# Efficient Sizing and Layout Optimization of Truss Benchmark Structures Using ISRES Algorithm
## 4 Benchmark Problems
### 25-Bar Truss Structure  
The 25-bar truss structure, displayed in Figure 11, is composed of a material with a mass density of 0.1 lb./in\({}^{3}\) and an elastic modulus of 10,000 ksi. Structural members have been grouped into eight design variables due to the tower's double symmetry along the x and y axes. Tables 3 and 4 outline the member groupings and their respective stress limitations. For consistency with other studies, the minimum and maximum cross-sectional values for all members were set to Amin = 0.01 in\({}^{2}\) and \(\mathrm{Amax}\) = 3.4 in\({}^{2}\), respectively. The permissible nodal displacement values for all free nodes were restricted to \(\pm\)0.35 in in the x, y, and z directions. The benchmark has 124 non-linear design constraints (50 tensile/ compressive stress and 12 displacement constraints) for each of the two loading conditions outlined in Table 4. Given the complexity of the problem, the ISRES algorithm is employed as the optimization tool of choice. With the careful selection of parameters and control settings, the algorithm undertakes the task of minimizing the truss weight while satisfying the extensive set of design constraints. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Element Group** & **Members** & **Comp. Stress (Kips)** \\ \hline A\({}_{1}\) & 1 & 35.092 \\ A\({}_{2}\) & 2,3,4,5 & 11.590 \\ A\({}_{3}\) & 6,7,8,9 & 17.307 \\ A\({}_{4}\) & 10,11 & 35.092 \\ A\({}_{5}\) &  
Multiple runs of the ISRES algorithm are conducted to explore the solution space and identify diverse candidate solutions. The best-performing solutions are selected based on the objective function, which seeks to minimize the truss weight. In addition to optimizing the truss sizing and layout, the ISRES results are compared with those obtained from various other optimization methods available in the literature. The comparative analysis encompasses a range of metaheuristic algorithms, gradient-based techniques, and other state-of-the-art approaches. The comparison considers the quality of the solutions and the computational efficiency of each method. Furthermore, a comprehensive structural analysis is performed on the optimized truss designs to validate their compliance with the imposed constraints. A finite element analysis is utilized to assess stress distributions, member sizes, and overall structural stability for each of the loading conditions. By conducting the comparison and verification, the study offers valuable insights into the performance of the ISRES algorithm in handling highly constrained truss-optimization problems. The findings contribute to a better understanding of the algorithm's strengths and weaknesses, particularly in the context of complex structural designs. Figure 12 provides a convergence plot of the ISRES algorithm applied to the 25-bar truss structure. The plot offers a graphical representation of the algorithm's convergence behavior over the optimization iterations. The gradual reduction of the objective function values indicates the algorithm's progress towards optimal solutions. Analyzing the plot  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Node**} & \multicolumn{2}{c}{**Condition I**} & \multicolumn{4}{c}{**Condition II**} \\ \cline{2-7}  & \(\mathbf{P_{x}}\) & \(\mathbf{P_{y}}\) & \(\mathbf{P_{z}}\) & \(\mathbf{P_{x}}\) & \(\mathbf{P_{y}}\) & \(\mathbf{P_{z}}\) \\ \hline
1 & 0 & 20 & \(-5.0\) & 1 & 10.0 &  
assists in understanding the convergence rate and identifying any convergence stagnation or rapid convergence trends, contributing to a deeper insight into the optimization process. Figure 13 illustrates the displacement constraints along with their allowable limits for load case 1 and load case 2. The figure provides a visual representation of how the optimized truss designs conform to the specified nodal displacement limits. By presenting the displacement distributions across the truss structure, this figure offers insights into the structural stability of the optimized designs.