# Topology Optimization with Explicit Components Considering Stress Constraints
## 2 MMC Topology Optimization Framework and Geometrical Description  
### MMC Topology Optimization Method  
The MMC method is a novel topology optimization framework proposed by Professor [PERSON] in 2014. The basic idea of this topology optimization method is illustrated in Figure 4. This method uses components described by explicit geometric information as structural primitives. By updating the design variables that control the movement and deformation of the components, the component layout is gradually optimized to achieve the optimal structural configuration. Due to the parametric description of structural components used in the MMC method, the structural boundaries are clear and explicitly extractable, facilitating easy integration of the optimization results with CAD/CAE systems. Additionally, the design variables are a set of parameters that control the movement and deformation of the components, independent of the finite-element mesh. Compared with the SIMP method, the number of design variables is greatly reduced, improving solution efficiency.  
Unlike the SIMP method, which describes material distribution through changes in element density, the MMC method describes material distribution through the topology description function (TDF) of the component.  
\[\begin{cases}\Phi(x)>0,&ifx\in\Omega^{s},\\ \Phi(x)=0,&ifx\in\partial\Omega^{s},\\ \Phi(x)<0,&ifx\in D\setminus\Omega^{s}.\end{cases}, \tag{1}\]  
In Equation (1), \(x\) is a point in the design domain \(D\), and \(\Omega^{s}\) is the area occupied by the solid material component. There are \(n\) components in the design domain, and the global topology description function can be defined as \(\Phi(x)=\max(\phi_{1}(x),\dots,\phi_{\text{i}}(x),\dots\phi_{n}(x))\), \(i=1,\dots,n\), representing the component number. The topology description function of the i-th component is expressed as follows:  
\[\begin{cases}\phi_{i}(x)>0,&ifx\in\Omega_{i},\\ \phi_{i}(x)=0,&ifx\in\partial\Omega_{i},\\ \phi_{i}(x)<0,&ifx\in D\setminus\Omega_{i}.\end{cases} \tag{2}\]  
In Equation (2), \(\Omega_{i}\) is the area occupied by the i-th component. Since the pioneering work of Professor [PERSON] in 2014, many scholars have proposed various component topology description functions [22; 23; 24; 25; 26]. These functions have the ability to deform in different directions and dimensions, enhancing the expression of geometric boundaries and enabling the algorithm to search for the optimal structure more freely.  
Figure 4: Schematic diagram of topology optimization of the short beam example based on the MMC method. (**a**) Initial layout of the components. (**b**) Movement and deformation of the components during the early stages of the optimization process. (**c**) Movement and deformation of the components during the later stages of the optimization process. (**d**) Optimized layout of the components.  
However, if the end of the component is a straight line, it will lead to some inevitable problems. As shown in Figure 5, in the intersection area of the components, the boundary can become difficult to express. Additionally, the internal angle where the components connect should ideally be a smooth curve to reduce stress concentration, but this is difficult to achieve with components that have a straight end.