Lattice Boltzmann methods for solids: Difference between revisions

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The '''Lattice Boltzmann methods for solids (LBMS)''' are specific methods based on the [[lattice Boltzmann methods]] (LBM). LBM are a group of numerical methods that are used to solve [[Partial differential equation|partial differential equations]] (PDE). These methods themselves relying on a discretization of the [[Boltzmann equation]] (BE). When the PDE at stake are related to solid mechanics, this subset of LBM is called lattice Boltzmann methods for solids. The main categories of LBMS are relying 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 toof describedescribing solidssolid 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 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} \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. Stress
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|finite element methods]] 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 be also be a tedious task since these methods are very different, common issues are:

* Meshes or lattice discretization

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<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 |page=046308}}</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—ApplicationSolids — Application to Static Equilibrium of Isotropic Materials |journal=Applied Sciences |date=2022 |volume=12 |page=4627}}</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= 01n02

}}</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

}}</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

}}</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

}}</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

|url= https://tel.archives-ouvertes.fr/tel-02955821

}}</ref>

}}