## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
Then, the evolution of the particles described by this equation is a propagation of the particles without further modifications induced by the collisions. The interesting idea behind the Vlasov equation for our purpose is that complex systems with few collisions and strong interactions can be modeled by a Vlasov equation. Plus, these strong interactions are incorporated through the mean-field external forces term, which is the concept used in Equation (8). Therefore, considering the Vlasov equation of Dirac distribution with a mean-field divergence of Cauchy's stress, we obtained the following mesoscopic model: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Computation** & \multirow{2}{*}{**Iterations**} & \multirow{2}{*}{**Dependencies**} & \multirow{2}{*}{**FSI Ready?**} & \multirow{2}{*}{**Is Dynamic?**} \\  & **Time (s)** & & & & \\ \hline LBMS & 150 & 49,000 & PaLaBoS & Yes & Yes \\ \hline FEA & 8 & N/A & FEM solver & No & No \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between LBMS and standard FEA methods to compute a solid static equilibrium. Because LBMS method is intrinsically dynamic, results are not competitive for static studies. \[\frac{\partial f}{\partial t}(x,\xi,t)+\xi\cdot\
abla_{x}(f)+g\cdot \
abla_{\xi}(f) =0, \tag{18}\] \[f(x,\xi,t) =\rho\delta_{v}(\xi),\] (19) \[g =\frac{1}{\rho}\
abla_{x}\cdot(\sigma). \tag{20}\]

Then, using the passage formula for the two firsts moments \((1,\xi)\), the desired macroscopic system of the undefined equation of motion for solids is obtained from a theoretical point of view:

\[\partial_{t}(\rho)+\
abla_{x}\cdot\rho v =0, \tag{21}\] \[\partial_{t}(\rho v)+\
abla_{x}\cdot(\rho v\otimes v) =\
abla_{x}\cdot(\sigma). \tag{22}\]

Thus, the Boltzmann-Vlasov equation leading to the previous system of equations seems suitable for rigid matter simulations. Several remarks arise from these previous equations. This approach remains general for any constitutive law. Thus, the use of the Dirac distribution is somehow closer to the conventional representations of solid lattices: particles are not moving freely through the solid. In addition, the current discussion about the Vlasov equation for solids, is certainly presenting a working framework; however, the founding of this Vlasov equation requires a mathematical proof which is left for future work. Instead, we test here numerically some of the assumptions relative to this framework that are restricted to static equilibrium. ### Global Outlook and Proposed Improvements

C++ programmed classes and methods integrated into LBMS library can be drastically enhanced, i.e., mostly for efficiency purposes. Although the library is robust, computation time might be substantially reduced if other guidelines are adopted. For presented cases, tens or hundreds of seconds are necessary to reach the illustrated static equilibrium assuming sequential runs on an Intel(R) Xeon(R) CPU E5-1650 v3 @ 3.50 GHz, thus, involving a few tens of thousands of LBMS iterations. In contrast, a couple of seconds are enough to COMSOL even if the mesh is modeled very fine, because a standard static resolution is requested. Effectively, the involved computation times are hardly comparable because the methods are very different. In fact, LBMS is naturally dynamic due to its roots coming from the LBM and thus it requires an evaluation of the solid state at each time iteration. Concerning lattice discretization, additional experiments should be conducted in order to discover space convergence properties and effects. Thanks to the pointed reproducibility of the depicted numerical results, complements to existing objects can be made freely in order to improve calculation efficiency. However, here stands one of the great strengths of the LBM: parallel computing can be seen as one of the main advantages of LBM. The LBM is intrinsically efficient and adapted to massive parallel computing, especially on Graphics Processing Unit (GPU) because of its architecture.