# Topology Optimization with Explicit Components Considering Stress Constraints
## 3 Problem Formulation  
### Problem Statement and Mathematical Formulation  
The compliance minimization problem with global stress constraints and global volume constraints can be expressed as shown in Equation (8).  
\[\begin{cases}Find\mathbf{D}_{g}=((\mathbf{D}_{1})^{T},(\mathbf{D}_{2})^{T},\ldots,(\mathbf{D}_ {n})^{T})^{T},\\ min:\mathcal{C}=\int_{D}H\big{(}\Phi\big{(}\mathbf{x},\mathbf{D}_{g}\big{)}\big{)}f^{j} \cdot\mathbf{u}dV+\int_{\Gamma_{t}}t\cdot\mathbf{u}dS,\\ s.t.\\ \sum_{j=1}^{n}\int_{D}H\big{(}\[PERSON]\big{(}\mathbf{x},\mathbf{D}_{g}\big{)}\big{)}\hat{E} \varepsilon(\mathbf{u})\colon\varepsilon(\mathbf{v})dV\\ =\int_{D}H\big{(}\[PERSON]\big{(}\mathbf{x},\mathbf{D}_{g}\big{)}\big{)}f\cdot\mathbf{v}dV+\int _{\Gamma_{t}}\big{(}\[PERSON]\big{(}\mathbf{x},\mathbf{D}_{g}\big{)}\big{)}t\cdot vdS,\forall \mathbf{v}\in u_{\text{\emph{a}nd}},\\ \int_{D}H\big{(}\Phi\big{(}\mathbf{x};\mathbf{D}_{g}\big{)}\big{)}dV\leq\tilde{V},\\ g(\mathbf{u},\mathbf{D}_{g})\leq 0,\\ D\subset u_{D},\mathbf{u}=\overline{\mathbf{u}},\text{\emph{on}}\Gamma_{u}\end{cases} \tag{8}\]  
In the above equation, \(\mathbf{D}_{g}\) is the vector of the global design variables for all components (\(n\) components) in the design domain. \(f^{j}\) and \(t\) are the volume force and surface traction, respectively, and \(\mathbf{u}\) and \(\mathbf{v}\) are the displacement fields. \(\overline{\mathbf{u}}\) is the prescribed displacement on the Dirichlet boundary \(\Gamma_{\mathbf{u}}\). The Neumann boundary condition is represented as \(\Gamma_{\mathbf{t}}\). \(\varepsilon\) represents the second-order linear strain tensor. In this formula, \(\hat{\mathbb{E}}=\mathbb{E}/(1+\
u)[\mathbb{I}+\
u/(1-2\
u)\delta\otimes\delta]\), where \(E\) is the [PERSON]'s modulus, \(\
u\) is the [PERSON]'s ratio, and \(\mathbb{I}\) and \(\delta\) represent the  
Figure 6: Parametric description of the variability of the shape of the components.  
Figure 7: Component deformation diagrams with different parameters.  
fourth-order and second-order unit tensors of the identity, respectively. \(\Phi\) is the topology description function of all components mentioned in Section 2, \(\bar{V}\) is the prescribed global volume constraint, and \(g(\mathbf{u},\mathbf{D}_{g})\) is the prescribed global stress constraint, which is discussed in detail in the subsequent sections. The Heaviside function used in Equation (8) is defined as follows:  
\[H(\mathbf{\phi})=\begin{cases}1&\phi>\lambda\\ \frac{3(1-\xi)}{4}\left(\frac{\phi}{\lambda}-\frac{\phi^{3}}{3\lambda^{3}} \right)+\frac{1+\xi}{2}&-\lambda\leq\phi\leq\lambda\\ \xi&\text{else}\end{cases} \tag{9}\]  
In Equation (9), \(\lambda\) and \(\xi\) are parameters that control regularization and singularity, respectively. The finite-element analysis unit in this paper adopts the widely used four-node rectangular element and uses the esratz material model [30] to obtain the [PERSON]'s modulus of the element by interpolating the TDF values at the four nodes of the element as follows:  
\[E^{e}=\frac{E\left(\sum_{i=1}^{4}\left(H(\phi_{i}^{e})\right)^{q}\right)}{4}, \tag{10}\]  
Equation (10) establishes the relationship between the geometric model and the finite-element model, where \(E^{e}\) is the [PERSON]'s modulus of the element \(e\). \(E\) is the [PERSON]'s modulus of the solid material. \(\phi_{i}^{e},i=1,...,4\), represents the values of the TDF at the four nodes of element \(e\), and \(q\) is the penalty parameter, typically chosen as \(q\geq 1\)[30]. In the present study, the value of \(q\) is set to 2 in accordance with established research precedents [24; 25; 31]. This treatment greatly enhances computational efficiency, thereby reducing computational costs.