Remarks on symmetry conditions in computational homogenisation problems

Abstract

Purpose

The purpose of this paper is to use symmetry conditions for the reduction of computing times in problems involving finite element‐based multi‐scale constitutive models of nonlinear heterogeneous media.

Design/methodology/approach

Two types of representative volume element (RVE) symmetry often found in practice are considered: staggered‐translational and point symmetry. These are analyzed under three types RVE of kinematical constraints: periodic boundary fluctuations (typical of periodic media), linear boundary displacements (which gives an upper bound for the macroscopic stiffness) and the minimum kinematical constraint (corresponding to uniform boundary tractions and providing a lower bound for the macroscopic stiffness).

Findings

Numerical examples show that substantial savings in computing times are achieved by taking advantage of such symmetries. These are particularly pronounced in fully coupled two‐scale analyses, where the macroscopic equilibrium problem is solved simultaneously with a large number of microscopic equilibrium problems at Gauss‐point level. Speed‐up factors in excess of seven have been found in such cases, when both symmetry conditions considered are present at the same time.

Originality/value

This paper extends the original considerations of Ohno et al. to account for other RVE kinematical constraints, namely, the linear boundary displacement and the minimum kinematical constraint (or uniform boundary traction model). Provides a more precise assessment of the impact of the use of such symmetries on computing times by means of numerical examples. In addition, for completeness, the direct enforcement of such constraints within a Newton‐based finite element solution procedure for the RVE equilibrium problem is detailed in the paper.

Citation

Saavedra Flores, E.I. and de Souza Neto, E.A. (2010), "Remarks on symmetry conditions in computational homogenisation problems", Engineering Computations, Vol. 27 No. 4, pp. 551-575. https://doi.org/10.1108/02644401011044612