## 1 Introduction  
As they have the advantage of a light weight, reinforced thin-walled structures are widely used in aviation, shipbuilding and construction engineering [1; 2; 3; 4; 5]. By reinforcing ribs, the stiffness or strength of thin-walled structures could be significantly raised without increasing the weight of the whole structure by much. On the other hand, if the reinforcing ribs are not rationally arranged, the reinforcement would not be effective and thus the whole weight of the structure would not be decreased. So, the rational design of reinforcing ribs for thin-walled structures is of significance in engineering. Various studies have been conducted on the rational design of reinforced structures. Buckling analysis and optimization by means of the finite element method (FEM) of hierarchical stiffened plates under axial compression suffer from heavy computational costs; [PERSON] et al. [6] proposed an efficient and simple hybrid framework for the buckling analysis and optimization of hierarchical stiffened plates. [PERSON] et al. [7] found that for steel plate shear walls with a partial plate-column connection (SPSW-PC), although removing the connection between the web-plate and columns can be beneficial in terms of decreasing the overall system demand on the vertical boundary members, based on the results and findings of this research study, such detachment can reduce the stiffness and strength capacities of steel shear walls by about 25% on average. To improve the mechanical properties of reinforced thin-walled structures, some design schemes and algorithms have been proposed [8]. [PERSON] et al. [9] studied two methods for the finite element analysis of stiffened plate structures and compared their calculation accuracy and efficiency. Based on the discrete finite element model of plate beams, the growth and removal of stiffeners were controlled by the ESO method, and a reasonable stiffener layout was obtained. The method of structural topology optimization is often used to optimize the design of the location and cross-sectional dimensions of the reinforcement for thin-walled structures. One of the earliest studies to determine the optimal reinforcement layout based on the plate thickness obtained through topology optimization to design reinforced plates was conducted by [PERSON] et al. [10]. [PERSON] et al. [11] adopted a homogenization method to obtain an optimized rib layout. The obtained results did not give a clear reinforcement layout, although the optimized thickness distribution was given. [PERSON] et al. [12] proposed a method to simultaneously optimize the geometry of the planes in the shell and the layout of the surface stiffeners. [PERSON] et al. [13] determined the optimal composite reinforcement layout based on the optimized plate thickness. [PERSON] et al. [14] proposed a strategy combining topology optimization and shape optimization to identify the optimal layout, in which the same height was used in the region with nearly the same or a higher thickness. [PERSON] et al. [15] proposed a new Heaviside-function-based directional growth topology parameterization (H-DGTP) of the casting constraints for simultaneously optimizing the layout and height of the stiffeners. [PERSON] [16] proposed a topology optimization method based on Opti-Struct. By specifying the thickness of the thin-plate foundation and extending the design space along the height of the ribs, the distribution results of the material reinforcement trajectories were obtained, and then the reinforcement layout design of the thin-walled structure was guided. The adaptive growth method and variable density method are often used in the design of stiffeners. [PERSON] et al. [17] proposed an adaptive growth design method based on the research on the optimality of the branch system form in nature and its growth mechanism, taking the stiffener distribution design of the thin-plate structure as an example. [PERSON] et al.