Dirichlet and Neumann boundary conditions in a Lattice Boltzmann Method for Elastodynamics

TL;DR

This paper introduces simple local boundary rules for the Lattice Boltzmann Method to accurately model Dirichlet and Neumann boundary conditions in elastodynamics, validated through crack and plate simulations.

Contribution

It proposes novel boundary treatment rules for LBM in elastodynamics, applicable to complex geometries, enhancing simulation accuracy for wave propagation problems.

Findings

LBM accurately models crack loading with analytical comparison

LBM performs well in transient tension of a plate with a circular hole

Boundary rules improve LBM's handling of curved geometries

Abstract

Recently, Murthy et al. [2017] and Escande et al. [2020] adopted the Lattice Boltzmann Method (LBM) to model the linear elastodynamic behaviour of isotropic solids. The LBM is attractive as an elastodynamic solver because it can be parallelised readily and lends itself to finely discretised dynamic continuum simulations, allowing transient phenomena such as wave propagation to be modelled efficiently. This work proposes simple local boundary rules which approximate the behaviour of Dirichlet and Neumann boundary conditions with an LBM for elastic solids. Both lattice-conforming and non-lattice-conforming, curved boundary geometries are considered. For validation, we compare results produced by the LBM for the sudden loading of a stationary crack with an analytical solution. Furthermore, we investigate the performance of the LBM for the transient tension loading of a plate with a circular hole, using Finite Element (FEM) simulations as a reference.