STAGGERED MESH GODUNOV (SMG) SCHEMES FOR

ALE HYDRODYNAMICS

Gabi Luttwak1

and

Joseph Falcovitz2

1Rafael, P.O. Box 2250, Haifa 31021, Israel 2Institute of Mathematics, The Hebrew University of Jerusalem, Israel

Hydro-Schemes Classical Lagrange (CL) & ALE

– staggered in space & time • even schemes are staggered in space with predictor-corrector time

integration– pseudo-viscosity used for shock capturing

Second Order Godunov (Eulerian) – zone centered variables & gradients– gradients limited to preserve monotonic profiles– Riemann Problems (RP) solved at zone faces

We seek the best approach for an MM-ALE code– It is required to treat the flow in both Lagrangian and

(Multi-material) Eulerian framework

Staggered Mesh Godunov - SMG

SMG schemes:– staggered in space & time

• or staggered in space and even-time integration

– use RP to capture shocks– bridge the Lagrange-to-Godunov

“conceptual gap”

(MM)ALE code Geometry Update (not necessary for pure Euler)

– compute areas, volumes etc. Lagrangian Phase

– CL/SMG Scheme– Definition of pressure and stress in a MM zone

Advection Phase (not present for pure Lagrange)– Second order advection for cell-centered variables– Staggered Mesh Momentum advection

• SALE• HIS (Benson)

– Single step /Partially-split advected-volume integration– Interface tracking: VOF (Luttwak 1984,2002) similar to Youngs scheme

Grid Motion Algorithm

Classical Lagrange (CL) Schemes

simple, efficient and robust– ~ second order accurate for smooth flow

• on a uniform mesh with constant time steps

– until grid gets distorted pseudo-viscosity is used to capture shocks

– over 3-4 zones

– + linear terms required for weak shocks

2vq

Pseudo-viscosity

form and coefficients somewhat arbitrary– not from first principles

unphysical heating in regions of isentropic flow– especially in Multi-D

second order Godunov-like methods help overcome deficiencies of classical Lagrange Q– e.g. Christensen, Benson, Caramana et al.

Second-order zone-centered Godunov

sharp shock resolution < 2 zone thick second order accurate physically intuitive BUT hard to add more physics

– Elastic-plastic flow, chemical reaction etc. Multi-D is well formulated only in Eulerian

coordinates (operator splitting)

Zone-Centered Godunov with Lagrange/ALE mesh

Vertex velocities obtained by interpolating either – zone-centered velocities (from momentum equation), or

– face velocities (from RP solution)

Result: Weak coupling between node motion and zone deformation Spurious grid deformation

– Remedy: SMG scheme

SMG schemes

SMG/L scheme– RP solved only at zone faces

• Zone-centered velocity obtained by interpolation of vertex velocities• Face-centered data extrapolated from the zone-centered data using the

(limited) zone–centered gradients– Additional impulse due to RP solution distributed to neighboring

vertices SMG/G scheme

– “collision” RP solved at corner-zone faces is used to update the momentum equation

– “contact discontinuity” RP solved at zone faces serves to update the (total) energy equation

SMG/Q scheme (closest to CL)

Staggered Mesh

Zone-centered but node-centered velocity velocity jump on corner-zone faces Jump in only at the zone faces

– zones k, edge-neighbor nodes i,i+1– Corner-zones (i,k).– The faces separate corner-zones (i,k) and (i+1,k)

pe,,

pe,,

kiiA 1,

SMG/Q scheme Data for “collision” RP at corner zone faces:

– velocity jump at face :

– Continuous in zone k

– depends on velocity gradient in the zone and on velocity in neighboring nodes. for first order accuracy (sharp velocity jump). in regions of smooth flow.

solution

kiiu 1, ii

kiik uuA

11,

ˆ1

kkp ,

kiiA 1,

110 k

kii

kii pu 1,1, ,

k0k1k

“collision” Riemann problem 1D symmetric collision RP (data jump only in velocity).

