## 3 Case Studies
### Clamped-Clamped Case
In the layout for the softening case in Figure 6a, a specific region leading to the overall nonlinear features of the structure is highlighted by a dashed box, and a schematic representation is provided. The actual layout in the box can be divided into two parts that are denoted by A and B. In the simplified layout, part A is represented by an arched beam, and part B is the foundation to support the part A through clamped boundary conditions. This type of structural configuration like an arched beam has a geometrically similar configuration to the softening spring [35; 36]. As stated in the reference, a softening spring is induced by combination of sources of positive and negative stiffness provided by a bistable mechanism. On the one hand, a bistable mechanism is characterized by a snap-through phenomenon. Therefore, it can be interpreted that the desired softening nonlinearity in the optimization process is induced by the snap-through phenomenon occurring in the dotted region. Here, the region in the dash-dot box acts as a prismatic joint in the reference. In addition, it is well known that an arched beam is related to the softening nonlinearity by the buckling phenomenon [37; 38]. On the other hand, in the hardening nonlinear case, it is not easy to find distinctive features in the configuration of the resultant layout, and it is considered that the entire structure is associated with the hardening nonlinearity. As shown in Figure 6, the structures have small displacements but a large strain. In this case, the void regions do not have a large deformation, and eventually the use of the energy interpolation method has little effect on the optimization process and the results. The case of large deformation is covered in the next section. Next, in order to investigate the effectiveness of the slope constraint proposed in this study, we proceeded with the optimal design process while increasing the number of target points without the slope constraint. An investigation of the effect of the slope constraint was done in the softening case. First, the resultant layout on the softening case, Figure 6a, was set as the reference layout. Then, the number of target points \(N_{TP}\) was increased from three to 15 in steps of three; therefore, five additional optimization design processes were performed. The magnitude of the force considered in the performed individual optimization process is as follows. \[\mathbf{f}_{N_{TP}}=0:\frac{150}{N_{TP}}:150\;\;,\;\;N_{TP}=3:3:15 \tag{17}\]  
For N\({}_{\text{TP}}\) = 3, for instance, the forces \(\mathbf{f}_{N_{TP}}\) considered in the optimization problem were 0 N, 50 N, 100 N, and 150 N. These forces are equal to the force magnitudes in the optimization processes performed in the preceding case studies for softening and hardening. For N\({}_{\text{TP}}\) = 6, the forces \(\mathbf{f}_{N_{TP}}\) considered in the optimization problem were 0, 25, 50, 75, 100, 125, and 150 N. Thereafter, the displacements were measured by applying defined forces \(\mathbf{f}_{N_{TP}}\) to the forcing position of the resultant layout in Figure 6a. The measured displacements were used as \(u_{i}^{*}\) in Equation (1), and the optimization design processes were performed without the slope constraints. The optimization design parameters were the same as those used previously. Figure 7 shows the relationship between the force-displacement curves obtained in the optimization design processes while increasing the number of target points without the slope constraints, and the force-displacement curve (--) measured from the optimal layout (i.e., the same as in Figure 5a (--)). We can verify the performance of the proposed slope constraint by confirming how the optimal F-D curves approach the reference curve (--) when the number of target points is increased. As seen in the figure, the trend of the F-D curve is close to the reference curve as the number of target points increases. This trend is also evident at around 150 N external force.