Tag Archives: homogenization

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},

which leads to the equation

-\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.

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