# Stability Time Step

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.

The characteristic length, $L$
, for computing the
critical time step, referring back to Figure 1, is defined by:(1)
(2)
(3)

$${L}_{1}=\frac{area}{\mathrm{max}\left(\overline{13},\overline{42}\right)}$$

$${L}_{2}=\mathrm{min}\left(\overline{12},\overline{23},\overline{34},\overline{41},\overline{13},\overline{42}\right)$$

$${L}_{c}=\mathrm{max}\left({L}_{1},{L}_{2}\right)$$

When the orthogonalized mode of the hourglass perturbation formulation is used, the
characteristic length is defined as:(4)
(5)
(6)

$${L}_{3}=\mathrm{max}\left({L}_{1},{L}_{2}\right)$$

$${L}_{4}=0.5\frac{\left({L}_{1}+{L}_{2}\right)}{\mathrm{max}\left({h}_{{m}^{\prime}}{h}_{f}\right)}$$

$${L}_{c}=\mathrm{min}\left({L}_{3},{L}_{4}\right)$$

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.