# Design Optimization of Truss Structures Using a Graph Neural Network-Based Surrogate Model
## 4 Results and Discussion
### Accuracy and Effectiveness  
The comparative analysis of the two optimization techniques, namely, the proposed GNN-based and the conventional FEM-based, presented in Sections 4.1-4.3 has demonstrated the accuracy and effectiveness of the developed GNN-based optimization in solving three distinct truss design problems. The results of the study reveal that the GNN-based optimization technique can find the optimal weights with a remarkable degree of accuracy without violating any constraints. Specifically, the achieved optimal weights are characterized by only a marginal deviation of 0.05%, 0.2%, and 0.9% from the results of the FEM-based optimization for the 10-bar planar truss, 72-bar space truss, and 200-bar planar truss, respectively. In addition to its high accuracy, the GNN-based optimization exhibits a higher degree of efficiency for more complex structures compared to the FEM-based optimization. This is evident from Figure 17, which shows that the GNN-based optimization was able to reach the optimal results faster for the 200-bar planar truss problem, outperforming the FEM-based optimization, which identified the optimal weights for the 10-bar planar truss and the 72-bar space truss problems more expeditiously.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Variables** & \multicolumn{2}{c}{**Cross-Sectional Areas (in2)**} \\ \hline \multirow{2}{*}{**Element Group**} & \multirow{2}{*}{**Members**} & **FEM-Based** & \multirow{2}{*}{**GNN-Based**} \\  & & **Optimization** & **(Benchmark)** & **Optimization** \\ \hline \multirow{4}{*}{\(11\)} & 44, 45, 47, 48, 50, 51, 53, 54, 65, 66, 68, & \multirow{4}{*}{0.100} & \multirow{4}{*}{0.100} \\  & 69, 71, 72, 74, 75 & & \\ \cline{1-1}  & 77–80 & & 0.100 & 0.100 \\ \cline{1-1}  & 13 & 81, 84, 87, 90, 93 & 0.929 & 0.910 \\ \cline{1-1}  & 14 & 95–100 & 0.100 & 0.100 \\ \cline{1-1}  & 15 & 102, 105, 108, 111, 114 & 1.135 & 1.095 \\ \cline{1-1}  & 82, 83, 85, 86, 88, 89, 91, 92, 103, 104, & 0.100 & 0.100 \\ \cline{1-1}  & 106, 107, 109, 110, 110, 112, 113 & 0.100 & 0.100 \\ \cline{1-1}  & 115–118 & & 0.100 & 0.100 \\ \cline{1-1}  & 119, 122, 125, 128, 131 & 1.220 & 1.065 \\ \cline{1-1}  & 133–138 & & 0.100 & 0.100 \\ \cline{1-1}  & 20 & 140, 143, 146, 149, 152 & 1.240 & 1.162 \\ \cline{1-1}  & 120, 121, 123, 124, 126, 127, 129, 130, & & \\ \cline{1-1}  & 141, 142, 144, 145, 147, 148, 150, 151 & 0.100 & 0.100 \\ \cline{1-1}  & 22 & 153–156 & 0.100 & 0.100 \\ \cline{1-1}  & 23 & 157, 160, 163, 166, 169 & 1.434 & 1.816 \\ \cline{1-1}  & 24 & 171–176 & 0.100 & 0.100 \\ \cline{1-1}  & 25 & 178, 181, 184, 187, 190 & 1.446 & 1.619 \\ \cline{1-1}  & 158, 159, 161, 162, 164, 165, 167, 168, & & \\ \cline{1-1}  & 179, 180, 182, 183, 185, 186, 188, 189 & 0.133 & 0.116 \\ \cline{1-1}  & 27 & 191–194 & 0.839 & 0.823 \\ \cline{1-1}  & 28 & 195, 197, 198, 200 & 1.497 & 1.496 \\ \cline{1-1}  & 29 & 196, 199 & 2.000 & 2.000 \\ \hline \(|\delta_{node}|^{\max}\)(in) & & 4.00 & 4.00 \\ \multicolumn{2}{c}{Weight (lb)} & 4166.81 & 4204.00 \\ \multicolumn{2}{c}{N\({}_{analyses}\)} & 11,250 & 3750 \\ \hline \end{tabular}
\end{table}
Table 11: \(Cont.\)