Combine Modal Reduction

A domain decomposition method allowing the combination of nonlinear sub-domains with linear modal sub-domains has been proposed ¹. With this technique, the displacement field in the linear sub-domains is projected on a local basis of reduction modes calculated on the detailed geometry and the kinematic continuity relations are written at the interface in order to recombine the physical kinematic quantities of reduced sub-domains locally. The method yields promising save of computing time in industrial applications. However, the use of modal projection is limited to linear sub-domains. In the case of overall rigid-body motion with small local vibrations, the geometrical nonlinearity of sub-domains must be taken into account. Therefore, the projection cannot be used directly even thought the global displacements may still be described by a small number of unknowns; for example six variables to express motion of local frame plus a set of modal coordinates in this frame. This approach is used in the case of implicit framework. ² In the case of direct integration with an explicit scheme an efficient approach is presented. ³ One of the main problems is to determine the stability conditions for the explicit integration scheme when the classical rotation parameters as Euler angles or spin vectors are used. A new set of parameters, based on the so-called frame-mass concept is introduced to describe the global rigid body motion. The position and the orientation of the local frame are given by four points where the distances between the points are kept constant during the motion. In this way, only the displacement type DOF is dealt and the equations of motion are derived to satisfy perfectly the stability conditions. This approach, which was integrated in Radioss V5, will be presented briefly here.

Linear Modal Reduction

A modal reduction basis is defined on one or more sub-domains of the decomposition. The definition of this basis is completely arbitrary. Any combination of eigen modes and static corrections can be used. All these modes are orthogonalized with respect to the finite element mass matrix in order for the projected mass matrix to be diagonal and suitable for an explicit solver.

The number of modal unknowns ${\alpha }_1$ chosen is much smaller than the original number of degrees of freedom of Sub-domain 1.

The structure of this system is strictly identical to that which existed before reduction. Therefore, use exactly the same resolution process and apply the multi-time-step algorithm.

The time step for a reduced sub-domain is deduced from the highest eigen frequency of the projected system in order to preserve the stability of the explicit time integration. This time step is often larger than that given by the Courant condition with the finite element model before reduction.

Model Reduction with Finite Overall Rotation

Since large rotations are highly nonlinear ⁴, the displacement field in a sub-domain undergoing finite rotations cannot be expressed as a linear combination of constant modes. However, the rigid contribution to the displacement field creates no strain. In the case of small strains and linear behavior, the local vibrating system can still be projected onto a basis of local reduction modes. Then, take into account the rotation matrix from the initial global coordinate system to the local coordinate system and its time derivatives. A classical parameterization of this rotation, for example, Euler angles, would introduce nonlinear terms involving velocities. Since these quantities, in the central difference scheme, are implicit, this would require internal iterations in order to solve the equilibrium problem, a situation you clearly want to reduce the computation time due to the reduction.

Classically, the displacement field of a rotating and vibrating sub-domain is decomposed into a finite rigid-body contribution and a small-amplitude vibratory contribution measured in a local frame. The large rigid motion is represented using the so-called four-mass approach. ³

Displacement Field Decomposition

Thus, the fields in Equation 6 have the following expression:(8)

Where, $\delta P=\left[\delta u_{\mathrm{A}}-\delta u_{\mathrm{O}};\delta u_B-\delta u_{\mathrm{O}};\delta u_{\mathrm{C}}-\delta u_{\mathrm{O}}\right]$

Again, the same assumption as above allows us to simplify this expression:(11)

Local Reduced Dynamic System

To introduce Equation 4 into this weak form of the equilibrium, you must express with the new parameterization the virtual works of both the internal and external forces and the virtual work, due to the rigid links among the points defining the local frame.

The internal forces can be calculated from the local part of the displacement, using Equation 5 and taking into account the rigid links, for example, the fact that displacement ${\text{u}}_E$ creates no strain.

Where, index $\mathrm{L}$ expresses that the coordinates and the spatial derivatives are taken in the local frame.

The integration by parts in the local frame introduces external surface forces ${\text{f}}_{ext}$ :(16)

Where, ${\text{D}}_{IJ}=\Vert \left({\text{x}}_I^0+{\text{u}}_I\right)-\left({\text{x}}_J^0+{\text{u}}_J\right)\Vert -\Vert {\text{x}}_I^0-{\text{x}}_J^0\Vert$ and ${\text{x}}_I^0$ are the initial coordinates of point $I$ and the rigid link between points $I$ and $J$ is given by: ${\text{D}}_{IJ}\left({\text{u}}_I,{\text{u}}_J\right)=0$ .

