Lattice Boltzmann methods for solids

The Lattice Boltzmann methods for solids (LBMS) are specific methods based on the lattice Boltzmann methods (LBM). LBM are a group of numerical methods that are used to solve partial differential equations (PDE). These methods themselves relying on a discretization of the Boltzmann equation (BE). When the PDE at stake are related to solid mechanics, this subset of LBM is called lattice Boltzmann methods for solids. The main categories of LBMS are relying on:

• Vectorial distributions
• Wave solvers^[1]
• Force tuning^[2]

The LBMS subset remains highly challenging from a computational aspect as much as from a theoretical point of view. Solving solid equations within the LBM framework is still a very active area of research. If solids are solved, this shows that the Boltzmann equation is capable to describe solids motions as well as fluids and gases: thus unlocking complex physics to be solved such as fluid-structure interaction (FSI) in biomechanics.

Proposed insights

Vectorial distributions

Wave solvers

Force tuning

Introduction

This idea consists of introducing a modified version of the forcing term^[3] into the LBM as a stress divergence force. This force is considered space-time dependent and contains solid properties^{[Note 1]}:

${\displaystyle {\vec {g}}={\frac {1}{\rho }}\mathbf {\nabla } _{x}\cdot {\overline {\overline {\sigma }}}}$ ,

where ${\displaystyle {\overline {\overline {\sigma }}}}$  denotes the Cauchy stress tensor. ${\displaystyle {\vec {g}}}$  and ${\displaystyle \rho }$  are respectively the gravity vector and solid matter density. Stress tensor is usually computed accross the lattice aiming finite difference schemes.

Some results

 

2D displacement magnitude on a solid system using force tuning. Obtained field is in accordance with finite element methods results.

Force tuning^[2] has recently proven its efficiency with a maximum error of 5% in comparison with standard finite element solvers in mechanics. Accurate validation of results can be also a tedious task since these methods are very different, common issues are:

• Meshes or lattice discretization
• Location of computed fields at elements or nodes
• Hidden information in softwares used for finite element analysis comparison
• Non-linear materials
• Steady state convergence for LBMS

Notes

1. Matter properties such as Young's modulus and Poisson's ratio.

References

1. Frantziskonis, George N. (2011). "Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids". Physical Review E. 83 (6): 066703. doi:10.1103/PhysRevE.83.066703.
2. Maquart, Tristan; Noël, Romain; Courbebaisse, Guy; Navarro, Laurent (2022). "Toward a Lattice Boltzmann Method for Solids—Application to Static Equilibrium of Isotropic Materials". Applied Sciences. 12: 4627.
3. Guo, Zhaoli; Zheng, Chuguang; Shi, Baochang (2002). "Discrete lattice effects on the forcing term in the lattice Boltzmann method". Physical review E. 65: 046308.