– Solution: either 2 shock waves or 2 centered rarefaction waves

SMG/Q scheme

momentum equation:– sum up “corner face forces” on all faces

surrounding node i

internal energy equation: – assume all work performed by against the

deformation of zone k is deposited in the zone

i

ii

kiii

kii

kii

kii m

FaFFApF

;; 1,1,1,1,

kiiA 1,

kiiF 1,

1,

1,11,1, ;/ii

kii

kkii

kii

kii eemuuFe

SMG/Q characteristics

As each depends only on the velocity difference the Lagrange scheme is Galilei-Invariant (like CL).

As in “traditional” Godunov schemes no pseudo-viscosity is required.

Needs solution to a simple “collision” RP. Formulated with internal energy.

– see also Christensen, Caramana et al.

kiie 1,

ii uu

1

SMG/Q and pseudo-viscosity

We can regard as a kind of pseudo-viscosity.

– as at a shock • except at phase changes or for reacting HE.

– define it as a uni-axial tensor acting along the velocity difference (Not along ).

kkii

kii ppq 1,1,

)0,max( 1,1,kii

kii qq

ii uu

1kiiA 1,

kkii pp 1,

Uni-axial tensor Q

Edge-defined Q is better if related lumped forces are aligned with the velocity difference, instead of the edge (Margolin, Caramana et al.).

– a moment opposing edge rotation is thereby generated (a grid stabilizing effect).

Summing up “corner face forces” over all faces separating edge-neighbor nodes, produces normal-to-face forces.

Alternately define as uni-axial tensor along the direction of :

kkii

kii ppq 1,1,

iiii uuuunnnqq

11ˆ;ˆˆ

ii uu

1

The Riemann Problem solutions

The RP solution is aimed to provide the necessary dissipation at shocks.

An approximate Riemann solution may be adequate for moderate shocks.

SMG/Q can be used for elastic-plastic and reactive flow even without solving the exact RP (with results at least similar to Q-based CL). – Using the specific RP solution in each case might

further improve the results.

“Collision” RP solution

The Hugoniot is and the RP can be solved assuming a linear shock-to-particle-velocity relationship

The exact RP can be solved numerically (or, for an ideal gas, analytically) at strong shocks or strong rarefactions.

Note: two waves, each with half the velocity jump !! Kuropatenko’s Q: single shock with full velocity jump!!

sH uUuP 0

ucU s 0 ucuuPH 00

Second-Order SMG/Q

Velocity jump at faces computed using the zone-centered velocity gradient limited to preserve a smooth velocity profile:

kiiA 1,

kiiu 1,

kk uu

lim

with the largest such that the velocities extrapolated to the center of every single-vertex-neighbor zone stay in the min/max range of nodes in surrounding zones.

In 1D: this algorithm (QG ) is similar to Christensen’s, which we denote QT

10

The slope limiter

This limiter extends van Leer 1D scheme to a multi-D staggered mesh– Uses average zone-centered velocity gradient– The limiter is defined on either structured or

unstructured mesh– In 1D it reduces to ,

while Christensen’s TVD-like limiter is: RLk RR 2,2,1modmin

kk

RLRLRLk xu

xuRRRRR

1

/;5.0,2,2,1modmin

Planar Noh Problem

with the data: and analytic solution:

The 1D calculation had 100 unit length zones An exact Rieman solver was used in the 1D

simulations Sharp and accurate shock capture and almost

identical results for QG and QT

60,35,1,1,0,, tup R

200;4

10020;1)(;3

1r

rrU s

Fig.1 Planar Noh problem. 60,35,1,1,0,, tup R

Centered Compression Wave (CCW)

The initial (t=0) smooth profile has steepened up to a sharp discontinuity at t=20 – The t=0 profile is the “inverse” of a centered

rarefaction with a linear velocity distribution

Results show an almost isentropic compression

Fig.2 Centered Compression Wave (CCW). 4.1 ,0,25.0,06997.0,,,1,1,4873.0,, RL upup

