1 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 asymptotic 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

The periodic extension to 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₊∈ 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:

Let ε = 1/n (n∈). The problem is to find u^ε(x) such that

(1)

(2)

(3)

(4)

where

The numbers x_ij^k represent the discontinuity points of a^ε(x) in the interval (0, 1), i.e.,

That is, there are l₁n+l₂n² discontinuity points, and β_ki^ε=ε^-kβ_ki, where β_ki∈(i=1, 2, …, l_k, and k = 1, 2) indicate the microscales where the related discontinuities take place.

Since the condition in Eq. (2) becomes as β_ki→∞, u^ε(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 β_ki→∞ in the more general expressions to be obtained for Eqs. (1)-(4).

In the context of heat conduction, u^ε(x) is the temperature distribution in the heterogeneous medium, a^ε(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 conductances β_ki^ε, at the discontinuity points x_ij^k, 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:

(5)

where u_i(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

and equating the terms corresponding to equal powers of ε, we can obtain the following recurrent family of partial differential equations:

(6)

(7)

(8)

(9)

(10)

and

(11)

for 1≤ i≤ m-4, where, with p, q∈{x, y, z}, the differential operators L_pq are defined as follows:

Analogously, the contact conditions can be expressed as follows:

(ⅰ) For ε^-2,

(12)

(13)

(14)

(15)

(ⅱ) For ε^-1,

(16)

(17)

(18)

(19)

(ⅲ) For ε⁰,

(20)

(21)

(22)

(23)

(ⅳ) For εⁱ (1≤ i≤ m-4),

(24)

(25)

(26)

(27)

The boundary conditions (4) for the unknown functions u_i(x, y, z) are

(28)

(29)

(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, …, ξ_n} (0 < ξ_j < 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

(31)

(32)

(33)

is

(34)

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-periodic solutions given by =N(ξ)+c, where c∈ is an arbitrary constant, i.e., the 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 u_i of the form

(35)

where N_i(x, y, z), M_i(x, y), and v_i(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

(36)

(37)

(38)

(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)+a(y, z) is independent of z, we can introduce the intermediate effective coefficient (the effective coefficient related to the first microscale of the original problem) as follows:

(40)

Some manipulations of this relation yield

(41)

Note that 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

(42)

where z∈(z_j, z_j+1), and j=0, 1, …, l₂.

The second local problem consists in finding the 1-periodic solution M(y) for the problem

(43)

(44)

(45)

(46)

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 is independent of y, we can introduce the global effective coefficient related to the macroscale of the original problem as follows:

(47)

Some manipulations in this relation give

(48)

Note that 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, 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

(49)

where y∈(y_j, y_j+1), and j=0, 1, …, l₁.

The so-called homogenized problem is

(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

(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

(52)

where N(y, z) is the solution of the first local problem. The function M₂(x, y) will be considered to have the form M₂(x, y)=M₂⁰(x, y)+, where M₂⁰(x, y) is the solution of the problem

(53)

(54)

(55)

(56)

The function will be introduced later. Note that this problem satisfies the hypotheses of Lemma 1 with

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

does not depend on y. Therefore, it is possible to define

(57)

The function v₂(x) will be determined later. For u₃(x, y, z), we have

(58)

where N₃⁰(x, y, z) is the 1-periodic solution of the following problem:

(59)

(60)

(61)

(62)

The problem (59)-(62) satisfies the hypotheses of Lemma 1 with

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

(63)

for z∈(z_j, z_j+1) and j=0, 1, …, l₂, where

(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.

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

for every x and y. Therefore, the existence of a solution u_i+4(x, y, z) of Eq. (11), which is 1-periodic in z, can be ensured by satisfying the necessary and sufficient condition

(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

(66)

Thus, equating Eq. (66) to zero leads to the equation for u_i+4(x, y, z), L_zzu_i+4=-L_zyu_i+3-L_zxu_i+2, and

This equation, together with the corresponding appropriate conditions, satisfies the hypotheses of Lemma 1 with

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

(67)

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

(68)

where M_i+2⁰(x, y) is the solution of the problem

(69)

(70)

(71)

(72)

This problem satisfies the hypotheses of Lemma 1 with

for every x. Therefore, the solution M_i+2⁰(x, y) exists, is 1-periodic in y, and, according to Remark 2, is unique.

