## 3 Numerical Implementations and Algorithms
### Additional Solid Routines
Figure 3 shows us the deformed shape and displacement field magnitude of the lattice computed with our LBMS approach. Figure 2: 2D results comparison of displacement field for case \(E=200\) GPa, \(\
u=0.15\) per field component. (**a**) Displacement along \(X\) axis, with LBMS. (**b**) Displacement along \(Y\) axis, with LBMS. (**c**) Displacement along \(X\) axis, with COMSOL. (**d**) Displacement along \(Y\) axis, with COMSOL. Figures 4 and A3 depict the Von Mises stress distribution along cutting curves \(C_{1}\) and \(C_{2}\), respectively (see Figure 1 for more details concerning \(C_{1}\) and \(C_{2}\)). Comparing stress values between different codes is often very sensitive. Because we solve for displacement in an FEA approach, stresses values are strongly dependent on the computed displacement field accuracy. Plotted curves show a similar global behavior and the same order of magnitude (no more than 15% of difference in Figure A3), although Figure 4 has a striking match with the reference study. These first results lead to the proposed method having an interesting potential. #### 4.2.2 Poisson's Ratio Sensitivity

To evaluate the reliability of the proposed method, parametric studies are led with various values of \(\
u\) with a fixed \(E\) = 200 GPa. For that purpose several curves are analyzed to illustrate the capability of our LBMS method to simulate a wide range of [PERSON]'s ratios. Figures 5, A4, A6 and A11 exhibit results with negative \(\
u\) whereas Figures A5, A7, A10 and A12 present positive ones. For \(\
u\in[-0.50,\ldots,0.35]\) results are satisfying and replicate expected displacement field with a good accuracy and credibility. Concerning highly auxetic materials and nearly incompressible solids, results are mixed. Among all ratios, the general trend still remains analogous. Detailed views along cutting curve

Figure 4: Von Mises stress along cutting curve \(C_{2}\) for case \(E=200\) GPa, \(\
u=0.15\). Figure 3: Deformed solid for case \(E=200\) GPa, \(\
u=0.15\) amplified 5 times with its distorted underlying lattice. Color mapping shows magnitude of displacement field. (**a**) Overall view of the deformed solid. (**b**) Detailed view along loading. are provided in Figures A8 and A9 for the first displacement direction and in Figures A13 and A14 for the second one. The results shown are acceptable in terms of matching with the reference study. Differences between curves are more pronounced when poisson's ratio is moving away from zero. Further detailed explanations are presented in Section 5. #### 4.2.3 Young's Modulus Sensitivity

Like in the previous part, we perform a parametric study involving [PERSON]'s modulus. Poisson's ratio is set to zero, leading to \(\lambda=0\). Thus, the isotropic law is modified and therefore the Poisson effect does not exist for this particular case. Figures A15 and A16 illustrate the first displacement coordinate along cutting curves \(C_{1}\) and \(C_{2}\) respectively. In the same way, Figures A17 and A18 depict the second displacement coordinate. At boundaries, displacement is imposed, thus yielding to identical curves no matter is the material stiffness. Indeed, \(E\) acts as a stress factor into studied isotropic law (see Equation (16)), i.e., only stresses are subject to change. [PERSON]'s modulus can be viewed as a convergence coefficient during iterations. Investigations are made to demonstrate the insensitivity and stability of LBMS with various modulus. To go further, we remark that the first component of displacement partly arises from intense shear stress at the frontiers of the velocity condition. A non-constant shear stress along second basis direction produces a non-zero stress divergence, i.e., contributing to its first component and hence yielding a displacement even if \(\
u=0\). #### 4.2.4 Triangular Loading: Case \(E=200\) GPa, \(\
u=0.15\)

To exhibit robustness of the LBMS method and its ability to compute solid static equilibrium, a last example is illustrated with a different boundary condition.