The Sod Problem

The initial (shock tube) data:

It was run with 100 unit-size zones. The results at t=20 compare favorably to

those of the GRP (a genuine second-order Godunov scheme see Ben Artzi & Falcovitz)

SMG with even-time integration gave virtually identical results for this problem

20t, 4.1],0,125.0,1.0[,,];0,1,1[,, RL upup

Sod Problem. SMG/QG and GRP 20t, 4.1],0,125.0,1.0[,,];0,1,1[,, RL upup

Sedov-Taylor blast wave problem

The initial data is: cold ideal gas with a “point source” represented by setting in a single zone at the origin (see Caramana et al.)

The physical space is a 253 box divided into a uniform 903 mesh

3D computation conducted in the octant with the symmetry planes

The exact solution has a post-shock density and the front radius is at .

0,3/5];0,1,0[,, 0 eup

7.5027e

]0,0,0[ zyx0,0,0 zyx

41R 1t

Fig.4 Sedov-Taylor blast wave. The 453 mesh spans a 1.253 box. The

velocity plot at t=1. (a) xy plane (b) 3D view

7.5027)1(;0;3/5];0,1,0[,, 0 eeup

Fig.5 The Sedov-Taylor blast wave. The xy plane mesh plot at t=1, (left) without NC, (right) with NC.

Fig. 6 The Sedov-Taylor blast wave. Isobar plot at t=1.

Fig. 7 The Sedov-Taylor blast wave. Isodensity plot at t=1.

Fig. 8 The Sedov-Taylor blast wave. Density profiles at t=1.

The Saltzman Problem

A piston moves with a velocity of u=10 into an almost cold ideal gas

This 1D problem is solved on a 2D mesh with planar symmetry

The 100x10 mesh shown above tests the propagation of a planar shock into an obliquely-inclined non-uniform mesh

3/5];0,1,00667[.,,];10,4,33.133[,, RL upup

Fig.9 The Saltzman Problem. 100x10 Lagrange mesh at t=0.7 (a) Scalar Q (b) SMG with uni-axial tensor q and NC

3/5];0,1,00667[.,,];10,4,33.133[,, RL upup

(a)

(b)

Fig.9 Saltzman Problem. 100x10 Lagrange mesh at t=0.7 Improved SMG/Q (RP solved along velocity vector difference of edge-neighbor pairs). It seems no NC is required!!!

Cylindrical Noh Problem

with the data: and analytic solution:

6.0,35,1,1,0,, tup R

3/0;16

13/;1)(;3

1tr

rtrt

rU S

Cylindrical Noh problem 1D Lagrangian mesh in 3D code

3D code can handle 1D,2D mesh with axial symmetry.– The B.C. and the stability condition must be adjusted

A 100 equal size grid was used. The over-shoot at the front might be caused by the use of the

approximate RP solver in the 3D code. An infinite strength shock in an ideal gas does not obey the linear shock to particle relationship.

–

t=0.6

Cylindrical Noh problem 2D ALE mesh in 3D code

We consider a cylinder with a body-fitted grid which has 32x32x1 zones

There are symmetry planes at z=0,z=0.05 The ALE grid motion keeps the mesh

smooth while the outer radius shrinks

Cylindrical Noh problem 2D ALE mesh in 3D code at T=0.6 (a) velocity plot (b) isobars

Cylindrical Noh problem 2D ALE mesh in 3D code at T=0.6. isodensity plot

Cylindrical Noh problem 2D ALE mesh in 3D code at T=0.6 (a) velocity plot (b) isobars

Spherical Noh Problem

with the data: and analytic solution:

see Noh, Birman et. al.

