# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 3 Parallel Direct Solution Scheme Based on Block Gaussian Elimination
### Feasibility Study of BGE on Sub-Block Jacobian Matrix  
Firstly, we demonstrate that the submatrix \(\mathbf{J}_{i}\) is expected to be symmetrically positive definite, and the block Gaussian elimination (BGE) method can be applied to it. We know that \(\mathbf{M}_{i}\) is symmetrically positive definite; since the component \(\frac{\partial(\mathbf{M}_{i}^{(j)}\bar{\xi}_{i}^{(j)})}{\partial\xi_{i}^{\text{f}}}\) in \(\mathbf{K}_{i}\) is much smaller than others, \(\mathbf{K}_{i}\) can be considered symmetrically semi-definite. Consequently, \(\mathbf{J}_{i}=\frac{\mathbf{M}_{i}}{\hbar^{2}}+\mathbf{K}_{i}\) is symmetrically positive definite. With the symmetrically positive definite nature of \(\mathbf{J}_{i}\), the BGE method can be applied, and numerical stability can be maintained without selecting principal elements, according to the stability theory of BGE proved in references [29; 34; 35].  
After the variable reordering in Section 3.1, the BGE process can preserve the sparsity within submatrices \(\mathbf{J}_{i}\) and the entire matrix \(\mathbf{J}\), thereby achieving linear computational complexities. Consider \(\mathbf{J}_{i}\) in Figure 3a, for example. We apply the BGE process on the following linear equations with Jacobian matrix \(\mathbf{J}_{i}\):  
\[\begin{bmatrix}\mathbf{J}_{i}^{\text{ff}}&\mathbf{J}_{i}^{\text{ff}}\\ \mathbf{J}_{i}^{\text{ff}}&\mathbf{J}_{i}^{\text{RR}}\end{bmatrix}\begin{bmatrix} \Delta\mathbf{\xi}_{i}^{\text{ff}}\\ \Delta\mathbf{\xi}_{i}^{\text{R}}\end{bmatrix}=-\begin{bmatrix}\mathbf{b}_{i}^{\text{ f}}\\ \mathbf{b}_{i}^{\text{R}}\end{bmatrix}, \tag{48}\]  
wherein the superscripts \({}^{\text{f}}\) and \({}^{\text{R}}\) denote the components of flexible modes and rigid body motion, respectively. Solving Equation (48) is equivalent to solving the following [PERSON]'s complement [16] equations within the rigid body motion DOFs:  
\[\begin{bmatrix}\mathbf{J}_{i}^{\text{RR}}-\mathbf{J}_{i}^{\text{R}}{(\mathbf{J}_{i}^{ \text{ff}})}^{-1}\mathbf{J}_{i}^{\text{RR}}\end{bmatrix}\Delta\mathbf{\xi}_{i}^{\text {RR}}=-\begin{bmatrix}\mathbf{b}_{i}^{\text{R}}-\mathbf{J}_{i}^{\text{ff}}{(\mathbf{J}_{i }^{\text{ff}})}^{-1}\mathbf{b}_{i}^{\text{f}}\end{bmatrix}, \tag{49}\]  
Because \(\mathbf{J}_{i}^{\text{ff}}\) is diagonal, its reverse \({(\mathbf{J}_{i}^{\text{ff}})}^{-1}\) consumes only \(n_{i}\) times of divisions. Since \(\mathbf{J}_{i}^{\text{ff}}\) is a \(6\times n_{i}\) submatrix and \(\mathbf{J}_{i}^{\text{RR}}\) is a \(n_{i}\times 6\) submatrix, \(\mathbf{J}_{i}^{\text{ff}}{(\mathbf{J}_{i}^{\text{ff}})}^{-1}\) consumes \(6n_{i}\) times of operations. Then, \(\mathbf{J}_{i}^{\text{ff}}\Big{[}{(\mathbf{J}_{i}^{\text{ff}})}^{-1}\mathbf{J}_{i}^{\text{ RR}}\Big{]}\) consumes another \(6\times 6\times 2n_{i}=72n_{i}\) times of operations, and \(\mathbf{J}_{i}^{\text{ff}}\Big{[}{(\mathbf{J}_{i}^{\text{ff}})}^{-1}\mathbf{J}_{i}^{\text{ RR}}\Big{]}\) consumes another \(6\times 2n_{i}=12n_{i}\) times of operations. At last, the subtraction \(\Big{[}\mathbf{J}_{i}^{\text{RR}}-\mathbf{J}_{i}^{\text{R}}{(\mathbf{J}_{i}^{\text{ff}})} ^{-1}\mathbf{J}_{i}^{\text{RR}}\Big{]}\) and \(\Big{[}\mathbf{b}_{i}^{\text{R}}-\mathbf{J}_{i}^{\text{ff}}{(\mathbf{J}_{i}^{\text{ff}})} ^{-1}\mathbf{b}_{i}^{\text{f}}\Big{]}\) consumes another \(36+6=42\) times of operations. So, to form [PERSON]'s complement Equation (49), the total computational complexity is  
\[6n_{i}+72n_{i}+12n_{i}+42=90n_{i}+42=O(n_{i}). \tag{50}\]The computational complexity of solving the \(6\times 6\) Equation (49) is merely \(O(1)\). Consequently, the computational complexity of BGE on \(\mathbf{J}_{i}\) is linear to \(n_{i}\). Similar applications of the [PERSON]'s complement have been reported in rigid multibody system simulations involving contact problems [20; 36].