Referential Domain

At any location in space x and time t, there is one material point, identified by its space coordinates x at time t=0, and one grid point identified by its coordinates ξ at time t=0. Figure 1 provides a pictorial representation and defines the velocities in each formulation.
Figure 1. ALE Formulation


The derivative of any physical quantity can be computed either following the material point or following the grid point. They can then be related to each other.

Given that F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ is a function f of space and time representing a physical property:
  • The spatial domain is given by f ( x , t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaaaaa@3B10@
  • The material domain is given by f * ( X , t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCa aaleqabaGaaiOkaaaakmaabmaabaGaamiwaiaacYcacaWG0baacaGL OaGaayzkaaaaaa@3BD5@
  • The mixed domain is given by f * * ( ξ , t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCa aaleqabaGaaiOkaiaacQcaaaGcdaqadaqaaiabe67a4jaacYcacaWG 0baacaGLOaGaayzkaaaaaa@3D69@

Therefore,

Also:

∂ f * ∂ t | x = ∂ f * * ∂ t | ξ + ( V j − W j ) ∂ f ( x , t ) ∂ x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiabgkGi2kaadAgadaahaaWcbeqaaiaacQcaaaaakeaa cqGHciITcaWG0baaamaaeeaabaWaaSbaaSqaaiaadIhaaeqaaaGcca GLhWoacqGH9aqpdaWcaaqaaiabgkGi2kaadAgadaahaaWcbeqaaiaa cQcacaGGQaaaaaGcbaGaeyOaIyRaamiDaaaadaabbaqaamaaBaaale aacqaH+oaEaeqaaaGccaGLhWoacqGHRaWkdaqadaqaaiaadAfadaWg aaWcbaGaamOAaaqabaGccqGHsislcaWGxbWaaSbaaSqaaiaadQgaae qaaaGccaGLOaGaayzkaaWaaSaaaeaacqGHciITcaWGMbWaaeWaaeaa caWG4bGaaiilaiaadshaaiaawIcacaGLPaaaaeaacqGHciITcaWG4b WaaSbaaSqaaiaadQgaaeqaaaaaaaa@5D90@

This relates to acceleration by:

γ → = d v → ∂ t | x = ∂ ∂ t v → | ξ + ( v j − w j ) ∂ ∂ x v → | t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacuaHZoWzgaWcaiabg2da9maalaaabaGaamizaiqadAhagaWcaaqa aiabgkGi2kaadshaaaWaaqqaaeaadaWgaaWcbaGaamiEaaqabaaaki aawEa7aiabg2da9maalaaabaGaeyOaIylabaGaeyOaIyRaamiDaaaa ceWG2bGbaSaadaabbaqaamaaBaaaleaacqaH+oaEaeqaaaGccaGLhW oacqGHRaWkdaqadaqaaiaadAhadaWgaaWcbaGaamOAaaqabaGccqGH sislcaWG3bWaaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaayzkaaWaaS aaaeaacqGHciITaeaacqGHciITcaWG4baaaiqadAhagaWcamaaeeaa baWaaSbaaSqaaiaadshaaeqaaaGccaGLhWoaaaa@5B79@

Where,
ν
Material velocity
w
Grid velocity