II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

The stable time increment is determined from Equation (3.5.2) as the minimum over all elements.
The estimate of the critical time increment given in the preceding equation is for the case where there is no
damping present in the system. If we define & as the fraction of critical damping in the highest element mode, the
stability criterion of Equation (3.5.2) becomes
AG.(- ) . (3.5.3)
Conventional estimates of the critical time increment size have been based on the transit time of the dilatational
wave over the shortest dimension of an element or zone. For the undamped case, this gives
At= ~/C (3.5.4)
where c is the dilatational wave speed and t is the shortest element dimension.
There are two fundamental and important differences between the time increment limits given by Equations
(3.5.2) and (3.5.4). First, our time increment limit is dependent on a characteristic element dimension, which is
based on the finite element gradient operator and does not require an ad hoc guess of this dimension. This
characteristic element dimension, t, is defined by inspection of Equation (3.5.2) as
(3.5.5)
Second, the sound speed used in the estimate is based on the current response of the material and not on the
original elastic sound speed. For materials that experience a reduction in stiffness due to plastic flow, this can result
in significant increases in the critical time increment.
It should be noted that the stability analysis performed at each time step predicts the critical time increment for
the next step. Our assumption is that the conservativeness of this estimate compensates for any reduction in the
stable time increment over a single time step.
3.6 Hourglass Control Algorithm
The mean stress-strain formulation of the uniform strain element considers only a fully linear velocity field. The
remaining portion of the nodal velocity field is the so-called hourglass field. Excitation of these modes may lead to
severe, unresisted mesh distortion. The hourglass control algorithm described here is taken directly from Ftanagan
and Belytschko (1981). The method isolates the hourglass modes so that they may be treated independently of the
rigid body and uniform strain modes.
A fully linear velocity field for the quadrilateral can be described by
li~in=iii+iii,j(Xj ‘~j) . (3.6.1)
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