Boundary conditions for macroscale waves in an elastic system with microscale heterogeneity

TL;DR

This paper develops a systematic method to derive accurate macroscale boundary conditions for wave propagation in heterogeneous elastic materials, improving model fidelity in multiscale simulations.

Contribution

The authors introduce a novel approach to derive macroscale boundary conditions from microscale properties, enhancing the accuracy of multiscale wave models.

Findings

Derived boundary conditions improve macroscale model accuracy

Method successfully applied to various boundary value problems

Numerical tests confirm the effectiveness of the approach

Abstract

Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments, potentially resulting in a macroscale model which fails to adequately capture the behaviour of the microscale system. We derive the macroscale boundary conditions of the macroscale model for longitudinal wave propagation on a lattice with periodically varying density and elasticity. We model the macroscale dynamics of the microscale Dirichlet, Robin-like, Cauchy-like and mixed boundary value problem. Numerical experiments test the new methodology. Our method of deriving boundary conditions significantly improves the accuracy of the macroscale models. The methodology developed here can be adapted to a wide range of multiscale wave propagation problems.