Gabi Luttwak1

and Joseph Falcovitz2

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

“Numerical Methods for multi-Material Fluid Flows”, Arcachon, France,5-9 Sept 2011

A novel VIP (Vector Image Polygon) slope limiter for scalar variables in

multidimensional grids

VIP-Convex Hull based limiters

What is VIP?– the natural extension of the monotonicity from scalar to vector variables

VIP => for the Lagrangian phase of the SMG/Q scheme (Pavia 2009)

VIP => for the advection phase of ALE/Euler SMG calculations (Paris 2010)

VIP => for advection of scalar variables ?– In the advection phase of ALE/Euler SMG/Q calculations

Test cases:– Noh 2D, Sedov 2D/3D, Standing wave in Saltzmann mesh– New “Noh Corner” Test

In Finite Volume Schemes

The conservation laws solved over a control volume :– (e.g. the computational zone - for zone centered

variables)

Fluxes evaluated on the control volume faces Face-centered values of variables are required

V V

g dsuudVdt

d)(

Slope Limiters

Second order Godunov and some other high resolution schemes use the gradients of the variables to get face-value:

In hyperbolic PDE discontinuities (shock or slip lines) are present, or can be generated, during the flow.

At discontinuities gradients are unbounded. Using them produces unphysical fluctuations in the solution Thus either gradients or the related fluxes must be limited

to preserve a monotonic flow field

cfccf rr

Slope Limiters in 1D

Limiters were first formulated in 1D:– Either the slope-extrapolated values of a variable at the zone faces

must lie in the range spanned by the values of that variable in neighboring cells:

Or the limited gradient defined as some kind of monotonic averaging of the gradients in the zone c and its neighbors . e.g. van Leer’s double minmod:

00

0)2,2,2

min(),(2abif

abifbaba

baave

)max()min(1 0lim0 nn rr

nb

;

a

Limiters for scalars in 2D/3D

In 1D the two approaches are equivalent Equation (1) is the usual way to extend the

limiters to 2D/3D– See Dukowicz, Barth&Jesperson

Equation (2) is van Leer’s double minmod– The gradient of scalars are vectors– What would be monotonic averaging of vectors

in 2D/3D ?

How is (1) extended to 2D/3D

Look at zone centered scalar variables Zone face in 1D=>either face/edge center or zone

vertex in 2D/3D Extrapolate using (1) from the zone center to each of its

vertices

Find a limited gradient– Such that all vertex extrapolated values of the density

remain within the density range spanned by values in the zones around that vertex

lim

Extending Equation (1) to 2D/3D(a) Scalar Limiting For each vertex v find the largest such that:

And take:

Straightforward but dissipative !! This was our default scheme for scalar variables

v

)max()min(3 00 nvn rr

vv

min4

lim

Extending Equation (1) to 2D/3D(a) Non scalar α

Find obeying Eq. (1) at each vertex ʋ “closest” in some sense to the unlimited . Limiting should be preferentially normal to

shock direction, so that the gradient direction may change:

Berger et.al. - Linear programming Ridder&Kothe – Least square reconstruction

lim

lim

lim

Let us extend Eq.(2) to 2D/3D !

We apply the VIP convex hull based monotonicity criterion to the slopes:

In 1D: In 2D/3D:

– If a slope is outside the VIP generated by twice the slopes of its neighbors=>move into the hull

• Factor 2 like in the double minmod

• e.g. we move it to the nearest point on the VIP

0)2,2,2

min(

abifbaba

What is the analog of ?

In 1D corresponds to an extremum– (a) In 2D/3D extremum is when .

• In the vector space of the slope-vectors, extremum exists if the origin (0,0,0) is inside the VIP => set slope to zero

• Difficulty: “Ridge configuration” a small slope component in some direction may prevent detecting a discontinuity normal to a major gradient

0ab

0ab0

VIP slope limiter for scalars

(b) At a surface discontinuity

is the normal to the surface

• changes sign along

• Take in the direction of the unlimited gradient

0

n

n̂

n

n̂

n̂

VIP slope limiter for scalars

Let the unlimited slopes in cell c and its k neighbors.

