# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 4 Problem Formulation
### Short Cantilever  
The short cantilever example has dimensions of \(L\times H=12\) m \(\times\) 24 m as shown in Figure 6. It is fixed at the top and bottom left nodes and subject to a concentrated vertical force applied at the right middle. The domain is discretised into 4 \(\times\) 8 three-meter-wide square cells. Similar to the compression column, a ground structure with two-level connectivity is employed, where members are split at their intersections, and each is subdivided into four elements for structural analysis. This ground structure comprises the same number of nodes and members as the compression column.  
Similarly, three optimisation problems are tackled: the first focuses solely on displacement and stress constraints, and the second incorporates additional global and local stability constraints, while the third considers all constraints and employs the hard-kill scheme to remove low-stiffness elements during the optimisation process. The resulting optimal structures are denoted as Layout-I, -II, and -III, respectively, and are depicted in Figure 7, along with their buckling modes. Layout-I presents a simple two-bar structure, with one compression and one tension bar, Layout-II and -III include additional elements concentrated at the lower part of the structure to reinforce the compression zone. Both Layout-II and -III exhibit similarities to the results presented in previous works such as [PERSON] and [PERSON] [76] and [PERSON] et al. [77].  
The material volume, mechanical performance metrics, and number of elements for Layout-I, -II, and -III are summarised in Table 2. All layouts demonstrate satisfactory mechanical performance when relevant constraints are incorporated in the optimisation problem. A separate sizing optimisation is performed on Layout-I with the inclusion of additional global and local stability constraints. The resulting layout consumes a material volume of 0.758 m\({}^{3}\), which is more than 1.2 and 1.1 times that of Layout-II and -III, respectively. This indicates that the member configurations in Layout-II and -III are more optimal than that in Layout-I by considering the stability properties during the optimisation. Furthermore, the material volume of Layout-II is less than that of Layout-III, despite the former presenting more elements. This demonstrates the effectiveness of the SIMP  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Layout** & \(\boldsymbol{V/\text{m}^{3}}\) & \(\boldsymbol{U_{\text{max}}^{y}/\text{m}}\) & \(\boldsymbol{\sigma_{\text{max}}^{vm}/\text{MPa}}\) & \(\boldsymbol{\lambda_{1}}\) & \(\boldsymbol{n}\) \\ \hline I & 0.267 & 0.024 & 300 & 0.08 & 64 \\ II & 0.431 & 0.017 & 300 & 4.94 & 1000 \\ III & 0.496 & 0.018 & 300 & 4.96 & 1032 \\ IV & 0.407 & 0.021 & 300 & 4.99 & 1368 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric and mechanical properties of the optimal layouts of the compression column.  
Figure 6: Design information of the short cantilever.  
based scheme over the \"hard-kill\" scheme when treating low-stiffness elements in the optimisation process.