Subspace dynamics

ProductsAbaqus/Standard

This method can be very effective for systems with mild nonlinearities that do not substantially change the mode shapes. The method cannot be used for contact problems. As with the other direct integration methods, it is more expensive in terms of computer time than the modal methods of purely linear dynamic analysis, but it is often significantly less expensive than the direct integration of all of the equations of motion of the model. This method is implemented in Abaqus/Standard using the explicit (central difference) operator to integrate the equations of motion projected onto the eigenmodes. In this procedure the explicit integration method is particularly effective because the eigenmodes are orthogonal with respect to the mass matrix so that the projected system always has a diagonal mass matrix and because the stability limit is determined by the highest eigenfrequency associated with the modes used in the analysis and not by the highest eigenfrequency of the structure.

Let us write the finite element approximation to the virtual work equations as

\begin{matrix} (1) & δ u^{N} (M^{N M} {\ddot{u}}^{M} + I^{N} - P^{N}) = 0, \end{matrix}

where

M^{N M} = \int_{V_{0}} ρ_{0} N^{N} \cdot N^{M} d V_{0}

is the consistent mass matrix,

I^{N} = \int_{V_{0}} β^{N} : σ d V_{0}

is the internal force vector, and

P^{N} = \int_{S} N^{N} \cdot t d S + \int_{V} N^{N} \cdot F d V

is the external force vector.

The nodal displacements, velocities, accelerations, and the variations in displacement are expressed in terms of eigenmodes:

\begin{matrix} (2) & \begin{matrix} u^{N} & = \sum_{α} Φ_{α}^{N} q_{α}, {\dot{u}}^{N} = \sum_{α} Φ_{α}^{N} {\dot{q}}_{α}, \\ {\ddot{u}}^{N} & = \sum_{α} Φ_{α}^{N} {\ddot{q}}_{α}, δ u^{N} = \sum_{α} Φ_{α}^{N} δ q_{α}, \end{matrix} \end{matrix}

where $q_{α}$ , ${\dot{q}}_{α}$ , ${\ddot{q}}_{α}$ , and $δ q_{α}$ represent generalized displacement, velocity, acceleration, and displacement variation, respectively. Substitution into the virtual work expression yields the formula for the acceleration associated with mode $α$ as

\begin{matrix} (3) & {\ddot{q}}_{α} = \frac{1}{m_{α}} Φ_{α}^{N} (P^{N} - I^{N}) (no sum on α), \end{matrix}

where $m_{α}$ is the generalized mass associated with mode $α$ :

m_{α} = Φ_{α}^{N} M^{N M} Φ_{α}^{M} .

From Equation 3 it is seen that the element residuals are projected onto the vector space spanned by the chosen number of eigenmodes. Having calculated the generalized acceleration for each mode, the generalized displacement and velocity are calculated with the central difference operator

\begin{matrix} (4) & \begin{matrix} q_{t + Δ t} & = q_{t} + {\ddot{q}}_{t} Δ t^{2}, \\ {\dot{q}}_{t + Δ t} & = Δ q_{t} / Δ t + {\ddot{q}}_{t} Δ t / 2. \end{matrix} \end{matrix}

The nodal values for all kinematic variables are obtained using the formulæ in Equation 2.

When initial velocities are applied, specified either explicitly by the user or implicitly by continuation of the previous dynamic step, the initial velocity vector ${\dot{u}}_{0}$ has to be projected into the eigenspace. This leads to an initial generalized velocity for the mode $α$ in the form

{\dot{q}}_{α}^{0} = \frac{1}{m_{α}} Φ_{α}^{N} M^{N M} {\dot{u}}_{0}^{M} .

As in standard explicit dynamic integration, the method is conditionally stable. The stability limit is determined by the highest eigenfrequency of the modes used in the analysis:

Δ t_{s t a b l e} = \frac{2}{ω_{\max}},

where $ω_{\max}$ is the highest circular frequency of the eigenmodes that are used as the basis of the solution. Since we will generally use a relatively small number of modes, this stability limit is usually much less restrictive than the stability limit for standard explicit integration.

Throughout the procedure a fixed time increment is used: the value is chosen as the smaller of the time increment specified by the user, or 80% of the stable time increment. The 80% factor is intended as a safety factor, so a small increase in the highest eigenfrequency caused by nonlinear effects will not cause the integration to become unstable.

The number of eigenvectors spanning the solution space for a subspace dynamic analysis can be specified by the user. The default number of vectors will be equal to the number of eigenvectors extracted in the eigenfrequency calculation.

It is possible to perform a subspace dynamic simulation for some time and then reextract the modes based on the current, stressed geometry (by using another eigenfrequency extraction step), followed by continuation of the analysis with the new modes as the subspace basis system. This can improve the accuracy of the method in certain cases.

Note that the method is noniterative; hence, there are no tolerances required for the procedure.