## 1 Introduction
Moreover, the iterative calculation cannot be completed under a certain load. The analytical solution to calculating the main cable curve is complex and has many iterations. With the rapid development of computer technology, finite element theory can be used to solve the problem of calculating the curve, and professional finite element software such as Midas has emerged. The calculation efficiency and accuracy of FEM depend not only on a high-precision element tangent stiffness matrix, such as a truss cable element, a catenary clue element, and a saddle element, but also on the efficient numerical algorithm of nonlinear equations, such as the load increment method, the Newton-Raphson (N-R for short) method, or the hybrid method. [PERSON] (1966) [PERSON] (1966) first proposed to use nonlinear finite element theory to solve the inclined suspender problem. With a tension only main cable in mind and considering the secondary influence caused by the initial axial force and large displacement, the linear member finite element stiffness matrix was modified; the nonlinear tangent stiffness matrix was derived; and then, the N-R method was used to solve the equations in incremental form. [PERSON] (1966) [PERSON] (1966) considered the initial bending moment, shear force, and the secondary effect caused by the interaction between the axial force and the bending moment but, without regard to the initial axial force and large displacement, deduced the nonlinear tangent stiffness matrix and formed a more comprehensive nonlinear finite element theory. Thus, the analysis of suspension bridges began to enter the era of finite displacement theory. [PERSON] et al. (1933, 1944) constructed the element stiffness matrix based on an analytical formula of elastic catenary in three-dimensional coordinates and conducted both static and dynamic analyses of the cable structure. Furthermore, taking a multi-tower suspension bridge as research background, the main cable curve was iteratively calculated, and the influence of the dead live load ratio was studied to predict the limits of suspension bridges. [PERSON] et al. (2020) [PERSON] et al. (2020) proposed an improved curve-searching method for suspension bridges based on FEM, and the method involved four steps, which are hanger force determination, determination of the initial equilibrium state of the cable in the mid-span,curve finding in the side span, and examination of the whole bridge. [PERSON] et al. (2020) [16] put forward an analytical method for finding the main cable curve using the framework of multi-segment catenary theory and applied the method on a three-pylon suspension bridge. [PERSON] et al. (2021) [17] developed the method based on the nonlinear finite element approach with a Eulerian description to find the main cable curve. Combining finite element analysis, analytical formula, and an optimization algorithm, [PERSON] et al. (2021) [18] proposed a new optimal form-finding method for spatial self-anchored hybrid cable-stayed suspension bridges. The FEM shows good calculation accuracy in the main cable curve calculation, while when solving nonlinear equations, it mostly depends on matrix iteration and complex programming. Additionally, it would often face the convergence problems of initial value selection and numerical divergence. For bridge designers, its calculation efficiency was still insufficient [19]. The particle swarm optimization (PSO) method was proposed by [PERSON] and [PERSON] (1995) [20], whose basic idea was to use a group of initial irregular particles to explore from disorder to order in space to obtain the optimal solution. PSO is a relatively easy-to-implement algorithm, for which the continuity of the objective function has little influence on the algorithm, and the algorithm can adapt well in various environments. Therefore, PSO is widely used in various fields. [PERSON] et al. (2021) [21] proposed a hybrid meta-heuristic algorithm based on PSO and the dynamic penalty method, to solve complex mathematical programming problems quickly. [PERSON] et al.