# Efficient Sizing and Layout Optimization of Truss Benchmark Structures Using ISRES Algorithm
## 4 Benchmark Problems
### 10-Bar Truss Structure  
The 10-bar planar truss structure has been a common problem studied in the literature and has been used as a sizing benchmark by many researchers. Figure 2 displays the layout and loadings of the truss. Two loading cases were investigated: Case I with a load value of P1 = 100 kips and P2 = 0, and Case II with a load value of P1 = 150 kips and P2 = 50 kips. The material used to construct the truss had a mass density of 0.10 lb./in\({}^{3}\) and elastic modulus of 10,000 ksi. The size of each member was taken from a range between Amin = 0.1 in\({}^{2}\) and Amax = 35.0 in\({}^{2}\). The allowable member stresses and nodal displacement limits were set as \(\pm\)25 ksi and \(\pm\)2 in for all free nodes in the x and y direction, respectively. In total, the problem had 32 non-linear design constraints (10 tension, 10 compression, and 12 displacement constraints) and 14 discrete design variables. In order to handle the design constraints effectively during the optimization process, penalty functions were introduced. Penalty methods are a certain class of algorithms for solving constrained optimization problems. They convert constrained optimization problems into unconstrained ones by modifying the objective function to penalize constraint violations.  
The optimization process involved formulating the single-objective optimization problem for the truss structure with the weight as the objective to be minimized. The design variables (member sizes) were optimized by using the ISRES algorithm, which efficiently explores the solution space and searches for the optimal design.  
For each loading case, the ISRES algorithm was executed multiple times with appropriate parameters and control settings to obtain diverse candidate solutions. The penalty  
Figure 2: Schematic of the 10-bar structure.  
functions were utilized to ensure that the final solutions adhered to the specified stress and displacement constraints.  
Through extensive experimentation and fine-tuning the algorithm, the ISRES approach effectively converged to optimal solutions that minimized the weight of the truss while satisfying all the design constraints. The optimization results revealed well-performing designs that met the structural requirements for both loading scenarios.  
In addition, the results obtained from the ISRES algorithm were compared to those achieved through other optimization methods reported in the literature. This comparison allowed for an assessment of the ISRES algorithm's performance in terms of convergence speed, solution diversity, and solution quality. After applying the ISRES algorithm to both load case 1 and load case 2, the convergence graph for each case, as depicted in Figure 3, provided a dynamic snapshot of the optimization process. The graph vividly portrays the algorithm's iterative journey towards optimal solutions. The descending trendlines demonstrate the algorithm's persistent effort in refining the truss designs by minimizing their weight while satisfying the allowable stress limits. The consistent downward trajectory signifies the algorithm's ability to progressively enhance solution quality through each iteration, reflecting its effectiveness in converging to design configurations that strike a balance between structural performance and material efficiency.  
Figure 4 shows the constraints graph for load case 1 and load case 2 with allowable stress, and it offers a comprehensive visual representation of the algorithm's ability to manage multiple design constraints. The graph elegantly illustrates how the ISRES algorithm strategically guides the optimization trajectory within the permissible stress regions, assuring that the final truss designs align with the designated allowable stress thresholds. The intricate interplay between the optimization path and the constraint boundaries highlights the algorithm's prowess in navigating the intricate trade-offs between design efficiency and structural integrity for both loading scenarios.  
Figure 3: Convergence plot for 10-bar structure.