Multiscale modeling using goal-oriented adaptivity and numerical homogenization. Part I: Mathematical formulation and numerical results

摘要

This paper is the first in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal-oriented finite element mesh adaptivity. We describe the physical application, Step and Flash Imprint Lithography, in brief and present the mathematical ideas and numerical verification. The method requires the Moore-Penrose pseudoinverse of element stiffness matrices. Algorithms for efficiently computing the pseudoinverse of sparse matrices will be presented in the second paper. The purpose of numerical homogenization is to reduce the number of degrees of freedom, find locally optimal effective material properties, and perform goal-oriented mesh refinement. Traditionally, a finite element mesh is designed after obtaining material properties in different regions. The mesh has to resolve material discontinuities and rapid variations in the solution. In our approach, however, we generate a sequence of coarse meshes (possibly 1-irregular), and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem. This upscaling is done using the Moore-Penrose pseudoinverse of the linearized fine-scale element stiffness matrices, and a material-independent interpolation operator. Numerical verification is done using a two dimensional conductivity problem with known analytical limit. Finally, we present results for two and three dimensional geometries. The results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine-scale problem.