# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 1 Introduction
### Precedent Work
#### 1.1.2 Local and Global Buckling  
Local buckling can be addressed in a manner similar to stress constraints, by limiting element stress below a threshold based on the Euler Buckling criterion [20; 28; 29; 30], or by adhering to design standards as demonstrated in the works of [PERSON] [31], [PERSON] et al. [32], and [PERSON] et al. [33]. Consequently, local buckling constraints encounter the same challenges as stress constraints, and can be mitigated through similar approaches. Alternatively, lateral deformation local to elements can be captured using linear buckling analysis by subdividing elements, which allows for the simultaneous treatment of local and global stability [34].  
The global stability of structures can be evaluated by linear buckling analysis, which involves solving a generalised eigenvalue problem, formulated with an elastic stiffness matrix and a geometric (stress) stiffness matrix [35]. The fundamental (lowest) eigenvalue obtained is taken as an indicator of global stability. The main challenges associated with global stability-based problems are pseudo (artificial/spurious) buckling modes and possible clustering and switching between the first several orders of buckling. Pseudo buckling modes mainly derive from the low-stiffness regions, where the geometric stiffness is higher than the elastic stiffness. In order to eliminate pseudo buckling modes, [PERSON] et al. [36] proposed to ignore the geometric stiffness of the low-stiffness material; [PERSON] and [PERSON] [37] suggested different interpolation schemes to calculate the material elastic moduli used in the elastic and geometric stiffness matrix. [PERSON] and [PERSON] [38] developed a method based on the modal energy ratio, to identify the pseudo buckling modes in the context of continuum-structure topology optimisation. The same strategy was extended to the stability design of truss and frame structures by [PERSON] and [PERSON] [39] and [PERSON] and [PERSON] [40], respectively. Recently, [PERSON] et al. [41] proposed a method to mitigate spurious buckling modes by integrating a pseudo mass matrix into the eigenvalue problem. This matrix is constructed based on the status of nodes, characterised by their \"nodal density\", which is determined as the maximum density among neighbouring elements.  
Aside from pseudo buckling modes, the emergence of repeated or closely spaced eigenvalues, and the possible switching between them, also pose great challenges for sensitivity analysis and optimisation. In the case of multiple eigenvalues, when more than one eigenvector is associated with an eigenvalue, these multiple eigenvalues usually do not possess direct differentiable properties, and their sensitivity analysis requires special care [42; 43; 44]. Moreover, the switching between the first few eigenvalues, and therefore eigenmodes, which may occur along with material re-distribution, can cause oscillations, and even divergence, to the optimisation problems [37]. For the purpose of ensuring the consistency of the order of eigenvalues (eigenmodes) and, therefore, avoiding multiplicity and rendering them differentiable, [PERSON] and [PERSON] [37] recommended imposing consecutive bounding constraints to eigenvalues. As an alternative, the eigenvalues of interest can be transferred into a global measure using aggregation functions [34; 45; 46], which simultaneously enables differentiability and resolves the dilemma of mode switching. Subsequently, the global measure of eigenvalues, being an approximation to the critical eigenvalue, can be used to guide the stability design of structures [15; 34; 40; 47].