Homogenization of a Continuously Microperiodic Multidimensional Medium

Marcos Pinheiro Lima,Leslie Darien Pérez Fernández,Julián Bravo Castillero

doi:10.5540/tema.2018.019.01.15

Abstract

The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.

Highlights

The heterogeneous medium is formed by distributions of occupied domains: (a) by different homogeneous materials called phase, constituting a composite; or (b) of the same material in different states, such as a polycrystal [21], or a functionally graded material [19]
The work is structured as follows: in section 2, we develop the Asymptotic Homogenization Method (AHM) in detail to build the f.a.s.; in section 3, we mathematically justify the AHM, i.e., we prove that uε − u0 H01(Ω) = O( ε), where H01(·) is the space of null-trace square-integrable functions whose generalized first-order derivatives are square-integrable; in sections 4 and 5, we show some properties preserved on apply of the AHM; an example is solved in section 6, showing the development analytical and numerical applied; in section 7 we describes the conclusions about development realized in the previous sections
This implies a high computational cost and compromises the convergence of the numerical method, which happened when we attempted to solve the original problem of this example by the finite difference method

Summary

INTRODUCTION

The heterogeneous medium is formed by distributions of occupied domains: (a) by different homogeneous materials called phase, constituting a composite; or (b) of the same material in different states, such as a polycrystal [21], or a functionally graded material [19]. We can find various applications of homogenization theory in literature, for example: obtaining properties coupled nonexistent in the constituents (magnetoelectric [4], pyroelectric and pyromagnetic [5]); topology optimization [3]; optimal design of heterogeneous materials [22]; biomechanics of bone [16], prediction of structural failures [17]; propagation of seismic waves [7]; physics of nuclear reactors [1]; transport of a chemical species [15] In this contribution, we study the following elliptic problem: Find uε ∈ C2(Ω), Ω = [0, 1]d, such that.

APLICATION OF THE AHM

PROXIMITY

PRESERVATION OF SYMMETRY

PRESERVATION OF POSITIVE-DEFINITENESS

EXAMPLE

CONCLUSIONS