A High-Order Dual-Distribution Lattice Boltzmann Method for Hyperelastic Dynamics

The lattice Boltzmann method (LBM) has been applied to dynamic simulations of hyperelastic solids. However, when a single distribution function is used, the nonlinear part of the hyperelastic stress is difficult to represent through the moment terms of the distribution function. As a result, the constitutive relation of hyperelastic materials cannot be completely recovered, which limits the computational accuracy of LBM simulations. To address this issue, a dual-distribution-function formulation is introduced in this work. In this formulation, an additional set of distribution functions is assigned specifically to the deformation-gradient tensor, so that the complete hyperelastic stress can be naturally obtained through the evolution of the deformation gradient. Based on the D2Q9 lattice and BGK relaxation, a LBM algorithm with third-order accuracy is developed for dynamic simulations of compressible Neo-Hookean materials through a fourth-order Chapman--Enskog expansion. The implementation of Dirichlet and Neumann boundary conditions under the dual-distribution-function formulation is also discussed. Numerical experiments demonstrate that the proposed algorithm achieves higher computational accuracy than existing methods, and the admissible range of Poisson's ratio is substantially extended.