## 3 Numerical Examples
### Topology Optimization of a U-Beam  
Considering the simplified 3D engine traverse model shown in Figure 8, the NP constraint was imposed on a 3D topology optimization problem using a solid isotropic material with a penalization (SIMP) approach [56]. The following multi-objective optimization problem given in Equations (11) and (12) was solved. The objective functions were \(f_{1}=\sum_{i=1}^{n}\left|G(\omega_{i})\right|\), \(f_{2}=\sum_{i=1}^{n}\left|G^{\prime}(\omega_{i})\right|\), \(f_{3}=\sum_{e=1}^{m}\rho^{e}V^{e}\), and \(f_{4}=\overline{D}^{DK}\). Discrete frequencies \(\omega_{i}\) were chosen as the natural frequencies. Note that it is common to use compliance minimization for the cost function in topology optimization problems in the literature. The compliance minimization was achieved equivalently by minimizing the term \(f_{1}=\sum_{i=1}^{n}\left|G(\omega_{i})\right|\) in Equation (11), since our formulations were based on the NP constraint, which is a function of the FRFs \(G(\omega_{i})\). The total volume was calculated using \(\int_{\Omega}\rho^{e}d\Omega^{e}=V\), where the artificial density of the elements is denoted by \(\rho^{e}\), bounded by \(\rho\leq\rho^{e}\leq\overline{\rho}\), \(\rho^{e}\epsilon\Omega\), \(\underline{\rho}=0.01\), and \(\overline{\rho}=1\). Note that there were attempts to solve Equation (11) by imposing the volume fraction ratio in addition to the other constraints; however, these efforts caused convergence difficulties. Therefore, the volume fraction ratio was not included in the solutions. Imposing the volume fraction ratio in a stable way will be examined in the future studies. Figure 6: Pareto optimal solutions for the truss example obtained by interior point method, (**a1,b1,c1**) with the NP constraint, and (**a2,b2,c2**) without the NP constraint. Figure 8: Traverse model of a 3D engine for topology optimization. Figure 7: Pareto optimal solutions for the truss example, obtained using the constrained PSO method and the NP constraint (**a**) Damage index vs. weight (**b**) Damage index vs. response amplitude (**c**) Amplitude of response derivative vs. response amplitude. The dimensions of the block material of the side walls in the x, y, and z directions were _700 mm_, _150 mm_, and _200 mm_, respectively. Loads were applied to the elements on their top faces, as shown in Figure 8, while the elements at the bottom corners were fixed, i.e., there was an unsymmetric loading condition. The material was assumed to be steel with a density of \(\rho_{steel}=7.7e^{-9}\ tonnes/mm^{3}\), [PERSON]'s modulus of _210,000 MPa_, and [PERSON]'s ratio of _0.29_. The S-N curve parameters of the material were the same as the material parameters given in Section 3.2. The applied random force was similar to that in Section 3.2: \(f=f(t)=F_{0}\Big{(}sin(3t)+0.2 sin(7t)+0.1 sin(13t)+0.05 rand_{uniform}(0,1)+0.02 rand _{Gaussian}(0,1)\Big{)}[N]\). Its magnitude was \(F_{0}=\frac{5000}{8}N\). The element formulations for 3D elastic solids in [57] were used, and the FRFs \(G(\omega_{i})\) in Equation (11) were calculated according to Equations (3)-(5). The half-symmetry of the model in the y-z plane was used and there were 160 elements in total. The first attempts to solve Equations (11) and (12) were conducted using the conventional gradient-based optimization algorithm of the interior point method; however, CPU times were too high and convergence difficulties were observed. Therefore, the PSO algorithm was used, which constitutes the basis for the numerical solutions presented here. The weightings \(\alpha\), \(\beta\), \(\gamma\), and \(\lambda\) in Equation (11) were used to scale the individual costs \(f_{i}\), so that their nominal values were approximately equal to each other. When the optimization problem was solved with the NP constraint, the constrained PSO algorithm [58] was used. Similar to Section 3.2, when the optimization problem was solved without the NP constraint, the unconstrained PSO algorithm in Matlab was used.