## 4 Verification Examples
### Comparative Analysis of 2D Beam and 3D Wheel Hub Optimization  
A comparative analysis of the 2D cantilever beam and 3D wheel hub optimization cases is critical to evaluating the framework's adaptability to varying geometric and computational complexities. Such a comparison highlights how structural features and mesh density influence optimization efficiency, guiding practical applications in industrial design.  
As summarized in Table 5, the 3D wheel hub model (15,600 elements) required nearly triple the computation time (3049 s) of the 2D beam (1124 s) despite identical iteration counts (60 cycles). This disparity arises from two factors:  
1. An increase in element count: A multiplicative increase in the number of elements substantially prolongs finite element computation time in ABAQUS, while cross-platform data exchange time also rises proportionally.
2. Geometric complexity: The hub's radial symmetry and fillets necessitated finer meshing to resolve stress gradients, whereas the beam's rectilinear geometry allowed for uniform coarser meshes.  
These results underscore that geometric intricacy and element count jointly dictate computational demands. While the framework successfully handles both cases, its efficiency for complex 3D structures relies on ABAQUS's parallelized solvers to mitigate scalability challenges. This insight informs its targeted use in industrial applications requiring high-resolution topology optimization.  
A Topology Optimization Engineering Application for a Hydraulic Transmission Test Bench Adapter Support  
Hydraulic transmissions, as core powertrain components in construction machinery and heavy-duty vehicles, require performance validation and factory testing via test benches to ensure product quality. As shown in Figure 14, the test bench employs a drive motor to simulate engine input and a load motor to simulate operational loads. The mechanical system adopts a modular design, with adapter boxes and driveline configurations, enhancing adaptability to meet testing requirements for various hydraulic transmission models.  
Figure 14: High-power hydraulic transmission test bench.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **2D Cantilever Beam** & **3D Wheel Hub** \\ \hline Element count & 3200 & 15,600 \\ Iterations to convergence & 60 & 60 \\ Total computation time & 1124 s & 3049 s \\ Key geometric features & Rectilinear & Radial symmetry, fillets \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance comparison between 2D beam and 3D wheel hub optimization.  
The adapter support, located between the test specimen and adapter box, is a critical load-bearing component subjected to complex alternating loads such as torque and vibrational impacts. The initial design, as shown in Figure 15, utilizes gray cast iron HT300, CNC-machined with a volume of 0.1866 m\({}^{3}\) and a mass of 1362 kg, ensuring load transmission stability, vibration reduction, and structural rigidity.  
Finite element analysis (FEA) of the initial design reveals displacement and stress distributions. The maximum displacement is 0.808 mm, as shown in Figure 16a, and the maximum stress is 148.2 MPa, as shown in Figure 16b, both well below the material's allowable limits, indicating significant material redundancy. To reduce manufacturing costs and improve lightweight performance, topology optimization is applied to remove low-stress material while preserving structural integrity.