Let and with If => zero the slope: Else if lies outside the VIP

(convex hull ) move it to the closest point on hull

kia ii ..1,0;

kinas iii ..1;ˆ

00ˆ aan

iki

mxiki

mn ssss

11max;min

)0( mxmnss 0lim

0lim0

a

0a k

iia 12

10

2

1

11

3

4

6

8

5

97

12

A Convex Hull (CH)

The VIP convex hull limiter

V1

Vc

V2

V3

V4

V5

V0

“Move to hull”

V0

V6

• if V0 inside VIP => no limiting: • if V0 outside, VIP limiter is

applied and V0 => Vc

The Staggered Mesh Godunov (SMG/Q) Scheme

SMG schemes for 3D Lagrangian and ALE hydrodynamics were presented at Oxford 2005, Prague 2007, Pavia 2009,APS 2003,2005.

Use Riemann problem (RP) solutions to capture shocks. Vertex-centered velocities have jumps on the in-cell corner zone faces. These are simplified, “impact” RP or (IRP) with continuous .

The IRP are solved in the normal to shock direction, i.e. along the velocity difference . This defines a uniaxial tensor pseudo-viscosity.

,p

RL uu

The Staggered Mesh Godunov (SMG/Q) Scheme

The velocities on either side of the face, serving as data for the RP, are evaluated from the vertex velocities using a cell centered velocity gradient, limited to preserve a monotonic velocity distribution.

A limited velocity gradient is also used to compute the momentum fluxes during the advection phase

The resulting scheme captures shocks with sharp monotonic profiles. It also has a strong inherent mesh stabilizing effect.

The rotation invariant VIP limiter is used to preserve the symmetries present

L R

u

C

Limited slope

Original slope

Velocity jump for

RP

Lu

Ru

u

u

u

u

QF

QF

Test Problems

Saltzman, Sedov, Noh problems test the propagation of 1D shocks over non-aligned or distorted meshes

Our corner Noh (a 2D/3D collision Riemann problem) tests 2D/3D sector-wise converging flows

VIP for scalars is used only in the advection phase– ALE could smooth the Saltzman mesh making it regular

– Sedov test will lack resolution at the shock in an Eulerian or smoothed out non-adaptive ALE calculation

In the examples VIP for scalars is compared with using the convex hull limiter only for vectors and “scalar limiting” (Eqs.1,3,4) for scalar variables

The Standing Shock Saltzman Test

In the original test, a piston moves with a velocity of u=10 into an almost cold ideal gas. This 1D problem is solved on a 2D 100x10 mesh as shown.

Instead, we look at the same mesh fixed in a frame of reference in which the shock is stationary with the gas flowing in from the right and out to the left

33.13;3/5];0,1,00667[.,,];10,4,33.133[,, SRL Uupup

0;3/5];33.13,1,00667[.,,];333.3,4,33.133[,, SRL Uupup

Saltzman with standing wave T=0.4 isodensity plot

(a) VIP vectors & scalars (b) VIP vectors

Saltzman with standing wave T=0.4 Profiles of density and pressure

(a) VIP vectors & scalars (b) VIP vectors

2D Noh Test

A disk of cold ideal gas

collapses to its axis with A 100x100x2 disk is meshed with 50x50x1 zones The analytic solution is a shock expanding

with and density profile: ALE motion is limited to preserve resolution at

and behind the shock

0,3/5 e

]1,1,0[],,[ rup

3/1SU

trtrt

trr

3/;/1

3/0;16)(

Noh 2D T60 iso-density plot(a) VIP vectors & scalars (b) VIP vectors

Noh 2D T60 iso-density plot inside(a) VIP vectors & scalars (b) VIP vectors

“Noh Corner” Test

A disk of cold ideal gas moves toward the origin with the velocity directed in each quadrant along the diagonal as shown

