# Topology Optimization with Explicit Components Considering Stress Constraints
## 1 Introduction
Figure 2: Schematic diagram illustrating the SIMP-MMC hybrid algorithm. (**a**) The initial force path, derived from the SIMP method, is extracted and mapped onto the component layout. (**b**) The final optimized result obtained using the MMC method [8]. topology [10]. Several effective methods have been proposed to address this issue, including \(\epsilon\)-relaxation [11] and the smooth envelope function (SEF) [12]. Secondly, the local nature of stress constraints poses a considerable computational burden. The typical practice is to use aggregation functions to consolidate local stress constraints into a global constraint, significantly reducing the number of constraints. The constraint functions commonly applied in the literature include the Kreisselmeier-Steinhauser (KS) function [13] and the P-norm [14]. Additionally, the optimization method based on the SIMP model faces the challenge of defining stress constraints for intermediate-density elements. While stress constraints can be clearly defined for elements with relative densities of 0 or 1, defining stress constraints for intermediate-density material elements remains a topic of discussion. Most studies address this by establishing a stress interpolation function, characterizing it as the relationship between element stress and design variables. However, the form of the stress interpolation function directly affects the effectiveness of stress constraints, and applying such functions may make the optimized design more sensitive to initial designs and parameter selection, potentially increasing the uncertainty of optimization results. To address these issues, researchers have extensively studied the L-shaped beam as a classic test case, as illustrated in Figure 3.  
[PERSON] et al. proposed a stress correction scheme based on the Stability Transformation Method (STM) and indicated that the KS stress function based on the STM exhibits a faster convergence rate in topology design [15], as demonstrated in Design A in the figure above. [PERSON] et al. introduced a method to effectively address stress constraint issues with a large number of constraints by modifying the penalty term and the objective function term of the augmented Lagrangian function [16], as shown in Design B. In Design C, [PERSON]  
Figure 3: The designs of the L-shape beam in recent research. (**a**) Optimized result obtained by [PERSON] et al. using the implicit topology optimization method in 2018 [15]. (**b**) Optimized result obtained by [PERSON] et al. using the implicit topology optimization method in 2020 [16]. (**c**) Optimized result obtained by [PERSON] et al. using the implicit topology optimization method in 2021 [17]. (**d**) Optimized result obtained by [PERSON] et al. using the implicit topology optimization method in 2021 [18]. (**e**) Optimized result obtained by [PERSON] et al. using the explicit topology optimization method in 2017 [19]. (**f**) Optimized result obtained by [PERSON] et al. using the explicit topology optimization method in 2018 [20]. (**g**) Optimized result obtained by [PERSON] et al. using the explicit topology optimization method in 2021 [21]. [PERSON] and colleagues adopted a similar concept and proposed an auxiliary stress variable linked to actual variables through equality constraints. The augmented Lagrangian format was utilized, combining the equality constraints into the objective function as linear and quadratic terms. The auxiliary stress variable and the density variable are treated as two sets of optimization variables, referred to as design and stress-alternating optimization [17]. [PERSON] et al. compared local and global stress constraint strategies in topology optimization.