## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
In fact, a GPU detains hundreds or thousands of cores capable of doing very fast executions despite management of few instructions and small caches. Finite difference schemes are playing an important role; in fact, they define external forces to be applied at each iteration for each cell. Upgrading abilities to treat mechanical singularities and enhancing accuracy under high field variation are part of the future work. For example, corners demand a special therapy because cell vicinity is restricted. Another possible way is to use robust mean-field forces as already implemented into the Shan-Chen approach [10]. Moreover, as mentioned above, WENO schemes could help us. Another advantage of the LBM compared to FEA, is this ease of moving from lower to higher dimensions. The transition from the 2D modeling to the 3D and 4D modeling is implied in the method. Then, LBMS extension to 3D should be straightforward and large-scale studies can be led from now on. Such computations permit a better comparison of the complex geometries with current methods usually dedicated to solving mechanical problems. The door is thus wide open in terms of further intensive analysis or significant studies. In addition, the parallelized strategy by blocks remains unchanged for solids. However, depending on the used finite difference schemes, some complementary properties are needed for block envelopes. Furthermore, for the broad range of [PERSON]'s modulus and [PERSON]'s ratios simulated, the presented method must be compared with non-linear constitutive laws, to confirm its potentiality and generality. Based on the strength of the presented results, a very straightforward extension is to investigate quasi-static dynamic problems. Of course, this eventuality could only be viable if the computation time of the LBMS is more deeply characterized. Only the static equilibrium has been compared to existing methods. Because this static equilibrium has been reached for different relaxation time, in particular for \(\omega=0.01\); solid dynamics with the Boltzmann-Vlasov equation seems a natural next step. The numerical stability for the relaxation parameter \(\omega<0.25\) is already unconventional LBM and supports the theoretical developments introduced here. Validation of the dynamic behavior requires additional work, especially for the reference mechanical study. Indeed, the transitional state before reaching static equilibrium is hard to confirm taking into account the actual state of our research and LBM groundwork for solids. As we mentioned before, the goal of the solid behavior with the LBM, might also be investigated through the research of an appropriate equilibrium distribution function. To go further with that idea, if \(\omega=1\), we also observe that (see Equation (15)):

\[\Omega\Big{(}f_{i}(G_{x},G_{y},t),f_{i}^{(eq),sld},s_{i}(G_{x},G_{y},t)\Big{)}= -\omega f_{i}(G_{x},G_{y},t)+\omega\Big{(}f_{i}^{(eq),sld}+s_{i}(G_{x},G_{y},t) \Big{)},\forall i\in[0,\ldots,8]. \tag{23}\]

This latter formulation yields a new form of solid equilibrium where solid body forces are integrated into the equilibrium function; thus, the stress divergence term may be integrated into the BGK operator if the relaxation parameter is equal to 1. Moreover, such an approach may looks like an artifact because with \(\omega=1\), the LBM is equivalent to a particular Finite Difference Method (FDM), with more complex boundary conditions. Thus, the previous equilibrium distribution deserves further dedicated investigations, beyond the work. Of course, many applications are plausible from the next development of the LBMS, especially regarding the interface and boundary conditions. By solving the solid interface, the motion and deformation of solids under the action of forces and contact problems between two solids could be achieved. Then, the solid-liquid interface leads to FSI in order to study the behavior of a solid product immersed in a liquid.