## 4 Verification Examples
### Lightweight Design of a 3D Wheel Hub  
To further validate the efficiency improvement of the proposed cooperative optimization method for the BESO algorithm, a comparative study is conducted between the proposed framework and the BESO method implemented solely via ABAQUS scripting [21]. A wheel hub model is selected for optimization. The hub dimensions are shown in Figure 9a, with a thickness of 20 mm. The inner and outer annular regions (white) are designated as non-design domains, while the intermediate annular region (blue) serves as the design domain. Radial concentrated loads of 1 N are applied at four equally spaced points on the outer ring, and the inner ring is fully fixed. The meshed model is illustrated in Figure 9b.  
The CAE model parameters and MATLAB program settings for the ABAQUS-MATLAB cooperative framework are detailed in Table 3. After submitting the topology optimization task, the iterative process is shown in Figure 10a-d, with the final configuration after 60 iterations (Figure 10d) exhibiting radially distributed stiffening ribs and symmetric void structures. The optimization result from the ABAQUS-only BESO implementation is presented in Figure 11. The compliance and volume fraction evolution histories of both methods are compared in Figure 12.  
\begin{table}
\begin{tabular}{c c c} \hline
**Number of Elements** & **MATLAB Platform** & **ABAQUS–MATLAB Framework** \\ \hline
3200 & 16 s & 812 s \\
12,800 & 738 s & 1851 s \\
51,200 & 6730 s & 3049 s \\ \hline \end{tabular}
\end{table}
Table 2: Computation time comparison between MATLAB and ABAQUS–MATLAB frameworks.  
Figure 8: Compliance and volume fraction evolution histories of the two methods.  
Figure 11: Optimization results from the ABAQUS-only platform.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mesh Type**} & **Design** & **Design** & **Elastic** & **Moulus** & **Poisson’s** & **Evolutionary** & **Penalization** & **Filter Radius** & **Volume Fraction** \\  & **Element** & **Element** & **Element** & **(MPa)** & **Ratio** & **Rate** & **Factor** & **(mm)** & **Fraction** \\  & **Size** (mm) & **Count** & & & & & & & \\ \hline C3D8 & 4 & 15,600 & 1 & 0.3 & 4\% & 3 & 10 & 20\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Optimization design parameters of 3D wheel hub.  
Figure 9: A 3D wheel hub model. (**a**) 3D hub model; (**b**) hub meshing.  
A comparison between the cooperative framework result (Figure 10d) and the ABAQUS-only result (Figure 11) reveals similar topological configurations. Table 4 quantifies the computational efficiency of both platforms, with a more intuitive comparison provided in Figure 13. The ABAQUS-only platform requires 1.65 times more total computation time than the cooperative framework, with grid filtering preparation consuming the majority of time (812 s). In contrast, the cooperative framework reduces grid filtering time to 25.2 s. This disparity arises because Python-based matrix operations in the ABAQUS-only platform iteratively traverse matrix elements, whereas the MATLAB-integrated framework leverages MATLAB's optimized matrix computation capabilities. Although data exchange between ABAQUS and MATLAB incurs minor overhead, this time is negligible compared to the overall efficiency gains. Notably, the computational gap widens with increasing mesh density.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Metric** & **ABAQUS Platform** & **ABAQUS–MATLAB Platform** \\ \hline Total iteration time (s) & 3702 & 2231 \\ Number of iterations & 51 & 53 \\ Single iteration time (s) & 72.6 & 42.1 \\ Grid filtering time (s) & 812 & 25.2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Computational time comparison between the two platforms.  
Figure 12: Compliance and volume fraction evolution histories of the two platforms.  
Figure 13: Computational time comparison between the two platforms.