Dirichlet and Neumann boundary conditions in a lattice Boltzmann method for elastodynamics

Recently, Murthy et al. (Commun Comput Phys 2:23, 2017. http://dx.doi.org/10.4208/cicp.OA-2016-0259 ) and Escande et al. (Lattice Boltzmann method for wave propagation in elastic solids with a regular lattice: theoretical analysis and validation, 2020. arXiv.doi:1048550/ARXIV.2009.06404. arXiv:2009.06404) 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 simulations of dynamic effects in continua, allowing transient phenomena such as wave propagation to be modeled efficiently. This work proposes simple local boundary rules which approximate the behaviour of Dirichlet and Neumann boundary conditions with an LBM for elastic solids. The boundary rules are shown to be consistent with the target boundary values in the first order. An empirical convergence study is performed for the transient tension loading of a rectangular plate, with a Finite Element (FE) simulation being used as a reference. Additionally, we compare results produced by the LBM for the sudden loading of a stationary crack with an analytical solution from Freund (Dynamic fracture mechanics. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511546761).