# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 4 Numerical Examples
### Sample of a Low-Speed Rotating Crank-Slider Mechanism  
Consider the elastic vibration response of a crank-slider mechanism at low speed (0.1 Hz), as depicted in Figure 8. The system consists of a crank, a linker, and a slider, with the elastic DOFs and interconnections detailed in Table 4. The FEM details of the flexible crank and flexible linker for modal analysis are listed in Table 5. We ignore friction in all joints. The physical quantity measured is the elastic displacement of point A in the \(y\) direction with respect to the body-fixed coordinate system Ox\(y\). Figure 8: Sample of a low-speed crank-slider mechanism: its components, constraint connections, and rotational driving. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Total CPU Time**} & \multicolumn{2}{c}{**Computational**} \\
**Method for Solving** & \multicolumn{2}{c}{**Consumption**} & \multicolumn{2}{c}{**Efficiency**} & \multicolumn{1}{c}{**Total Number of**} \\
**Equation (45)** & **Serial** & **Parallel with 4 Cores** & **Improvement by** & **Solving System** \\  & **(ms)** & **(ms)** & **Parallelization 2 (\%)** & **Equation (45)** \\ \hline Traditional Pivoting GE & 666 & 897 & \(-\)34.7\% & 2733 \\ Block GE (presented) & 632 & 405 & 35.9\% & 2733 \\ Computational efficiency & & & & \\ improvement by block GE 1 (\%) & 5.1\% & 54.8\% & 39.2\% & - \\ \hline \hline \end{tabular}  
* Computational efficiency improvement by block GE (\%) = (block \(-\) pivoting)/pivoting \(\times\) 100%; 2 Computational efficiency improvement by parallelization (\%) = (parallel \(-\) serial)/serial \(\times\) 100%.  
\end{table} 
Table 3: Computational efficiency comparison between the traditional pivoting GE and the presented parallel BGE method (using the aeroengine stator system as an example). The parallel block GE and the serial pivoting GE methods are executed for computational comparison. The elastic part of the vibration response \(y_{A}\), which is the vibration displacement of point A in the body-fixed coordinate system \(Oxy\), within one period is shown in Figure 9. The results from both methods are highly coincident, which verifies the numerical stability of the presented parallel block GE method for the FMB system with low-speed rotating parts. As indicated in Table 6, the overall computational efficiency increased by 1.9% in the low-speed crank-slider sample, where the total CPU time consumption decreased from 4089 ms to 4013 ms. The parallelization of the presented block GE method accelerates the FMB system with low-speed rotating parts, albeit not to the extent demonstrated in the example of the aeroengine stator system. This is due to the presence of five flexible components in the aeroengine example compared to only two in the current crank-slider sample. This phenomenon suggests that the parallelization efficiency of the presented \begin{table}
\begin{tabular}{c c c c} \hline
**Part Number** & **Part Name** & **Total Number of the CB** & **Connections with Other** \\
**(in Order of Assembly)** & **Part Name** & **Modes** & **Components** \\ \hline \⃝1 & Crank & 50 & Rotation to ground \\ \⃝2 & Linker & 50 & Spin to \⃝1, Spin to \⃝3 \\ \⃝3 & Slider & 0 & Translation to ground \\ \hline \end{tabular}
\end{table} 
Table 4: Elastic modal DOFs and constraint connections between components in the low-speed crank-slider mechanism. Figure 9: Numerical accuracy comparison of the elastic vibration displacement to verify numerical stability of the presented algorithm-level parallel method (using the low-speed crank-slider mechanism as a sample). method is related to the number of flexible components in the system. In other words, the parallelization is executed between sub-blocks of flexible components. It is also noteworthy that the number of solving the linearized systemic Equation (45) rose from 30,214 to 31,546, an increase of approximately 4.4%. This is due to the fact that the gyroscopic component of the Jacobian matrix \(\mathcal{C}\) is neglected in Equation (45) using our method.