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:
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
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]
,
where denotes the
Cauchy stress tensor. and are respectively the gravity vector and solid matter density.
The stress tensor is usually computed across the lattice aiming
finite difference schemes.
Some 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:
^
abFrantziskonis, 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.
PMID21797512.
^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.
doi:
10.1785/0120110191.
^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.
PMID12006014.
^Noël, Romain (2019).
"4". The lattice Boltzmann method for numerical simulation of continuum medium aiming image-based diagnostics (PhD). Université de Lyon.