/ALE/GRID/STANDARD

Block Format Keyword Describes the standard formulation for ALE grid velocity computation.

It is an improved /ALE/GRID/SPRING formulation based on edge springs and anti-shear springs. 1

ale_standard
Figure 1.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/ALE/GRID/STANDARD
α γ η l c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGJbaabeaaaaa@37FC@  
Blank Format

Definition

Field Contents SI Unit Example
α Scale factor for maximum stiffness. 2

Default = 0.9 (Real)

 
γ Nonlinearity factor for edge spring stiffness. 3

Default = 1e-2 (Real)

 
η Damping coefficient. 4

Default = 1e-2 (Real)

 
l c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGJbaabeaaaaa@37FC@ Characteristic length.

(Real)

[ m ]

Comments

  1. Fictitious springs are introduced on solid elements to control grid velocities.
    These springs are nonlinear elastic viscous. To ensure stability, their stiffness is computed from time step. The two types of springs are edge and anti-shear springs.
    1. Edge springs
      The forces for an edge spring are a function of its length variation during time.(1)
      Δ F e d g e = k ( h ) ( w 2 w 1 ) d t

      Where,

      w 1 , w 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4DamaaBa aaleaacaaIXaaabeaakiaacYcacaWH3bWaaSbaaSqaaiaaikdaaeqa aaaa@3A80@ are grid velocities on nodes N1 and N2, respectively.

      h is the N1 distance from opposite face

      d t is the time step

      and k ( h ) is the spring stiffness k ( h ) = k c r i t i c a l

      If h is inferior to the characteristic length l c and N1 is moving toward the opposite face then,(2)
      k ( h ) = 1 λ 2 [ γ + ( γ 1 ) ( h l c l c ) 3 ] k c r i t i c a l
      1 λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGaaeaaca aIXaaabaGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaaaaa@3960@ is the stability factor taking into account the damping factor β , the scale factor α , and time step d t 4

      ale_standard_towards
      Figure 2.
      otherwise, k ( h ) = γ λ 2 k c r i t i c a l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Aamaabm aabaGaamiAaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeq4SdCga baGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaakiaadUgadaWgaaWcba Gaam4yaiaadkhacaWGPbGaamiDaiaadMgacaWGJbGaamyyaiaadYga aeqaaaaa@4751@

      ale_standard_eq
      Figure 3.
    2. Anti-shear springs

      The anti-shear forces F s h e a r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGZbGaamiAaiaadwgacaWGHbGaamOCaaqabaaaaa@3B9E@ are computed from node penetration. gap is l c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGaaeaaca WGSbWaaSbaaSqaaiaadogaaeqaaaGcbaGaam4Caaaaaaa@3910@ from opposite face.

      The value of F s h e a r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGZbGaamiAaiaadwgacaWGHbGaamOCaaqabaaaaa@3B9E@ is:

      (3)
      F s h e a r = g a p k ( h ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGZbGaamiAaiaadwgacaWGHbGaamOCaaqabaGccqGH9aqp caWGNbGaamyyaiaadchacqGHflY1ciGGRbWaaeWaaeaacaWGObaaca GLOaGaayzkaaaaaa@4526@
      and (4)
      k ( h ) = γ λ 2 [ γ + ( γ 1 ) ( h l c l c ) 3 ] k c r i t i c a l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Aamaabm aabaGaamiAaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeq4SdCga baGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaakmaadmaabaGaeq4SdC Maey4kaSYaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzk aaWaaeWaaeaadaWcaaqaaiaadIgacqGHsislcaWGSbWaaSbaaSqaai aadogaaeqaaaGcbaGaamiBamaaBaaaleaacaWGJbaabeaaaaaakiaa wIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaakiaawUfacaGLDbaaca WGRbWaaSbaaSqaaiaadogacaWGYbGaamyAaiaadshacaWGPbGaam4y aiaadggacaWGSbaabeaaaaa@5929@

      ale_standard_critical
      Figure 4.
    3. Viscous Damping
      Viscous forces are computed from a critical damping corresponding to the upper bound for stiffness: 1 λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGaaeaaca aIXaaabaGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaaaaa@3961@ (5)
      F v i s c o u s = β α ( 1 + β 2 β ) ( w 2 w 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWG2bGaamyAaiaadohacaWGJbGaam4BaiaadwhacaWGZbaa beaakiabg2da9iabek7aIjabeg7aHnaabmaabaWaaOaaaeaacaaIXa Gaey4kaSIaeqOSdi2aaWbaaSqabeaacaaIYaaaaaqabaGccqGHsisl cqaHYoGyaiaawIcacaGLPaaadaqadaqaaiaahEhadaWgaaWcbaGaaG OmaaqabaGccqGHsislcaWH3bWaaSbaaSqaaiaaigdaaeqaaaGccaGL OaGaayzkaaaaaa@50A2@
    4. Grid Velocity
      The grid velocity is then updated according to:(6)
      w n + 1 = w n + ( Δ F e d g e + F s h e a r + F v i s c o u s ) d t m

      Where, m is fictitious mass on node from springs (automatically computed during Starter).

  2. Increasing α = 1 , the maximum stiffness will be increased. The scale factor α determines the maximum stiffness for a given spring at zero length. The scale factor ensures that the critical stability value is not exceeded (to avoid time step decrease).
  3. This flag is acting on stiffness shape. Stiffness is linear with γ = 0. Moreover, increasing γ , the lower bound stiffness for edge spring will be increased. Springs have a critical stiffness at zero length (this corresponds to a unitary factor). For a length greater than or equal to the characteristic length, the spring stiffness is the critical stiffness multiplied by γ .
  4. It is recommended to use small values for β , otherwise damping may become over critical. The stability factor is:(7)
    λ = d t α 0 ( 1 + β 2 β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey ypa0ZaaSaaaeaacaWGKbGaamiDaaqaaiabeg7aHnaaBaaaleaacaaI WaaabeaakmaabmaabaWaaOaaaeaacaaIXaGaey4kaSIaeqOSdi2aaW baaSqabeaacaaIYaaaaaqabaGccqGHsislcqaHYoGyaiaawIcacaGL Paaaaaaaaa@458A@
  5. l c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGJbaabeaaaaa@37FC@ defines the length below which:
    • Edge spring stiffness is increased h < l c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiabgY da8iaadYgadaWgaaWcbaGaam4yaaqabaaaaa@39ED@
    • Anti-shear spring is activated: h < l c 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiabgY da8maaliaabaGaamiBamaaBaaaleaacaWGJbaabeaaaOqaaiaaiwda aaaaaa@3AC8@
  6. All these parameters can be modified during an Engine restart (/ALE/GRID/STANDARD).
  7. Mesh auto correction. It is possible to give more weight to anti-shear forces by either:
    • Setting l c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGJbaabeaaaaa@37FC@ parameter close to the mesh size
    • Setting a negative value for α parameter (elastic forces on edges are set to 0 at the first cycle of current run)
  8. This method assumes a homogeneous spring repartition around each node. This is not the case when connecting two meshes, where topologies are different.