The first glance at homogenization

Leveraging on practical applications such as composite materials with periodic heterogeneity, the homogenization (upscaling, averaging, etc.) is a typical framework to deal with problems in a class of complicated perforated media, and enables us to obtain good overall approximations.

In fact, we usually characterize the properties of the aforementioned materials by means of a local scale, i.e. taking each component into account. In reality, a global scale is much more attractive to gain considerable attention of many researchers. In principle, the global behavior by homogenization is driven by replacing the heterogeneity the so-called homogenized material whose behavior must be as close as possible to the composite’s behavior.

Another reason comes from numerics. For example, when one takes into consideration a superconducting multifilamentary composite (SMC) with several thousands of fibers whose diameter is very small, a direct numerical computation is impossible. As one could readily expect, one needs to have an extremely fine discretization mesh in hands, and obviously, it requires a long and long time elapsed for computation. Concerning more complex materials, let us also mention the honeycomb structures, the reinforced structures and their generalized versions.

In order to have a lucid mathematical explanation, a one-dimensional equation with oscillating coefficients is presented in this note as the first glance in the study of homogenization.

The problem here is governed by

$-(a^{\varepsilon}u_{x}^{\varepsilon})_{x} = f,$

in $\Omega=(0,1)$, and associated with with zero Dirichlet boundary conditions, i.e. $u^{\varepsilon} = 0$ on $\partial\Omega$, and $f \in L^{2}(\Omega)$ is a given function.

Included here, the oscillating coefficient $a^{\varepsilon}$ is supposed to be periodic with period $T$ and generated by a bounded positive function $a\in L^{\infty}(\Omega)$ satisfying

$0<\alpha\le a(x) \le \beta <\infty.$

The coefficient is thus assigned by

$a^{\varepsilon}(x) = a\Big(\dfrac{x}{\varepsilon}\Big).$

For each $\varepsilon$, the considered problem admits a unique solution. Moreover, an important thing shows that by the Poincaré inequality, we obtain the a priori estimate of $u^{\varepsilon}$ over $H^{1}_{0}$-norm. In addition, if we define the stress function

$\xi^{\varepsilon} = a^{\varepsilon}u^{\varepsilon}_{x},$

then the a priori estimate also holds.

We are led to the fact that there exists a subsequence, ignored the sub-index, such that $u^{\varepsilon}\to u^0$ in $H^{1}_{0}$, $\xi^{\varepsilon}\to \xi^0$ in $H^1$, $a^{\varepsilon}\to a^0$ in $L^{\infty}$ in the weak or weak-* sense.

By the average convergence for periodic functions, $a^0$ is exactly the mean value of $a$, i.e.

$a^0 = \mathcal{M}(a)\equiv \dfrac{1}{T}\displaystyle{\int}_{0}^{T} a(x)dx.$

Moreover, we also obtain $-\xi^{0}_{x} = f$ in $\Omega$, and go through by the Rellich-Kondrachov theorem the fact that the convergence $\xi^{\varepsilon}\to \xi^0$ is strong in $L^2$.

Notice that $\xi^{\varepsilon}$ is the product of two weakly converging quantities. In general, this does not imply that the limit $\xi^0$ does not coincide with the product of the limits $a^0$ and $u^{0}_{x}$, saying that $\xi^0\equiv a^0 u^{0}_{x}$ does not hold. In fact, we show that

$u^{0}_{x}=\mathcal{M}\Big(\dfrac{1}{a}\Big)\xi^{0}$,

$-\dfrac{1}{\mathcal{M}\Big(\dfrac{1}{a}\Big)}u^{0}_{xx}=f.$
In conclusion, we considered here the periodic case with $a^{\varepsilon}$ a bounded sequence (not necessarily periodic) satisfying $\dfrac{1}{a^{\varepsilon}} \to A$ weak-* in $L^{\infty}$. The solution $u^{\varepsilon}$ of the considered problem is shown to be weakly convergent to $u^0$ in $H^{1}_{0}$. And the function $u^0$ is proved to be a unique solution of the problem given by
$-\Big(\dfrac{1}{A}u^{0}_{x}\Big)_{x}=f,$
in $\Omega$ and along with the zero Dirichlet boundary conditions.