Stability Time Step

The characteristic length, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@ , for computing the critical time step, referring back to 図 1, is defined by:
図 1.
L 1 = a r e a max ( 13 ¯ , 42 ¯ )
図 2.
L 2 = min ( 12 ¯ , 23 ¯ , 34 ¯ , 41 ¯ , 13 ¯ , 42 ¯ )
図 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:
図 4.
L 3 = max ( L 1 , L 2 )
図 5.
L 4 = 0.5 ( L 1 + L 2 ) max ( h m ′ h f )
図 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.