6.0,35,1,1,0,, tup R

3/0;64

13/;1)(;31

2

tr

rtrt

rU S

3D ALE mesh for Spherical Noh problem

The mesh is Cartesian. The ALE grid is uniformly contracting at a rate given by the grid velocity:

where is the velocity of a grid point at . is the spherical boundary at time t.

The 3D mesh has 503 zones. The material boundary cuts through mesh zones.

gu

gr

tR0

tRtRtRrrtRu ggg

0;;1

0000

Spherical Noh problem 3D (Eulerian-like) ALE mesh at T=0

Spherical Noh problem 3D (Eulerian-like) ALE mesh at T=0.6. isobar plot

Spherical Noh problem 3D (Eulerian-like) ALE mesh at T=0.6. isodensity plot

Concluding Remarks

SMG is a ‘dual scheme’. It is both a Godunov method and a classical-Lagrange method.

SMG/Q bridges the Lagrange-to-Godunov “conceptual gap”.

The second-order extension produces sharp shock capturing while minimizing entropy errors in regions of smooth compression.

The method can be applied to structured mesh, as well as to unstructured mesh.

Future Work

Improve the velocity limiter for multi-D structured and unstructured mesh

Explore alternate Q formulations:– Zone-centered “directional scalar Q”– Zone-centered “directional uni-axial tensor Q”

Elastic-plastic flow

References Wilkins M. L., "Calculation of Elastic-Plastic Flow”, Meth. Comp.

Phys., 3, p211, B. Alder et al. eds., Academic Press (1964). van Leer B., J. Comp. Phys., 32, p101, (1979). Ben Artzi M., Falcovitz J., "Generalized Riemann Problems in

Computational Fluid Dynamics", Cambridge Univ. Press, London, (2003).

Luttwak, G., "Comparing Lagrangian Godunov and Pseudo-Viscosity Schemes for Multi-Dimensional Impact Simulations”, p255-258, Shock Compression of Condensed Matter-2001, Furnish M. D. et al Eds, (2002), AIP, CP620.

Luttwak, G. ”Staggered Mesh Godunov (SMG) Schemes for Lagrangian Hydrodynamics”, presented at the APS Topical Conference on Shock Compression of Condensed matter,Baltimore,MD,July31-Aug5,2005

References Christensen R.B., "Godunov Methods on a Staggered Mesh. An

Improved Artificial Viscosity", L.L.N.L report UCRL-JC-105269, (1990).

Benson D. J., Schoenfeld S., Comp. Mech., 11, p107-121, (1993).

Caramana E.J., Shaskov M.J., Whalen P.P., J. Comp. Phys. 144, p70, (1998).

Margolin L.G., LLNL report UCRL-5382,(1988) Noh F. W.,J. Comp. Phys., 72,p78,(1987) Sod G.,A., J. Comp. Phys., 27,p1,(1978) Birman A., Har’el N. Y. ,Falcovitz J. , Ben-Artzi M., Feldman U.,

22nd Int. Symp. on Shock Waves, Imperial College, London, UK, July 18-23, (1999)

More References

Luttwak G., Cowler M. S., "Remapping Techniques for Three Dimensional Meshes", presented at the 1999 Int. Workshop for New Models and Hydro-codes for Shock Wave Processes, held at the Univ. of Maryland, College Park, MD,(1999), to appear

Luttwak G., "Interface Tracking in Eulerian and MMALE Calculations",p283-286, Shock Compression of Condensed Matter-2001,Furnish M. D., et al. eds., AIP, CP620, (2002)

Luttwak G., "On Rules for Grid Motion in 3D ALE Codes", presented at the "New Models and Hydro-codes for Shock Wave Processes 2002", Edinburgh, Scotland, UK, May 2002, ed. by Klimenko V.Y., to appear

Luttwak G., "Partially Split Volume Integration scheme for the Advection Phase of Eulerian and MMALE Simulations", presented at the 2004 Int. Conf. for New Models and Hydro-codes for Shock Wave Processes, Univ. of Maryland, College Park, MD, 16-21 May,2004