# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 4 Problem Formulation
### Compression Column  
The design information of the compression column is illustrated in Figure 4. The design domain measures \(L\times H=8\) m \(\times\) 16 m, fixed at the bottom and with a concentrated force acting at the mid-point of the upper surface. The domain is discretised into a grid of \(4\times 8\) two-meter wide square cells. Candidate members are formed by interconnecting the internal nodes to their neighbour nodes, and the neighbours of their neighbours, generating a ground structure with two-level connectivity. Additionally, members are split at their intersections as depicted in the amplified view in the middle of Figure 4. This ground structure consists of, in total, 501 nodes and 1292 candidate members. To capture the lateral deformation of members during local buckling in the linear buckling analysis, each member is further subdivided into four elements as shown in the right-end amplified view in Figure 4. This results into a total of 4377 nodes and 5168 elements in the structural analysis. Four optimisation problems are addressed for the compression column. The first problem considers only displacement and stress constraints, while the remaining three problems incorporate additional global and local stability constraints. These latter problems are differentiated by their methods of treating low-stiffness elements. The second problem employs the SIMP-based scheme with the \(\omega_{E}\) initially set to 1.5 and gradually increased up to 4 as described in Section 4. The third problem uses a \"hard-kill\" scheme, directly removing low-stiffness elements (\(\tilde{d}_{e}<d_{\text{th}}\)) during the optimisation. The fourth case disregards the SIMP-based scheme by adopting \(\omega_{E}=1\) consistently throughout the entire optimisation process. The optimisation results for these four problems are denoted as Layout-I through Layout-IV, respectively. The optimal structures and the corresponding first three buckling modes are depicted in Figure 5, where red lines signify compression elements, blue lines represent tension elements, and black lines denote grey elements with diameters (\(\tilde{d}_{e}\)) falling into the range of \((0,d_{\text{th}})\). As shown in Figure 5a, Layout-I presents a simple vertical bar. In this layout, stress constraints dominate over displacement constraints, resulting in a uniform cross-section with a diameter of \(\tilde{d}=0.334\) m. The corresponding cross-sectional area and moment of inertia are \(A=0.0167\) m\({}^{2}\) and \(I=2.106\times 10^{-4}\) m\({}^{4}\), respectively. The buckling load factors and modes align with the results of the theoretical Euler buckling analysis for a column  
Figure 4: Design information of the compression column. with fixed-free ends. The critical load is given by \(P_{\rm cr}=\dfrac{\kappa^{2}\pi^{2}EI}{4H^{2}}\), where \(\kappa=1,3,5\) for the first, second, and third buckling modes, respectively. As depicted in Figure 5b-d, incorporating global and local stability constraints leads to Layout-II, -III, and -IV, which feature additional elements providing lateral support. The buckling modes exhibit concurrent nodal movement and in-member lateral displacement, demonstrating the feasibility of using linear buckling analysis with element refinement to capture both local and global buckling. Moreover, these modes mostly correspond to solid elements (\(d_{e}\geq d_{\rm th}\)), highlighting the effectiveness of imposing distinct penalisation to the elastic and geometric stiffness in suppressing pseudo buckling modes. Furthermore, Layout-II, -III, and -IV share similarities with the designs presented in the works of [PERSON] and [PERSON] [39], [PERSON] and [PERSON] [40], and [PERSON] et al. [71], where the stability properties were also considered. However, quantitative comparisons are not feasible due to differences in the optimisation problems, as this study integrates multiple mechanical constraints that are not addressed in the aforementioned works.