Convergence and Error Analysis of FE-HMM/FE$^2$ for Energetically Consistent Micro-Coupling Conditions in Linear Elastic Solids

TL;DR

This paper analyzes the convergence and error properties of FE-HMM/FE$^2$ methods under various micro-coupling conditions in linear elastic solids, focusing on error estimates, regularity effects, and mesh refinement strategies.

Contribution

It provides a detailed error and convergence analysis for FE-HMM/FE$^2$ with different coupling conditions, including error propagation and mesh refinement strategies.

Findings

Error estimates depend on regularity and coupling conditions.

Micro-macro error propagation is characterized.

Optimal mesh refinement strategies are identified.

Abstract

A cornerstone of numerical homogenization is the equivalence of the microscopic and the macroscopic energy densities, which is referred to as Hill-Mandel condition. Among these coupling conditions, the cases of periodic, linear displacement and constant traction conditions are most prominent in engineering applications. While the stiffness hierarchy of these coupling conditions is a theoretically established and numerically verified result, very little is known about the numerical errors and convergence properties for each of them in various norms. The present work addresses these aspects both on the macroscale and the microscale for linear as well as quadratic finite element shape functions. The analysis addresses aspects of (i) regularity and how its loss affects the convergence behavior on both scales compared with the a priori estimates, of (ii) error propagation from micro to macro and of (iii) optimal micro-macro mesh refinement strategy. For constant traction conditions two different approaches are compared. The performance of a recovery-type error estimation based on superconvergence is assessed. All results of the present work are valid for both the Finite Element Heterogeneous Multiscale Method FE-HMM and for FE$^2$.