In addition, the equation of the problem for M_i+2⁰(x, y) states that

does not depend on y. Therefore, it is possible to define

and Eq. (70) becomes

Now, the deduced equation for u_i+4(x, y, z) can be written as

which leads to an expression that suggests

(73)

With these considerations, u_i+4(x, y, z) takes the following form:

(74)

where N_i+4⁰(x, y, z) is the 1-periodic solution for the problem

This problem satisfies the hypotheses of Lemma 1 with

for every x and y. Therefore, the solution N_i+4⁰(x, y, z) exists, is 1-periodic in z, and, according to Remark 2, is unique. The solution of this problem is

(75)

for z∈(z_j, z_j+1) and j=0, 1, …, l₂, where

(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 u_i+4 = N_i+4 + M_i+4 + v_i+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 N_i, M_i, and v_i. 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 N_i, M_i, and v_i 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:

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, …, x_l} (l∈), and denote x₀=0 and x_l+1=1. Then, the generalized (weak) derivative of a function u∈L₂((0, 1)) can be defined as follows:

where

for k∈. 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, φ〉_{L2((0, 1))} for all φ∈L₂((0, 1)). Then, it is said that the weak derivative belongs to L₂((0, 1)), and =v. In the particular case of u∈(Ω^*), is the usual derivative of u in each interval (x_j, x_j+1).

Now, the space H¹(Ω^*) can be defined as the space of all functions on L₂(Ω^*) with weak derivatives belonging to L₂(Ω^*) as follows:

An equivalent definition of H¹(Ω^*) is

The space H¹(Ω^*) is equipped with the usual norm, denoted as ‖·‖₁, and is defined as follows:

where ‖·‖_L2(Ω^*) = ‖·‖_{L2((0, 1))}. Therefore, it is necessary to equip H¹(Ω^*) with a norm accounting for the behavior of functions in the neighborhood of the discontinuity points x = x_j (j = 1, 2, …, l).

Proposition 1  Consider u∈H¹(Ω^*), and define ‖·‖_1* as follows:

Then, ‖·‖_1* is a norm in H¹(Ω^*) equivalent to the norm ‖·‖₁, and H¹(Ω^*) equipped with ‖·‖_1* is a Banach space.

In fact, one can prove that

where

Such an equivalence of norms implies that, since H¹(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 H¹(Ω^*) defined as follows:

The seminorm

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 H¹(Ω^*).

Proposition 3  (Ω^*)={u∈(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∈(Ω^*). Then,

Taking the absolute values and using the Cauchy-Schwarz inequalities for the integral and the sum on the right-hand side, we have

(77)

Squaring both sides, using the inequality (x+y)²≤ 2(x²+y²), and integrating with respect to x∈(0, 1), we have

Adding

to both sides of the above equation leads to

Then, the equivalence can be obtained from the relation

Since (Ω^*) is dense in H₀¹(Ω^*), the equivalence holds in H₀¹(Ω^*).

Remark 6  H₀¹(Ω^*) equipped with the norm |·|_1* is a closed subspace of a Banach space. Hence, it is also a Banach space. Moreover, the norm |·|_1* is associated with the scalar product

Therefore, H₀¹(Ω^*) is a Hilbert space.

Remark 7  It follows from Eq. (77) that, for any u∈H₀¹(Ω^*),

Then,

Remark 8  Both constants in the equivalence relation of the norms ‖·‖_1* and |·|_1* and in the relation from Remark 7 depend on l, which is the number of the discontinuity points of functions in H¹(Ω^*).

