# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 1 Introduction
### Contribution of This Work  
To address the aforementioned design challenges, this study develops a comprehensive framework for the layout optimisation of frame structures. By simultaneously considering multiple mechanical constraints and controlling geometric complexity, this framework enhances both the design and manufacturability of frame structures. The constraints addressed include displacement, stress, and both local and global stability. Geometric complexity is managed by minimising low-stiffness elements and reducing the number of elements in the resulting layouts. The primary innovation of this study lies in the concurrent handling of multiple mechanical constraints and geometric complexity. Moreover, the proposed method is developed within a gradient-based framework specially tailored for large-scale structures, which involve a large number of degrees of freedom and candidate elements. This approach is particularly advantageous, as such problems are typically challenging to solve using non-gradient based methods like meta-heuristic algorithms. Regarding the mechanical constraints, displacement and stress are evaluated through linear static analysis, and local and global stability are assessed using linear buckling analysis and element subdivisions. While non-linear structural analysis can provide a more accurate estimation of a structure's load-carrying behaviour, it often requires complex iterative calculations to determine the equilibrium state [60; 61]. The sensitivity of mechanical performance metrics, such as the stress, strain, displacement, and buckling load factor, is typically path dependent, necessitating the calculation and storage of structural tangential stiffness and mechanical responses at each load increment [39; 62; 63; 64]. Additionally, when considering nodal displacement and element stress constraints, the number of constraints usually matches the number of nodes and elements in the structure, leading to a significant computational burden. The high cost of non-linear structural analysis, combined with significant non-linearities in constraints and the complex sensitivity analysis, poses substantial challenges for problem formulation and achieving convergence, especially in large-scale structures with fine grid divisions. Consequently, this study employs linear static and buckling analyses as the primary methods for evaluating structural performance. These linear analyses are expected to provide valuable insights for optimising material distribution, making them particularly suitable for the conceptual design of frame structures. However, relying on linear analyses may compromise the accuracy and reliability of the optimisation results, especially in scenarios where non-linear effects are significant. To mitigate these risks, a post-optimisation non-linear analysis is recommended to verify structural performance and ensure the safety and robustness of the final designs. To circumvent the computational demands of analysing numerous node-wise and element-wise constraints, an aggregation technique is employed for each constraint type. This not only streamlines sensitivity analysis but also addresses numerical challenges arising from the non-differentiability of multiple eigenvalues and potential mode switching that can occur in linear buckling analysis. Low-stiffness elements are penalised using the SIMP-based approach [65], therefore driving their removal. To mitigate stress singularities, a \(pq\) relaxation scheme [26; 66] is implemented when formulating stress constraints. Additionally, pseudo buckling modes are eliminated by imposing more severe penalisation to the geometric stiffness matrix compared to the elastic stiffness matrix for low-stiffness elements [47]. Furthermore, an auxiliary component reflecting structural fabrication cost is heuristically combined with material volume to form the objective function. This approach is strategically designed to impose additional penalties on elements, thereby favouring their removal.