A new lattice Boltzmann scheme for linear elastic solids: periodic problems

TL;DR

This paper introduces a second-order accurate lattice Boltzmann scheme for solving linear elasticity problems in two dimensions, emphasizing simplicity, stability, and computational efficiency without finite difference approximations.

Contribution

The paper presents a novel lattice Boltzmann scheme that uses a single distribution function and standard velocity set for linear elasticity, improving simplicity and stability over previous methods.

Findings

The scheme achieves second-order accuracy and stability across a wide range of Poisson's ratios.

Numerical experiments confirm the theoretical convergence and stability analysis.

The method avoids finite difference approximations, leveraging full lattice Boltzmann benefits.

Abstract

We propose a new second-order accurate lattice Boltzmann scheme that solves the quasi-static equations of linear elasticity in two dimensions. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The novel scheme is systematically derived using the asymptotic expansion technique and a detailed analysis of the leading-order error behavior is provided. As demonstrated by a linear stability analysis, the method is stable for a very large range of Poisson's ratios. We consider periodic problems to focus on the governing equations and rule out the influence of boundary conditions. The analytical derivations are verified by numerical experiments and convergence studies.