Lattice Boltzmann methods for solids: Difference between revisions

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{{Short description|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 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.

The lattice Boltzmann methods (LBM) are a group of numerical methods that are used to solve partial derivatives equations (PDE).

These methods themselves relying on a discretization of the Boltzmann equation.

When the PDE at stake are related to solid mechanics, the subset of LBM is called lattice Boltzmann methods for solids (LBMS).

LBMS methods are categorized by their reliance on:

The main categories of LBMS are: vectorial distributions, wave solvers, force tuning, etc.

* Vectorial distributions<ref name="Marconi_2003"/>

The LBMS subset remains highly challenging from a computational aspect as much as from a theoretical point of view.

* 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 ==

<!-- Inline citations added to your article will automatically display here. See en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations. -->

{{reflist|refs=

{{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}}

<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 |url-access= subscription}}</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

|url-access= subscription

}}</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]]