# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 5 Conclusions
\begin{tabular}{l l}
**0** & Zero vector or zero matrix \\ \(a\) & Constraints scale factor for numerical conditioning \\ \(\mathbf{A}_{i}\) & Transformation matrix from body-fixed frame to global frame \\ \(\mathbf{a}_{i}=[\alpha_{i};\beta_{i};\gamma_{i}]\) & Rotation parameters from global frame to body-fixed frame \\ \(\mathbf{b}\) & Residual of body general force equations \\ \(\text{B}_{i}\) & Body-fixed coordinate frame (also the FFR) of flexible body \(i\) \\ \(\mathbf{c}\) & Complete constraints in system \\ \(\mathbf{C}\) & System damping matrix \\ \(\mathbf{C}_{i}\) & Damping matrix of body \(i\) \\ \(c_{\xi}\) & Partial derivative of \(\mathbf{c}\) corresponding to \(\mathbf{\xi}\), i.e., \(\partial\mathbf{c}/\partial\mathbf{\xi}\) \\ \(\mathbf{D}_{i}\) & Coefficient matrix before \(\mathbf{\dot{\alpha}}_{i}\) to obtain \(\mathbf{\omega}_{i}\) \\ \(\Delta\mathbf{\lambda}^{(n,j)}\) & Increment of \(\mathbf{\lambda}^{(n,j)}\) in the \(j^{\text{th}}\) iteration at \(n^{\text{th}}\) time step \\ \(\Delta\mathbf{\xi}^{(n,j)}\) & Increment of \(\mathbf{\xi}^{(n,j)}\) in the \(j^{\text{th}}\) iteration at \(n^{\text{th}}\) time step \\ \(\Delta\mathbf{\lambda}\) & Abbreviation of \(\Delta\mathbf{\lambda}^{(n,j)}\) when there is no ambiguity \\ \(\Delta\mathbf{\xi}\) & Abbreviation of \(\Delta\mathbf{\xi}^{(n,j)}\) when there is no ambiguity \\ \(\text{diag}(\cdots)\) & Block-diagonal matrix composed of submatrices \\ \(e_{\xi},e_{\lambda},e_{b},e_{c}\) & Error thresholds for displacement variables, constraint forces, \\  & body force equation residuals, and constraint equation residuals \\ \(\mathbf{E}_{n_{i}}\) & Unit matrix with dimension \(n_{i}\) \\ \(f_{\text{v}}\) & Velocity's algebraic function for time integration with BDFs \\ \(f_{\text{a}}\) & Acceleration's algebraic function for time integration with BDFs \\ \(\widehat{f}_{i}\) & External force applied to the boundary DOFs in FE equations \\ \(\mathbf{F}_{\text{iner}}\) & General inertial forces \\ \(\mathbf{F}_{\text{elas}}\) & General elastic forces \\ \(\widehat{\mathbf{\varphi}}_{i,j}^{\text{A}}\) & The \(j^{\text{th}}\) low-frequency mode of flexible body \(i\) on coordinates \(\mathbf{u}_{i}^{\text{A}}\) \\ \(\mathbf{\Phi}_{i}\) & Elastic modes of the CB method (orthogonal and non-zero frequency) \\ \(\widehat{\mathbf{\Phi}}_{i}\) & Preserved modes matrix of body \(i\) \\ \(\widehat{\mathbf{\Phi}}_{i}\) & Non-zero-frequency reorthogonalized modal matrix \\ \(\widehat{\mathbf{\Phi}}_{i}^{\text{oun}}\) & Boundary modes matrix of body \(i\) \\ \(\widehat{\mathbf{\Phi}}_{i}^{\text{other}}\) & Preserved vibration modes matrix of body \(i\) \\ \(\mathbf{\Phi}_{i,p}\) & Components of the modes matrix at node \(p\) \\ \(\mathbf{\Phi}_{i,p}^{\text{t}}\) & Translational components of \(\mathbf{\Phi}_{i,p}\) \\ \(\mathbf{\Phi}_{i,p}^{\text{t}}\) & Rotational components of \(\mathbf{\Phi}_{i,p}\) \\ G & Global coordinate frame \\ \(\gamma\) & Proportional modal damping coefficient \\ \(h\) & Time step size \\ \(\mathbf{H}_{i,p}\) & Moment of inertia lumped at node \(p\) of body \(i\) \\ \(i\) & Serial number for bodies in the system \\ \(I_{i,1}\sim I_{i,8}\) & Modal invariant submatrices in the modal neutral file \\ \(J\) & System Jacobian matrix \\ \(J_{i}\) & Jacobian matrix of body \(i\) \\  & Submatrix of \(J_{i}\) corresponding to rows DOFs \"\" and column DOFs \"#\", \\ \(J_{i}^{\text{\#}}\) & \"\(d^{\text{\#}}\) for flexible modes, \"\(d^{\text{\#}}\)\" for rigid translation, \"\(d^{\text{\#}}\)\" for rigid rotation, \"\(d^{\text{\#}}\)\" for rigid translation and rotation \\ \end{tabular}