Efficient computational homogenisation of 2D beams of heterogeneous elasticity using the patch scheme

TL;DR

This paper presents a multiscale patch scheme that efficiently and accurately predicts the macroscale behavior of 2D heterogeneous elastic beams using microscale computations on small patches, supported by dynamical systems theory.

Contribution

It introduces a non-intrusive, systematic multiscale patch scheme for homogenizing 2D heterogeneous beams, enabling accurate macroscale predictions with finite scale separation.

Findings

The patch scheme accurately predicts macroscale beam behavior.

The method is stable and efficient for various boundary conditions.

Supports large-scale engineering simulations with microscale detail.

Abstract

Modern 'smart' materials have complex heterogeneous microscale structure, often with unknown macroscale closure but one we need to realise for large scale engineering and science. The multiscale Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately predict the large scale, system level, solutions through computations on only small sparse patches of the given detailed microscale system. Here the microscale system is that of a 2D beam of heterogeneous elasticity, with either fixed fixed, fixed-free, or periodic boundary conditions. We demonstrate that the described multiscale Patch Scheme simply, efficiently, and stably predicts the beam's macroscale, with a controllable accuracy, at finite scale separation. Dynamical systems theory supports the scheme. This article points the way for others to use this systematic non-intrusive approach, via a developing toolbox of functions, to model and compute accurately macroscale system-levels of general complex physical and engineering systems.