## 4 Design Scheme of SIMP Method in the Abaqus-Matlab Topology Optimization Integration Framework
### Multi-Condition Optimization Design  
In multi-condition topology optimization, the optimization results of each condition often differ, leading to conflicts during multi-objective optimization. To reconcile the conflicts of goals and integrate elements such as dimensions, weights and orders of magnitude, researchers often employ the compromise programming method to handle such difficult problems. This method can simplify complex multi-objective problems, and its calculation formula is as follows:  
\[\min\left\{\left[\sum_{i=1}^{k}\lambda_{i}^{P}\left(f_{i}(x)-y_{i}^{0}\right) ^{P}\right]^{\frac{1}{P}}\right\} \tag{12}\]  
Figure 7: Overall design scheme.  
\[\Lambda=\left\{\lambda\in R^{k}\mid\lambda\geq 0,\sum_{i=1}^{k}\lambda_{i}=1\right\} \tag{13}\]  
In the formula, \(f_{i}(x)\) represents the i-th sub-objective, \(y_{i}^{0}\) is the optimal solution of the i-th objective function, and \(p\in[1,\infty)\). When the value of p varies, the formula is endowed with different practical meanings. This study focuses on improving the performance of subframe components and transforms the structural stiffness optimization problem into the solution of minimizing flexibility. Taking the minimum unit strain energy as the objective function, combined with the volume ratio and stress distribution constraints, and using the multi-objective optimization method, a multi-condition topology optimization design model was established:  
\[\begin{split}&\min C(\rho)=\left\{\sum_{k=1}^{m}\omega_{k}^{q} \bigg{[}\frac{\left(C_{k}(\rho)-C_{k}^{\min}\right)}{\left(C_{k}^{max}-C_{k}^{ \min}\right)}\bigg{]}^{q}\right\}^{\frac{1}{q}}\\ &\text{s.t.}\begin{cases}\frac{V(\rho)}{V_{0}}\leq\text{f}\\ \sigma_{k}\leq[\sigma]\end{cases}\\ \end{split} \tag{14}\]  
Here, \(\rho\) represents the density of the material; \(m\) represents the number of load conditions (\(m=6\)); \(\omega_{k}\) is the weight coefficient of the kth working condition; \(q\) is the penalty factor (\(q=2\)); \(C_{k}(\rho)\) is the total strain energy function of the unit under the load condition. \(C_{k}^{max}\) and \(C_{k}^{\min}\) are, respectively, the maximum and minimum values of the total strain energy function of the structure under the kth load condition; \(V(\rho)\) is the optimized effective volume; \(V_{0}\) is the initial volume; \(f\) is the volume fraction limit value; \(\sigma_{k}\) is the maximum stress value; \([\sigma]\) is the allowable stress of the material.  
In the multi-condition structural optimization design, determining the weight coefficients of each condition is the current research focus. At present, there is still a lack of systematic investigations on the frequency of vehicle usage under extreme conditions. Coupled with the diverse design requirements of different vehicle models, there are differences in the attention paid to each condition. Therefore, it is necessary to conduct a comprehensive assessment of the working condition weights by combining literature and design practice. Generally, the setting of weights needs to take into account the degree of mechanical influence and the actual probability of occurrence under working conditions. The specific settings are shown in Table 3.