# A Level-Set-Based Density Method for Buckling Optimization of Structure with Curved Grid Stiffeners
## 1 Introduction  
Stiffeners can effectively enhance structures in bending stiffness, strength. Therefore, they have been widely utilized in aerospace, automotive, architectural, and other fields [1; 2; 3]. Curved grid stiffeners typically consist of many curved and crossed ribs, and they can adjust the load paths according to the different requirements of different local regions of the structure. Therefore, they have greater potential in reducing structural weight and improving structural performance [4; 5; 6; 7; 8; 9; 10], and there has been growing interests in the layout optimization of curved grid stiffness [6; 8; 11; 12; 13; 14; 15]. The stability of a structure is an important indicator of its performance. When subjected to axial compression and the load reaches a certain value, structures often experience buckling failure [16]. Therefore, the buckling performance of structures becomes a key factor in design and optimization. In related studies, [PERSON] et al. [17] employed generalized beam theory to analyze buckling in thin-walled structures. [PERSON] et al. [18] presented a topology optimization of structures under plane stress condition for linear buckling with 250 lines of code. [PERSON] and others [19] presented an innovative linear interpolation approach to improve the buckling resistance of structures in topology optimization. In related experimental research of buckling response, [PERSON] [20] performed both linear and nonlinear buckling analyses, as well as experimental tests, on a thin-walled composite plate with a central cutout under axial compression. In latest developments [21; 22], asymmetric systems and their mechanical couplings have been utilized to create elements that can function as elastic components, and comprehensive experimental and numerical studies have been conducted using multiple detection techniques and interdisciplinary approaches. For stiffened structures, the introduction of stiffeners significantly improves the buckling resistance. In addition, optimizing the layout of stiffeners can further amplify this advantage [4; 7; 8; 23]. The design optimization of stiffened structures considering buckling characteristic has become an important research topic. [PERSON] et al. [4] optimized stiffened plates primarily governed by buckling phenomena and demonstrated that, compared to straight stiffeners, curved stiffeners provide enhanced design space, resulting in better design outcomes. A buckling analysis of composite plates was proposed [8],indicating that arranging curved stiffeners for composite plates can improve buckling load compared to straight stiffeners. [PERSON] and co-workers [24] explored the static, vibration, and buckling behaviors of stiffened plates under in-plane compressive and tensile stresses. [PERSON] et al. [25] and [PERSON] et al. [26] utilized the smeared stiffener method (SSM) and the Rayleigh-Ritz method to forecast and determine the buckling load of hierarchical stiffened plates. [PERSON] and [PERSON] [5] significantly improved the buckling performance for elliptical lattice cylinders by adjusting the distribution of circumferential stiffeners. An effective optimization strategy for cylindrical structure with reinforced cutouts was proposed by [PERSON] et al. [7], employing curved stiffeners to enhance the load distribution and local stiffness of cylindrical stiffened shells. [PERSON] et al. [27] proposed an approach for optimizing the arrangement of stiffeners in slender structure, considering the maximum critical buckling load under geometric uncertainties that vary spatially. [PERSON] et al. [13; 28] calculated the global and local buckling loads of periodically grid-stiffened composite plates and shells and optimized the layout of curved stiffeners. A data-driven optimization approach was presented to maximize the critical BLF [29]. These research works on the design optimization considering buckling of stiffened structures further promote the progress and development of structural optimization, especially in the field of structures with curved stiffeners.