# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 3586-Bar Truss Tower  
The last design example is the weight minimization of the spatial 3586-bar tower with 897 nodes shown in Figure 6. This large-scale optimization problem, originally presented in [51], has 280 sizing variables. Material properties are the same as in the 1938-bar tower problem. Structural symmetry allows to categorize bars into 280 groups (see Figure 1 and  
Figure 5: Convergence curves obtained for the 1938-bar tower design example. Table A1 of the Appendix A) each of which includes elements with the same cross-sectional area taken as a sizing variable. The tower is 415 m tall, includes five segments and three junction modules; its layout section is a regular hexadecagon at the ground level and a square at the top segment. In Figure 6b, the segments of the structure are represented in different colors. Figure 6c-f clarifies the storey numbering, progressing from the top to the bottom of the structure. The layout view of Figure 6g indicates the nodes limiting the tower cross-sections for the different segments. Node and element group numbering increases from the top to the bottom of the structure. The square-based pyramid \"Segment 1\" includes element groups 1 through 57; the square-based prismatic \"Segment 2\" groups 58 through 97; group 98 connects segments 1 and 2; the octagon-based prismatic \"Segment 3\" includes groups 99 through 143; group 144 connects segments 3 and 4; the dodecagon-based prismatic \"Segment 4\" includes groups 145 through 204; group 205 connects segments 4 and 5; the hexadecagon-based prismatic \"Segment 5\" includes groups 206 through 280. Figure 6: Schematic of the spatial 3586-bar tower (representative story and junction numbers also are shown): (**a**) Assembly view; (**b**) Color view of the five segments and three junction modules included in the structure; (**c**) Transition from the top of the tower to segment 1 (top of the structure); (**d**,**e**) Transitions between segments 2 and 3, and between segments 3 and 4 (center of the structure); (**f**) Transition between segments 4 and 5 (bottom of the structure); (**g**) Layout view indicating key-nodes. It can be seen from Figure 6g that the node numbering is the same as for the 1938-bar tower up to node 481. The tower is then completed by 25 sets of 16 nodes each, from 482-483-484-485-486-487-488-489-490-491-492-493-494-495-496-497 to 882-883-884-885-886-887-888-889-890-891-892-893-894-895-896-897, belonging to Segment 5. The tower must carry the three independent loading conditions listed in Table 5. The explanation given in Section 4.4 holds true also for the 3586-bar structure. However, the total load acting on the tower (i.e., about 4862 ton) now is about 15 times larger than the gravity load corresponding to the target structural weight. In the second loading condition, forces acting in the positive X-direction on the left side of the tower are applied, for example, at nodes 98, 101, 228, 479, etc. ; forces acting in the negative X-direction on the right side of the tower are applied, for example, at nodes 99, 100, 224, 473, etc. In the third loading condition, forces acting in the positive Y-direction on the front side of the tower are applied, for example, at nodes 100, 101, 226, 476, etc. ; forces acting in the negative Y-direction on the rear side of the tower are applied, for example, at nodes 98, 99, 222, 470, etc. The optimization problem includes 37,374 non-linear constraints on nodal displacements, member stresses and critical buckling loads using the same limits defined for the  
1938-bar tower problem. Side constraints on sizing variables also coincide with those of the previous design example. Table 6 presents the results obtained for the 3586-bar tower design example. LSSO-HHSA's optimization runs were carried out for \(N_{\text{POP}}\) = 20 and \(N_{\text{POP}}\) = 500 for two reasons: (i) to have statistically more significant results than Ref.