## 3 Case Studies
### Clamped-Clamped Case
Furthermore, the softening tendency of the force-displacement curve in the cases where the number of target points increases is shown to be strong. In the case of 15 target points, the resultant layout with the desired nonlinear force-displacement characteristic defined as the reference curve was obtained. The CPU of the computer used in this study was Intel (R) Core i9-9900k and the memory (RAM) was 64 GB. Figure 5a shows the procedure for the optimization design process performed considering three target points and slope constraints. Figure 6a shows the resultant layout. The analysis time required was 5 h and 45 min. Figure 6: Deformed layouts of optimal topology at target points. (**a**) Softening; (**b**) Hardening. The time duration for optimization design process to obtain the resultant F-D curve shown in Figure 7 was 15 days and 22 h, which was obtained by considering only the objective function using 15 target points without slope constraints. In conclusion, the proposed slope constraint not only shortens the optimization process time by using only a small number of target points in the optimization problem but also improves the convergence in the optimization design process. In summary, in the absence of a slope constraint, a large number of target points are required to obtain a layout that follows the prescribed target points. Of course, however, the computational analysis takes a long time in this case. In order to overcome this, this study proposed a slope constraint, which makes it possible to obtain the layout of a structure with a nonlinear F-D curve of the desired with just three target points. Lastly, we tried to verify if the desired nonlinear behavior would indeed appear in the fabricated structure based on the resultant layout. To do this, every element whose design variable \(\overline{\overline{\rho}_{e}}\) is less than 0.5 is set to 0 and whose design variable \(\overline{\overline{\rho}_{e}}\) is larger than 0.5 is set to 1. The reference value 0.5 is decided based on the threshold value \(\eta\) which was used in the projection filter. Next, the void elements with zero design variables are removed and the structure layout is composed solely of the solid elements with the design variable of 1. The F-D curve for modified layout is finally computed using the commercial software COMSOL, at the same external force levels as those used in the optimization problem. Figure 8 shows that the F-D curves, which are measured from the modified layout in which only the solid elements are composed, exhibit the same nonlinearities as the curves obtained in the optimization process, and as the curves obtained using COMSOL. Interestingly, one would expect the layout () in which there are void elements to be stiffer than the layout () in which the void elements are removed, but the opposite observation is made. This is because the void elements also affect the stiffness of the entire structure. However, the results show that the effect of stiffness reinforced in the solid region is greater than the stiffness reduced by eliminating the void elements. This is natural because in the optimization design, the void region represents a low sensitivity region and the solid region represents a high sensitivity region. Therefore, it is obvious that the stiffening effect reinforced in the high sensitivity region is larger than the stiffness effect removed in the low sensitivity region. Figure 2:0 Force–displacement curves measured from the optimization processes with increasing the number of target points without the slope constraints. For three target points case ()–), for six target points case ()–), for nine target points case ()–), for twelve target points case ()–), for fifteen target points case ()–), and measured curve from the optimal layout in Figure 6a ().