Formal Asymptotic Homogenization

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Walter Park
6 years ago
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1 September 21 25, 2015

2 Formal asymptotic homogenization Here we present a formal asymptotic technique (sometimes called asymptotic homogenization) based on two-spatial scale expansions These expansions allow to move away the oscillations and successfully lead to upscaled model equations and effective coefficients, under the basic assumption of periodicity The introduced expansions are only formal; analysis work is required to justify the validity of the two-scale expansion

3 Model Problem: Setting of the problem Consider a model problem of diffusion or conductivity in a periodic medium (e.g., an heterogeneous domain obtained by mixing periodically two different phases, one being the matrix and the other the inclusions) Let a domain Ω (a bounded open set in R d, d 1) be periodic with period diam(ω) Denote Y = (0,1) d. For a given source term f(x) L 2 (Ω) scalar function defined in Ω, consider the following BVP: Stationary diffusion in perforated media { (a(x) u (x)) = f(x), x Ω u (x) = 0, x Ω As before, assume a (x) = a ( x ), where a( ) is a Y-periodic function, bounded and s.t. d α, β > 0 : α ζ 2 a ij(y)ζ iζ j β ζ 2, ζ R d, y Y i=1 RK. Here we don t make any assumptions on symmetry of a( ) Function u (x) H 1 (Ω) is the unknown potential (temperature) (1)

4 Asymptotic Homogenization Method Remark: Well-possedness of (1) follows from assumption f(x) L 2 (Ω) and the Lax-Milgram Lemma:! u (x) H 1 0(Ω) : u H 1 (Ω) C, where C > 0 is independent of Definition (homogenization) Averaging the solution of (1) and finding the effective properties of Ω is called homogenization Ansatz: We seek for the solution of (1) in the form of the series: u = k u k (2) k=0

5 Asymptotic Homogenization Method (cont.) We search for the behavior of the system as 0 Since coefficient of (1) are Y-periodic in x, it is reasonable to assume that all terms of (2) are periodic functions of x For x Ω, introduce y = x Y, then variables x,y are called x slow micro global y fast macro local Hence, assume u (x) = u 0(x,y)+u 1(x,y)+ 2 u 2(x,y)+..., (3) where u k (x,y) is Y-periodic in the second variable y Now scale separation is exploited: we treat x and y as independent variables Using = x + 1 y (4) i.e. for G (x) := g ( x, ) x one has ( G (x) = xg x, x ) + 1 ( y= x yg x, x ) (5) y= x

6 Asymptotic Homogenization Method (cont.) Introduce the following notations for operators: A 0 := y [a(y) y] A 1 := y [a(y) x] x [a(y) y] A 2 := x [a(y) x] Then substituting (3) into (1), using (4) and (5), and collecting terms of the same powers of, we obtain (0) A 0u 0 := 0 (1) A 0u 1 := A 1u 0 (6) (2) A 0u 2 := A 2u 0 A 1u 1 +f(x) supplied with periodicity condition of u i, i = 0,1,2, w.r.t. y-variable Problem (6-0) with periodicity condition for u 0 yields u 0(x,y) = u 0(x) (7) Note that y is the variable here, while x plays role of a parameter

7 Asymptotic Homogenization Method (cont.) Co back to (6-1) with periodicity condition for u 1. With (7) it becomes A 0u 1 = y [a(y) xu 0(x)] (8) The RHS of (8) is a multiple of xu 0(x), hence, (by linearity) one may seek for the solution of (6-1) in the form u 1(x,y) = λ(y) xu 0(x)+û 1(x), (9) where the function λ(y) = (λ 1(y),...,λ d (y)) R d is vector-valued Substituting (9) into (6-1) we obtain equations for λ i(y), i = 1,...,d with periodicity of λ i(y), called cell problems (will be given below). But now let us turn to (6-2) for which we apply the following Lemma (Compatibility condition) Let f(y) L 2 #(Y) then there exists a unique u H 1 #(Y) solving { (A(y) w(y)) = f, y Y iff f(y)dy = 0 Y u is Y periodic

8 Asymptotic Homogenization Method (cont.) Application of this compatibility condition to the RHS of (6-2) yields { } x a(y)[ yu 1(y)+ xu 0(x)]dy = f(x), x Ω Y u 0(x) = 0, x Ω (10)

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