# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 5 Conclusions  
In this paper, the CB method is utilized to model large, complex, flexible, multibody systems. System Jacobian matrices with favorable block symmetry and sparsity properties are formed by appropriately ordering the generalized coordinates of bodies and joints. A local-global two-layer parallel algorithm is constructed to directly solve the linearized systemic equations. The algorithm also circumvents the disruption of matrix topological sparsity in the traditional pivoting GE process of direct solution and achieves algorithm-level parallelism. The algorithm ensures numerical stability by employing the physical assumption of component modal synthesis methods and the specific format of the Craig-Bampton method. The numerical stability and high efficiency of the proposed block parallel algorithm are validated by comparing the results with those from commercial software Abaqus 2020 and with the pivoting GE algorithm, using the examples of aeroengine stator system vibration and crank-slider mechanism's low-speed rotation. The method detailed in this paper is unsuitable for multibody systems that contain high-speed rotating parts directly, as in such cases, the gyroscopic matrix may become the primary component of the Jacobi matrix, leading to instability in the presented scheme. A potential solution is to construct an appropriate preconditioning matrix [40] to ensure numerical stability and prevent the need for pivoting operations. This approach could be explored in future papers. Conceptualization, C.Y. and Z.X. ; methodology, C.Y. ; software, B.X., Y.W. and Y.X. ; validation P.Y. ; writing--original draft preparation, C.Y. ; writing--review and editing, P.Y., B.X. and Y.W. ; supervision, Z.X. All authors have read and agreed to the published version of the manuscript. This research received no funding. **Institutional Review Board Statement:** Not applicable. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method for Solving Equation (45)** & \
\begin{tabular}{c} **Total CPU Time** \\ **Consumption (ms)** \\ \end{tabular} &
\begin{tabular}{c} **Total Number of Solving System** \\ **Equation (45)** \\ \end{tabular} \\ \hline Serial pivoting GE & 4089 & 30,214 \\ Parallel block GE & 4013 & 31,546 \\ (presented, with 4 cores OpenMP) & \\ Computational efficiency improvement & 1.9\% & - \\ by block GE (\%) 1 & 1.9\% & - \\ \hline \hline \end{tabular}  
* Computational efficiency improvement by block GE (\%) = (block \(-\) pivoting)/pivoting \(\times\) 100\%.  
\end{table} 
Table 6: Computational efficiency comparison between the serial pivoting GE method and the presented parallel block GE method (using the low-speed crank-slider mechanism as a sample). **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available on request from the corresponding author because the research project involved in this article involves relevant confidentiality agreements, and the overall project has not been completed yet. **Conflicts of Interest:** The authors declare no conflicts of interest. **Nomenclature**