In this paper an exposition of the asymptotic method for periodic
composite structures is given

I. Aganović

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139Mathematical analysis of composite structures∗Ibrahim Aganovic´†Abstract. In this paper an exposition of the asymptotic methodfor periodic composite structures is given.Key words: composite material, homogenization, asymptoticexpansionSazˇetak. Matematicˇka analiza kompozitnih struktura.U ovom cˇlanku dan je prikaz asimptoticˇke metode za kompozitnestrukture.Kljucˇne rijecˇi: kompozitni materijal, homogenizacija, asimp-toticˇki razvojExamples of a composite structure are porous media, perforated materials,layered media, lattice elements in civil engineering, etc. A common propertyin all these cases is a rapid variability of either the geometric or the physicalstructure. Therefore, a boundary-value problem for a process in a compositemedium cannot be solved by classical methods. In order to obtain a well posedproblem, one has to average (in some sense) a microscopic model and formu-late a correspondent macroscopic model, describing a process “in mean”. Suchmacroscopic laws appeared in engineering during the last century; a good ex-ample is the well known Darcy’s law, connecting the velocity and pressure ofa fluid in a porous medium and taking the place instead of the Navier-Stokesequation.A mathematical justification of macroscopic equations belongs to the ho-mogenization theory, having the origin in last seventies. Definite results have sofar been obtained for composites with a small periodicity in the structure. Weshall describe the basic idea of the homogenization method on a simple example:stationary heat conduction in a bar with a periodic conduction coefficient of thematerial.∗The lecture presented at the Mathematical Colloquium in Osijek organized by CroatianMathematical Society - Division Osijek, March 29, 1996.†Department of Mathematics, University of Zagreb, Bijenicˇka 30, HR-10 000 Zagreb, Croa-tia, e-mail: aga@cromath.math.hr140 I. Aganovic´Let for ε > 0 a sufficiently small conduction coefficient aε(x) (x ∈ (0, 1)) bedefined asaε(x) = a(xε), (1)where a(y) > 0 is a 1-periodic function on R; then aε is an ε-periodic function.Let uε(x) denote the temperature of the bar. Assuming the density of an exter-nal heat flux in the form f(x) − kuε(x), where f(x) is given function and k =const. ≥ 0, we have the following equation for the temperatureddx(aε(x)ddxuε(x))− kuε(x) + f(x) = 0, x ∈ (0, 1). (2)We take the homogeneous boundary conditionsuε(0) = (uε)′(1) = 0. (3)For each ε > 0 a sufficiently small problem (2), (3) has a unique solution uε. Asthe macroscopic temperature of the bar we consider the limit of uε, as ε → 0.The purpose of the homogenization theory is to identify this limit. A formalmethod for it is a two-scale expansion.One can expect that the function uε has a property of local ε-periodicity, i.e.that in a small neighbourhood of an interior point it holds uε(x + ε) ≈ uε(x).An example of such a function is the mapping x → u(x, x|ε), where u(x, y) isa smooth function, 1-periodic in the variable y. A more general example is theseriesu0(x, y) + εu1(x, y) + ε2u2(x, y) + . . . , (4)where (uk(x, y), k = 0, 1, 2, . . .) are a smooth functions, 1-periodic in the vari-able y. It is plausible to look for a solution uε(x) of the problem (2), (3) in theform of a two-scale asymptotic expansion (4) (see [2], [3]). One can expect that(in some sense) uε → u0 as ε → 0. Inserting (4) into (2) and taking the termsof the order ε−2, we conclude that the function u0 does not depend on y:u0 = u0(x). (5)Taking the terms of the order ε−1, we obtain thatu1(x, y) = (u0(x))′w(y), (6)where w(y) is a 1-periodic solution of the equation(a(y)w′(y))′ + a′(y) = 0. (7)One easily obtainsw(y) = a0∫ y0dya(y)− y + const., a0 =(∫ 10dya(y))−1. (8)Mathematical analysis of composite structures 141Finally, the terms of the order ε0 give the equationa(y)∂2∂x2u0(x) + a(y)∂2∂x∂yu1(x, y) +∂∂y(a(y)∂∂xu1(x, y))+∂∂y(a(y)∂∂yu2(x, y))− ku0(x) + f(x) = 0. (9)Integrating this equation due to the variable y and taking into account (6) andthe periodicity conditions, we obtain the equation for the function u0(x)a0(u0(x))′′ − ku0(x) + f(x) = 0. (10)According to (3) , the admissible boundary conditions for this equation areu0(0) = (u0)′(1) = 0. (11)The main mathematical problem is to establish a convergence uε → u0, ε→0. From (2) and (3) one can easily obtain the a priori estimate∫ 10(( ddxuε)2 + (uε)2) dx ≤ C, (12)where C > 0 does not depend on ε. From this estimate it follows that, up to asubsequence, uε has a limit in L2(0, 1), as ε→ 0. For proving the fact that thelimit is the function u0 one can use either the energy method (see [4]) or thetwo-scale convergence method (see [1]). Taking into account that the problem(10), (11) has a unique solution, we conclude that the limit does not dependon a choice of a subsequence. In other words, the homogenized (macroscopic)solution is identified.References[1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math.Anal. 23(1992), 1482–1518.[2] A. Bensoussan, J-L. Lions, G. Papanicolau, Asymptotic Analysis ofPeriodic Structures, North-Holland, Amsterdam, 1978.[3] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory,Springer, Berlin, 1980.[4] L. Tartar, Convergence of the homogenization process, Appendix in [3].

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