## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
Furthermore, throughout the iterative LBMS process, error accumulation is prevented by using a linear behavior implicating linearized strains: 1% of deformation is, thus, the maximum amplitude that we allow ourselves for model validation. #### 4.1.4 Comsol: Setup

A benchmark, i.e., a mechanical study which is accurate under assumptions presented above, has to be carefully established. For that purpose, we choose COMSOL [33] as the FEA preprocessor and solver for its simplicity of use. However, some simulation options remain important to set even to resolve a standard static problem. As mentioned above, linearized strains are assumed and in the way forced by default. Because displacement is considered non-existent, all stresses are supposed to be true. Hence, computed Cauchy stress can be directly compared with the tensor field \(\Sigma\) from the proposed LBMS approach. A plane stress assumption is made so as to simulate a very thin plate. Although, stress-strain relation of this hypothesis is slightly different than the Hooke's law presented previously; we keep this in mind when investigating results. A standard extra fine triangular mesh is generated with fully integrated elements using cubic shape functions. This setup enables accurate computations of the solution at boundary interfaces or areas subject to stress concentration. More formally, we want a reliable solution that can be examined in detail, e.g., near discontinuities produced by the imposed displacement. Figure 1: Mechanical study for LBMS approach validation. Black frontiers are fixed boundary conditions. A displacement is imposed across the vector \(u\) described with black, as is depicted on the left. Solid material properties are defined through Young’s modulus \(E\) and Poisson’s ratio \(\
u\). Blue oriented curves \(C_{1}\), \(C_{2}\) and \(C_{3}\) are used for post-processing. All dimensions are given in meters. Concerning nearly incompressible materials [37; 38; 43] later presented in this work, we use a hybrid element approach, i.e., a mixed formulation that avoids volumetric locking for fully integrated elements. Other methods exist to solve this issue, see e.g., [44]. Indeed, due to the constant pressure inside triangles, this numerical phenomenon can arise and may distort the solution. In any cases, the mesh discretization and nature of shape functions further secure the simulation reliability avoiding other locking difficulties. Used triangular elements are rich enough to represent a correct result. Another solution would be to mesh with reduced integration elements where volumetric locking problem does not exist; however, an hourglass control then becomes necessary. ### Results Comparison with Commercial Software

Reference FEA studies and associated LBMS simulations are compared with 2D plots and curves. Parametric numerical analysis is led so as to show performance details among a wide range of isotropic materials, i.e., \(\
u\ \in\ [-0.99,\ldots,0.49]\) and \(E\ \in\ [2\times\ 10^{3},\ldots,2\times\ 10^{11}]\). They show the undeniable capability of the LBM to solve solid mechanics cases. #### 4.2.1 Case \(E=200\) GPa, \(\
u=0.15\)

Firstly, we decide to illustrate a specific case, i.e., with material properties \(E=200\) GPa, \(\
u=0.15\). This example shows typical results of our presented framework. Figure 2 contrasts the reference study with the proposed LBMS study. Strong similarities and likenesses are observed between the two different approaches. Displacement extremum values remain consistent across the lattice and are directly correlated with the related FEA study. More locally, field values are practically identical and same solid behavior is exhibited. However, we notice unwanted small oscillations of the displacement field arising from the used finite difference scheme. As we mentioned above, WENO schemes may certainly improve the solution quality. In addition, these perturbations can be drastically reduced by setting a smoother loading profile.