# Topology Optimization with Explicit Components Considering Stress Constraints
## 6 Conclusions  
This paper proposes a two-dimensional topology optimization scheme that considers global stress constraints within the framework of the MMC method. A novel topology description function for the component is proposed, enhancing its deformation ability. Compared to a component with a straight-line end, this function provides a better and more reasonable representation of the geometric boundary at the intersection of the components. The global stress constraint is managed using the lower-bound KS aggregation function, and the proposed method is validated through two classical numerical examples. In the two classical numerical examples, while maintaining structural stiffness, the increase in the compliance value does not exceed 8%, and the maximum von Mises stress is reduced by about 50%, successfully addressing the stress concentration issue. In addition, the overall stress distribution is more uniform. However, the authors observed some unsmooth, sawtooth-shaped boundary phenomena in the fillet expression, which may require introducing functions within the framework to smooth the boundary. As described in the literature [8], the minimum size of the structure is precisely controlled by setting the minimum spacing between the end features of the components to ensure their complete connection. Furthermore, the use of non-uniform rational B-spline (NURBS) curves or other advanced curve-fitting techniques to describe component boundaries, along with improvements to existing topology description functions to further enhance the geometrical descriptive capability of the components, should also be considered. Additionally, during the debugging process, the authors found that the MMC method is highly dependent on the initial values of the component variables, the initial layout, and certain parameters in the MMA algorithm. Some empirical changes may worsen the convergence of the algorithm. Combining parameter selection with existing intelligent algorithms, such as deep learning, might achieve better results. Similar ideas can be found in the literature [38]. The current research aims to preliminarily address stress constraints within the explicit topology optimization framework, demonstrating the substantial potential of explicit topology optimization methods in tackling stress-related challenges through two classical examples. Future research should focus on effectively mitigating stress concentration within the MMC framework to achieve smoother boundary representations and on developing a rational parameter selection mechanism to enhance optimization efficiency, both of which remain challenging and significant topics. From the perspective of the authors, addressing the challenges identified in existing studies by introducing advanced boundary description methods, optimization algorithms, and parameter-tuning techniques could be highly promising. Future research will further explore these aspects  
Figure 15: Some key iterative steps in the optimization process of the T-shaped beam.  
to enhance practical applicability in engineering, thereby advancing the field of structural topology optimization.  
Conceptualization, Y.M.; methodology, Y.M.; software, Y.M.; validation, Y.W. and K.Y.; formal analysis, Z.L.; investigation, Y.M.; resources, Z.L.; data curation, Y.M. and K.Y.; writing--original draft preparation, Y.M.; writing--review and editing, Y.M. and K.Y.; visualization, Y.M.; supervision, Z.L. and Y.W.; project administration, Y.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data that support the findings of this study are available from the authors upon reasonable request.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
SIIMP Solid isotropic material with penalization  
LSM Level-set method  
ESO Evolutionary structural optimization  
MMC Moving morphable components  
MMA Method of moving asymptotes  
TDF Topology description function