## 3 Numerical Examples
### Topology Optimization of a U-Beam
Figure 9 shows the optimization solutions obtained both with and without the NP constraint. The minimum total cost function with the NP constraint was _0.0228_, obtained at iteration number _1965_ with a total artificial density of _84.161_, while _0.0229_ was obtained at iteration number _3887_ with a total artificial density of _96.468_. In short, the global minimum was reached in fewer iterations for the NP-constrained case, and its optimum solution indicated thinner elements. Furthermore, it was observed that the solutions obtained without the NP constraint showed greater scatter during the iterations, which is reflected in the cost function plots in Figure 9. The reason for this is as follows: without using the NP constraint, the optimization algorithm attempts to reduce both the transfer function gain and its derivative simultaneously at each iteration; however, they are not independent. Such an attempt results in greater scatter in the cost functions during the iterations. On the other hand, by using the NP constraint we implicitly consider the relationship between the transfer function gain and its derivative; thus, the parameter vector search takes place in the physically realizable region and the global minimum can be achieved in a smaller number of iterations. The distributions of the artificial densities are shown in Figure 10 for both cases, with and without the NP constraint. Note that the solutions obtained with the NP constraint show a smoother density distribution with a lighter design and exhibit lower-density regions with slopes of approximately 45 degrees. Additionally, the optimum topology solutions without the NP constraint did not converge. This phenomenon is illustrated by the sample artificial density distribution in Figure 10b. In fact, Figure 9 shows the scattering behavior in cost terms in Equation (11) without the NP constraint. In Figure 9, all cost terms obtained without the NP constraint show large scattering across iterations and the results did not converge at all. Figure 9: Cont. Figure 10: Side view of the distribution of artificial densities for a topology optimization problem in the x–z plane obtained (**a**) with the NP constraint, and (**b**) without the NP constraint. Figure 9: Pareto optimal solutions for the topology optimization problem (**a1**,**b1**,**c1**,**d1**,**e1**) with the NP constraint, and (**a2**,**b2**,**c2**,**d2**,**e2**) without the NP constraint. Similar to the case reported in [59], unsymmetric topological solutions were observed in our optimum topologies in Figure 10a. In [59], the authors claimed that nonlinear and non-convex constraints were the contributors to unsymmetric optimum solutions for the topology optimization problems in their study. We believe that the same reasons also caused our unsymmetric topology solutions when the NP constraint was used. This feature requires further investigation in future studies. To reveal the reason for the oscillatory behavior of the cost function when solving Equations (11) and (12) without the NP constraint, the following two cases were also solved: first, \(\beta=0\) was selected in Equation (11), the results of which are shown in Figure 11a; then, both \(\alpha=0\) and \(\beta=0\) were selected in Equation (11), the results of which are shown in Figure 11b. In short, only when \(\alpha=0\) and \(\beta=0\) were selected was convergence achieved without the NP constraint for solving Equation (11); however, the corresponding optimization formulation was the solution to a different problem than the one with non-zero \(\alpha\) and \(\beta\) values.