## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
Concerning numerical correlations, the standard FEA solution is completely approved by \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Loading** & **Curve** & **Field** & \(E\) **(GPa)** & \(\
u\) & RMSE & NRMSE \\ \hline Rectangular & \(C_{2}\) (Figure 4) & Von Mises stress & 200 & 0.15 & \(7.37\times 10^{7}\) & 3.77\% \\ \hline Rectangular & \(C_{1}\) (Figure 5) & Displacement Y & 200 & \(-0.99\) & \(1.52\times 10^{-5}\) & 5.01\% \\ \hline Rectangular & \(C_{1}\) (Figure 5) & Displacement Y & 200 & \(-0.5\) & \(4.41\times 10^{-6}\) & 1.66\% \\ \hline Rectangular & \(C_{1}\) (Figure 5) & Displacement Y & 200 & 0.0 & \(1.23\times 10^{-5}\) & 5.50\% \\ \hline Triangular & \(C_{2}\) (Figure 7) & Displacement X & 200 & 0.15 & \(1.52\times 10^{-5}\) & 2.01\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE and NRMSE of the displacement and Von Mises stress obtained for different study cases. the mechanical community and hence serves as a reference solution. We further point out that both methods (i.e., LBMS and FEA) are truly different and must be compared carefully, see Table 2 for more details. Our proposed LBMS method is not very competitive in terms of computational time, because it was developed to evolve to solve complex FSI dynamic problems. In these mechanical cases, FEA runs are time-consuming calculations and the previously explained trend could be drastically reversed. On the other hand, the LBM approach can trade its \"locality\" for an improved computational time in steady-state cases as it has been illustrated in [40]. However, because our focus is on FSI dynamic problems, the latter approach was not used here. However, it contributes to the explanation of the limited computational time competitiveness of the suggested method. ### Considerations about the Vlasov Equation

Despite the previous attempts of LBM for solids mentioned in Section 1.1, to the best of the authors' knowledge, none of them use the physical representation of the particles of the Boltzmann equation. Indeed, the method previously used to derive an LBM-like method is considering the LBM equation as a pure mathematical framework to solve a set of Partial Differential Equations (PDE), without relying on the original fluid-based historical method. These previous mathematical methods found a set of advection-relaxation equations for new variables that are equivalent to a given macroscopic model. Such mathematical approaches contrast with the suggested method. We propose keeping this link between particles and the Boltzmann method. To do so, we find the similarities between plasma and solids. In both cases, the motion of particles is almost not subject to collisions: in the plasma case, the Knudsen number is large \(Kn\gg 1\) and particles are collisionless in the free-flight regime; whereas in solids \(Kn\to 0\) and particles are collisionless, because a particle does not move freely without interacting with the surrounding particles. Even if these two regimes should not be confused, they share some physical and mathematical concepts resulting in the fact that in both cases the collision term in the Boltzmann equation vanishes. Such a collisionless Boltzmann equation is called a Vlasov equation [46]. In order to find a suitable framework for the use of the Boltzmann equation for solids, it is possible to note that, at rest (at thermodynamic equilibrium), a particle distribution having a low temperature could be described by a Dirac distribution (omitting all quantum effects). Then, the solid matter at rest condition can be seen as a combination of motionless point mass, and without a large energy perturbation, the molecular interactions can be linearized. Starting from this state, a small vibration of one edge of the network will propagate freely. This situation can be described by the Vlasov equation of Dirac distribution centered on the velocity of the the surrounding particles, \(f(x,\xi,t)=\rho\delta_{v}(\xi)\).