View source for Lattice Boltzmann methods for solids

{{Short description|Class of computational fluid dynamics methods}}
{{Orphan|date=December 2024}}

The '''Lattice Boltzmann methods for solids (LBMS)''' are a set of methods for solving [[partial differential equation]]s (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<ref name="Marconi_2003"/>
* Wave solvers<ref name="geo2011wave"/>
* Force tuning<ref name="mnnclbms"/>

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|fluid-structure interaction]] (FSI) in biomechanics.

== Proposed insights ==

=== Vectorial distributions ===
The first attempt<ref name="Marconi_2003"/> 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.<ref name="geo2011wave"/><ref name="xia07"/><ref name="Guangwu_2000a"/><ref name="obr12"/>

=== Force tuning ===

==== Introduction ====
This idea consists of introducing a modified version of the forcing term:<ref name="guo2002force"/> (or equilibrium distribution<ref name="noel2019"/>) into the LBM as a stress divergence force. This force is considered space-time dependent and contains solid properties<ref group="Note" name="notesolidproperties"/>

::<math>\vec{g} = \frac{1}{\rho} \vec{\mathbf{\nabla}_{x}} \cdot \overline{\overline{\sigma}}</math>,

where <math>\overline{\overline{\sigma}}</math> denotes the [[Cauchy stress tensor]]. <math>\vec{g}</math> and <math>\rho</math> are respectively the gravity vector and solid matter density. 
The stress tensor is usually computed across the lattice aiming [[Finite difference method|finite difference schemes]].

==== Some results ====
[[File:LBMS solid displacement.png|thumb|2D displacement magnitude on a solid system using force tuning. Obtained field is in accordance with [[finite element method]]s results.]]
Force tuning<ref name="mnnclbms"/> has recently proven its efficiency with a maximum error of 5% in comparison with standard [[Finite element method|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 method|finite element analysis]] comparison
* Non-linear materials
* Steady state convergence for LBMS

==Notes==
{{reflist|group=Note|refs=
<ref name="notesolidproperties">Matter properties such as Young's modulus and Poisson's ratio.</ref>
}}

== References ==

{{reflist|refs=
<ref name="geo2011wave">{{cite journal |last1=Frantziskonis |first1=George N. |date=2011 |title=Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids |journal=Physical Review E |volume=83 |issue=6 |pages=066703 |doi=10.1103/PhysRevE.83.066703|pmid=21797512 |bibcode=2011PhRvE..83f6703F }}</ref>

<ref name="guo2002force">{{cite journal |last1=Guo |first1=Zhaoli |last2=Zheng |first2=Chuguang |last3=Shi |first3=Baochang |title=Discrete lattice effects on the forcing term in the lattice Boltzmann method |journal=Physical Review E |date=2002 |volume=65 |issue=4 Pt 2B |page=046308|doi=10.1103/PhysRevE.65.046308 |pmid=12006014 |bibcode=2002PhRvE..65d6308G }}</ref>

<ref name="mnnclbms">{{cite journal |last1=Maquart |first1=Tristan |last2=Noël |first2=Romain |last3=Courbebaisse |first3=Guy |last4=Navarro |first4=Laurent |title=Toward a Lattice Boltzmann Method for Solids — Application to Static Equilibrium of Isotropic Materials |journal=Applied Sciences |date=2022 |volume=12 |issue=9 |page=4627|doi=10.3390/app12094627 |doi-access=free |hdl=20.500.11850/548477 |hdl-access=free }}</ref>

<ref name="Marconi_2003">{{cite journal 
	|last1= Marconi |first1= Stefan |last2=Chopard |first2=Bastien 
	|title= A Lattice Boltzmann Model for a Solid Body 
	|date= 2003
	|volume= 17
	|pages= 153–156 
	|issn= 0217-9792
	|doi= 10.1142/S0217979203017254
	|url= http://www.worldscientific.com/doi/abs/10.1142/S0217979203017254
	|journal= International Journal of Modern Physics B
	|number= 1n02
|bibcode= 2003IJMPB..17..153M }}</ref>

<ref name="Guangwu_2000a">{{cite journal 
	|last1= Guangwu |first1= Yan
	|title= A Lattice Boltzmann Equation for Waves
	|date= 2000
	|volume= 161
	|pages= 61–69
	|issn= 0021-9991
	|doi= 10.1006/jcph.2000.6486
	|url= http://www.sciencedirect.com/science/article/pii/S0021999100964866
	|journal= Journal of Computational Physics
	|number= 1
|bibcode= 2000JCoPh.161...61G
	}}</ref>

<ref name="xia07">{{cite journal 
	|last1= Xiao |first1= Shaoping
	|title= A lattice Boltzmann method for shock wave propagation in solids
	|journal= Communications in Numerical Methods in Engineering
	|volume= 23
	|number= 1
	|pages= 71–84
	|date= 2007
	|publisher= Wiley Online Library
|doi= 10.1002/cnm.883
	}}</ref>

<ref name="obr12">{{cite journal 
	|last1= O’Brien |first1= Gareth S |last2= Nissen-Meyer |first2= Tarje |last3= Bean |first3= CJ 
	|title= A lattice Boltzmann method for elastic wave propagation in a poisson solid
	|journal= Bulletin of the Seismological Society of America
	|volume= 102
	|number= 3
	|pages= 1224–1234 
	|date= 2012
	|publisher=Seismological Society of America
|doi= 10.1785/0120110191 |bibcode= 2012BuSSA.102.1224O }}</ref>

<ref name="noel2019">{{cite thesis 
	|last= Noël |first= Romain 
	|date= 2019
	|title= The lattice Boltzmann method for numerical simulation of continuum medium aiming image-based diagnostics
	|type= PhD 
	|chapter= 4 |publisher= Université de Lyon 
	|chapter-url= https://tel.archives-ouvertes.fr/tel-02955821
}}</ref>

}}

[[Category:Biomechanics]]
[[Category:Fluid dynamics]]
[[Category:Thermodynamics]]