# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 4 Numerical Examples
### Example of an Aeroengine stator System  
We consider the vibration response of an aeroengine stator system under the excitation of a rotor's eccentric periodic force, as depicted in Figure 5. The system comprises five components, a connector, compressor, combustion chamber, turbine, and nozzle, in the order of their modeling (i.e., the sequence of parts in the linearized equations). Each component, treated as an individual entity, is analyzed using finite element software Abaqus 2020 to determine the fixed interface modes, with the lower-order orthogonal elastic modes retained. The part connection relationships, finite element mesh, and number of the CB modes are illustrated in Table 2. A sinusoidal force with a frequency of 20 Hz and an amplitude of 100,000 Newtons is applied at the same position of the fixed connection between part 3 and part 4 (i.e., at the red star point) to simulate the excited vibration of the stator system by the eccentric rotor through the bearing.  
The topology of the Jacobian matrix derived from the aeroengine stator system is depicted in Figure 6. We can observe very appealing block sparsity in the submatrices and the whole matrix, which exhibit self-similarity.  
Figure 5: Example of an aeroengine stator system: its structural composition, finite element meshing, constraint connections, and position of acting forces.  
For the purpose of comparing accuracy and verifying stability, the numerical results of the aeroengine stator system were computed using the present method, commercial software ADAMS 2020, and the traditional serial Gaussian algorithm. The results are compared in Figure 7, which shows highly consistent outcomes, thereby verifying the numerical stability of the proposed BGE-based algorithm-level parallel method.  
As shown in Table 3, when employing 4-core OpenMP parallelism to solve linear Equation (45), the computational efficiency of pivoting GE decreases by 34.7% compared  
\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 \hline \multirow{2}{*}{1} & Connector & 50 & (Ball hinge to ground) \(\times\) 2 \\  & Compressor & 32 & Fixed to \⃝1 \\  & Combustion chamber & 26 & Fixed to \⃝2 \\  & Turbine & 32 & Fixed to \⃝3 \\  & Nozzle & 26 & Ball hinge to ground \\ \hline \hline \end{tabular}
\end{table}
Table 2: Elastic modal DOFs and constraint connections between components in the aeroengine stator system example.  
Figure 6: Block sparsity and self-similarity of the Jacobian matrix derived from the aeroengine stator system (nz = none zeros).  
Figure 7: Numerical accuracy comparison of the elastic vibration displacement to verify numerical stability of the presented algorithm-level parallel method (using the aeroengine stator system as an example).  
with the serial case, primarily due to the consumption of parallel initialization. This phenomenon illustrates the inadaptability of the pivoting GE to parallelism. In contrast, the efficiency of the block GE parallelism increases by 35.9% compared to its serial counterpart.  
In the serial case comparison, the block GE efficiency increases by 5.1% compared to the pivoting GE, primarily due to saving of pivoting operations between rows and columns. In contrast, in the parallel case comparison, the block GE efficiency improves by a total of 54.8% over the pivoting GE parallelism. This demonstrates the high parallelism adaptability of the block GE. In summary, the computational efficiency of the presented parallel block GE method increases by 39.2% in total compared to the original serial pivoting GE method.