# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 1 Introduction  
The layout optimisation of discrete structures can be traced back to the foundational design theorem proposed by [PERSON] [1], wherein an analytical method for deriving optimal truss layouts was presented. Later, [PERSON] et al. [2] proposed the ground structure method (GSM) as a numerical approach for determining the optimal layout of discrete structures. The ground structure consists of candidate elements formed by connecting nodes within a specified design domain. Throughout the optimisation process, the cross-sectional areas of these candidate elements can be taken as design variables. An element is eliminated from the ground structure if its cross-sectional area falls below a predefined cut-off threshold, enabling the concurrent optimisation of both size and topology. The resulting optimal layouts have practical applications in guiding the design of diverse engineering structures such as high-rise buildings [3], large-scale roofs [4], long-span bridges [5], etc.  
As an alternative, topology optimisation has been applied to continuum structures to achieve optimal material distribution. Unlike discrete-structure layout optimisation, working within a continuum domain broadens the design space, allowing for greater flexibility in shaping the structural components. However, the resulting designs often exhibit varying cross-sections, posing challenges for manufacturing using conventional methods. To mitigate this issue, various strategies have been developed to control design features, such as enforcing the minimum, maximum, and uniform sizes of structural members as demonstrated in the studies of [PERSON] et al. [6], [PERSON] et al. [7], and [PERSON] et al. [8]. Furthermore, optimised continuum designs can be converted into discrete structures using topology-preserving skeleton extraction techniques [9; 10]. When combined with subsequent shape and sizing optimisation, these methods can significantly enhance constructibility without excessively compromising structural optimality [11]. However, optimising long-span and large-scale structures can be computationally prohibitive due to the fine discretisation required to generate detailed structural elements [12]. Given thedistinct advantages of discrete structures, such as their explicit predefined members in the ground structure and easily controllable cross-sections, this work focuses on layout optimisation of discrete structures, addressing both the manufacturing challenges and the barriers to large-scale applications.  
To meet the practical functional requirements, structures must demonstrate satisfactory mechanical performance, particularly in terms of displacement, stress, and stability, while ensuring constructibility by avoiding overly complex designs [13]. This work presents a comprehensive framework for optimising the layout of large-scale frame structures, which offer greater flexibility in forming load paths due to their rotational rigidity compared to trusses [14; 15]. The approach integrates multiple mechanical constraints and strategies for controlling geometric complexity. Constraints are applied to displacement, stress, and both local and global stability, while geometric complexity is managed by minimising low-stiffness elements and reducing the number of elements in the resulting layouts. Given that stress and stability (both local and global) involve complex mathematical modelling and exhibit highly non-linear relationships with material configurations, this section reviews current studies on these challenges. Additionally, it provides an overview of the existing methods for controlling geometric complexity in the layout optimisation of discrete structures.