# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 5 Conclusions
\begin{tabular}{l l} \(k\) & Total number of constraint equations in the system \\ \(\mathbf{K}\) & Stiffness matrix of system \\ \(\mathbf{K}_{i}\) & Stiffness matrix of body \(i\) \\ \(\overline{\mathbf{K}}_{i}\) & Stiffness matrix of finite element equations \\ \(L\) & Lagrange function of system \\ \(\mathbf{\lambda}\) & Lagrangian multipliers of constraints \\ \(\mathbf{\lambda}^{(n)}\) & Value of \(\mathbf{\lambda}\) at discrete time \(t=nh\) \\ \(m\) & Total number of bodies in the system \\ \(m_{i,p}\) & Lumped mass at node \(p\) of body \(i\) \\ \(\mathbf{M}\) & Mass matrix of system \\ \(M_{i}\) & Number of boundary modes of body \(i\) \\ \(\mathbf{M}_{i}\) & Mass matrix of body \(i\) \\ \(\overline{\mathbf{M}}_{i}\) & Mass matrix of finite element equations of body \(i\) \\  & Submatrix of \(\mathbf{M}_{i}\) corresponding to rows DOFs \"*\" and column DOFs \"\#\", \\ \(\mathbf{M}_{i}^{*\#}\) & \({}^{\shortleftarrow}\)for flexible modes, \({}^{\shortleftarrow}\)for rigid translation, \({}^{\shortleftarrow}\)\" for rigid rotation, \\  & \({}^{\shortleftarrow}\)R\" for rigid translation and rotation \\ \(N_{i}\) & Number of preserve low-frequency modes of body \(i\) \\ \(n_{i}\) & Total number of flexible modes after rigid modes \(i\) \\ \(\mathbf{\Omega}_{i}^{2}\) & Diagonal matrix of modal circle frequencies of body \(i\) \\ \(\omega_{i,ij}\) & The \({}^{\shortleftarrow}\)h\({}^{\shortleftarrow}\) modal circle frequency of flexible body \(i\) \\ \(\mathbf{\omega}_{i}\) & Local angular velocity of the body-fixed frame \\ \(p\) & An FE node on the flexible body \\ \(\overline{\mathbf{\psi}}_{i,ij}^{\Lambda}\) & The \({}^{\shortleftarrow}\)h boundary mode of flexible body \(i\)on coordinates \(\mathbf{u}_{i}^{\Lambda}\) \\ \(\mathbf{q}_{i}\) & Aoda coordinates of the flexible body \(i\) (excluding \\  & zero-frequency modes and ensuring orthogonality) \\ \(\overline{\mathbf{q}}_{i}\) & Redundant modal coordinates of the flexible body \(i\) \\ \(\mathbf{r}_{i,p}\) & Global position of node \(p\) of flexible body \(i\) \\ \(s_{i,p}\) & Original position of node \(p\) of flexible body \(i\) in B\({}_{i}\) \\ \(t\) & Simulation time \\ \(\mathbf{\tau}\) & General force vector of external loads \\ \(T\) & Kinetic energy of system \\ \(\mathbf{u}_{i}\) & Finite element DOFs of flexible body \(i\) \\ \(\mathbf{u}_{i,p}^{\Lambda}\) & Translational displacement of node \(p\) due to flexible deformation of flexible body \(i\) \\ \(\mathbf{u}_{i}^{\Lambda}\) & Inner DOFs of flexible body \(i\) within the FE formulation \\ \(\mathbf{u}_{i}^{\Gamma}\) & Boundary DOFs of flexible body \(i\) within the FE formulation \\ \(V\) & Potential energy of system \\ \(\mathbf{v}_{i,p}\) & Global velocity of node \(p\) of flexible body \(i\) \\ \(\mathbf{x}_{i}=\{x_{i};y_{i};z_{i}\}\) & Position vector from global frame to body-fixed frame in global frame \\ \(\xi\) & General displacement of bodies \\ \(\xi\overline{\xi}\) & General velocity and acceleration of bodies \\ \(\xi^{(n,0)}\) & Initial value of \(\xi\) in Newton iteration for the \(n^{\text{th}}\) time step \\ \(\xi^{(-1)}\) & Value of \(\xi\) at discrete time \(t=-h\) \\ \(\xi_{i}\) & General displacement of body \(i\) \\ \(\xi^{(n)}\) & Value of \(\xi\) at discrete time \(t=nh\) \\ \({}^{\shortleftarrow}\),\" & Separator of components in a \"column\" vector \\ \(\square^{(0)}\) & Variable values at initial time \(t=0\) for \(\square\) \\ \(\square^{(n)}\) & Variable values at discrete time \(t=nh\) for \(\square\) \\ \(\square^{(n,0)}\) & Initial values of the \(n^{\text{th}}\) time step for \(\square\) in the Newton iteration \\ \(\square^{(n,0)}\) & Results of the \(j^{\text{th}}\) iteration in the \(n^{\text{th}}\) time step for \(\square\) \\ \(\square^{\Gamma}\) & Vector transpose or matrix transpose of \(\square\) \\ \(\square^{\Lambda}\) & Inner DOFs components in the Craig-Bampton method for \(\square\) \\ \(\square^{\Gamma}\) & Boundary DOFs components in the Craig-Bampton method for \(\square\) \\ \end{tabular}