Asymptotic homogenization

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In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients,[1] [2] [3] such as

$\nabla\cdot\left(A\left(\frac{\vec x}{\epsilon}\right)\nabla u_{\epsilon}\right) = f$

where $\epsilon$ is a very small parameter and $A\left(\vec y\right)$ is a 1-periodic coefficient: $A\left(\vec y+\vec e_i\right)=A\left(\vec y\right)$, $i=1,\dots, n$.

It turns out that the study of these equations is also of great importance in physics and engineering, since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics. Under this assumption, materials such as fluids, solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc.

Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are subjected to loads or forcings which vary on a lengthscale which is far bigger than the characteristic lengthscale of the microstructure. In this situation, one can often replace the equation above with an equation of the form

$\nabla\cdot\left(A^*\nabla u\right) = f$

where $A^*$ is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed as

$A^*_{ij}=\int_{(0,1)^n} A(\vec y)\left( \nabla w_j(\vec y)+\vec e_j\right) \cdot\vec e_i\, dy_1\dots dy_n , \qquad i,j=1,\dots,n$

from 1-periodic functions $w_j$ satisfying:

$\nabla_y\cdot\left(A(\vec y)\nabla w_j\right)= -\nabla_y\cdot\left(A(\vec y)\vec e_j\right).$

This process of replacing an equation with a highly oscillatory coefficient one with a homogeneous (uniform) coefficient is known as homogenization. This subject is inextricably linked with the subject of micromechanics for this very reason.

In homogenization one equation is replaced by another if $u_\epsilon\approx u$ for small enough $\epsilon$, provided $u_\epsilon\to u$ in some appropriate norm as $\epsilon\to 0$.

As a result of the above, homogenization can therefore be viewed as an extension of the continuum concept to materials which possess microstructure. The analogue of the differential element in the continuum concept (which contains enough atom, or molecular structure to be representative of that material), is known as the "Representative Volume Element"[4] in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous medium in order to be representative of the material. Therefore averaging over this element gives an effective property such as $A^*$ above.

The method of asymptotic homogenization

Mathematical homogenization theory dates back to the French, Russian and Italian schools.[1][2][3][5] The method of asymptotic homogenization proceeds by introducing the fast variable $\vec y=\vec x/\epsilon$ and posing a formal expansion in $\epsilon$:

$u_\epsilon(\vec x) = u(\vec x,\vec y) = u_0(\vec x,\vec y)+ \epsilon u_1(\vec x,\vec y)+\epsilon^2 u_2(\vec x,\vec y)+O(\epsilon^3)\,$

which generates a hierarchy of problems. The homogenized equation is obtained and the effective coefficients are determined by solving the so-called "cell problems" for the function $u_1(\vec x,\vec x/\epsilon)$.

Notes

1. ^ a b Sanchez-Palencia, E: "Non-homogeneous media and vibration theory, Springer Verlag, 1980
2. ^ a b Bakhvalov, N. and Panasenko, G: "Homogenization: Averaging Processes in Periodic Media", Kluwer, Dordrecht, 1989
3. ^ a b Bensoussan, A., Lions, J.L. and Papanicolaou, G., "Asymptotic Analysis for Periodic Structures", North-Holland, Amsterdam, 1978
4. ^ Ostoja-Starzewski, M., Microstructural randomness and scaling in materials, Chapman and Hall/CRC Press, 2007
5. ^ Dal Maso, G., "An introduction to $\Gamma$-convergence", Birkhauser, 1992