## 1 Introduction
### The Need for a LBM for Solids  
The efforts to construct top-down collision operators able to simulate many physical complex behaviors started when the LBM raised from the LGCA, with works from [PERSON] et al. [16; 17]. Some pioneering attempts, by [PERSON] et al. for example [18], in solid simulations using the LBM have been performed, with good application results on fractures and fragmentation for a solid body. These simulations used a distribution of forces rather than a distribution of particles as is usually the case with the LBM. Such distributions of forces bring two disadvantages. The mass conservation can not be recovered anymore. Secondly, since distributions are no more scalars but composed of vectors, it brings a large computational cost. Despite these limitations and the inability to fully recover the linear elastic equations, this work showed that the LBM is a good candidate when it comes to simulating the decohesion of the material.  
Alternative approaches for solid simulations exist with the LBM. They can be classified into two main categories: those dealing with velocity distribution functions [19; 20; 21] and those working with displacement distribution functions [22] in order to recover the Sophie-Germain equation [23]. In these previous works, the LBM is more often used as a numerical solver and the underlying link with the particles' dynamic is lost.  
Other works combined the lattice-spring method and the LBM for Fluid Structure Interaction (FSI) [24]. For instance, Ref. [25] is dedicated to the hemodynamic simulation in deformable blood vessels. However, the combination of two different methods with different coordinate systems complicates the simulation. In addition, pure viscoelastic wave propagation simulations using the LBM have been proposed [26; 27; 28]. All these works highlight the potential of the LBM for simulating FSI in a lot of applications requiring the use of an Eulerian grid.  
Indeed, simulating solids in an Eulerian framework, as in the LBM, is intricate and limited. The main limitation is certainly the loss of the initial configuration, which is not required for fluid simulations, but mandatory for solid simulations. [PERSON] et al. [29] proposed the so-called \"reference map\" to compute finite-difference simulations of large solid-like deformations. They highlighted the fact that this reference map can simplify the simulation of fluid/solid interactions, as both materials have a similar Eulerian expression.  
The benefits of being able to simulate solids are keys in solving FSI problems and simulation of specific constitutive laws in the LBM framework. The advantage of Eulerian framework is principally the absence of a deformable mesh. Moreover, interest in the LBM for the simulation based on medical imaging data [30] and biological phenomena [31] has been highlighted. It appears that the use of the LBM framework allows for the simulation of various problems based on what constitutes raw data in many biomedical applications, i.e., medical images.  
This work proposes an LBM for the simulation of solids, retaining the link with particles. It is mainly based on the use of the Boltzmann-Vlasov equation (see Section 5.2), which is a special case of the Boltzmann equation. To be more specific, only its restriction to static equilibrium, through dedicated force terms, is tested here. The aim is to introduce the newly proposed method in detail and to present some numerical implementations for its first validation.  
The work is organized as follows. Section 2 describes our model of LBM for solid aiming static equilibrium simulations. Section 3 presents numerical implementations and key algorithms of the method, and the integration in the Parallel Lattice Boltzmann Solver (PaLaBoS) [32] framework. Section 4 is dedicated to benchmarks and comparisons with Finite Element Analysis (FEA) using COMSOL software [33], in order to validate the method. Section 5 discusses the results and their openings, and Section 6 concludes the paper.