## 3 Case Studies  
### Clamped-Clamped Case  
For validation purposes, let us consider the clamped-clamped rectangular structure presented in Figure 2. The tested structure is 0.6 m long (L = 0.6 m), 0.05 m wide (B = 0.05 m), and 1e-3 m thick (t = 1e-3). [PERSON]'s modulus and Poisson's ratio are 20 GPa and 0.33, respectively. An external force \(f\) is exerted on two-thirds of the total length L from the left end in the upward direction. The structure is discretized using a 20 by 240 mesh of four-node elements. Four amplitudes, namely 0 N, 50 N, 100 N, and 150 N, were defined as the external forces of interest on the F-D curve. Target points were defined by setting displacements at these forces on the external force location. For the parameters in the optimization problem, the volume is set as 70% as compared with the design where all the elements are solid. In the initial design model, the design variable \(\rho\) was set as 0.7 over the entire design domain. The slopes are set to the minimum and maximum values with a marginal value of \(\pm\)10%. The weighting function \(w_{i}^{f}\) is set to the corresponding external force levels (i.e., 50 N, 100 N, and 150 N). The value for the projection parameter \(\beta\) in Equation (9) is doubled at every 50 iterations of topology optimization or at convergence, with a maximal value of \(\beta_{\max}~{}=~{}64\) and an initial value of \(\beta~{}=~{}1\). The threshold \(\eta\) is fixed as \(\eta~{}=~{}0.5\). The threshold  
Figure 2: Clamped-clamped test problem. parameters of the energy interpolation scheme in Equation (12) are fixed as \(\beta_{1}=500\) and \(\rho_{0}=0.01\). The filter radius \(r\) is fixed as \(r_{\text{filter}}=2\times\) the element size. Let us first consider the case of a targeted softening nonlinearity. First, we measured displacements at the external force levels on the initial design layout. On the basis of the measured F-D curve from the initial design layout, the prescribed target points for the softening nonlinearity on the F-D plane are arbitrary determined around the F-D curve for the initial design layout, as shown in Figure 3. In order to study the influence of the slope constraint proposed in this work, we apply the optimization processes with and without the slope constraints. The curve measured from the initial design layout is depicted by a dash and circle line (-O-) in Figure 3. The line seems straight, but it is slightly curved upward, meaning that the initial design behaves with hardening nonlinearity. As shown in Figure 3, on the one hand, the measured curve (-R-) from the optimization process, in which the slope constraint was incorporated, and the prescribed target curve (-V-) are in good agreement with each other. On the other hand, the measured curve (-D-) from the optimization process, in which the slope constraint was not incorporated, obviously differs from that of the target curve. In addition, it seems straight as for the initial layout case but is also slightly curved upward, meaning that the inherent hardening nonlinearity that was in the initial layout remained during the optimization process even though the targeted nonlinearity was softening. It can thus be seen that the optimization process has tried to reduce only the objective value while maintaining the inherent nonlinearity in the initial layout. Therefore, it can be concluded that the slope constraint is necessary if different nonlinearity behaviors from the inherent nonlinearity in the initial layout is desired in the prescribed target curve. In addition, we found it interesting to analyze the process of obtaining results based on with and without slope constraints in the optimization process.