Stability Time Step

The characteristic length,

L

, for computing the critical time step, referring back to Figure 1, is defined by:(1)

L_{1} = \frac{a r e a}{\max (\bar{13}, \bar{42})}

(2)

L_{2} = \min (\bar{12}, \bar{23}, \bar{34}, \bar{41}, \bar{13}, \bar{42})

(3)

L_{c} = \max (L_{1}, L_{2})

When the orthogonalized mode of the hourglass perturbation formulation is used, the characteristic length is defined as:(4)

L_{3} = \max (L_{1}, L_{2})

(5)

L_{4} = 0.5 \frac{(L_{1} + L_{2})}{\max (h_{m^{'}} h_{f})}

(6)

L_{c} = \min (L_{3}, L_{4})

Where, $h_{m}$ is the shell membrane hourglass coefficient and $h_{f}$ is the shell out of plane hourglass coefficient, as mentioned in Hourglass Modes.