Lattice Boltzmann methods for solids: Difference between revisions

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{{Short description|definitionClass of thecomputational latticefluid Boltzmanndynamics methods for solids}}

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The '''Lattice Boltzmann methods for solids (LBMS)''' are specific methods based on the [[lattice Boltzmann methods]] (LBM). LBM are a groupset of numerical methods thatfor are used to solvesolving [[Partialpartial differential equation|partial differential equations]]s (PDE) in solid mechanics. TheseThe methods themselves relying onuse a discretization of the [[Boltzmann equation]] (BEBM)., Whenand thetheir PDEuse atis stakeknown areas related to solid mechanics, this subset of LBM is calledthe lattice Boltzmann methods for solids. The main categories of LBMS are relying on:

LBMS methods are categorized by their reliance on:

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

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=== 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.

The approach requires more computational memory but obtained results 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.

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==== Some results ====

[[File:LBMS solid displacement.png|thumb|2D displacement magnitude on a solid system using force tuning. Obtained field is in accordance with [[Finitefinite element method|finite element methods]]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 softwaressoftware used for [[Finite element method|finite element analysis]] comparison

* Non-linear materials

* Steady state convergence for LBMS

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== 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=

<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 reviewReview 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

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|date= 2003

|volume= 17

|pages= 153--156153–156

|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= 01n021n02

|bibcode= 2003IJMPB..17..153M |url-access= subscription}}</ref>

}}</ref>

<ref name="Guangwu_2000a">{{cite journal

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|date= 2000

|volume= 161

|pages= 61--6961–69

|issn= 0021-9991

|doi= 10.1006/jcph.2000.6486

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|journal= Journal of Computational Physics

|number= 1

|bibcode= 2000JCoPh.161...61G

}}</ref>

|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 numericalNumerical methodsMethods in engineeringEngineering

|volume= 23

|number= 1

|pages= 71--8471–84

|date= 2007

|publisher= Wiley Online Library

|doi= 10.1002/cnm.883

}}</ref>

}}</ref>

<ref name="obr12">{{cite journal

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|volume= 102

|number= 3

|pages= 1224--12341224–1234

|date= 2012

|publisher=Seismological Society of America

|doi= 10.1785/0120110191 |bibcode= 2012BuSSA.102.1224O }}</ref>

}}</ref>

<ref name="noel2019">{{cite thesis

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|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]]