# Truss Structure Optimization with Subset Simulation and Augmented Lagrangian Multiplier Method
## 4 Test Problems and Optimization Results
### Spatial 25-Bar Truss Structure  
The second test problem regards the spatial 25-bar truss shown in Figure 4. The modulus of elasticity and material density are the same as in the previous test case. The 25 truss members are organized into 8 groups, as follows: (1) \(A_{1}\), (2) \(A_{2}\)-\(A_{5}\), (3) \(A_{6}\)-\(A_{9}\), (4) \(A_{10}\)-\(A_{11}\), (5) \(A_{12}\)-\(A_{13}\), (6) \(A_{14}\)-\(A_{17}\), (7) \(A_{18}\)-\(A_{21}\), and (8) \(A_{22}\)-\(A_{25}\). The cross-sectional areas of the bars are defined as the design variables and vary between 0.01 inch\({}^{2}\) and 3.5 inch\({}^{2}\). The weight of the structure is minimized, subject to all members satisfying the stress limits in Table 3 and nodal displacement limits in three directions of \(\pm\)0.35 inch. There is a total of 110 inequality constraints. The truss is subject to the two independent loading conditions listed in Table 4, where \(P_{x}\), \(P_{y}\) and \(P_{z}\) are the loads along \(x\)-, \(y\)- and \(z\)-axis, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(N\) & **Best** & **Mean** & **Worst** & **SD** & **NSA** \\ \hline \multirow{3}{*}{ALSSO} & 100 & 5060.931 & 5064.559 & 5079.894 & 3.363699 & 48,064 (44,710) \\  & 200 & 5060.931 & 5062.391 & 5065.715 & 0.889472 & 94,540 (108,600) \\  & 500 & 5060.885 & 5061.713 & 5062.291 & 0.360457 & 247,828 (253,400) \\ \hline HPSACO [33] & & 5056.56 & 5057.66 & 5061.12 & 1.42 & 10,650 \\ ABC-AP [36] & & 5060.88 & N/A & 5060.95 & N/A & 500,000 \\ SAHS [31] & 5061.42 & 5061.95 & 5063.39 & 0.71 & 7081 \\ TLBO [34] & 5060.96 & 5062.08 & 5063.23 & 0.79 & 16,872 \\ MSPSO [27] & 5061.00 & 5064.46 & 5078.00 & 5.72 & N/A \\ HPSSO [41] & 5060.86 & 5062.28 & 5076.90 & 4.325 & 14,118 \\ VEO [39] & 5060.99 & 5062.09 & 5975.41 & 2.05 & 19,540 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of statistical performance in the 10-bar truss problem. Figure 3: Convergence curve obtained for the 10-bar truss problem with \(N=500\). This truss problem has been previously studied by using deterministic global methods [3] and many stochastic optimization methods [21; 27; 28; 31; 34; 35; 36; 37; 38; 39; 41; 45]. Table 5 compares the best designs found by ALSSO and the above-mentioned stochastic methods. Among the deterministic global methods, the Pareto-Lipschitzian Optimization with Reduced-set (PLOR) algorithm is better than the three DIRECT-type algorithms [3]. Therefore, only the optimized result obtained by PLOR is listed in Table 5. All the published results satisfy the limits on the stress. To the best of our knowledge, CA [37] generated the best design for this test problem. The best designs from HM, HPSSO, improved TLBO, FPA, and WEO slightly violated the nodal displacement constraints. It can be seen from Table 5 that the proposed algorithm produced better designs than PLOR, HS, HPSSO, SSO ABC-AP, SAHB, and MSPSO, and comparable with TLBO. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Load Cases**} & \multirow{2}{*}{**Nodes**} & \multicolumn{3}{c}{**Loads**} \\ \cline{3-5}  & & \(\mathbf{P_{x}}\) (kips) & \(\mathbf{P_{y}}\) (kips) & \(\mathbf{P_{z}}\) (kips) \\ \hline
1 & 1 & 0.0 & 20.0 & \(-\)5.0 \\  & 2 & 0.0 & \(-\)20.0 & \(-\)5.0 \\
2 & 1 & 1.0 & 10.0 & \(-\)5.0 \\  & 2 & 0.0 & 10.0 & \(-\)5.0 \\  & 3 & 0.5 & 0.0 & 0.0 \\  & 6 & 0.5 & 0.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Loading conditions acting on the 25-bar truss. Figure 4: Schematic of the spatial 25-bar truss structure.