## 1 Introduction  
Structural topology optimization, an engineering technique that seeks optimal material distribution through mathematical methods, aims to maximize structural performance with minimal material consumption. Its essence lies in globally optimizing material layouts to establish efficient load transfer paths rather than performing local dimensional adjustments. Since [PERSON] and [PERSON] pioneered the homogenization theory [1], several representative methodologies have emerged: the Solid Isotropic Material with Penalization (SIMP) method achieves material distribution through continuous density variables [2, 3, 4]; the Bi-directional Evolutionary Structural Optimization (BESO) method eliminates intermediate densities using discrete 0-1 variable [5]; the Level Set Method (LSM) describes structural boundaries via implicit surface evolution [6,7]; and the Moving Morphable Component (MMC) method employs geometric parameterization for explicit topology representation [8,9]. Originating from [PERSON] and [PERSON]'s Evolutionary Structural Optimization (ESO), BESO overcomes the unidirectional limitations of traditional ESO through bidirectional material addition/removal strategies. It features distinct topological boundaries and rigorous mathematical convergence [10; 11; 12], having been successfully applied to compliance minimization, frequency-constrained optimization, and multi-scale design, thereby emerging as a mainstream approach for complex structural optimization. Despite rapid theoretical advancements in topology optimization, significant challenges persist in its practical implementation. The engineering practice of topology optimization demonstrates trends toward multi-objective integration and dynamic coupling. For static structures, [PERSON] et al. [2] systematically reviewed synergistic design pathways for lightweight satellite structures and additive manufacturing, emphasizing the potential of multi-material optimization in aerospace applications. [PERSON] et al. [13] proposed a hybrid integer programming framework integrating multi-cut optimization and adaptive trust regions, reducing finite element iterations by approximately tenfold and providing efficient solutions for large-scale variable optimization. In dynamic applications, [PERSON] et al. [14] validated the generalizability of the Bi-directional Evolutionary Structural Optimization (BESO) algorithm in 2D flexible mechanisms through explicit finite element analysis, while [PERSON] et al. [15] developed a nonlinear structural power spectral density optimization method based on Moving Isocontour Surfaces (MIST), effectively suppressing dynamic response peaks. [PERSON] et al. [4] integrated wing aerodynamics with structural optimization, achieving over 50% weight reduction through concurrent size and topology optimization. Simultaneously, the field increasingly prioritizes uncertainty quantification in practical engineering. Traditional deterministic methods often neglect uncertainties in material properties, loading conditions, and geometric imperfections, leading to suboptimal designs or reduced robustness. To address these limitations, multidimensional innovations have emerged: [PERSON] et al. [16] proposed an Extended Finite Element Method (XFEM)-based framework for continuum structures under geometric uncertainty. The team [17] subsequently introduced a robust topology optimization method using Moving Morphable Components (MMC), balancing robustness, boundary quality, and computational efficiency via adaptive sparse grid techniques. [PERSON] et al. [18] significantly enhanced structural robustness under uncertainty through polynomial chaos expansion and sparse grid quadrature for statistical moment computation. [PERSON] et al. [19] established a reliability-based framework incorporating load position uncertainty, integrating geometric nonlinear analysis with normal distributions to validate probabilistic constraint regulation on optimized configurations. However, challenges persist in multi-scale optimization efficiency [20].