Topology Optimization for Minimum Compliance with Material Volume and Buckling Constraints under Design-Dependent Loads  
[PERSON]  
1 State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 1  
[PERSON]  
1 State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 1  
[PERSON]  
1 State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 1  
[PERSON]  
1 State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 122222222222222222222222222222222222222  
problem when solving the eigenvalue problem. They proposed that the geometric stiffness matrix of the low-stress element should be set to zero. [PERSON] [5] believed that this method would cause the algorithm to oscillate, and they proposed an improved SIMP interpolation model that interpolated the element stiffness matrix and the geometric stiffness matrix separately. [PERSON] [6] believed that the method required a reasonable selection of parameters to avoid artificial modes, so the same interpolation method was adopted for the element stiffness matrix and the geometric stiffness matrix. [PERSON] et al. [7] used the ICM method to study the topology optimization problem with consideration of stability constraints. Based on the eigenvalue translation technique, [PERSON] and [PERSON] [8; 9] proposed a new method for dealing with artificial modes. In combination with a two-stage algorithm, [PERSON] and [PERSON] studied buckling and material volume constrained compliance minimization problems. The coating topology optimization method was used for additively manufactured infill components by [PERSON] et al. [10]. The buckling performance of the optimized structure was significantly improved compared to that of the standard minimum compliance optimized structure, and it was pointed out that the improvement in the impact resistance of the porous structure was due to the increased number of structural components. In terms of stiffness and improving the specific bending stiffness, [PERSON] and [PERSON] [11] studied the problem of compliance minimization subjected to material volume constraints and buckling constraints and discussed the effects of different finite elements, sensitivities, and aggregation functions instead of a single buckling constraint on the optimization results. After that, [PERSON] and [PERSON] [12] proposed a method to approximate the calculation eigenvalue problem. Their method resulted in a remarkable reduction in the computational cost of eigenvalue calculation, and they applied the method applied to topology optimization problems subjected to buckling constraints of cantilever beams in 2D and 3D. Recently, [PERSON] et al. [13] published an educational article considering buckling constraints, in which the MATLAB code was open source. [PERSON] et al. [14] adopted the feature-driven optimization method and the finite cell method to solve buckling-constrained topology optimization problems. [PERSON] et al. [15] adopted the topology optimization of binary structures method to solve structural optimization problems under buckling constraints and design-dependent loads. This paper describes an investigation of the effects of buckling constraints on problems with design-dependent loads. For these problems, the structural loading changed in terms of magnitude, direction, and location as the design changed. In pressure loading, the most crucial task is to identify the material boundary location at which the pressure acts. At present, research on the loading surface search algorithm is generally divided into two categories. The first category involves achieving boundary identification based on the element or node density.