## 1 Introduction  
With rapid progress in architectural design and construction technology, architecture is developing towards complexity, diversity, and novelty. Shape -resistant spatial structures, whose mechanical behaviors are closely related to architectural performance, are highly favored by engineers. As representatives of such structures, free-form and branching structures are widely used in modern large-span architecture [1; 2; 3; 4; 5; 6]. There are also branch-supported free-form surface buildings that incorporate the geometric characteristics of these two structural forms [7; 8]. The free-form surface structure has a flexible shape that is difficult to express with a single or a few analytical functions. Additionally, its geometry plays a crucial role in conveying the architectural intent as well as determining the structural mechanical performance [9; 10]. A branching structure, also known as a tree-like structure, is a multi-level branching and three-dimensional expanding bar system. It is often used as a building support structure, which can effectively reduce the span of the roof without affecting the function of the lower space [11]. The shape and supporting position are important considerations in its design process [12]. The issue of creating a reasonable form for structures has garnered significant attention from scholars. In terms of creating a reasonable form of free-form surface structure, scholars have developed numerous effective shape generation methods based on the principle of inverted hanging models and the theory of structural optimization. Before the widespread application of computers, the inverted hanging model experiment was the main means of determining the reasonable form of the free-form surface structure [2; 13; 14; 15; 16]. This method used flexible materials that could only withstand tensile forces to simulate the actual structural stress state and obtain a stable equilibrium shape under specified design conditions. Representative cases include the Deitingen motorway BP service station in Switzerland [17], the Heimberg Tennis Hall (also located in Switzerland) [18], and the Mannheim Multihalle in Germany [19]. While the concept behind the model experiment method is clear, the implementation process can be cumbersome when comparing multiple design options and may be affected by many uncertain factors during the operation [1; 10; 20]. As computer performance improves and numerical theories develop, numerical methods are gradually becoming the mainstream approach in the field of morphological generation, covering form-finding and structural optimization techniques. Form-finding methods mainly include the geometric stiffness method [21; 22], the dynamic equilibrium method [23], and the stiffness matrix method [24; 25; 26]. Structural optimization is a systematic process that utilizes optimization algorithms to drive design variables and obtain an optimal structural shape, which is reflected by the objective function [1; 20]. Common objective functions include minimizing structural weight [27], uniforming stress state [28; 29], and minimizing strain energy (maximizing stiffness) [30; 31; 32; 33]. Studies have shown that minimizing structural strain energy can effectively reduce structural bending moments, resulting in structures with higher structural efficiency [10; 28; 29]. Compared to form-finding, structural optimization has a wider range of applications [1; 20]. In terms of creating a reasonable form of branching structure, its development process is similar to that of free-form surface structure. Early scholars also sought a reasonable form of branching structure based on the principle of model experiments, relying on axial force transmission to bear loads, mainly including the inverse hanging model, thread model, etc. [4; 6; 11; 14; 15; 16], which were successfully applied to the design process of the Riyadh government center, the Stuttgart Airport, and other architectural projects [34].