Stability

The Courant condition (neglecting viscosity effects) is used to determine the stability of an ALE process.

The maximum time step is calculated by:(1)
Δ t k Δ l c + ν w
Where,
k
Coefficient
Δl
Smallest characteristic length of an element
c
Material speed of sound
ν
Material velocity
w
Grid velocity
The speed of sound is determined by:(2)
c = 1 ρ p ρ + 4 3 μ ρ
Where,
ρ
Density
μ
Dynamic viscosity
p
Pressure
The relative velocity between the material and grid motion ( ν - w ) is computed by:(3)
ν w = 1 N i = 1 3 I = 1 N ( ν i I w i I ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH9oGBcqGHsislcaWG3bGaeyypa0ZaaOaaaeaadaWcaaqaaiaa igdaaeaacaWGobaaamaaqahabaWaaabCaeaadaqadaqaaiabe27aUn aaDaaaleaacaWGPbaabaGaamysaaaakiabgkHiTiaadEhadaqhaaWc baGaamyAaaqaaiaadMeaaaaakiaawIcacaGLPaaaaSqaaiaadMeacq GH9aqpcaaIXaaabaGaamOtaaqdcqGHris5aaWcbaGaamyAaiabg2da 9iaaigdaaeaacaaIZaaaniabggHiLdGcdaahaaWcbeqaaiaaikdaaa aabeaaaaa@5481@
Where,
N
Number of nodes of the considered element (usually N =8)