## 4 Optimization of Diagrid Performance Based on the Geometrical Features
Further developments in the external diagrid patterns are carried out by [PERSON] et al. [32]. In addition to the consideration of uniform- and varying-angle (VA) solutions, the authors also propose diagrid patterns with variable-density (VD) in the diagonal layout (Figure 9a). FE calculations are performed on a 90-storey tall building with an aspect ratio of 6.62, under gravity and wind loads, and the structural responses are analyzed in terms of top lateral deflection, inter-story drifts and diagonal DCR. For each solution, an efficiency parameter is proposed as \(1/D_{top}w\), \(D_{top}\) and \(w\) being the top lateral displacement and the employed steel weight per total floor area, respectively. The lower the lateral displacement and the amount of steel, the greater the efficiency of the investigated solution. The obtained efficiency parameters are shown in Figure 9b for all the considered solutions. From the results, it is found that the 80\({}^{\circ}\) and VA3 solution results in lower efficiency in the investigated case, whereas VA1 is the most efficient one. Uniform-angle solutions with 60\({}^{\circ}\) and 70\({}^{\circ}\), as well as the variable patterns VA2, VD1 and VD2, show similar efficiency values [32]. Additional analyses regarding different pattern configurations can be found in the work of [PERSON] and [PERSON] [9]. After exploring the effectiveness of stiffness-based approaches for a 351-m tall diagrid with optimal (69\({}^{\circ}\)) and non-optimal (82\({}^{\circ}\)) diagonal angles, in order to evaluate whether common approaches lead to optimized member sizes, the authors propose additional variable-density (VD) patterns for the diagonal arrangement (Figure 10). Two VD strategies are suggested for the non-optimal (82\({}^{\circ}\)) diagrid tube in order to meet the stiffness requirements: a localized pattern, resembling one-outrigger-like (Figure 10a) or two-outrigger-like (Figure 10b) schemes; and a more uniform VD pattern, which provides additional distributed stiffness over the building's elevation (Figure 10c,d). The outcomes from FE calculations show that the local density increments (Figure 10a,b) are not efficient strategies for meeting the stiffness and strength requirements. Conversely, the solutions involving a more uniform VD pattern (Figure 10c,d), where the diagonal concentration rarefies towards the top of the building, turn out to be appropriate solutions for limiting the lateral displacements, while obtaining notable material savings [9]. The previous work of [PERSON] et al. [32] has been subsequently developed by [PERSON] et al. [33], who propose additional diagonal patterns for the 90-story tall diagrid building (Figure 11a). Figure 10: Variable-density (VD) patterns proposed by [PERSON] and [PERSON] [9]: (**a**) and (**b**) concentrated outrigger-like VD pattern; (**c**) and (**d**) distributed VD pattern. Used with permission from [PERSON] and [PERSON] [9]. Figure 9: (**a**) Different geometrical patterns from [PERSON] et al. [32]: uniform-angle patterns (60\({}^{\circ}\), 70\({}^{\circ}\), 80\({}^{\circ}\)), varying-angle patterns according to Moon approach (VA1, VA2) [12; 30], varying-angle pattern according to Zhang approach (VA3) [31], variable-density patterns (VD1, VD2). (**b**) Efficiency parameters for the investigated solutions. Used with permission from [PERSON] et al. [32]. Besides considering the usual uniform- and varying-angle patterns, the authors also suggested a double-density pattern (DD) (where the diagonal layout is doubled and mirrored over the diagrid facade), a variable-density pattern (VD) (generated by starting from the DD pattern with further topology optimization), and a diagrid-like pattern (ISO) (where the diagonals follow the principal stress lines obtained from the equivalent building cantilever). Stiffness- and strength-based preliminary designs are carried out, together with optimization procedures based on Genetic Algorithms through the use of commercial software.