂N2 ∂y

for every x and y. Therefore, the solution N30 (x, y, z) exists, is 1-periodic in z, and, according to Remark 2, is unique. The solution is N30 (x, y, z) = a3 (x, y)

z

0

ds −1 + − β2i a(y, s) i=0 j

z

0

∂N2 ds ∂y

(63)

for z ∈ (zj , zj+1 ) and j = 0, 1, · · · , l2 , where

∂N ∂N 0 ∂N2 2 3 + = a(y) a3 (x, y) = a(y, z) . ∂z ∂y ∂y z

(64)

The functions M3 (x, y) and v3 (x) will be determined later. Remark 3 Until now, the main results of the homogenization process have been stated with a fourth-order asymptotic expansion. Constructing higher-order formal asymptotic solutions is an induction process that requires the consideration of the asymptotic expansions of orders higher than the fourth in order to provide the information which completes Table 1. In Table 1, all the previous results are summarized, constituting the basis for an induction process developed next. Table 1

Functions involved in the asymptotic expansion

i

Ni0 (x, y, z)

Ni (x, y, z)

Mi0 (x, y)

Mi (x, y)

vi (x)

ai (x, y) b

e ai (x)

0

0

0

0

0

Eq. (50)

0

0

1

0

0

0

Eq. (51)

–

0

0

2

0

Eq. (52)

Eqs. (53)–(56)

–

–

0

Eq. (57)

3

Eq. (63)

–

–

–

–

Eq. (64)

–

4

–

–

–

–

–

–

–


1128

For the sake of mathematical completeness, from now on, for m 5, the recurrent family of problems formed by Eq. (11) with Eqs. (24)–(27) will be studied. For each i, Eqs. (11), (25), and (27) constitute a problem satisfying the hypotheses of Lemma 1 with ⎧ ⎪ ⎨ F0 = −Lyz ui+3 − Lyy ui+2 − Lyx ui+1 − Lxz ui+2 − Lxy ui+1 − Lxxui , ∂u ∂ui+2 i+3 ⎪ ⎩ ξ = z, F1 = −a(y, z) + ∂y ∂x for every x and y. Therefore, the existence of a solution ui+4 (x, y, z) of Eq. (11), which is 1-periodic in z, can be ensured by satisfying the necessary and sufficient condition Lyz ui+3 + Lyy ui+2 + Lyx ui+1 + Lxz ui+2 + Lxy ui+1 + Lxx ui z = 0.

(65)

In fact, the condition (65) is satisfied if F0 = 0 as it does not depend on z. Indeed, proceeding as above, it follows that ∂M ∂Mi+1 dvi+1 d ai+1 d2 vi ∂ i+2 ai+3 (x, y) + + + + + a 2. a(y) (66) −F0 = ∂y ∂y ∂x dx dx dx Thus, equating Eq. (66) to zero leads to the equation for ui+4 (x, y, z), Lzz ui+4 = −Lzy ui+3 − Lzx ui+2 , and ∂M ∂ ∂Mi+2 ∂ dvi+1 d ai+1 d2 vi i+1 a(y) =− ai+3 (x, y) + + − − a 2. a(y) ∂y ∂y ∂y ∂x dx dx dx This equation, together with the corresponding appropriate conditions, satisfies the hypotheses of Lemma 1 with ∂M d2 vi ∂Mi+1 dvi+1 d ai+1 i+2 − a 2 , F1 = − + + ξ = y, F0 = − ai+3 (x, y) − a(y) dx dx ∂y ∂x dx 2

i+1 for every x. The necessary and sufficient condition deadx + a ddxv2i y = 0 is satisfied if F0 = 0 as F0 does not depend on y, which together with the boundary conditions (30) lead to the problem for vi (x) as follows:

− a

d ai+1 d2 vi , = dx2 dx

x ∈ (0, 1),

vi (0) = vi (1) = 0.

