Comparing Lagrangian Godunov and Pseudo-Viscosity Schemes for Multi-Dimensional Impact Simulations. Gabi Luttwak

Transcription

1 Comparing Lagrangian Godunov and Pseudo-Viscosity Schemes for Multi-Dimensional Impact Simulations Gabi Luttwak Rafael, P.O.Box 2250, Haifa 31021,Israel

2 Godunov or Pseudo-viscosity Modern Eulerian codes use Second Order Godunov schemes Lagrangian calculations done with first order pseudo-viscosity codes Why? And what to use in a multi-material ALE code which should handle both?

3 Lagrangian Godunov schemes Are there 2D,3D Lagrangian Godunov schemes? Vertex velocities has to be defined interpolating the face velocities interpolating the zone centered velocities Or, consider a new staggered mesh Godunov (SGM) scheme

4 Which scheme behaves better? The schemes: 1. Lagrangian 2. Godunov 3. SMG Test problems: 1D - SOD shock tube 3D-Normal impact of a square rods into a wall The code AUTODYN-3D with some external user routines

5 The Equations of Motions Conservation laws of mass,momentum and energy + the material model In integral form they also apply at discontinuities They yield the differential equations as the control volume 0 The difference equations obtained when the control volume is the computational zone They yield the Rankine-Hugoniot equations at discontinuities

6 The Conservation laws s p e p p u e e ds u ds u u ue dv e dt d ds ds u u u udv dt d ds u u dv dt d T V V g T V T V V g V V g V ; ), ( ; 2 2

7 The Differential Equations u t du dt de dt : u 0

8 The pseudo-viscosity Von-Neuman pseudo-viscosity: added to the pressure captures shock over 2-3 zones R-H relations satisfied over a shock Kuropatenko-reversed this process pseudo-viscosity obtained from R-H: Q Uu 0 (1)

9 The pseudo-viscosity most solids obey a linear relationship between shock to particle velocity: U S 0 su 2 inserting the shock velocity into (1) : c Q s 2 0u 0c0u 3 This is the form used by Wilkins and in most codes

10 The Godunov scheme The conservation laws hold at discontinuities If there is a jump at the faces between zones a Riemann problem has to be solved there In the two-shock approximation => R-H relations has to be solved Assuming a linear shock to particle velocity we get for the pressure jump (3) again Riemann solution yields p* and u* at the face

11 Second order Godunov scheme Piece-wisely linear variable distribution The zone-centered values projected to the faces using the slopes The slopes and the face centered values limited to prevent instabilities (VanLeer and Hancock) At the faces Riemann solution at T n+1/2 and T n+1 The T n+1/2 p* integrated over the zone to get new momentum and velocity. The T n+1 solutions used to update the slopes

12 For Euler, u* at T n+1/2 used in the flux terms For 1D face and vertex coincide u* natural choice for Lagrangian vertex velocity For 2D or 3D vertex velocity found by interpolation: The vertex velocity 4 1,,, 1,4 1,4 * A u u k j i

13 The Staggered Mesh Godunov (SMG) scheme The idea to use p* from Riemann solution but to integrate the momentum over a staggered mesh E A D O H C F B G Figure 1. The staggered mesh We use the zone pressures around the control volume * The impulse of p p p is added temporary to the neighbor zones and distributed to the nodes

14 The Staggered Mesh Godunov (SMG) scheme The zone centered velocity found by averaging over the vertices The velocity gradient in the zone utilized too u 8 1 u ( u r r 0 ) i 1 i 0 i 0 8

15 הלגרנג י במעבד לגרנג י חישוב

16 חישוב לגרנג י במעבד הלגרנג י

17 הגודונוב חישוב לגרנג י במעבד

18 . החיצוני בשיפווע וגם מדרגות

19 חישוב לגרנג י במעבד גודונוב מהירות בקודקוד כממוצע מהירויות בתא

20 עם ממוצע המהירויות בתאים

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23 Second order Godunov scheme with Lagrangian mesh T=6, 500m/s impact of Cu rod

24 T=8 Second order Godunov scheme for 500m/sec Cu rod impact with no strength

25 T=8 SMG scheme for 500m/sec 10x10x50mmCu rod on wall

26 T=8 Lagrangian 500m/s impact of 10x10x50mm Cu rod on wall

27 Sod problem SMG Lagrange mesh density profile

28 Sod problem SMG scheme Euler mesh density profile

29 Sod problem 1st order Lagrange scheme density profile

30 Second order Godunov with Lagrange mesh for Sod problem

31 Sod problem 2nd order Godunov Euler mesh density profile