Lattice Boltzmann methods for solids

Class of computational fluid dynamics methods

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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.

## Proposed insights

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### Vectorial distributions

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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

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

### Force tuning

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#### Introduction

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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 → = 1 ρ ∇ x → ⋅ σ ¯ ¯ {\displaystyle {\vec {g}}={\frac {1}{\rho }}{\vec {\mathbf {\nabla } _{x}}}\cdot {\overline {\overline {\sigma }}}} ,

where  σ ¯ ¯ {\displaystyle {\overline {\overline {\sigma }}}} denotes the Cauchy stress tensor.  g → {\displaystyle {\vec {g}}} and  ρ {\displaystyle \rho } are respectively the gravity vector and solid matter density. The stress tensor is usually computed across the lattice aiming finite difference schemes.

#### Some results

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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 software used for finite element analysis comparison
* Non-linear materials
* Steady state convergence for LBMS

## Notes

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

## References

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1. ^ a b Marconi, Stefan; Chopard, Bastien (2003). "A Lattice Boltzmann Model for a Solid Body". International Journal of Modern Physics B. 17 (1n02): 153–156\. Bibcode:2003IJMPB..17..153M. doi:10.1142/S0217979203017254. ISSN 0217-9792.
2. ^ a b Frantziskonis, George N. (2011). "Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids". Physical Review E. 83 (6): 066703. Bibcode:2011PhRvE..83f6703F. doi:10.1103/PhysRevE.83.066703. PMID 21797512.
3. ^ a b 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. hdl:20.500.11850/548477.
4. ^ Xiao, Shaoping (2007). "A lattice Boltzmann method for shock wave propagation in solids". Communications in Numerical Methods in Engineering. 23 (1). Wiley Online Library: 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\. Bibcode:2000JCoPh.161...61G. doi:10.1006/jcph.2000.6486. ISSN 0021-9991.
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. 102 (3). Seismological Society of America: 1224–1234\. Bibcode:2012BuSSA.102.1224O. 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. Bibcode:2002PhRvE..65d6308G. 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.