## 1 Introduction  
The combination of topology optimization (TO) and additive manufacturing (AM) is one of the most effective tools for structural weight reduction [1, 2]. In the design process, the TO is at the initial conceptual design stage, and the quality of its structural configuration directly influences the performance of the final structure. At present, TO-based methodologies include the ground structure approach [3], the homogenization method [4], the variable thickness method [5], the variable density method [6], the evolutionary structural optimization [7], bi-directional evolutionary structural optimization [8], the moving morphable component [9], the level-set method [10], floating projection topology optimization (FPTO) [11, 12, 13], smooth-edged material distribution optimization for topology (SEMDOT) [14], and the independent continuous mapping method [15]. These methods exhibit distinct advantages in mitigating numerical instability issues (e.g., the variable thickness method demonstrates suitability for shell structure optimization with limited design freedom; the BESO enables bidirectional material addition/removal optimization and adapts well to elastoplastic material design). However, their integration within commercial finite element analysis (FEA) software remains limited, and they have not been extensively implemented in mainstream engineering platforms. Among them, the variable density method, due to its well-established theoretical framework, has been extensively utilized in major FEA software such as HyperWorks, Ansys, Abaqus, SolidThinking Inspire, and Nastran [16]. However, engineers inevitably encounter numerical instability issues (such as intermediate density, mesh dependency, checkerboard patterns, and local extrema) when implementing TO via existing commercial software. These issues are primarily caused by numerical instability or singular solutions in the field functions of the FEA software, and the numerical instability of TO directly affects the accuracy, convergence, and manufacturing of the numerical results [17] and thereby leads to features such as isolated bodies, thin bars, and disconnected branches that negatively affect the performance of the structure. Specifically, isolated bodies refer to features that are not connected to any other parts of the structure; thin bars refer to features that are close to the minimum focal diameter of the laser spot; and disconnected branches refer to features that are connected to each other only at one end of the structure. Precise control of the structural configuration could help designers generate topological features with clear definitions and excellent performance. To achieve this goal, researchers have proposed a diverse range of methodologies such as the higher-order element method [17], the perimeter constraint method [18], the mesh filtering method [19], the optimality criteria method [20], and the method of moving asymptotes (MMA) [21] to control the structural configuration. Additionally, subsequent investigations have further analyzed the influence of various parameters on the structural configuration. [PERSON] et al. [22], based on the solid isotropic material with penalization (SIMP) method, employed the volume preservation projection method to calculate the genus or number of holes, combined with the filtered function calculated using persistence homology and the design space progressive restriction method, to analyze the minimum compliance optimization problem of a two-dimensional static cantilever structure. The obtained results revealed that with the growth of the filter radius, the number of holes gradually decreased, with a mesh size of 100 \(\times\) 160, a penalty factor of three, and a maximum number of iterations of 100. Since the filter radius was set as 1.1, many thin bars appeared in the structure. In addition, the influences of various mesh sizes and volume fractions on the structural configuration subjected to a five-hole constraint were also analyzed.