## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
We resume the previous case in Section 4.2.1, but a right-angled triangular imposed displacement is applied (with the maximum displacement localized at the right of the considered problem) instead straight one (see Figure 1). Figure 6 presents the 2D results compared with the reference study. In general, the displacement field is consistent and replicates the expected solid behavior with fidelity. In detail, minor errors are detected in field values leading to the same conclusions as above. Additional curves are given in Figures A19 and A20 for Von [PERSON] stress and in Figures 7, A21-A23 for displacement components along cutting

Figure 5: Second displacement components along cutting curve \(C_{1}\) for negative Poisson’s ratios. precision illustrates the fidelity of the LBMS. The numerical errors obtained are far lower than the amplitude of the computed fields, which yields only a few percents of error. These errors values tend to quantitatively validate the presented approach. The numerical tests highlight the stability and accuracy of the suggested methods in a very broad range of mechanical parameters. This large range allows us to conclude that the current method could be applied to numerous constitutive laws, and not only to the isotropic linear elasticity. This wide range tends to support the theoretical development presented in this study, especially the incorporation of the stress divergence as a mean-field force. Concerning the introduced boundary conditions for solids, some work remains in order to enhance adaptation for solids. Currently, it has been pointed out that fixed conditions are far from perfect and suitable for accurate computation near concerned areas: impreciseness on fields can arise due to this modeling issue. Light differences with reference plots may have several explanations. First and foremost, the LBMS method obtains an iterative process so as to arrive at solid static equilibrium; thus, the finite difference scheme accuracy could play an important role concerning quality of the calculated results. Moreover, we point out some post-processing issues which may be at the origin of the errors. Indeed, along cutting curves, the FEA displacement field is mainly interpolated whereas the LBMS field hardly is. All of these considerations can cause additional deviations from the reference study and may explain some of illustrated differences. Other slight errors may be generated by the non-differentiation between Eulerian and Lagrangian frameworks. Several iterations are needed to reach the equilibrium state (i.e., when stress divergence tends towards a null vector), yielding completely different pipelines between FEA and LBMS. Furthermore, as mentioned above in Section 4.1.4, the used plane stress assumption for two-dimensional reference studies might trigger some of the shown deviations in curve plots, especially for non-zero Poisson's ratios. Indeed, thin plates are usually modeled with a plane stress hypothesis, but this approach is softly inconsistent with the one used to produce illustrated LBMS runs, i.e., a plane strain assignment. For all of these reasons, extra investigations seem to be mandatory to fit calculated reference fields with better accuracy. All these achievements seem promising in terms of the use of the Boltzmann-Vlasov approach for solid dynamics. It is also worth noting that the mass of the system is always conserved. Indeed, the intrinsic advantages of the LBM are preserved by the LBMS (macroscopic equations recovered from the statistical moments, locality, simplicity, etc.). In contrast to previous attempts to cope with solid behavior with the LBM, our approach does not require statistical distribution of forces [18, 45] nor springs [24]. Both bring more complexity, heavier computations, are further away from the Boltzmann theory, and are less prone to FSI coupling. For our validation case, we do not have an analytical solution due to boundary conditions issues. A simpler test case is not easily achievable in a two dimensional domain.