## 3 Case Studies
### Clamped-Clamped Case
Figure 4 shows the history plot for the normalized objective function with respect to the objective function value on the initial design (c\({}_{0}\)) and volume fraction of each iteration in the optimization process; Figure 4a is the case where the slope constraint is incorporated in the optimization process and Figure 4b is the case where the slope constraint is not incorporated. The total optimization process is repeated until 185 iterations and 268 iterations, respectively. Figure 3: Four force–displacement curves measured from the initial layout (-O-), prescribed in the optimization problem as target points (-V-), measured from the optimal layout without slope constraints (-D-) and measured from the optimal layout with slope constraints (-R-). On the one hand, it is observed that the slope constraint helped to approach the optimal result layout faster in the optimization process. The objective function curve is mostly smooth except for a specific iteration range (120 \(\sim\) 170 iteration). On the other hand, when there is no slope constraint, it can be seen that the objective function curve fluctuates in most of the iteration range. In general, the Newton-Raphson method is based on the previous iteration results, therefore, for a smooth curve case, the desired convergence result in the Newton-Raphson process is close to the values of the previous iteration. If the fluctuation is severe in the objective and volume fraction curves, however, a large number of analyses should be performed internally in the Newton-Raphson process to find the convergent point. According to Figure 4, it was observed that, for the case without slope constraint, the objective function curve fluctuates in most of the iteration range. Figure 4: History plots of objective function (––) and volume fraction (––) for the optimization procedure. (**a**) With slope constraint; (**b**) Without slope constraint. the Newton-Raphson process takes 1.5-2 times longer for a single force level as compared with the case with slope constraint. In conclusion, although an additional constraint condition was added to the optimization problem, the total analysis time required for the optimization design process did not differ between the two cases, and in some cases, it took a shorter time in the case of constraint. The same tendency was also observed in the hardening nonlinearity case. As illustrated in Figure 5, the outlined optimization procedure could also be validated for a hardening nonlinearity. As for the softening case, target points are arbitrary decided based on the measured F-D curve from the initial design layout. In order to verify the nonlinear trend of the obtained designs for forcing values other than the targeted one, the F-D curves in the range of 0 to 150 N were measured at intervals of 2 N for both hardening and softening layouts. Figure 5: Force–displacement curves measured from the initial design domain (\(\cdots\)) and optimal layouts (**—**) with circles (\(\Circle\)) representing the target points. (**a**) Softening; (**b**) Hardening. Both curves increase monotonically, which means that they do not exhibit a snap-through or snap-back phenomenon in the region of the external force level of interest. In the softening case Figure 5a, the F-D curve shows a softening trend and the optimal curve passes through the target points, which confirms the effectiveness of the proposed method. In hardening cases, as shown in Figure 5b, it does not pass through the target points exactly. This means that the objective function is located at a local minimum, and the proposed method does not always guarantee to give perfectly optimized results for all types of defined target points. A similar case was introduced in a previous study [12]. Figure 6 shows the optimal layouts without deformation at 0N and with deformations at the target points in Figure 5. The upward arrows \(\uparrow\) indicate the forcing location. Gray regions are hardly seen between the solid and void regions, which shows that the projection filter worked well.