## 4 Verification Examples
### Topology Optimization of a 2D Cantilever Beam  
A 2D cantilever beam is first tested as a benchmark. The design domain measures 80 mm \(\times\) 40 mm, with the left end fully fixed and a vertical concentrated load of magnitude 1 applied at the bottom-right corner, as shown in Figure 5.  
The CAE model parameters and MATLAB program settings for the proposed ABAQUS-MATLAB cooperative framework are summarized in Table 1. After submitting the topology optimization task, the iterative process is illustrated in Figure 6a-d. The final configuration after 60 iterations is shown in Figure 6d.  
Figure 4: Python scripting functions.  
Figure 5: Original design domain of the cantilever beam.  
The optimized structure exhibits a typical truss-type load transfer path. Compared to the result from the original MATLAB-based program (without ABAQUS integration) [12] in Figure 7, the discrepancies in computational results between the MATLAB framework and the proposed ABAQUS-MATLAB framework stem from differences in element type precision. Consequently, the compliance values obtained by the proposed method are marginally higher than those derived from the MATLAB framework alone. However, the topological configurations of both methods are essentially consistent, with similar evolutionary histories. The final compliance error between the two results is 4%, while the proposed method achieves an iteration time of 18.7 s per cycle and a total optimization duration of 1124 s. The compliance and volume fraction evolution histories of both methods are compared in Figure 8, confirming the validity of the cooperative optimization framework.  
To further investigate computational efficiency, we compared the time costs of the MATLAB-only approach and the ABAQUS-MATLAB framework using the cantilever beam model illustrated in Figure 5, with varying element counts. The results are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mesh Type**} & **Element Size (mm)** & **Number of Elements** & **Elastic Modulus (MPa)** & **Poisson’s Ratio** & **Evolutionary Rate** & **Penalization Factor** & **Filter Radius (mm)** & **Volume Fraction** \\ \hline CPS4 & \(1\times 1\) & 3200 & 1 & 0.3 & 2\% & 3 & 3 & 40\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimization design parameters of 2D cantilever beam.  
Figure 6: Iterative optimization process of the cantilever beam. (**a**) Iteration 14; (**b**) Iteration 25; (**c**) Iteration 38; (**d**) Iteration 60.  
Figure 7: Optimization result from the original MATLAB-based program.  
For coarse meshes (e.g., 3200 elements), the MATLAB platform demonstrates significantly higher efficiency than the ABAQUS-MATLAB framework. This disparity arises from the substantial overhead of cross-platform data exchange in the collaborative framework. However, as the element count exceeds 50,000 (e.g., 51,200 elements), the ABAQUS-MATLAB framework outperforms MATLAB due to ABAQUS's highly efficient solvers--specifically, the standard implicit solver and explicit solver--which dominate the computational workload at scale. These solvers leverage advanced parallelization and optimized algorithms to drastically reduce finite element analysis (FEA) time for large-scale models. The computational overhead associated with cross-platform data exchange becomes negligible when handling models with large-scale element counts.