The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE~2 method

摘要

AbstractThe 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 FE2method, 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 FE2by numerical results.Highlights•Aspects of FE-HMM implementation for the vector-valued field problem of linear elasticity.•Explanation of FE-HMM by direct links to asymptotic homogenization.•Unified conceptual and numerical comparison of FE-HMM with FE2.•Assessment and verification of a priori error estimates and of superconvergence.•Assessment of optimal uniform micro-/macromesh refinement strategies.