Lattice Boltzmann methods for solids

Short description: Class of computational fluid dynamics methods

The Lattice Boltzmann methods for solids (LBMS) are a set of methods for solving partial differential equations (PDE) in solid mechanics. The methods use a discretization of the Boltzmann equation(BM), and their use is known as the lattice Boltzmann methods for solids.

LBMS methods are categorized by their reliance on:

* Vectorial distributions[1]
* Wave solvers[2]
* Force tuning[3]

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 of describing solid motions as well as fluids and gases: thus unlocking complex physics to be solved such as fluid-structure interaction (FSI) in biomechanics.

## Contents

* 1 Proposed insights
* 1.1 Vectorial distributions
* 1.2 Wave solvers
* 1.3 Force tuning
* 1.3.1 Introduction
* 1.3.2 Some results
* 2 Notes
* 3 References

## Proposed insights

### Vectorial distributions

The first attempt[1] of LBMS tried to use a Boltzmann-like equation for force (vectorial) distributions. The approach requires more computational memory but results are obtained in fracture and solid cracking.

### Wave solvers

Another approach consists in using LBM as acoustic solvers to capture waves propagation in solids.[2][4][5][6]

### Force tuning

#### Introduction

This idea consists of introducing a modified version of the forcing term:[7] (or equilibrium distribution[8]) into the LBM as a stress divergence force. This force is considered space-time dependent and contains solid properties[Note 1]

g→=1ρ∇x→⋅σ‾‾,

where σ‾‾ denotes the Cauchy stress tensor. g→ and ρ are respectively the gravity vector and solid matter density. The stress tensor is usually computed across 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[3] 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 also be 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. ↑ 1.0 1.1 Marconi, Stefan; Chopard, Bastien (2003). "A Lattice Boltzmann Model for a Solid Body". International Journal of Modern Physics B 17 (1n02): 153–156. doi:10.1142/S0217979203017254. ISSN 0217-9792. http://www.worldscientific.com/doi/abs/10.1142/S0217979203017254.
2. ↑ 2.0 2.1 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. PMID 21797512.
3. ↑ 3.0 3.1 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 (9): 4627. doi:10.3390/app12094627.
4. ↑ Xiao, Shaoping (2007). "A lattice Boltzmann method for shock wave propagation in solids". Communications in Numerical Methods in Engineering (Wiley Online Library) 23 (1): 71–84. doi:10.1002/cnm.883.
5. ↑ Guangwu, Yan (2000). "A Lattice Boltzmann Equation for Waves". Journal of Computational Physics 161 (1): 61–69. doi:10.1006/jcph.2000.6486. ISSN 0021-9991. http://www.sciencedirect.com/science/article/pii/S0021999100964866.
6. ↑ O’Brien, Gareth S; Nissen-Meyer, Tarje; Bean, CJ (2012). "A lattice Boltzmann method for elastic wave propagation in a poisson solid". Bulletin of the Seismological Society of America (Seismological Society of America) 102 (3): 1224–1234. doi:10.1785/0120110191.
7. ↑ Guo, Zhaoli; Zheng, Chuguang; Shi, Baochang (2002). "Discrete lattice effects on the forcing term in the lattice Boltzmann method". Physical Review E 65 (4 Pt 2B): 046308. doi:10.1103/PhysRevE.65.046308. PMID 12006014.
8. ↑ Noël, Romain (2019). "4". The lattice Boltzmann method for numerical simulation of continuum medium aiming image-based diagnostics (PhD). Université de Lyon.

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