# Topology Optimization with Explicit Components Considering Stress Constraints
## 2 MMC Topology Optimization Framework and Geometrical Description
### A New Topology Description Function  
To address the above-mentioned problems, this paper proposes a new component topology description function. Compared with those proposed by [PERSON] and [PERSON] (2018), the straight line at the end of the component in this function can be bent and deformed, greatly enhancing the component's deformation ability. The mathematical model of this component topology description function can be expressed as follows:  
\[\phi_{i}(x,y)=\left(\frac{x^{\prime}}{g(y^{\prime})}\right)^{m}+\left(\frac{y ^{\prime}}{f(x^{\prime})}\right)^{m}-1, \tag{3}\]  
\[\left\{\begin{matrix}x\\ y\end{matrix}\right\}=\left[\begin{matrix}cos\theta_{i}&sin\theta_{i}\\ -sin\theta_{i}&cos\theta_{i}\end{matrix}\right]\left\{\begin{matrix}x-x_{0i} \\ y-y_{0i}\end{matrix}\right\}, \tag{4}\]  
In Equation (3), \(\phi_{i}(x,y)\) represents a superellipse level-set function, where \(m\) is a parameter controlling the shape of the superellipse. By adjusting the value of \(m\), various shapes, ranging from a circle to a rectangle, can be generated and described. As \(m\) approaches infinity, the superellipse approximates a rectangle ([PERSON], 2018). Building upon the research of various pioneers ([PERSON], 2018; [PERSON] and [PERSON], 2018), we select \(m=6\) in this paper. Equation (4) is used for coordinate transformation to convert the global coordinate to the local coordinate of the component. \((x_{0i},y_{0i})\) is the central coordinate of the i-th component, and \(\theta_{i}\) is the inclination angle of the i-th component. The shape of the component is jointly controlled by \(f(x^{\prime})\) and \(g(y^{\prime})\), as shown in Equations (5) and (6).  
\[f(x^{\prime})=\frac{t_{i1}+t_{i2}-2t_{i3}}{2l_{i1}^{2}}x^{\prime 2}+\frac{t_{i2}-t _{i1}}{2l_{i1}}x^{\prime}+t_{i3}, \tag{5}\]  
\[g(y^{\prime})=\begin{cases}\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}+ x^{\prime}_{0i},&if&l_{i1}<l_{i2},\\ l_{i1},&if&l_{i1}=l_{i2},\\ -\sqrt{r_{i}^{2}-(y^{\prime}-y^{\prime}_{0i})^{2}}+x^{\prime}_{0i},&if&l_{i1}> l_{i2}.\end{cases} \tag{6}\]  
In Equation (5), \(t_{i1},t_{i2}\), and \(t_{i3}\) are half of the thicknesses at three positions of the component, \(l_{i1}\) is half of the length on the central axis of the component, and \(l_{i2}\) is half of the length on both sides of the component, as shown in Figure 6. Here, we assume that the lengths on both sides of the component are equal. \((x^{\prime}_{0i},y^{\prime}_{0i})\) and \(r_{i}\) are the central coordinates and radius of the arc at the end of the i-th component, respectively. They can be calculated as follows:  
Figure 5: Schematic diagram of component connection at the end of a straight line.  
\[\begin{cases}x_{0i}^{r}=\frac{l_{0i}^{2}}{2(l_{2}-l_{1i})},\\ y_{0i}^{r}=\frac{l_{0i}^{2}+l_{1i}^{2}}{2},\end{cases}r_{i}^{2}=(x_{0i}^{r}+l_{1i })^{2}+(y_{0i}^{r}+t_{1i})^{2}. \tag{7}\]  
In this way, each component has eight variables \((x_{0},y_{0},l_{1},l_{2},t_{1},t_{2},t_{3},\theta)\), and by optimizing the values of these variables, the movement and deformation of the component can be controlled. As shown in Figure 7, different variable values result in various component shapes (the variables \([l_{1},l_{2},t_{1},t_{2},t_{3}]\) controlling the length and thickness of the component take values of (a) [0.09, 0.05, 0.07, 0.11, 0.09], (b) [0.09, 0.05, 0.09, 0.09, 0.11], (c) [0.09, 0.05, 0.11, 0.07, 0.09], (d) [0.05, 0.09, 0.09, 0.05], and (e) [0.09, 0.09, 0.09, 0.09, 0.09]). It can be observed that this TDF effectively enhances the deformation ability of the component.