# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 1938-Bar Tower  
The second design example considered in this study is the weight minimization of the spatial 1938-bar truss tower with 481 nodes shown in Figure 4. The tower is 285 m tall; its layout section is a regular dodecagon at the ground level and a square at the top segment. The numbering of nodes proceeds from structure's top to bottom as follows: (i) Segment 1 ends into the top node 1; (ii) 25 sets of 4 nodes each, from 2-3-4-5 to 98-99-100-101, belonging  
Figure 3: Convergence curves obtained for the 200-bar truss design example. to segments 1 and 2; (iii) 16 sets of 8 nodes each, from 102-103-104-105-106-107-108-109 to 222-223-224-225-226-227-228-229, belonging to Segment 3; (iv) 21 sets of 12 nodes each, from 230-231-232-233-234-235-236-237-238-239-240-241 to 470-471-472-473-474-475-476-477-478-479-480-481, belonging to Segment 4. The nomenclature of segments included in the tower is detailed in the appendix. This large-scale optimization problem included 204 sizing variables and was originally presented in [53; 54]. The Young's modulus E of the material is 68.971 GPa, while the mass density is 2767.991 kg/m\({}^{3}\). Structural symmetry allows to categorize bars into 204 groups (see Table A1 of the Appendix A) each of which includes elements with the same cross-sectional area taken as a sizing variable. The tower must carry the three independent loading conditions listed in Table 3. The first loading condition presents concentrated forces at all free nodes acting downward; force magnitude increases from top to bottom of the structure: the total load acting on the tower is more than 20 times larger than the gravity load corresponding to the target structural weight. The second loading condition includes concentrated forces acting in the X-direction; forces applied to the left side of the tower (for example, at nodes 98, 101, 228 etc.) act in the positive X-direction while forces applied to the right side of the tower (for example, at nodes 99, 100, 224 etc.) act in the negative X-direction: hence, the resultant forces of this loading condition bend the tower rightwards. The third loading condition includes concentrated forces acting in the Y-direction; forces applied to the front side of the tower (for example, at nodes 100, 101, 226 etc.) act in the positive Y-direction while forces applied to the rear side of the tower (for example, at nodes 98, 99, 222 etc.) act in the  
Figure 4: Schematic of the spatial 1938-bar tower: (**a**) Assembly view; (**b**) Color view of the four segments and two junction modules included in the structure; (**c**) Layout view indicating key-nodes. negative Y-direction: hence, the resultant forces of this loading condition tend to compress the tower about the XZ symmetry plane of the structure. The optimization problem has 20,070 non-linear constraints on nodal displacements, member stresses and critical buckling loads. Displacements of free nodes in the X, Y, Z directions must not exceed \(\pm\)40.64 cm (i.e., \(\pm\)16 in). The allowable tensile stress is 275.9 MPa (i.e., 40,000 psi). The buckling strength of the jth element of the structure is \(-100.01\pi\)EA\({}_{\text{j}}\)/8l\({}_{\text{j}}^{2}\) as the tower includes tubular elements with a nominal diameter to thickness ratio of 100. Cross-sectional areas vary between 0.64516 and 1290.32 cm\({}^{2}\) (i.e., between 0.1 and 200 in\({}^{2}\)). The optimization results obtained for the 1938-bar tower design example are presented in Table 4. LSSO-HHSIA's optimization runs were carried out for \(N_{\text{POP}}\) = 20 and 500 in order to have a fair comparison with the hybrid HS, hybrid BBBC and HFSA algorithms of Ref. [53] and the improved JAYA algorithm of Ref. [54]. No data are listed in the table for sinDE [58] and SQP-MATLAB as those algorithms could not find satisfactory designs.