Second-order two-scale analysis for heating effect of periodical composite structure

Abstract

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method.