## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
A recent publication [47], illustrates the fact that a unified theory is needed to solve complex FSI phenomena, taking into consideration both fluid and solid characteristics with LBM. ## 6 Conclusions

We have introduced in this work a new method to solve solid static equilibriums from standard LBM. Our strategy depicts promising results concerning the static state of isotropic materials. The presented plots and figures illustrate well the efficiency of the presented method. Such a task is achieved thanks to detailed developments and a robust framework. In fact, the presented method is secured with a theory groundwork and then implemented using classical programming tools. A first key contribution was introduced thanks to theoretical developments around the Boltzmann-Vlasov equation. Furthermore, the stress tensor divergence was introduced as a mean-field force term. These two considerations allowed for the building of a theoretical framework to deal with solids, with the LBM. This framework is unprecedented and is very unique compared to previous works on the subject. Furthermore, we produced new programmed objects included in a user-friendly research library. Although this implementation is not optimal, the developed classes and methods are robust enough and trustworthy concerning the conclusions given on plots and figures. On the other hand, the adopted general implementation approach detseveral advantages. Non-invasive coding work permits the reproduction of results with a bewildering ease. Likewise, a light program environment enables easy-way additions for future improvements. Moreover, the standard LBM main loop is still not modified, i.e., only additional routines to treat solids are added. Basically, this means the colliding-streaming strategy is kept without compromise; only the dynamic related to solids is adjusted. Previous abilities, among others, allowed for modeling of fluids and solids in the same framework. The shown results are validated through comparative plots including a large range of parametric studies. They clearly prove the credibility of the proposed approach. Furthermore, for a significant number of plotted data, the matching with reference analysis is impressive. In all cases, exhibited mapped fields are globally consistent and replicate a real solid matter's comportment. The great precision of the results, not only highlights the capacity of the method to capture isotropic elasticity but is also allows us to imagine a complete confirmation of the general theory developed. Moreover, the reliable and stable results obtained at very high relaxation time (\(\omega\leq 0.01\)) show the strength of the theory and the perspectives for solid dynamics. Despite the promising illustrated results contrasted with reference studies, a lot of work still needs to be done to exploit the presented potential of the LBMS. Several aspects still require further afterthought so as to simulate proper solid behaviors. Indeed, modeling solids with LBM is still a challenge among scientific communities even though advances are presented in this work. This work also tries to answer to increasing expectations in numerical simulation fields. Beyond all these obtained results, LBMS seems very promising to simulate solids in an Eulerian framework. This work aims to bring new tools to push the limits of computational mechanics even further. Biomechanics is facing difficulties such as large deformations, non-linear behaviors, complex geometries and FSI. Standard approaches have several drawbacks in solving previously involved physics. LBM methods are, thus, naturally better suited to tackle these mentioned difficulties. The solid-solid contact problem is another perspective of the LBMS. It seems that the formulation of contacts is less difficult on the geometrical aspect in an Eulerian framework but will be more complex regarding formulation of the interaction equations between different objects. A large spectrum of perspectives is thus opened by the presented method.