Automatic Grid Computation

There are three different grid velocity formulations that can be used in an ALE simulation. New keywords define the type of method used. The different formulations are:
  • 0 - J. Donea Grid Formulation: use keyword /DONEA

    (NWALE =0 for version < 4.1)

  • 1 - Average Displacement Formulation: use keyword /DISP

    (NWALE =1 for version < 4.1)

  • 2 - Nonlinear Spring Formulation: use keyword /SPRING

    (NWALE =2 for version < 4.1)

J. Donea Grid Formulation (/DONEA)

This formulation 1 2 computes grid velocity using:(1)
W I ( t + Δ t / 2 ) = 1 N J W j ( t Δ t / 2 ) + 1 N 2 a Δ t J L I J ( t ) J u J ( t ) u I ( t ) L I J ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGxbWaaSbaaSqaaiaadMeaaeqaaOWaaeWaaeaacaWG0bGaey4k aSIaaeiLdiaadshacaGGVaGaaGOmaaGaayjkaiaawMcaaiabg2da9m aalaaabaGaaGymaaqaaiaad6eaaaWaaabuaeaacaWGxbWaaSbaaSqa aiaadQgaaeqaaOWaaeWaaeaacaWG0bGaeyOeI0IaaeiLdiaadshaca GGVaGaaGOmaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaaGymaaqa aiaad6eadaahaaWcbeqaaiaaikdaaaaaaOGaaGjbVpaalaaabaGaam yyaaqaaiaabs5acaWG0baaamaaqafabaGaamitamaaBaaaleaacaWG jbGaamOsaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaadaaeqb qaaiaaykW7daWcaaqaaiaadwhadaWgaaWcbaGaamOsaaqabaGcdaqa daqaaiaadshaaiaawIcacaGLPaaacqGHsislcaWG1bWaaSbaaSqaai aadMeaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaabaGaamit amaaBaaaleaacaWGjbGaamOsaaqabaGcdaqadaqaaiaadshaaiaawI cacaGLPaaaaaaaleaacaWGkbaabeqdcqGHris5aaWcbaGaamOsaaqa b0GaeyyeIuoaaSqaaiaadQeaaeqaniabggHiLdaaaa@73E7@
Where,
1 γ w v 1 + γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTiabeo7aNjabgsMiJoaalaaabaGaam4DaaqaaiaadAhaaaGaeyiz ImQaaGymaiabgUcaRiabeo7aNbaa@41FA@
N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36C9@
Number of nodes connected to node I
L IJ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGjbGaamOsaaqabaaaaa@3890@
Distance between node I and node J
α, γ
adimensional factors given in input
Therefore, the grid displacement is given by:(2)
u ( t + Δ t ) = u ( t ) + w ( t + Δ t / 2 ) Δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG1bWaaeWaaeaacaWG0bGaey4kaSIaeuiLdqKaamiDaaGaayjk aiaawMcaaiabg2da9iaadwhadaqadaqaaiaadshaaiaawIcacaGLPa aacqGHRaWkcaWG3bWaaeWaaeaacaWG0bGaey4kaSIaeuiLdqKaamiD aiaac+cacaaIYaaacaGLOaGaayzkaaGaeuiLdqKaaGOmaaaa@4F48@

Average Displacement Formulation (/DISP)

The average displacement formulation calculates average velocity to determine average displacement.(3)
u ( t + Δ t ) = 1 N J w j ( t )

Nonlinear Spring Formulation (/SPRING)

Each grid node is connected to neighboring grid nodes through a nonlinear viscous spring, similar to that shown in 図 1.


図 1. Spring Force Graph
The input parameters required are:
Δ T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiLdiaads fadaWgaaWcbaGaaGimaaqabaaaaa@38CF@
Typical time step (Must be greater than the time step of the current run.)
0 < γ < 1
Nonlinearity factor
η
Damping coefficient
V
Shear factor (stiffness ratio between diagonal springs and springs along connectivities)

This formulation is the best of the three, but it is the most computationally expensive.

1 Donea J., 「An Arbitary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions」, Computer methods in applied mechanics, 1982.
2 Brooks A.N. and Hughes T.J.R., 「Streamline Upwind /Petrov-Galerkin Formulations for Convection Dominated Flows with particular Emphasis on the Incompressible Navier-Stokes Equations」, Computer Methods in Applied Mechanics and Engineering, Vol. 32, 1982.