Functionally Gradient Material Multi-scale Modeling

Functionally graded materials have many applications, for example, the shellof space ships,which requires the character of both high temperature resistanceand high degree of hardness. In this paper , we will build the multiscale modelof functionally graded materials. In order to obtain the macro character of thesolutions of the multiscale problem, many multiscale methods need to solve anarray of microscale models over small domains. Modeling error arises due to themismatch between the local microstructures and the artificial boundary conditionover the domains for solving the microscale problem. In this paper, we will useheterogeneous multiscale method to model thermoelasticity equations with highlyoscillating coe?cients,and estimate the error of the modeling.In chapter 1, we introduce the background of Functionally graded materials.In chapter 2, we give the basic equations of thermoelasticity, and derive its homog-enized equations. In chapter 3, we give the multiscale model of thermoelasticityequations. In chapter 4, we mainly discuss the modeling error.