# Generalized Optimality Criteria Method for
## 3 Numerical Comparisons
### Performance Comparison for Multiple Constraints  
In this section, an optimization problem with multiple constraints is used to compare the performance of different algorithms. As mentioned before, when multiple constraints are present, the OCM cannot be used. Therefore, only MMA and GOCM algorithms can be used. The same gripper-arm model in the previous section was used for the comparison of MMA and GOCM when there were multiple constraints. The design optimization problem is defined as  
Figure 5: Optimization history of compliance using three optimization algorithms.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **Iteration** & **Compliance** & **Volume Fraction** & **Total Time (s)** & **Algorithm Time (s)** & **Algorithm/Total Time (\%)** \\ \hline OCM & 102 & \(1.728\times 10^{-2}\) & 0.09 & 554 & 33.16 & 6.0 \\ MMA & 102 & \(1.710\times 10^{-2}\) & 0.09 & 788 & 166 & 21.0 \\ GOCM & 102 & \(1.741\times 10^{-2}\) & 0.09 & 566 & 0.10 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of optimum designs using different optimization algorithms.  
\[\begin{array}{ll}\text{Minimize}&c(\mathbf{x})=\sum\limits_{c=1}^{N_{c}}(x_{c})^{ p}\{\mathbf{d}_{c}\}^{\text{T}}[\mathbf{k}_{c}]\{\mathbf{d}_{c}\}\\ \text{subject to}&\dfrac{\
u(\mathbf{x})}{\
u_{0}}\leq 0.09\\ &(\sigma_{v\text{M}})_{\text{max}}\leq 4.5\times 10^{7}\\ &u_{\text{max}}\leq 0.0001\\ &v_{\text{max}}\leq 0.0001\\ &w_{\text{max}}\leq 0.0001\\ &[\mathbf{K}]\{\mathbf{D}\}=\{\mathbf{F}\}\\ &\mathbf{x}_{\text{min}}\leq\mathbf{x}\leq\mathbf{x}_{\text{max}}\\ \end{array} \tag{16}\]  
The compliance objective and volume fraction constraint are identical to the previous example. In addition, the maximum von Mises stress and the maximum displacement constraints were added. For the maximum von Mises stress, the aggregated P-norm stress was used [20]. For the displacement constraints, a similar P-norm was used to calculate the maximum displacement in each coordinate direction.  
Figure 6a,b shows the optimum geometry using GOCM and MMA algorithms, respectively. Both methods took 102 iterations to converge, as shown in Table 4. In this optimization problem, GOCM found an optimum design whose compliance was about 8% less than that of MMA. Among five inequality constraints, the volume fraction, maximum y-displacement, and maximum von Mises stress were active at the optimum design. The MMA algorithm took 32% of the total time. Considering that the remaining 68% of the time was used for calculating structural response, adjoint load, adjoint response, and adjoint sensitivity, the optimization algorithm took a significant portion of the time. Compared to the MMA, the GOCM algorithm took a fraction of a second, which provides a significant advantage over MMA. It is interesting to note that the MMA algorithm became computationally expensive when constraints became active. As shown in Figure 7, initially the MMA algorithm took about 1.5 s for each iteration, but after the 40 th iteration, it took about 15 s for each iteration, where constraints became active. This is because the MMA subproblem can be expensive when multiple constraints are active.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **MMA** & **GOCM** \\ \hline Iteration & 102 & 102 \\ Compliance & \(1.855\times 10^{-2}\) & \(1.709\times 10^{-2}\) \\ Volume fraction & 0.09 & 0.09 \\ Displacement (y) & \(1.0\times 10^{-4}\) & \(0.96\times 10^{-4}\) \\ Stress & \(4.5\times 10^{7}\) & \(4.3\times 10^{7}\) \\ Total time (s) & 2541 & 1580 \\ Algorithm time (s) & 820 & 0.15 \\ Algorithm/total time (\%) & 32.3 & 0.01 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of optimum designs with multiple constraints.  
Figure 6: Optimum design of the gripper-arm model with multiple constraints.