Divergence Structure Homogenization

Check the recent reference:

Discrete Geometric Structures in Homogenization and Inverse Homogenization Mathieu Desbrun, Roger D. Donaldson, Houman Owhadi http://arxiv.org/abs/0904.2601

Notes on computing homogenized coeffs in the divergence structure case

Here there are two points of view which can be taken advantage of:

the variational point of view: minimizing a quadratic functional of the gradient
the PDE point of view: have a cell problem to solve, but it's for a vector function.

Homogenization theory for divergence form operators.

Gamma convergence theory and the homogenization theorem

Recall that, the time-dependent divergence structure PDE

$u_t = \nabla\cdot(A(x)\nabla u)$

is the gradient descent Euler-Lagrange equation for the corresponding variational problem

$\inf_{u} \frac{1}{2}\int \nabla u ^T A\nabla u.$

Now suppose $A (x) = A ε (x)$ , an $ε$ periodic function in $R n$ . To analyze how the divergence form $\nabla\cdot (A_\epsilon(x)\nabla u)$ converges in the limit $\epsilon\rightarrow 0$ , it is advantageous to approach it in the variation form and apply the theory of $Γ$ convergence. The theory of Gamma convergence give necessary conditions for the convergence of minimizers of an integral functional.

The main tool at hand is the Homogenization Theorem. It states that that, given $f (x, p)$ , 1-periodic in $x$ and has the growth condition,

$\alpha|p|^q \le f(x,p) \le \beta(1+|p|^q)$

for some $q\ge 1$ , then on a bounded open set $Ω$ the functional

∫	f(x / ε,Du)dx
Ω

converges in the sense of $Γ$ -convergence to the functional

∫	f_hom(Du)dx
Ω

where $f h o m (p)$ is given by the asymptotic homogenization formula:

$f_{hom}(p) = \lim_{t\rightarrow\infty}\inf\{ \int_{(0,t)^n} f(x,p+Du(x))dx : u\in W^{1,q}_0\}$

It turns out, for convex $f (x, p)$ , the asymptotic homogenization formula simplifies to a cell problem; that is, a minimization over a single cell $(0,1) n$ :

$f_{hom}(p) = \inf\{ \int_{(0,1)^n} f(y,p+Du(y))dy : u\in W^{1,q} \mbox{ and periodic}\}.$

The divergence structure case

Now we return to our original problem. For the divergence form, assume first that,

$\alpha' |x|^2 \le x^TAx \le \beta' |x|^2$

for some $0 < α' < β'$ and that $A$ is symmetric. Then, we have,

$f (x, p) = p T A (x) p$

which satisfies the growth condition with $q = 1$ . Then it is known that the homogenized integral is of the form $f h o m (p) = p T A 0 p$ for some constant symmetric matrix $A 0$ . Indeed, if we can compute $A 0$ , then the "homogenized" divergence form is $\nabla\cdot(A_0\nabla u)$ .

To compute $A 0$ , consider the case $n = 2$ , where (with abuse of notation) $x = (x, y)$ . Then the cell problem formula implies,

$(A_0)_{1,1} = \inf \{\int_{(0,1)^2} \nabla (x + u(x,y)) A(x,y) \nabla (x+u(x,y)) dxdy : u\in W^{(1,2)}\mbox{ and periodic}\}$ $(A_0)_{2,2} = \inf \{\int_{(0,1)^2} \nabla (y + u(x,y)) A(x,y) \nabla (y+u(x,y)) dxdy : u\in W^{(1,2)}\mbox{ and periodic}\}$

Also, if $P 1, P 2$ are the minimizers to the two cell problems above, respectively, the off diagonal elements can also be computed:

$(A_0)_{1,2} =\int_{(0,1)^2} \nabla (x + P^1(x,y)) A(x,y) \nabla (x+P^2(x,y)) dxdy$ $(A_0)_{2,1} =\int_{(0,1)^2} \nabla (y + P^2(x,y)) A(x,y) \nabla (y+P^1(x,y)) dxdy$

Computeing the minimizers $P 1, P 2$

Thus the difficulty, from a numerical standpoint, is to compute the minimizers $P 1, P 2$ . The Euler-Lagrange equation for the minimization is,

$\nabla (A(x)\nabla P^k) = \nabla A_k \mbox{ and } P^k\in W^{1,2} \mbox{ and periodic}$

for $k = 0,1$ and $A k$ is the k-th column of $A$ . Of course, the solution to the Euler-Lagrange equation unique only up to a additive constant, so we must add another condition. For the finite difference method, a convenient way to impose uniqueness is by setting a point boundary condition $P k (0,0) = 0$ .

Numerical approach

The finite difference approach to solving the Euler-Lagrange equation for $P k$ with the point boundary condition at the origin amounts to solving a $(n^2-1)\times (n^2-1)$ linear system, similar to the 9-point Laplacian. The integral to evaluate $(A 0) i, j$ was computed by the midpoint rule.

Software tools

Finite differences in MATLAB
FreeFEM http://www.freefem.org/

Examples

We first tested a case where the exact solution is known. Computations were done on a 80 by 80 grid. Let:

$A (x) = (s i n (2π x) + 2)(c o s (2π x + 2)) * I$

where $I$ is the identity matrix. The exact solution is:

$(A 0) 1,1 = 3.464101616$

$(A 0) 1,2 = 0$

$(A 0) 2,1 = 0$

$(A 0) 2,2 = 3.464101616$

The computed solution was:

$(A 0) 1,1 = 3.464103141$

$(A 0) 1,2 = - 5.685650478 e - 16$

$(A 0) 2,1 = 8.260059303 e - 16$

$(A 0) 2,2 = 3.464103141$

The auxiliary functions $P 1, P 2$ are plotted below.

References

1. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford, 1998. (Chapter 14, especially section 14.5)

2. G. Dal Maso, An Introduction to \Gamma-convergence, Birkhauser, 1993. (Chapter 24)

the majority of this page is contributed by Ryo Takei

Divergence Structure Homogenization - Adam Oberman Math

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