3.2 A variational problem for equation -L_ξξw=f₀-f'₁

Consider Ω^* = (0, 1)\{ξ₁, …, ξ_l} (l∈), ξ₀ = 0, and ξ_l+1 = 1. Then, the problem is to find w∈(Ω^*) such that

(78)

(79)

(80)

(81)

where a, f₁∈(Ω^*), and f₀∈(Ω^*), such that α_-≤ a(ξ)≤α₊ for every ξ∈Ω^*. In the above equations, α_-, α₊, β_j∈, c_j¹, c_j²∈, and j = 1, 2, …, l.

The variational formulation of Eqs. (78)-(81) is to find w∈H₀¹(Ω^*) such that

(82)

for all φ∈H₀¹(Ω^*), where

(83)

(84)

A solution of Eqs. (82)-(84) is called a generalized or weak solution of Eqs. (78)-(81).

The Lax-Milgram lemma (Theorem A.3 in Ref. [17]) allows proving that Eq. (82) is well-posed if b(w, φ) given by Eq. (83) is a bounded coercive bilinear form and L(φ) given by Eq. (84) is a bounded linear form, both in H₀¹(Ω^*).

Proposition 5   The bilinear form b(w, φ) given by Eq. (83) is coercive and bounded in H₀¹(Ω^*).

Proof  Since a(ξ)≥α_-,

(85)

where

Then, b(w, φ) is coercive.

Since a(ξ)≤α₊, from the Cauchy-Schwarz inequality, we have

(86)

where

Then, b(w, φ) is bounded.

Proposition 6   The linear form L(φ) given by Eq. (84) is bounded in H₀¹(Ω^*).

Proof  Remark 7 ensures the boundedness of the lateral limits and contrast operators over H₀¹(Ω^*) through the relations

Furthermore, from the Cauchy-Schwarz inequality and the equivalence of the norms ‖·‖₁, ‖·‖_1*, and |·|_1*, we have

where

Consequently, |L(φ)|≤ C₂|φ|_1*, where

Remark 9  It follows from Propositions 5 and 6 that Eqs. (82), (83), and (84) satisfy the hypotheses of the Lax-Milgram lemma. Then, there exists a unique function w∈H₀¹(Ω^*) such that Eqs. (82), (83), and (84) hold for all φ∈H₀¹(Ω^*), i.e., there exists a unique solution for the variational formulation of Eqs. (82), (83), and (84). Additionally, |w|_1*≤ C₂/C₀ holds for such a solution that is an immediate consequence of the chain of inequalities C₀|w|_1*²≤ b(w, w)=L(w)≤ C₂ |w|_1* and, as noted above, depends on the number l of the function discontinuity points in H₀¹(Ω^*).

Remark 10  If the solution w of Eqs. (82), (83), and (84), which is a weak solution of Eqs. (78)-(81), belongs to H₀¹(Ω^*)∩(Ω^*), and satisfies Eqs. (79) and (80), it is the solution of Eqs. (78)-(81).

Remark 11  Note that, for large values of β_j (j=1, 2, …, l), |w|_1*≤ C₂/C₀ is independent of β_j, since C₀=min{α₀, β_-}=α_-. Then, to ensure that Eq. (79) holds in the limit as β_j→∞, it is necessary that .

3.3 An auxiliary problem

Recall the statement of the original problem in Section 2, and consider the problem to find v^ε(x) such that

(87)

(88)

(89)

(90)

(91)

where δ_k1 is Kronecker's delta, c_ij^ε∈ are such that |c_ij^ε|≤εC, and C∈ does not depend on ε.

