A Lattice Boltzmann Method for Elastic Solids Under Plane Strain Deformation

TL;DR

This paper extends the Lattice Boltzmann Method to solve plane strain problems in elastic solids, implementing boundary conditions and comparing its performance to the Finite Element Method.

Contribution

It introduces an LBM scheme for general plane strain problems in elastic solids, including boundary condition implementation and performance evaluation.

Findings

LBM effectively models plane strain elastic problems.

The method accurately handles Dirichlet and Neumann boundary conditions.

LBM shows competitive performance compared to Finite Element Method.

Abstract

The Lattice Boltzmann Method (LBM), e.g. in [ 1] and [2 ], can be interpreted as an alternative method for the numerical solution of partial differential equations. Consequently, although the LBM is usually applied to solve fluid flows, the above interpretation of the LBM as a general numerical tool, allows the LBM to be extended to solid mechanics as well. In this spirit, the LBM has been studied in recent years. First publications [3], [4] presented an LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. For so-called anti-plane shear deformation, the only non-zero displacement component is governed by a two-dimensional wave equation. In this work, an existing LBM for the two-dimensional wave equation is extended to more general plane strain problems. The proposed algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non-zero component of the rotation vector, respectively. A particular focus is on the implementation of types of boundary conditions that are commonly encountered in engineering practice for solids: Dirichlet and Neumann boundary conditions. Last, several numerical experiments are conducted that highlight the performance of the new LBM in comparison to the Finite Element Method.