## 3 Numerical Examples
### Optimization of a Truss System  
As a second example, a randomly loaded truss structure was considered, shown in Figure 5, and a frequency domain fatigue damage index was included in the cost function. The principal objective was to test the effect of the NP constraint on multi-objective optimization problems with stress minimization and multiple load cases. Using the truss element formulations in [51], the corresponding mass matrix \(\mathbf{M}\), stiffness matrix \(\mathbf{K}\), and damping matrix \(\mathbf{C}\) were calculated using finite element method (FEM) equations. Then, the FRFs were used to calculate the Pick matrix in Equation (1). The cost function was formulated as follows:  
\[\underset{h_{n}}{Minimizef}=\alpha f_{1}+\beta f_{2}+\gamma f_{3}+\lambda f_{4} \tag{11}\]  
\[\underset{h_{l}}{Subject\ to} \begin{cases}-\Lambda_{j}+\varepsilon\leq 0\\ h_{l}\leq h_{n}\leq h_{ub}\end{cases}  \tag{12}\]  
where the objective function terms are \(f_{1}=\sum_{i=1}^{n}\lvert G(\omega_{i})\rvert,\ f_{2}=\sum_{i=1}^{n}\lvert G ^{\prime}(\omega_{i})\rvert,\ f_{3}=\sum_{i=1}^{n}h_{i}\), and \(f_{4}=\overline{D}^{DK}\). The FRFs \(G(\omega_{i})\) are calculated using Equations (3)-(5). The notation \(\overline{D}^{DK}\) refers to the Dirlik damage index [52]. Discrete frequencies \(\omega_{i}\) are chosen as natural frequencies. Minimizing \(f_{4}\) in Equation (11) corresponds to maximizing the fatigue life in the frequency domain, and \(\alpha\), \(\beta\), \(\gamma\), and \(\lambda\) are the weightings, defined as \(0\leq\alpha\leq 1\), \(0\leq\beta\leq 1,0\leq\gamma\leq 1\), \(0\leq\lambda\leq 1\), and \(\alpha+\beta+\gamma+\lambda=1\). These are changed by an increment of \(0.1\) in the search for Pareto optimal sets. In addition, \(h_{n}\) denotes the nth design variable, and \(h_{lb}\) and \(h_{ub}\) are its lower and upper bounds, respectively. The design parameter \(h_{n}\) is the weight of element number n and \(f_{3}\) is the total cross-sectional area of the truss members, which is proportional to the total weight. Note that \(\gamma=0\) corresponds to no penalty on the weight; therefore, this was excluded in the Pareto optimal solutions. The random excitation force was arbitrarily chosen to be \(f(t)=10^{3}(sin(3t)+0.2 sin(7t)+0.1 sin(13t)+0.05 rand_{uniform}(0,1)+0.02 rand_{ Gaussian}(0,1))\big{\lvert}N\big{\rvert}\). The system FRFs were calculated in response to this random transient force. The lower and upper limits for the cross-sectional areas were \(h_{lb}=10\) and \(h_{ub}=100\ [nm^{2}]\). To calculate the fatigue damage in the frequency domain, the Dirlik damage index \(\overline{D}^{DK}\)[52] was utilized to maximize the fatigue life of the structure and the weight was minimized under other constraints. In addition, \(rand_{uniform}(0,1)\) and \(rand_{Gaussian}(0,1)\) were uniform and Gaussian random numbers were generated in the interval \([0,1]\), respectively. The material  
Figure 5: The truss structure. parameters for the S-N curve were given by a fatigue strength constant of \(A=1298.5\) MPa, inverse slope of \(b=-0.125\), \(k=8\), and an ultimate tensile strength of \(UTS=510\) MPa [53]. Using the cost function of Equation (11) and the constraints of Equation (12) with the interior point method, Pareto optimal solution sets were computed with and without the use of the NP constraints, and are shown in Figure 6. It is observed that the Dirlik fatigue damage index \(\overline{D}^{DK}\) reduces as the optimum weight increases; thus, there is a trade-off between the fatigue damage index \(\overline{D}^{DK}\) and the weight. In addition, the frequency domain response gain \(|G(\omega)|\) falls as the fatigue damage index \(\overline{D}^{DK}\) is reduced, which indicates that heavier structures result in less deformation and a smaller fatigue damage index. The result of the total cost function Equation (11) was \(2.6144\)\(e^{-4}\) with the NP constraint, while it was \(0.0270\) without the NP constraint.