A Lattice Boltzmann Method for nonlinear solid mechanics in the reference configuration

TL;DR

This paper introduces a novel Lattice Boltzmann Method for large deformation nonlinear solid mechanics, formulated in the reference configuration, enabling efficient and versatile simulation of complex solid behaviors.

Contribution

It presents the first LBM algorithm for large strains in solid mechanics, formulated in the reference configuration, overcoming limitations of previous small strain, Eulerian approaches.

Findings

Efficiently simulates large deformation nonlinear solids.

Enables versatile boundary condition modeling.

Demonstrates applicability to complex solid mechanics problems.

Abstract

With a sufficiently fine discretisation, the Lattice Boltzmann Method (LBM) mimics a second order Crank-Nicolson scheme for certain types of balance laws (Farag et al. [2021]). This allows the explicit, highly parallelisable LBM to efficiently solve the fundamental equations of solid mechanics: the conservation of mass, the balance of linear momentum, and constitutive relations. To date, all LBM algorithms for solid simulation - see e.g. Murthy et al. [2017], Escande et al. [2020], Schl\"uter et al. [2021] - have been limited to the small strain case. Furthermore, the typical interpretation of the LBM in the current (Eulerian) configuration is not easily extensible to large strains, as large topological changes complicate the treatment of boundary conditions. In this publication, we propose a large deformation Lattice Boltzmann Method for geometrically and constitutively nonlinear solid mechanics. To facilitate versatile boundary modelling, the algorithm is defined in the reference (Lagrangian) configuration.