From Eqs. (87) and (90), we have

(92)

where c^ε∈. Then, Eq. (92) in combination with Eq. (91) yields

(93)

The condition (88) implies that

(94)

The substitution of Eqs. (94) into (93) followed by some manipulations yields

(95)

As a(y, z)≤α₊, it follows that

(96)

which is a bound that does not depend on ε. Moreover, since |c_ij^ε|≤εC and ε=1/n, it follows that

(97)

which is also a bound that does not depend on ε. Thus, the combination of Eqs. (96) and (97) with Eq. (95) yields

(98)

As a(y, z)≥α_-, the combination of Eq. (98) with Eq. (92) leads to

(99)

Moreover, since |c_ij^ε|≤εC, the combination of Eq. (98) with Eq. (94) leads to

(100)

which implies that

(101)

(102)

Let x∈(0, 1). Then, the condition (91) implies that

where D_x^ε is the set of discontinuity points of v^ε(x) contained between 0 and x. Taking absolute values on both sides yields

Applying the Cauchy-Schwarz inequality on the integral, with Eqs. (99), (101), and (102), we have

(103)

Squaring both sides and integrating from 0 to 1 with respect to x, we have

(104)

The relation (103) implies pointwise convergence of v^ε to 0 when ε→0⁺, while Eq. (104) implies strong convergence on L₂((0, 1)).

In what follows, the convergence of the solution u^ε(x) of the original problem (1)-(4) to the solution u₀(x) of the homogenized problem (50) as ε→0⁺ will be proven.

3.4 Proximity between u^ε(x) and u₀(x)

Let ε=1/n be fixed. Let u^ε(x) be the solution of the original problem (1)-(4) and u^(m)(x, ε) be its formal asymptotic solution constructed in Section 2. With the results from the homogenization process, we have that the function w^ε(x)=u^ε(x)-u^(m)(x, ε) is the solution of the following problem:

(105)

(106)

(107)

(108)

(109)

where

which is given by

is bounded in L₂(0, 1) as

In addition, the constants c_ij^k1 and c_ij^k2 given by

are bounded as

Moreover, the second term on the right-hand side of Eq. (106) is also bounded. Indeed,

where

Since K_r (r=1, 2, 3, 4) do not depend on ε, recalling Subsection 3.2, we can see that Eqs. (105)-(109) are well posed with the solution w∈H₀¹(Ω_ε^*). Consequently, the auxiliary problem

(110)

(111)

(112)

(113)

(114)

satisfies the hypotheses of Eqs. (87)-(91) in Subsection 3.3. Therefore, ‖v^ε‖_{L2((0, 1))}→0 as ε→0⁺.

From Eqs. (105)-(109) and (110)-(114), we have that the function defined as χ^ε=w^ε-v^ε is the solution of the problem

Now, it is possible to apply Remark 9 to obtain the estimate

(115)

where K=α_-^-1max{K₁, K₂+K₃}, which holds for small values of ε. Then, with Remark 7 and Eq. (115), we have

(116)

The right-hand side of Eq. (116) is of order ε^m-5. Then, choosing an asymptotic expansion u^(m)(x, ε) with m≥6 and taking the limit as ε→0⁺, we have

(117)

Moreover, the triangular inequality leads to

(118)

Then, using Eq. (117), recalling that ‖v^ε‖_L2((0, 1))→0 as ε→0⁺, and taking the limit as ε→0⁺ in Eq. (118), we have

(119)

Remark 12  Consider ε to be fixed. Then, Eq. (117) follows from Eq. (116) by taking the limit as m→∞.

Moreover, it follows from Eq. (5) that

Consequently,

Then, taking the limit as ε→0⁺, we have

(120)

Finally, the combination of Eqs. (117) and (120) with the triangular inequality

as ε→0⁺ leads to

(121)

i.e., u^ε→u₀ in L₂((0, 1)) as ε→0⁺, which completes the mathematical justification of the reiterated homogenization process in Section 2.

