## 3 Case Studies
### Clamped-Free Case  
In this section, we consider the clamped-free rectangular structure presented in Figure 9. The tested structure is the same as the structure tested in the previous section, and external force \(f\) is exerted on the upper side of the free end in the upward direction. The structure is also discretized using a 20 by 240 mesh of four-node elements. Similar to the previous section, the design variable \(\rho\) was set as 0.7 over the entire design domain. Let us first examine the F-D curve in the initial design model. The curve measured from the initial design layout is depicted by a dotted line (\(\cdots\)) in Figure 10. The line seems straight, but it is slightly curved upward, meaning that the initial design behaves with hardening nonlinearity. Three cases were tested with different types of nonlinearity, which are prescribed by defining target points at 0, 5, 100, and 150 N. We first consider softening nonlinearity. As for the previous case, the displacements in each external force level, i.e., the target points, are arbitrary decided based on the measured F-D curve from the initial design layout. The circles (\(\circ\)) in the figure represent the target points in the optimization design process. All the parameters in the optimization problem are set to the same values used in the  
Figure 8: Force–displacement curve comparisons between optimal layouts from the optimization process (\(\blacklozenge\)) and commercial software COMSOL (\(\blacklozenge\)). (a) Softening; (b) Hardening. previous section, except the filter radius \(r\) which is fixed as \(r_{\text{filter}}=2\times\) the element size for softening case and \(r_{\text{filter}}=6\times\) the element size for the hardening case. Figure 10 also shows the F-D curves obtained with the optimal layouts, for forcing amplitudes other than the targeted ones. All optimal curves increase monotonically, which means that they do not exhibit a snap-through and snap-back phenomenon in the region of the external force level of interest. In both cases, as shown in Figure 10, the optimal curves pass through the target points, which confirms the effectiveness of the proposed method. Figure 11 shows the optimal layouts with deformations at the target points in Figure 10. 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. In the results of the softening cases in Figure 11a, the outer frame is composed of a relatively thick layer, but the inside is composed of thin layers. Magnified parts of these thin layers in the softening results are shown in the dashed boxes. When a large external force is imposed (i.e., 150 N), buckling occurs. Hence, it can be concluded that softening nonlinearity is induced in the thin layers by buckling, which show macroscopic bending behavior when a large force is applied. Figure 10: _Cont._  
Figure 9: Clamped-free test problem. The difference between the case studies in the previous section and the cases covered in this section is that this section deals with large-strain and large-deformation cases. As stated in the previous section, for a small deformation but large strain case in the clamped-clamped boundary condition, almost the same results were obtained regardless of whether the energy interpolation was used or not. However, when we do not use energy interpolation for the large-deformation case discussed in this section, the Newton-Raphson process performed to measure the displacement during the optimization  
Figure 11: Deformed layouts of optimal topology at target points. (**a**) Softening; (**b**) Hardening. process often fails to converge by exceeding the maximum number of iteration set. Finally, the F-D curve result at the final iteration of the optimization process does not converge to the target points. In conclusion, energy interpolation is a necessary tool to mitigate the effect caused by large deformation in the void region.