## 4 Analysis of Results  
In this flat plat case study, the short edge has less impact on deformation. Clamps are only applied on the long edge of the workpiece. However different sheet metal materials are used by researchers. [PERSON] et al. [28] used workpiece material with a density of 2.8 \(\times\) 10\({}^{-6}\) kg/mm\({}^{3}\). [PERSON] et al. [17] and [PERSON] et al. [42] used workpiece material with a density of 7.85 \(\times\) 10\({}^{-6}\) kg/mm\({}^{3}\); the same material is used in this research. In subcase 1 and subcase 2, the first and fourth quadrants have the highest maximum deformation, so these were considered as the design quadrant. In stage 1 of subcase 1, six clamps were found as the initial number of clamps. In stage 2 of subcase 1, the design quadrants' population was reduced from 1600 to 840. In stage 3 of subcase 1, five clamps were found as the optimal number of clamps. Similarly, for subcase 2 the initial number of clamps was determined to be 5. The design quadrants' population was reduced from 1600 to 560. In stage 3 of subcase 2, the optimal number of clamps was found to be 4. The workpiece, after topology optimization, and the final optimum results of subcase 1 and subcase 2 are shown in Figures 7 and 8, respectively. The spacer grid has inclined long edges, so clamps cannot be placed on long edges due to geometric constraint. Therefore, only the short edge of the spacer grid was considered for placing clamps. In subcase 3 and subcase 4, the second and third quadrants have highest maximum deformation after stage 1. Therefore, these quadrants were taken as the design domain for topology optimization. Subcase 3, stage 1 results eight clamps as the initial number of clamps. In stage 2, the population was reduced from 2025 to 195. Stage 3 results four as the optimal number of clamps for subcase 3. In subcase 4, ten clamps were determined as the initial clamps in stage 1. The population was reduced from 2025 to 390 in stage 2. Stage 3 results in 5 clamps as the optimal number of clamps. Figure 9 illustrates the workpiece after topology optimization and maximum deformation obtained from fixture layout optimization in the spacer grid case study. Figure 8: **(a)** Subcase 2: population reduction after topology optimization, **(b)** Subcase 2: optimum layout maximum deformation. Figure 9: _Cont._  
No. of clamps, maximum deformation, total deformation, and population reduction of both case studies is presented in Table 2.  
The total deformation value in the final stage is optimum, which is less than the initial total deformation value. Similarly, the maximum deformation value in the initial layout and optimum layout is lowered. The number of clamps in both case studies are also optimized from stage 1 to stage 3. Maximum deformation is occurring in the design region, which indicates that topology optimization is accurate and that the criteria of the design region identification is validated. The topology optimization technique has proved to be helpful in reducing computational effort for GA. The maximum deformation values of each subcase, in the flat plate and spacer grid case studies, are shown against each iteration in Figure 10.  
The iteration number is represented on the \(x\)-axis and maximum deformation against each iteration on the \(y\)-axis. As shown in Figure 10a,b, optimum results of subcase 1 and subcase 2 were obtained on the 64 th and 46 th iterations, respectively. Similarly, as shown in Figure 10c,d, optimal results of subcase 3 and subcase 4 were obtained on the 4 th and 14 th iterations, respectively.