# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 4 Problem Formulation
### Compression Column
Notably, the optimal layouts generated using discrete structures are generally different from those using continuum structures. The latter usually manifest as two primary inclined components augmented with secondary lateral bracing as presented in the works of [PERSON] and [PERSON] [38], [PERSON] et al. [47], and [PERSON] et al. [72]. Investigating the reasons behind these differences and conducting a quantitative comparison of these layouts is of interest but is left for future study. The material volume (\(V\)), number of elements (\(n\)), maximum \(y\) displacement (\(U_{\rm max}^{y}\)), maximum von Mises stress \(\sigma_{\rm max}^{em}\), and fundamental buckling load factor (\(\lambda_{1}\)) for the four structural layouts are summarised in Table 1. All layouts exhibit satisfactory mechanical performance when the relevant constraints are included in the optimisation problems. To further illustrate the superior stability of Layout-II, -III, and -IV over Layout-I, a separate sizing optimisation is conducted on Layout-I while incorporating both the global and local stability constraints. In this context, the upper bound limit on element diameters is relaxed to allow for increases as necessary to satisfy the stability constraints. This optimisation results in a structure that retains the same layout but features a cone-shaped cross-section, with element diameters tapering from 1.01 m at the bottom to 0.433 m at the top. Consequently, the material volume of the resulting structure is 1.820 \(\,\mathrm{m}^{3}\), which is more than 4.2, 3.6, and 4.4 times that of Layout-II, -III, and -IV, respectively. Among the designs with stability constraints, Layout-IV, which was developed without penalising low-stiffness elements, exhibits a considerable number of grey elements, thereby substantially impairing its manufacturability. This observation aligns with findings from prior research on discrete-structure layout optimisation, which have separately explored various constraints such as stress [1; 49; 50; 73], displacement [51], compliance [52], and stability [39; 40; 71]. The presence of a significant number of elements, including those of low stiffness, in the resultant layouts can be attributed to the inherent tendency of the ground structure method to produce non-unique maximum stiffness designs [74]. Nevertheless, upon visually examining Layout-II and -IV and comparing their material usage, it becomes evident that implementing the SIMP-based scheme effectively reduces the occurrence of grey elements without excessively compromising economic feasibility, although some residual grey elements may persist. Notably, addressing these residual elements could potentially benefit from the density projection scheme developed in continuum-structure topology optimisation [75]. Furthermore, the discrepancy in material volume between Layout-II and -III highlights that employing the SIMP-based scheme to handle low-stiffness elements leads to a more efficient layout with decreased material consumption compared to the \"hard-kill\" approach. Therefore, it can be concluded that the SIMP-based scheme is appropriate for minimising low-stiffness elements, thereby improving manufacturability while maintaining the optimality of the resulting layouts. Figure 5: Optimal layouts and buckling modes of the compression column. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).