ALE Grid Calculation

In the ALE formulation, the freedom of moving the mesh is very appealing as it helps to combine the respective advantages of Lagrangian and Eulerian formulations. However, it is not easy to specify a grid velocity well-suited to the particular problem under consideration. As a consequence, the practical implementation of the ALE description requires that an automatic mesh-displacement prescription algorithm be supplied.

In Radioss, the following automatic grid computations exist.

/ALE/GRID/DONEA

This is the standard method applicable to the most of problems. It is based upon a combination of the material and grid velocities of the neighboring nodes:(1)
W I ( t+ Δt 2 )= 1 N J W J ( t Δt 2 ) + 1 N 2 α Δt J L IJ ( t ) J u J ( t ) u I ( t ) L IJ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaWGjbaabeaakmaabmaabaGaamiDaiabgUcaRmaaliaabaGa euiLdqKaamiDaaqaaiaaikdaaaaacaGLOaGaayzkaaGaeyypa0ZaaS aaaeaacaaIXaaabaGaamOtaaaadaaeqbqaaiaadEfadaWgaaWcbaGa amOsaaqabaGcdaqadaqaaiaadshacqGHsisldaWccaqaaiabfs5aej aadshaaeaacaaIYaaaaaGaayjkaiaawMcaaaWcbaGaamOsaaqab0Ga eyyeIuoakiabgUcaRmaalaaabaGaaGymaaqaaiaad6eadaahaaWcbe qaaiaaikdaaaaaaOWaaSaaaeaacqaHXoqyaeaacqqHuoarcaWG0baa amaaqafabaGaamitamaaBaaaleaacaWGjbGaamOsaaqabaGcdaqada qaaiaadshaaiaawIcacaGLPaaaaSqaaiaadQeaaeqaniabggHiLdGc daaeqbqaamaalaaabaGaamyDamaaBaaaleaacaWGkbaabeaakmaabm aabaGaamiDaaGaayjkaiaawMcaaiabgkHiTiaadwhadaWgaaWcbaGa amysaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaaaeaacaWGmb WaaSbaaSqaaiaadMeacaWGkbaabeaakmaabmaabaGaamiDaaGaayjk aiaawMcaaaaaaSqaaiaadQeaaeqaniabggHiLdaaaa@6E3E@
Where,(2)
1γ W v 1+γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTiabeo7aNjabgsMiJoaalaaabaGaam4vaaqaaiaadAhaaaGaeyiz ImQaaGymaiabgUcaRiabeo7aNbaa@41DB@
N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36C9@
Number of nodes connected to node
L I J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGjbGaamOsaaqabaaaaa@3890@
Distance between node I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamOsaaaa@36C6@ and node J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamOsaaaa@36C6@
α and γ
The adimensional factors given in input
Therefore, the grid displacement is given as:(3)
u( t+Δt )=u( t )+w( t+Δt 2 )Δt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaabm aabaGaamiDaiabgUcaRiabfs5aejaadshaaiaawIcacaGLPaaacqGH 9aqpcaWG1bWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaey4kaSIaam 4DamaabmaabaWaaSaaaeaacaWG0bGaey4kaSIaeuiLdqKaamiDaaqa aiaaikdaaaaacaGLOaGaayzkaaGaeuiLdqKaamiDaaaa@4C02@

/ALE/GRID/DISP

The average displacement formulation calculates the average velocity to determine the average displacement.(4)
w( t+Δt )= 1 N J W J ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Damaabm aabaGaamiDaiabgUcaRiabfs5aejaadshaaiaawIcacaGLPaaacqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGobaaamaaqafabaGaam4vamaaBa aaleaacaWGkbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaWc baGaamOsaaqab0GaeyyeIuoaaaa@46AF@

/ALE/GRID/SPRING

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

The stiffness of each spring is given by M Δ t 02 (where, M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C9@ is the mass of the node and Δ t 02 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iDamaaBaaaleaacaaIWaGaaGOmaaqabaaaaa@39F8@ is a user input typical time step), a viscosity and a ratio between shear spring stiffness and traction-compression stiffness of the springs can be defined.

It must be noted that those springs only affect the grid node velocity; they have no influence on the material velocity.

This method is very accurate and robust, but highly expensive in terms of CPU time.


Figure 1. Spring Force Graph

/ALE/GRID/ZERO

No automatic grid calculation is performed for the grid. The grid velocity is either constant (0 if no initial grid velocity is specified, the formulation is therefore Eulerian) or imposed by Property TYPE15 for parts with a rigid body movement.