Due to the symmetry only a quadrant has to be considered. We carry out this simulation in a frame in which the gas is initially stationary and the walls are moving

In a “hot gas” corner flow and

0,3/5 e]1,1,0[],,[ nup

]1,1,2.0[],,[ nup 3.0e

u

u

u

u

“Cold” Noh Corner Test Iso-densities at T=75.0100x100x1 zones

(a) VIP vectors and scalars (b) VIP vectors only

a b

“Cold” Noh Corner Test Iso-densities at T=75.0VIP vectors and scalars

(a) 200x200x1 zones (b) 100x100x1 zones

“Hot” Noh Corner Test Iso-densities at T=70.0100x100x1 zones

(a) VIP vectors and scalars (b) VIP vectors only

Sedov-Taylor blast wave problem

The initial data is: cold ideal gas with a “point source” in a single zone at the origin

The physical space is a 253 box divided into a uniform 903 mesh with 3D computation conducted in the octant with the symmetry planes at

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

To keep the mesh fine near the shock, the ALE motion was restricted to start at densities

0,3/5];0,1,0[,, 0 eup 7.5027e

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

41R 1T

25.0

Sedov-Taylor 3D Test T=1 VIP vectors & scalars isodensity plot

2D Sedov-Taylor test

The initial data is: To remove any effect of boundary conditions we

ran the simulations in all 4 quadrants:

We set at the 4 central zones The stronger cylindrical shock reaches

at The ALE motion was again restricted to

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

7.5027e1R

24.0T

25.0

25,25 yx

Sedov-Taylor 2D Test T=0.24isodensity plot

(a) VIP vectors and scalars (b) VIP vectors only

Sedov-Taylor 2D Test T=0.24isodensity plot inside

(a) VIP vectors and scalars (b) VIP vectors only

Conclusions

We have devised a new VIP based limiter for the gradient of scalar zone centered variables

This limiter is applied if either the unlimited gradient lies outside the VIP spanned by (twice) the gradients in the neighbor zones or if changes sign

Near a shock the second criterion dominates, so that we do not have to compute the convex hull and we zero the limited gradient. Thus for this case the results are identical to the scalar limiter (Eq. (1),(3),(4) )

n

More Conclusions

In principle the new limiter should be less dissipative than the “scalar limiter”

While we see some differences in the finer details pointing to this, in the test cases shown the results using both limiters are quite similar, virtually identical.

On one hand the new limiter performs well in all the test cases, on the other hand we will consider more test cases where it may have a clear advantage over scalar limiters.

Future Perspectives

This limiter is applied once to the zone centered gradient. The same limited gradient is used for fluxes at all zone faces.

Other variants, e.g. a directional limiter, should also be considered

References Luttwak G. , Falcovitz J., ”Slope Limiting for vectors: A Novel Vector

Limiting Algorithm”, Int.J.Num.Meth.Fluids,65,p1365-1375,(2011), presented at the Conf. Num. Meth. for Multi-Material Flows , Pavia, Italy, (2009)

Luttwak G. , Falcovitz J., “Vector Image Polygon (VIP) limiters in ALE Hydrodynamics”, EPJ Web of Conferences,10,00020,(2010), presented at the 8th Int. Conf. on New Models and Hydro-codes, Paris, France, (2010)

M.Berger, M.J.Aftosmis, ”Analysis of slope limiters on Irregular Grids”, NAS Technical Report NAS-05-007,[2005]

W. J. Rider, D. B. Kothe, “Constrained Minimization for Monotonic Reconstruction,” AIAA-97–2036, 1997.

Dukowicz J. K. and Kodis J. W. “Accurate conservative remapping (rezoning) for ALE computations, SIAM J. Scient. Stat. Comp., (8),p305-321,1987