4 Some numerical examples 4.1 Comparison of solutions

In order to illustrate the theoretical results presented above, consider the original problem (1)-(4), in which f(x) =0, β_1i=β₁, β_2i=β₂, t₀=0, t₁=1, and a(y, z) given by

where

and

In the above equations,

The expression (41) for the intermediate effective coefficient leads to

whereas Eq. (48) yields the global effective coefficient

(122)

Moreover, the solution of the first local problem (36)-(39) is

Furthermore, the solution M(y) of the second local problem (43)-(46) is

The solution of the homogenized problem (50) is v₀(x)≡u₀(x)=x. The functions N(y, z) and M(y) allow to obtain the function N₂(x, y, z), Eq. (52), which in the present case is

In this case,

which leads to N_i(x, y, z)=0 for i≥3, M_i(x, y)=0 for i≥2, and v_i(x)=0 for i≥1. Therefore,

with y=x/ε and z=x/ε² for all m≥ 2 is the exact solution of the problem. Figure 1 illustrates the convergence of the exact solution u^ε(x) to the solution u₀(x) of the homogenized problem as ε→0⁺ for β₁=β₂=1. From the figure, we can see that, for ε=1/64, there are 2(64²+64)=8 320 discontinuity points.

4.2 Gain enhancement

In this section, a problem with only one micro-scale will be homogenized through conventional homogenization, and the corresponding effective coefficient will be compared with the one obtained after applying the RHM to an equivalent problem with two micro-scales. Besides, it will be illustrated that there is a gain of macroscopic properties from the effective coefficient of the second problem with respect to the first one (see Ref. [18]). Consider h₁(y)=(2+sin(2πy))^-1, β₁, β₂, h₂∈, y₀∈(0, 1), Φ∈[0, 1-y₀), and

where

and

Apply the conventional homogenization method to the problem

(123)

(124)

(125)

(126)

where

and x_1j=ε(y₀+j) and x_2j=ε(y₀+Φ+j) are discontinuity points from a^ε(x). Then, we have

Consider now y₁, z₁∈(0, 1), Φ_Y∈[0, 1-y₁), Φ_Z∈[0, 1-z₁), and

with

where

Then, we have

and

It is important to remark that if y₁=y₀, Φ_Y=Φ, and Φ_Z=0, the functions b^ε(x)=b(x/ε, x/ε²) and a^ε(x) coincide. Let κ_i¹=β_i, and κ_i²=∞. Then,

(127)

(128)

(129)

(130)

where κ_ki^ε=ε^-kκ_i^k (i, k=1, 2), and x_1j¹=ε(y₁+j), x_2j¹=ε(y₁+Φ_Y+j), x_1j²=ε(z₁+j), and x_2j²=ε(z₁+Φ_Z+j) are discontinuity points from b^ε(x), which coincide with Eqs. (123)-(126). For Φ_Z=0, there are no discontinuity points at z=z₁, and the discontinuity conditions from z₁ and z₁+Φ_Z overlap each other and must be fulfilled simultaneously. In that case, the problem will only have a solution if κ₁²=κ₂². Applying the RHM to it gives exactly the same effective coefficient from Eqs. (123)-(126).

The aimed goal is to investigate the effect on that effective coefficient of the parameters Φ_Y, Φ_Z, and κ_i^k with some constrains so as to ensure the equivalence between the problems (123)-(126) and (127)-(130), i.e., to determine the effect of the cell geometry on the effective coefficient.

To this end, the concept of volumetric fraction for each one of the phases plays an important role. In the periodic cell from the function a(y), there is a set, in which the function is equal to h₂, i.e., Ω₂. The volume fraction for this phase is defined as the total length of the phase divided by the total length of the periodic cell which is equal to 1. Therefore, the volumetric fraction for this phase is Φ. To ensure certain equivalence between the problems (123)-(126) and (127)-(130), the volumetric fraction of the phase, where b(y, z) and a(y) are equal to h₂, must be the same. Consequently, |Ω₂^YZ|=|Ω₂|, or Φ=Φ_Y+Φ_Z-Φ_YΦ_Z. Moreover, it is also necessary to analyze how the coefficients κ_i^k affect the effective coefficient. Those constants must fulfill certain conditions to ensure that Eqs. (123)-(126) can be obtained as a limit case of Eqs. (127)-(130). Beside, Eqs. (127)-(130) have a solution for that limit case. As mentioned before, it is important that, if Φ_Z=0 (and therefore Φ_Y=Φ), κ₁²=κ₂², and κ_i¹=β_i. In the same way, if Φ_Y=0, κ₁¹=κ₂¹. The constants κ_i^k depend on the parameters Φ_Y and Φ_Z, and also have effects on the conductivity. It will be ruled here that