Weak Form of Equilibrium

Now, express the displacement field using Equation 5 and the local field projected on a Ritz basis:(20)

where ${\text{φ}}_X^i=X{\text{e}}_i$ , ${\text{φ}}_Y^i=Y{\text{e}}_i$ , ${\text{φ}}_Z^i=Z{\text{e}}_i$ , ${\text{φ}}_{1-X-Y-Z}^i=\left(1-X-Y-Z\right){\text{e}}_i$ , $\left({\text{e}}_1,{\text{e}}_2,{\text{e}}_3\right)$ is a basis of the global frame, ${\left\{{\text{φ}}_L^i\right\}}^{}$ is a basis of local Ritz vectors obtained, for example, by finite element discretization or by modal analysis, $\widehat{\text{U}}$ is the vector of the discrete unknowns:

$\widehat{\text{U}}=\left[\begin{array}{c}{}^{}\left[{\text{u}}_A^i\right]{}_{}\\ {}^{}\left[{\text{u}}_B^i\right]{}_{}\\ {}^{}\left[{\text{u}}_C^i\right]{}_{}\\ {}^{}\left[{\text{u}}_O^i\right]{}_{}\\ {}^{}\left[{\text{y}}^i\right]{}_{}\end{array}\right]=\left[\begin{array}{c}{\widehat{\text{U}}}_E\\ {\widehat{\text{U}}}_L\end{array}\right]$ , with ${\widehat{\text{U}}}_E=\left[\begin{array}{c}{}^{}\left[{\text{u}}_A^i\right]{}_{}\\ {}^{}\left[{\text{u}}_B^i\right]{}_{}\\ {}^{}\left[{\text{u}}_C^i\right]{}_{}\\ {}^{}\left[{\text{u}}_O^i\right]{}_{}\end{array}\right]$ and ${\widehat{\text{U}}}_L={\left[{\text{y}}^i\right]}^{}$ ,

${\text{Φ}}_P$ is the projection basis: ${\text{Φ}}_P={\left[\left\{{\text{φ}}_X^i\right\},\left\{{\text{φ}}_Y^i\right\},\left\{{\text{φ}}_Z^i\right\},\left\{{\text{φ}}_{1-X-Y-Z}^i\right\},\left\{{\text{Pφ}}_L^i\right\}\right]}^{}=\left[{\text{Φ}}_E,{\text{PΦ}}_L\right]$ .

Where, $\text{G}\left(\dot{\widehat{\text{U}}}\right)$ is the gyroscopic contribution to the acceleration, given by:

$\text{G}\left(\dot{\widehat{\text{U}}}\right)=2\dot{\text{P}}\left(\left[{\dot{\text{u}}}_A^i\right],\left[{\dot{\text{u}}}_B^i\right],\left[{\dot{\text{u}}}_C^i\right],\left[{\dot{\text{u}}}_O^i\right]\right){\text{Φ}}_L{\dot{\widehat{\text{U}}}}_L$

Where,

${\Phi }_L=\left\{{\varphi }_L^i\right\}$

${\text{K}}_L={\overline{\Psi }}_P{}^T\text{K}{\overline{\Phi }}_L$ , with $\text{K}$ being the sub-domain's local stiffness matrix and ${\overline{\Psi }}_P$ and ${\overline{\Phi }}_L$ deduced (as was ${\overline{\Phi }}_L$ ) from ${\overline{\Psi }}_P$ , ${\overline{\Phi }}_L$ and the mesh, ${\text{F}}_{ext}{}_P={\overline{\Phi }}_P{}^T{\text{F}}_{ext}$ , with ${\text{F}}_{ext}$ being the classical vector of the external forces assembled on the sub-domain.

Where, ${\text{M}}_{\text{E}}$ is the constant mass matrix corresponding only to the global displacement field given by $X{\dot{\text{u}}}_{\text{A}}+Y{\dot{\text{u}}}_{\text{B}}+Z{\dot{\text{u}}}_{\text{C}}+\left(1-X-Y-Z\right){\dot{\text{u}}}_{\text{O}}$ , ${\text{M}}_{\text{V}}$ is the constant mass matrix corresponding to the local vibration given by ${\text{u}}_{\text{L}}$ , ${\text{M}}_{\text{C}}$ is a coupling matrix, variable with overall rotation, arising from the interaction between the local vibratory acceleration field expressed in the global frame $\text{P}{\ddot{\text{u}}}_L$ and the overall virtual displacement field $X\delta u_{\text{A}}+Y\delta u_{\text{B}}+Z\delta u_{\text{C}}+\left(1-X-Y-Z\right)\delta u_{\text{O}}$ ; ${\text{M}}_{{}_{\text{C}}}^{\text{T}}$ naturally comes from the symmetric interaction between virtual local displacement field and the overall acceleration field.

The rigid body motion component of the displacement increment is computed in unconditionally stable way by the use of Lagrange Multiplier to impose the rigid links. The deforming part is generated by the local vibration modes retained in the reduction basis. Therefore, you can conclude that the stability condition is the same as that given by the local vibrating system. The critical time step is constant throughout the calculation and can be derived from the highest eigen frequency of the local reduced stiffness matrix with respect to the local reduced mass matrix.

Where, ${\Phi }_L={\left\{{\varphi }_L^i\right\}}^{}$ and $\text{f}=\frac{\omega }{2\pi }$ .