## 6 Conclusions  
A workpiece is constrained by the proper arrangement of fixture elements, known as fixture layout. The flexible nature of sheet metal requires the N-3-2-1 fixturing method, which requires at least four clamps normal to the plane to restrain workpiece excessive deformations. The genetic algorithm requires high computational effort for fixture layout optimization due to the large number of the population. In this research, a new method for fixture layout optimization of sheet metals is proposed. The proposed method combines topology optimization and GA. Topology optimization is used to reduce populations for GA. The objective function is to reduce populations for GA and minimize total deformation normal to plane of the workpiece, while maintaining maximum deformation of individual nodes up to 2 mm. The proposed method consists of three stages. The initial number of clamps are determined in stage 1. In stage 2, the population is reduced using topology optimization. In stage 3, the number and position of clamps, earlier identified in stage 1, are optimized using GA. The proposed method is implemented on two case studies: flat plate and spacer grid. Convergence criteria for GA is to solve 25% of the reduced population \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Maximum Deformation (mm)**} \\ \cline{2-7}
**Subcase** & \multicolumn{3}{c}{**Initial**} & \multicolumn{3}{c}{**Final**} \\ \cline{2-7}  & **Sim.** & **Exp.** & **\% Diff** & **Sim.** & **Exp.** & **\% Diff** \\ \hline
1 & \(-8.53\) & \(-9.2\) & \(-7.6\) & \(-1.77\) & \(-1.86\) & \(-4.5\) \\
2 & \(-10.5\) & \(-11.5\) & \(-8.3\) & \(-1.79\) & \(-1.91\) & \(-6.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of simulation (Sim.) and experimental (Exp.) results of case study 1—flat plate. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Subcase**} & \multirow{2}{*}{**Load (N)**} & \multicolumn{3}{c}{**Distance from Origin (0,0,0)**} & \multicolumn{3}{c}{**Maximum Deformation Point**} \\ \cline{2-7}  & & **Initial** & **Stage 1** & **Stage 3** & **Initial** & **Stage 1** & **Stage 3** \\ \hline \multirow{10}{*}{1} & \multirow{6}{*}{20} & \multirow{6}{*}{800, 0, 0} & 400, 0, 0 & 400, 0, 0 & & & \\ \cline{3-7}  & & & 800, 0, 0 & 800, 0, 0 & 640, 0, 0 & & & \\ \cline{3-7}  & & & 800, 600, 0 & 800, 600, 0 & 700, 600, 0 & & & \\ \cline{3-7}  & & & 0, 600, 0 & 400, 600, 0 & 600, 600, 0 & & & \\ \cline{3-7}  & & & 600, 600, 0 & 400, 600, 0 & & & & \\ \cline{3-7}  & & & 620, 600, 0 & & & & & \\ \hline \multirow{10}{*}{2} & \multirow{6}{*}{30} & \multirow{6}{*}{800, 0, 0} & 400, 0, 0 & 400, 0, 0 & & & & \\ \cline{3-7}  & & & 800, 0, 0 & 800, 0, 0 & 740, 0, 0 & & & \\ \cline{3-7}  & & & 800, 600, 0 & 800, 600, 0 & 0, 600, 0 & 510, 600, 0 & 670, 390, 0 & 650, 400, 0 \\ \cline{1-1} \cline{2-7}  & & & 0, 600, 0 & 660, 600, 0 & 680, 600, 0 & & & \\ \cline{1-1} \cline{2-7}  & & & 0, 600, 0 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coordinates of clamp positions and maximum deformation point case study 1—flat plate. in each subcase. The results show that in subcase 1 and subcase 2 the population was reduced by 47.5% and 65%, respectively. Whereas, in subcase 3 and subcase 4, 90% and 80% of the population is reduced using proposed method. Total deformation normal to the plane was also reduced in each subcase. The experiment is also performed on case study 1--flat plate and results are compared with simulation results. It concludes that the optimal number and position of clamps are determined using less computational effort by integrating topology optimization into GA. The proposed algorithm can also be used in different industrial scenarios to find optimum positions, especially for respot welding operations of sheet metals. Conceptualization, S.A.H. and Z.A. ; methodology, S.A.H., Z.A. and M.Z. ; software, S.A.H., T.S. and M.R.Y. ; verification, M.R.Y. and T.N.D. ; formal analysis, S.A.H. and Z.A. ; investigation, T.S., K.H. and M.R.Y.