Multiscale Analysis And Application For Composites With Periodic Structure

Recently, composite materials with periodic structure have been widely used in space-crafts, automobile, medicine, archi-tectural industry and others. The calculation and anal-ysis for the performance of composite materials is an interdisciplinary subject. Besidesmacro mechanical method and micro mechanical method, from the mathematical point ofview, many physical and mechanical problems can be represented as partial diferentialequations with rapidly oscillating coefcients for the materials with high degree of het-erogeneity. If we solve these problems by classical finite element method(FEM) or finitediference method, it should be solved in microscopic scale to ensure the accuracy of thenumerical solution because of locally rapid oscillation of coefcient, which will lead totremendous amount of computation and memory. Thereby, we primarily use the multiscaleasymptotic analysis method. The details are presented as follows:In the first part, we study the general second order elliptic equations whose zero orderterm of solution function and right hand is related toε. It is a general form of the parabolicequation with rapidly oscillating coefcients by Laplace transform. A multi-scale finiteelement method with high accuracy is presented. We obtain the first order and the secondorder multiscale asymptotic expansion solutions and prove theoretically the multiscale con-vergence rate with the orderε12. Compared with classic results, second order multiscaleasymptotic expansion solution is introduced a cell problem to eliminate the efects of thezero order term of solution function and right hand. On the basis of theoretical analysisand the finite element technique, we proposed the multiscale finite element algorithm andnumerical simulations are carried out to validate the proposed method of this part. Compar-ing with the classical methods, the results show our method can efectively reduce elementnumbers and node numbers, and save tremendous computational resources.In the second part, we expand the results of the first part to a kind of general periodicstructure, N dimension parallelotope periodic structure. At first, we prove the fundamentaltheorem of multiscale asymptotic analysis method for parallelotope periodic structure, andobtain the multiscale asymptotic expansion solutions with the multiscale convergence rateofε12order. And then the corresponding algorithm is constructed. Numerical simulationsare carried out to validate the proposed algorithm of this part. At last, based on that thehomogenized coefcient is independent of the isotropic scaling factor of periodic cell, wealso focus on the efect of structural parameters θ and the volume fraction of constituentson equivalent coefcients of thermal conductivity.Applying multiscale finite element algorithm presented in this paper, the computationof the partial diferential equations with rapidly oscillating coefcients by FEM in a fine mesh is substituted by the computation of homogenized problems in a coarse mesh andlocal cell problems in a fine mesh by FEM. And then, we overlay them together accordingto the algorithm. Considering the amount of calculation, numerical simulations in2Dare carried out to validate the proposed algorithm. Then, making use of PHG(ParallelHierarchical Grid), a toolbox for developing parallel adaptive finite element programs, wepresent the parallel multiscale finite element program for a general second order ellipticequations with rapidly oscillating coefcients.