# Topology Optimization with Explicit Components Considering Stress Constraints
## 1 Introduction  
Since [PERSON] and [PERSON] proposed the topology optimization method based on homogenization theory [1] in 1988, the theoretical framework of continuum topology optimization has garnered significant attention from scholars. A series of effective topology optimization methods have subsequently emerged. Topology optimization has gradually become one of the most challenging subjects in structural mechanics. Currently, topology optimization is primarily categorized into two main approaches. The first approach is based on material distribution, known as implicit topology optimization methods, and includes the solid isotropic material with penalization (SIMP) method [2] and the evolutionary structural optimization (ESO) method [3]. These methods utilize finite elements as independent design variables, which often result in the emergence of gray elements and jagged boundary issues, as illustrated in Figure 1a. Additionally, because the geometric description differs significantly from that of CAD modeling systems, the optimization results usually necessitate complex model reconstruction. The second approach is based on boundary descriptions and includes the level-set method (LSM) [4] and the moving morphable components (MMC)/voids (MMV) method [5, 6]. These methodsdirectly capture structural boundary characteristics, thereby facilitating the acquisition of topological structures with smooth, well-defined boundaries, as depicted in Figure 1b. However, although the level-set method is based on boundary descriptions, it utilizes a node-based implicit level-set function to describe the geometry, which significantly differs from the explicit geometric representation used in CAD modeling systems. Therefore, it is also considered an implicit topology optimization method. In 2014, Professor [PERSON]'s team introduced the moving morphable components method, which employs a set of deformable components as building blocks for topology optimization. By optimizing the shape, orientation, and layout of these components, the optimal structural topology is identified. Since it utilizes the same explicit geometric representation as CAD modeling systems, the boundaries are explicitly extractable, and the optimization results can be directly integrated with CAD systems, pioneering the explicit topology optimization method. For further details on the development status of the MMC method, the authors recommend reading the review article in [7]. In recent years, many scholars have integrated implicit and explicit topology optimization methods to leverage the strengths of both, proposing a SIMP-MMC hybrid algorithm [8; 9], as illustrated in Figure 2.  
By first utilizing the fast search capability of the SIMP method, the primary force transfer path of the structure can be identified after a few initial iterative steps. Subsequently, a contour extraction algorithm is employed to map the SIMP optimization results to the initial layout of the MMC components, thus addressing the issue of initial layout dependency in the MMC method and enhancing its solution speed and convergence stability. However, incorporating stress constraints has consistently been a challenging issue in the field of topology optimization. Introducing stress constraints into the optimization model can lead to several significant challenges. Firstly, there is the phenomenon of stress singularities, which has been observed for many years in truss topology optimization designs involving stress constraints. [PERSON] and [PERSON] explained this phenomenon as due to the discontinuous nature of stress constraints, where optimization algorithms cannot completely eliminate some low-density areas, thus failing to achieve a genuinely optimal  
Figure 1: Comparison of two types of topology optimization methods applied to the short beam example. (**a**) Optimization results obtained using the implicit topology optimization method. (**b**) Optimization results obtained using the explicit topology optimization method.