Lattice Boltzmann for linear elastodynamics: periodic problems and Dirichlet boundary conditions

TL;DR

This paper introduces a second-order accurate lattice Boltzmann method for linear elastodynamics, capable of handling arbitrary material parameters with stability and accuracy verified through numerical experiments.

Contribution

It develops a novel vector-valued lattice Boltzmann scheme for elastodynamics, ensuring second-order accuracy and stability for complex boundary conditions and material parameters.

Findings

Second-order accuracy demonstrated through convergence studies.

Stable for arbitrary material parameters under CFL-like conditions.

Effective boundary treatment for Dirichlet conditions in 2D domains.

Abstract

We propose a new second-order accurate lattice Boltzmann formulation for linear elastodynamics that is stable for arbitrary combinations of material parameters under a CFL-like condition. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that all computational benefits of the algorithm are preserved. Using the asymptotic expansion technique and the notion of pre-stability structures we further establish second-order consistency as well as analytical stability estimates. Lastly, we introduce a second-order consistent initialization of the populations as well as a boundary formulation for Dirichlet boundary conditions on 2D rectangular domains. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests.