A Lattice Boltzmann Method for Elastic Wave Propagation in a Poisson Solid

The lattice Boltzmann (LB) method is a numerical method that has its origins in discrete mechanics. The method is based on propagating discrete density distributions across a fixed lattice and implementing conservation laws between the density distributions at lattice intersections. The method has been successfully applied to a wide variety of problems in fluid dynamics but has yet to be applied to elastic wave propagation. In this article we outline a new 2D and 3D LB solution to the elastic wave equation in a Poisson solid using a regular lattice, in 2D a square geometry and in 3D a cubic geometry. We outline the theory behind the method and derive the elastic wave equation from a Chapman-Enskog expansion about the Knudsen number. The scheme is shown to give rise to the elastic wave equation with a fixed Poisson ratio of 0.25 with a Knudsen number truncation error of order two. We have performed a von Neumann plane-wave analysis and found that the numerical dispersion is comparable to other discrete methods for modelling wave propagation. We have compared the numerical method to two problems, a 3D infinite homogeneous medium and a 2D heterogeneous block model. In both cases, we found the solutions agreed, thus showing that the LB method can be used to model elastic wave propagation. The scheme offers the potential to model the interaction of several continuum equations within one solver as the continuum equation is solely dependent on the equilibrium distribution.