Dirichlet and Neumann boundary conditions in a Lattice Boltzmann Method for Elastodynamics

TL;DR

The paper develops simple local boundary rules to impose Dirichlet and Neumann boundary conditions in a Lattice Boltzmann Method for elastodynamics, formulating the Lamé-Navier equation as a moment chain with $\rho$, $\mathbf{j}$, and $\mathbf{P}$. It derives Dirichlet/Neumann boundary mappings in the moment framework and implements lattice-conforming and non-lattice-conforming rules using modified bounce-back and Bouzidi-type interpolation on a D2Q9 lattice. Validation against analytical and FEM benchmarks for plates with holes and dynamic cracks shows that the proposed boundary treatments capture key elastodynamic features and yield good agreement, while highlighting stability and accuracy limitations. The work suggests that LBM, with these boundary rules, can serve as an efficient, parallelizable tool for transient solid problems, with clear avenues for improvements and extensions to nonlinear regimes and complex materials.

Abstract

Recently, Murthy et al. [2017] and Escande et al. [2020] adopted the Lattice Boltzmann Method (LBM) to model the linear elastodynamic behaviour of isotropic solids. The LBM is attractive as an elastodynamic solver because it can be parallelised readily and lends itself to finely discretised dynamic continuum simulations, allowing transient phenomena such as wave propagation to be modelled efficiently. This work proposes simple local boundary rules which approximate the behaviour of Dirichlet and Neumann boundary conditions with an LBM for elastic solids. Both lattice-conforming and non-lattice-conforming, curved boundary geometries are considered. For validation, we compare results produced by the LBM for the sudden loading of a stationary crack with an analytical solution. Furthermore, we investigate the performance of the LBM for the transient tension loading of a plate with a circular hole, using Finite Element (FEM) simulations as a reference.

Figures (10)

• Figure 1: (Part of) a D2Q9 lattice for the LBM.
• Figure 2: Lattice-conforming boundary (dotted line) with lattice velocities leading out of the domain (blue) and lattice velocities for which distribution functions are missing (red) for a boundary lattice site at $\bm{x}$.
• Figure 3: Non-lattice conforming boundary for one lattice link with length $l_i$. Adapted from bouzidi, figure by the authors.
• Figure 4: Square plate with circular hole subjected to an in-plane tension load.
• Figure 5: $x_2$-components of dimensionless displacements at $P_1$ post-processed from LBM (blue) and FE (red) simulations of a Poisson- and non-Poisson solid under tension loading.