# Optimizing Support Locations in the Roof-Column Structural System
## 1 Introduction  
The roof-column structural system creates a simple form of shelter. It is utilized across many architectural and engineering applications due to its effectiveness in supporting environmental loads, transmitting the structure's weight, and reducing solar heat gains [1]. However, determining locations of supporting columns under the roof is a challenging task; they have direct influence on both architectural appearance and structural performance.  
The integrated structural-architectural design was suggested to be an effective strategy for capturing complex requirements from both engineering and architectural perspectives [2]. This includes the utilization of structural optimization, where the optimization design process achieves specific objectives by changing defined design variables. For example, a curved frame's minimal size and shape can be designed with a given applied load through optimization [3]. Many other applications were all enabled by extensive preceding work in structural optimization using diverse structural optimization methods [4], including continuum and discrete techniques.  
Continuum methods obtain optimal designs by determining the shapes and locations of cavities in continuous geometries [5, 6]. Many popular topology optimization methods are part of continuum methods, including the homogenization approach [7], the solid isotropic material with penalization (SIMP) method [8, 9, 10], the bidirectional evolutionary structural optimization (BESO) method [11, 12, 13, 14], and the level set method [15, 16, 17]. Although those continuum methods can maximize the stiffness of a structure with specified designconstraints, it is difficult to apply them in roof-column structural systems to find optimal locations of predefined columns, as they include the generation of complex 3D shapes. For example, the iconic Qatar National Convention Center, designed using the extended ESO method, has complex treelike columns supporting a simple rectangular roof [18].  
Discrete methods obtain optimal designs by determining the optimal spatial order and connectivity of line-based representations, such as trusses, frames, and equilibrium configurations of tensegrity structures [19; 20; 21; 22]. They typically involve the utilization of computational algorithms, such as the genetic algorithm (GA) [23], the simulated annealing (SA) algorithm [24], and particle swarm optimization (PSO) [25]. The key benefit of using discrete methods is that components in optimal designs can be predetermined, thereby being widely utilized in structural designs such as space frames with standard member types [26]. However, existing discrete methods can be computationally expensive due to many candidate elements, and require a user-defined objective function for optimization problems that can be difficult to formulate and implement.  
This paper proposes a new integrated structural-architectural design method for automatically finding optimal column locations in roof-column structural systems. The new method combines the objective function from topology optimization for compliance minimization with a discrete design domain comprising candidate-supporting columns, thus capturing architectural requirements and ensuring structural performance. Section 2 describes an optimization algorithm for implementing the new method. Section 3 introduces three substrategies for controlling the optimization tendency in order to obtain improved results. Section 4 validates the proposed method using a square-roof example. Section 5 presents research implications using two additional examples with nonregular roof shapes, followed by a conclusion in Section 6.