The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE2 method

The Heterogeneous Multiscale Finite Element Method (FE-HMM) is a two-scale FEM based on asymptotic homogenization for solving multiscale partial differential equations. It was introduced in [W. E, B. Engquist, Commun. Math. Sci. 1 (2003) 87-132]. The objective of the present work is an FE-HMM formulation for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, of a vector-valued field problem. A key ingredient of FE-HMM is that macrostiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Beyond this coincidence with the Hill-Mandel condition, which is the cornerstone of the FE2 method, we elaborate a conceptual comparison with the latter method. After developing an algorithmic framework we (i) assess the existing a priori convergence estimates for the micro- and macro-errors in various norms, (ii) verify optimal strategies in uniform micro-macromesh refinements based on the estimates, (iii) analyze superconvergence properties of FE-HMM, and (iv) compare FE-HMM with FE2 by numerical results.