A total-Lagrangian vectorial lattice Boltzmann method for finite-strain hyperelastic dynamics

Abstract

Inspired by the vectorial lattice Boltzmann method for linear elastodynamics \citep{boolakee2025linear}, we construct a total-Lagrangian vectorial lattice Boltzmann formulation for two-dimensional finite-strain hyperelastic dynamics. The governing equations are first written as a conservative first-order system for the material velocity and the full deformation gradient. This representation separates the kinematic part of the dynamics from the constitutive closure: the first Piola--Kirchhoff stress is evaluated locally from the current deformation gradient and enters the lattice only through nonlinear flux moments. A D2Q4 stencil with six-component vector populations is then used to match the state and the two material-coordinate fluxes. The formulation includes a second-order population initialization, trapezoidally centered body forcing, displacement reconstruction by velocity quadrature, and half-way reconstructions for velocity Dirichlet and Neumann traction boundaries on grid-aligned domains. The resulting method preserves the local collide--stream structure of standard lattice Boltzmann schemes while adapting the vectorial first-order strategy from linear elastodynamics to hyperelastic finite-strain dynamics.