Conservation of Momentum

Conservation of momentum, expressed in terms of a finite element formulation, is given by:

∫ V Φ I ( ρ ∂ ν i ∂ t − ∂ σ i j ∂ x j − ρ b i ) d V = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWdrbqaaiabfA6agnaaBaaaleaacaWGjbaabeaakmaabmaabaGa eqyWdi3aaSaaaeaacqGHciITcqaH9oGBdaWgaaWcbaGaamyAaaqaba aakeaacqGHciITcaWG0baaaiabgkHiTmaalaaabaGaeyOaIyRaeq4W dm3aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kaadIhada WgaaWcbaGaamOAaaqabaaaaOGaeyOeI0IaeqyWdiNaamOyamaaBeaa leaacaWGPbaabeaaaOGaayjkaiaawMcaaiaadsgacaWGwbGaeyypa0 JaaGimaaWcbaGaamOvaaqab0Gaey4kIipaaaa@5A72@

Where,
Φ I
Weight functions
ρ
Material density
ν
Velocity
σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39C2@
Stress matrix
b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaaaaa@37F7@
Body acceleration vector
V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaaaa@36D1@
Volume

This can be rewritten in a form similar to the explicit Lagrangian formulation with the addition of a new nodal force F t r m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCa aaleqabaGaamiDaiaadkhacaWGTbaaaaaa@39D0@ , accounting for transport of momentum:

M ∂ ∂ t ν = { F e x t } − { F int } + { F b o d } + { F h g r } + { F t r m } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGnbWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiabe27a Ujabg2da9maacmaabaGaamOramaaCaaaleqabaGaamyzaiaadIhaca WG0baaaaGccaGL7bGaayzFaaGaeyOeI0YaaiWaaeaacaWGgbWaaWba aSqabeaaciGGPbGaaiOBaiaacshaaaaakiaawUhacaGL9baacqGHRa WkdaGadaqaaiaadAeadaahaaWcbeqaaiaadkgacaWGVbGaamizaaaa aOGaay5Eaiaaw2haaiabgUcaRmaacmaabaGaamOramaaCaaaleqaba GaamiAaiaadEgacaWGYbaaaaGccaGL7bGaayzFaaGaey4kaSYaaiWa aeaacaWGgbWaaWbaaSqabeaacaWG0bGaamOCaiaad2gaaaaakiaawU hacaGL9baaaaa@61FD@

Where,
{ F t r m } = ∑ f t r m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaGadaqaaiaadAeadaahaaWcbeqaaiaadshacaWGYbGaamyBaaaa aOGaay5Eaiaaw2haaiabg2da9maaqaeabaGaamOzamaaCaaaleqaba GaamiDaiaadkhacaWGTbaaaaqabeqaniabggHiLdaaaa@45F5@
Transport of momentum forces

Momentum Transport Force

By default, since version 2018.0 a streamline upwind technique is used to computed transportation forces (SUPG). This formulation enables to get rid of false diffusion (numerical issue with classical upwind technique). Nevertheless, SUPG method can be disabled in order to retrieve numerical solution obtained with classical upwind technique from versions prior to 2017. For this purpose, Engine keyword must be define: /ALE/SUPG/OFF. Using this keyword, momentum transport forces are computed using the classical upwind technique which was the default method up to version 2017.

F i I t r m = ( 1 + η I ) ρ Φ I ( w j − v j ) ∂ v i ∂ x j V

The classical upwinding technique is introduced to add numerical diffusion to the scheme, which otherwise is generally under diffuse and thus unstable.

η I = η s i g n [ ∂ Φ I ∂ x j ( v j − w j ) ]

0 ≤ η ≤ 1 Upwind coefficient, given in input.

Full upwind η = 1 (default value) is generally used and can be tuned with η 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS baaSqaaiaaigdaaeqaaaaa@3889@ parameter from /UPWIND keyword (now obsolete with SUPG formulation).
Figure 1.