# Generalized Optimality Criteria Method for
## 3 Numerical Comparisons
### Performance Comparison with OCM and MMA  
The next example is the gripper-arm model, as shown in Figure 3(a). The design domain of \(127.6\times 43.1\times 10.3\) mm\({}^{3}\) was modeled by \(135\times 46\times 11\approx 68,000\) elements. All nodes on the two holes were fixed, and a total of \(222.41\) N force was uniformly distributed on the upper-right edge as shown in the figure. For material, stainless steel 426 L was used with [PERSON]'s modulus of 193 GPa and [PERSON]'s ratio of 0.25.  
In order to have a comparison using all three optimization algorithms, the optimization problem still uses the minimization of the compliance with volume fraction constraint, i.e., single objective and single constraint. Therefore, the optimization problem is defined as  
\[\begin{array}{ll}\text{Minimize}&c(\mathbf{x})=\sum\limits_{e=1}^{N_{e}}(x _{e})^{p}\{\mathbf{d}_{e}\}^{\text{T}}[\mathbf{k}_{e}]\{\mathbf{d}_{e}\}\\ \text{subject to}&\frac{V(\mathbf{x})}{V_{0}}\leq 0.09\\ &[\mathbf{K}]\{\mathbf{D}\}=\{\mathbf{F}\}\\ &\mathbf{x}_{\text{min}}\leq\mathbf{x}\leq\mathbf{x}_{\text{max}}\end{array} \tag{15}\]  
It is noted that the volume fraction constraint allows only 9% of the material; therefore, most materials will be removed. The OCM uses the equality constraint, but both MMA and GOCM use the less-than-or-equal-to type constraint. Since the volume fraction constraint is active at the optimum design, both formulations would be identical.  
The optimization problem was solved using Autodesk Nastran 2020 (Autodesk Inc., San Rafael, CA, USA) [19]. At each optimization iteration, using the current design variables, Autodesk Nastran calculated the structural responses, adjoint loads, adjoint responses, and sensitivity of objective and constraints. Then, using this information, the optimization algorithm calculated new design variables. This iteration was repeated until the convergence criteria were satisfied.  
Figure 4: The initial design domain and optimum designs using different methods for the gripper-arm model.  
Figure 4b-d shows the optimum designs using three algorithms. Even if the optimum designs were slightly different, the objective and constraint were close to each other, as shown in Table 3. The MMA algorithm tended to make smaller features than other algorithms, but this could be different if the optimization problem started from different initial densities. This was the fundamental limitation of gradient-based optimization, wherein only local optima could be found. An important distinction could be observed in computational times in the last two columns of Table 3. The total time is the time in seconds to solve the optimization problem, while the algorithm time is the time that is used in the optimization algorithm. While the MMA took 21% of computational time, the OCM and GOCM took 6% and 0.02%, respectively. The history of the compliance objective displayed in Figure 5 shows that the OCM showed a smooth variation of the objective, but none of the methods provided a monotonic change of the objective. In this particular example, both OCM and MMA converged to the optimum design from the feasible domain, while GOCM converged from the infeasible domain. It is also noted that the optimization history of MMA and GOCM showed a saw-tooth type pattern, which was not related to the optimization algorithm. Rather, it was related to the progressive increase of the penalty parameter in the SIMP algorithm.