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add references and description about vectorial distribution and wave solver sections |
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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.
=== Wave solvers ===
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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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