though this is not the only possible consideration. Note that, if Φ_Z=0, κ₁²=κ₂²=∞; if Φ_Z=Φ, κ_i²=β_i. Finally, to obtain Eqs. (123)-(126) from Eqs. (127)-(130) when Φ_Y=Φ and Φ_Z=0, y₁ will be chosen in such a way that, when Φ_Y=Φ, y₁=y₀. This choice is not unique. Here, y₁=y₀ will be used for any value of Φ_Y.

With these conditions, the global effective coefficient obtained with the RHM to Eqs. (127)-(130) is

As Φ_Y depends on Φ_Z, the effective coefficient depends on this parameter as well. This makes it possible to analyze the effective coefficient gain k_gain= as a function of Φ_Z and to find the optimal values of Φ_Z∈[0, Φ].

Figures 2, 3, and 4 show the behaviors of k_gain as a function of Φ_Z with the parameters Φ=0.5, y₀=0.25, and the variable h₂ for β₁=β₂=1, β₁=β₂=3.5, and β₁=β₂=40, respectively. From Figs. 2 and 4, we can see that, the absolute value of the effective coefficient gain is different from those for all the variation range of Φ_Z. Figure 3 also shows that there exist only two values of Φ∈(0, 0.5) so that the effective coefficient gain is equal to 1. It is interesting to mention that a similar behavior has also been found in Ref. [16] for the effective gain thermal conductivity of two-dimensional (2D) and three-dimensional (3D) composites with perfect contact conditions at the interfaces.

Figures 5, 6, and 7 show the behaviors of k_gain as a function of Φ_Z with different values of variables. From the figures, we can see that, in all cases, there exists enhancement in the gain. In particular, for large values of β₁, β₂, y, and h₂, the gain could be greater than 10%.

In Figs. 2, 3, and 4, the value of Φ_Z, where there is no gain (k_gain=1), is the same for all values of h₂. The reason is that, for those values of y₀ and Φ, the equation k_gain=1, after some manipulation, becomes

which does not depend on h₂. Consequently, its solution is independent of h₂. In the same way, the minimum and maximum points of k_gain are independent of h₂. The stationary points of k_gain can be obtained by solving k'_gain=0, where the derivative is taken with respect to Φ_Z.

After some manipulation, the equation k'_gain=0 becomes

(131)

This equation does not depend on h₂, neither do its solutions. Since k_gain has no critical points for Φ_Z∈(0, 0.5), all maximum and minimum points are independent of h₂.

5 Concluding remarks

The RHM is applied to a family of two-point boundary-value problems with periodic and rapidly oscillating coefficients depending on two microscopic scales. A necessary and sufficient condition for the existence of periodic solutions is proven to guarantee the construction of a formal asymptotic solution of arbitrary order. A variational formulation is derived to obtain an estimate for the solution so as to provide a mathematical justification for the homogenization process. An example is presented to illustrate the possibilities of obtaining a gain in the macroscopic response of heterogeneous materials provided by different microscopic scales as well as under the effects of imperfect contact conditions at the interfaces.

Acknowledgements The authors acknowledge the Cátedra Extraordinaria IIMAS-UNAM, México and PREI-DGAPA, UNAM, and are grateful to Ana Pérez Arteaga and Ramiro Chávez for the computational support. We would also like to thank to the reviewers for their valuable comments and to the editors for their inestimable assistance and suggestions.