(67)

Then, the solution Mi+2 (x, y) exists, is 1-periodic in y, and, according to Remark 2, is unique. Similarly, 0 Mi+2 (x, y) = Mi+2 (x, y) + M (y)

dvi+1 , dx

(68)

0 where Mi+2 (x, y) is the solution of the problem

0 ∂Mi+2 ∂Mi+1 ∂ a(y) + a(y) + ai+3 (x, y) = 0, ∂y ∂y ∂x

y = yj ,

(69)

0 ∂Mi+2 ∂Mi+1 + a(y) + ai+3 (x, y) = β1j [[M20 (x, y)]]yj , a(y) ∂y ∂x yj

(70)

0 ∂Mi+2 ∂Mi+1 + a(y) + ai+3 (x, y) = 0, a(y) ∂y ∂x yj

(71)

0 Mi+2 (x, 0) = 0.

(72)


1129

This problem satisfies the hypotheses of Lemma 1 with ξ = y,

F0 = 0,

F1 = − a(y)

∂Mi+1 − ai+3 (x, y) ∂x

0 (x, y) exists, is 1-periodic in y, and, according to for every x. Therefore, the solution Mi+2 Remark 2, is unique. 0 In addition, the equation of the problem for Mi+2 (x, y) states that

a(y)

0 ∂Mi+2 ∂Mi+1 + a(y) + ai+3 (x, y) ∂y ∂x

does not depend on y. Therefore, it is possible to define ai+2 (x) = a(y)

0 ∂Mi+2 ∂Mi+1 + a(y) + ai+3 (x, y), ∂y ∂x

and Eq. (70) becomes 0 (x, y)]]yj = ai+2 (x). β1j [[Mi+2

Now, the deduced equation for ui+4 (x, y, z) can be written as Lzz Ni+4 = −Lzy (Ni+3 + Mi+3 ) − Lzx (Ni+2 + Mi+2 + vi+2 ), which leads to an expression that suggests ∂M ∂Mi+2 dvi+2 i+3 0 Ni+4 (x, y, z) = Ni+4 + + . (x, y, z) + N (y, z) ∂y ∂x dx

(73)

With these considerations, ui+4 (x, y, z) takes the following form: ∂M ∂Mi+2 dvi+2 i+3 0 + + ui+4 (x, y, z) = Ni+4 (x, y, z) + N (y, z) ∂y ∂x dx + Mi+4 (x, y) + vi+4 (x),

(74)

0 (x, y, z) is the 1-periodic solution for the problem where Ni+4

∂N 0 ∂Ni+4 ∂Ni+2 ∂ i+3 a(y, z) + a(y, z) + = 0, ∂z ∂z ∂y ∂x

a(y, z)

z = zj ,

∂N 0 ∂Ni+4 ∂Ni+2 i+3 0 + a(y, z) + = β2j [[Ni+4 (x, y, z)]]zj , ∂z ∂y ∂x zj

∂N 0 ∂Ni+4 ∂Ni+2 i+3 + a(y, z) + = 0, a(y, z) ∂z ∂y ∂x zj 0 Ni+4 (x, y, 0) = 0.

This problem satisfies the hypotheses of Lemma 1 with ξ = z,

F0 = 0,

∂N ∂Ni+2 i+3 + F1 = −a(y, z) ∂y ∂x


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0 for every x and y. Therefore, the solution Ni+4 (x, y, z) exists, is 1-periodic in z, and, according to Remark 2, is unique. The solution of this problem is

0 Ni+4 (x, y, z) = ai+4 (x, y)

0

z

ds −1 + − β2k a(y, s) j

k=0

z

∂N

i+3

∂y

0

+

∂Ni+2 ds ∂x

(75)

for z ∈ (zj , zj+1 ) and j = 0, 1, · · · , l2 , where

∂N ∂Ni+2 i+3 + ai+4 (x, y) = a(y) . ∂y ∂x z

(76)

Remark 4 In the previous section, the basis of the induction process developed here is established (see Remark 3), which corresponds to the consideration i = −4, −3, · · · , 0 in Eq. (11). In particular, the necessary and sufficient condition ensuring that the existence of the 1-periodic solutions ui+4 = Ni+4 + Mi+4 + vi+4 is satisfied if the sum of the last six terms of Eq. (11) is equal to zero, i.e., F0 = 0, leading to problems (or expressions) to obtain some of the functions Ni , Mi , and vi . Now, note that, for i = 1, 2, · · · , m − 4 in Eq. (11), such a condition also leads to Eqs. (73), (68), and (67) for the functions Ni , Mi , and vi and that F0 = 0. This ensures that if F0 = 0 for i, F0 = 0 for i + 1. Thus, the process of constructing higher-order formal asymptotic solutions is justified by mathematical induction, which follows a recurrent cycle as follows: 0 0 Ni0 → ai → Mi−1 → ai−1 → vi−2 → Mi−1 → Ni → Ni+1 ,

where negative-index functions are considered to be null.

