## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
Thus, stress can be determined from the strain with only two scalar values, and these two tensors are symmetric:

\[\Sigma[G_{x},G_{y}]=C\ :\ \mathbb{E}[G_{x},G_{y}]=\lambda\operatorname{Tr} \left(\mathbb{E}[G_{x},G_{y}]\right)I+2\mu\mathbb{E}[G_{x},G_{y}]. \tag{16}\]

where \(C\) is the corresponding fourth-order isotropic stiffness tensor. Moreover, \(\lambda\) and \(\mu\) are the Lame coefficients. As mentioned above, \(\mathbb{E}\) and \(\Sigma\) are the strain and stress tensor fields defined over the whole lattice. Secondly, we want to superimpose the deformed configuration and the initial one. In order to do that, the involved displacement field must be small in terms of its norm. The material solid structure is then considered motionless during its transformation whereas the body remains deformable. In fact, all mechanical tensors such as displacement, stress, and strain are computed considering initial configuration, thus simplifying the problem. Furthermore, to be able to consider linearized strains, deformation must represent less than 1% of a solid characteristic dimension. Thirdly, in our process, distinction between the two different descriptions, i.e., Eulerian description (_a.k.a._ spatial coordinates) and Lagrangian description (_a.k.a._ material coordinates) does not apply. The Lagrange approach is widely preferred for solids where materials move with the coordinate system including elements. Basically, it can be explained by the fact that no matter passes from one element to another. On the other hand, in CFD, the mesh or lattice is fixed. Because the lattice structure is supposed to be stationary throughout time, this no-distinction assumption can be made freely. In substance, the considered trajectories of solid particles can be insignificant, thus allowing for the merging of these two descriptions. Furthermore, all stresses are assumed to be true, i.e., no differences arise from the deformation gradient, which is reduced to the identity matrix [43]. #### 4.1.2 Mechanical Study

The comparison study setup is quite simple. All borders are affected with fixed boundary conditions except one where displacement is imposed; see Figure 1 for more details. This enforced displacement is formulated as follows:

\[u=0\text{X}-0.002\text{Y}, \tag{17}\]

where numerical values are expressed in SI base units. X and Y represent first and second vectors of the space basis respectively. The Lame coefficients are classically retrieved frommaterial properties, mostly thanks to [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\). Then, we define three cut lines in order to post-process results with parametric curves so as to compare fields with thoroughness, i.e., thanks to \(C_{1}\), \(C_{2}\) and \(C_{3}\). #### 4.1.3 Lbms: Setup

Taking into account the previously targeted mechanical study, a lattice characterized by a \(D2\)\(\
eq\)\(9\) scheme is sampled with a resolution of 200 nodes per meter, yielding a suitable discretization. In this spirit, a mesh convergence study was made and it shows an improvement of the solution quality when lattice becomes finer; see Figures 1 and 2 for more details. A forced lattice descriptor is adopted in order to manage discrete force values. In all the following cases, a density of \(\rho=1000\) is affected to any cells regardless of the matter's nature. To simulate an imposed displacement condition, a velocity boundary condition is applied during a certain amount of time and then reduced to zero. The previous process allows solid equilibrium to be reached which is established after a couple of supplementary iterations following the change in value of imposed macroscopic speed. Displacement field is then stable and no longer evolves, producing next presented results. These results can be all retrieved and re-simulated freely from the C++ main file _Article.cpp_ included in the sources of the LBMS library. We aim to keep the initial configuration as close to the deformed one in order to compute fields onto the same lattice geometry regardless of the transformation.