3

Mathematical justification

In this section, some of the results of Chapter 2 of Ref. [17] are extended to the case of piecewise differentiable coefficients in order to provide a mathematical justification of the homogenization process. 3.1 Preliminaries Consider Ω∗ = (0, 1) \ {x1 , x2 , · · · , xl } (l ∈ N), and denote x0 = 0 and xl+1 = 1. Then, the generalized (weak) derivative of a function u ∈ L2 ((0, 1)) can be defined as follows: 1

dϕ dϕ =− u(x) dx, Du(ϕ) = − u, dx L2 ((0,1)) dx 0 where

∀ϕ ∈ C01 (Ω∗ ),

⎧ ⎨ C01 (Ω∗ ) = {ϕ ∈ C 1 (Ω∗ ) : ϕ(xj ) = 0, j = 0, 1, · · · , l + 1}, ⎩ C k (Ω∗ ) = {ϕ : ϕ|

[xj ,xj+1 ]

∈ C k ([xj , xj+1 ]), j = 0, 1, · · · , l}

for k ∈ Z+ . If Du(ϕ) is bounded for all u ∈ L2 ((0, 1)), it follows from the Riesz representation theorem[17] that there exists a unique function v ∈ L2 ((0, 1)) such that Du(ϕ) = v, ϕL2 ((0,1)) for all ϕ ∈ L2 ((0, 1)). Then, it is said that the weak derivative belongs to L2 ((0, 1)), and du dx = v. is the usual derivative of u in each interval (x , xj+1 ). In the particular case of u ∈ C 1 (Ω∗ ), du j dx Now, the space H 1 (Ω∗ ) can be defined as the space of all functions on L2 (Ω∗ ) with weak derivatives belonging to L2 (Ω∗ ) as follows: du ∈ L2 (Ω∗ ) . H 1 (Ω∗ ) = u ∈ L2 (Ω∗ ) : dx


1131

An equivalent definition of H 1 (Ω∗ ) is H 1 (Ω∗ ) = {u : u|(xj ,xj+1 ) ∈ H 1 ((xj , xj+1 )), j = 0, 1, · · · , l}. The space H 1 (Ω∗ ) is equipped with the usual norm, denoted as · 1 , and is defined as follows: du 2 12 ,

u 1 = u 2L2 ((0,1)) + dx L2 ((0,1)) where · L2 (Ω∗ ) = · L2 ((0,1)) . Therefore, it is necessary to equip H 1 (Ω∗ ) with a norm accounting for the behavior of functions in the neighborhood of the discontinuity points x = xj (j = 1, 2, · · · , l). Proposition 1 Consider u ∈ H 1 (Ω∗ ), and define · 1∗ as follows: l du 2 12 + [[u]]2xj .

u 1∗ = u 2L2 ((0,1)) + dx L2 ((0,1)) j=1

Then, · 1∗ is a norm in H 1 (Ω∗ ) equivalent to the norm · 1 , and H 1 (Ω∗ ) equipped with · 1∗ is a Banach space. In fact, one can prove that

u 1 u 1∗ C l u 1, where

l

C = max

0jl

8l + 1. xj+1 − xj

Such an equivalence of norms implies that, since H 1 (Ω∗ ) equipped with the norm · 1 is a Banach space, it is also a Banach space if it is equipped with the norm · 1∗ . Remark 5 Consider the space H01 (Ω∗ ), which is the subspace of H 1 (Ω∗ ) defined as follows: H01 (Ω∗ ) = {u ∈ H 1 (Ω∗ ) : u(0) = u(1) = 0}. The seminorm l 12 du 2 |u|1∗ = + [[u]]2xn dx L2 ((0,1)) n=0

is a norm in H01 (Ω∗ ). The following three propositions ensure that H01 (Ω∗ ) equipped with | · |1∗ is a Banach space. Proposition 2 H01 (Ω∗ ) is a closed subspace of H 1 (Ω∗ ). Proposition 3 C01 (Ω∗ ) = {u ∈ C 1 (Ω∗ ) : u(0) = u(1) = 0} is dense in H01 (Ω∗ ). Proposition 4 The norms · 1∗ and | · |1∗ are equivalent in H01 (Ω∗ ). The following is a proof for Proposition 4. Proof Consider x ∈ Ω∗ and u ∈ C01 (Ω∗ ). Then, x du ds + u(x) = [[u]]xi . 0 ds 0