Luttwak G. , Falcovitz J., ”Applying the SMG Scheme to Reactive Flow”, J. Energetic Materials, 28(1), p279 - 302,(2010), presented at the 7th Int. Conf. on New Models and Hydro-codes, Lisbon,Portugal,2008

References Luttwak, G. ”Staggered Mesh Godunov (SMG) Schemes for

Lagrangian Hydrodynamics”, p339-342, Shock Compression of Condensed Matter-2005, Furnish M. D. et al Eds, (2006), AIP, CP

Luttwak G., “ Sliding and Multifluid Velocities in Staggered Mesh (MMALE) Codes ", Conference/Workshop On Numerical Methods For Multi-Material Flows, Prague , Sept. 2007, www-troja.fjfi.cvut.cz/~multimat07

Luttwak G. , Falcovitz J. , "Staggered Mesh Godunov (SMG) Schemes for ALE Hydrodynamics", Workshop On Numerical Methods For Multi-Material Flows , Oxford, UK, Sept. 2005 , www.extra.rdg.ac.uk/ifcd/Multimaterial-Workshop.htm

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.

Christensen R.B., "Godunov Methods on a Staggered Mesh. An Improved Artificial Viscosity", L.L.N.L report UCRL-JC-105269, (1990).

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

D.J.Benson,”Computational Methods in Lagrangian and Eulerian Hydrocodes”, Comp.Meth.Appl.Mech. Engn.,99 (1992) ,p235-391

References

How does a slope limiter work?

Familiar algorithms are formulated for scalar variables. The slope extrapolated values of a variable must lie in the range

defined by the values of that variable in neighboring cells ν:

For vectors, limiting is usually applied separately to each component.

Such procedure is frame-dependent:– Rotating the coordinate axes produces different results– Component limiters do not preserve problem symmetry– They do not transform like vector entities should

)max()min( 0lim0 rr

The SMG VIP Vector Limiter Both for the Lagrangian and Advection phases of the SMG scheme

(in 2D/3D). The extension to a general connectivity (FEM) mesh is

straightforward The limiting is done separately along each edge of a cell (e.g. 1-2 in

the figure) :– The outward extrapolated velocities must lie in the VIPs of edge nodes

1,2– The limited gradient assumed to lie along the velocity difference

2,12,12,1122,1 /ˆ; vvvvvv

5

4

12

12 312

12

8

6

7

SMG VIP vector limiter

The velocities of 1,2 are extrapolated outward along

the edge. e.g. velocity of 2 extrapolated toward middle of 23:

The cell centered velocity gradient is used, taking its component along the velocity difference :

1212222,2 ˆˆ vvrvvv ie

1212232 ˆ)ˆ(5.0 rrrr

2,1v

5

4

12

12 312

12

8

6

7

SMG VIP limiter

The value of chosen to keep inside the VIP of the edge neighbors (1,5,3,4)

The same way chosen to keep inside the VIP of the edge neighbors (2,6,7,8)

The VIP limited slope along 12 generates a velocity jump between the corner zones :

where:

The above jump serves as the data for the IRP in the Lagrange phase

10 2 ev ,2

1 ev ,1

1212122112 ˆ)(ˆ,min vrvvv red

redvvv 121212

12v

5

4

12

12 312

12

8

6

7

Momentum Advection on Staggered mesh

Momentum integrated around corner zones around vertex O (12345678 in the figure)

132

4

56 7

8O

st stV V

g dsuuudVudt

d)(

The momentum advection phase

Momentum fluxes at the mid-faces between the corner zones found using the limited velocity profile

The vertex mass flux between these corner zones found from the mass flux through the zone faces (similar to Benson HIS)

The upstream weighted velocity at the middle of the fluxed volume is:

here d,a are the donor and acceptor sides (1or 2) And the momentum flux =>

redda

d

dadf v

m

mvv

15.0

fdavm

The Staggered Mesh Godunov (SMG/Q) Scheme

Let p* be the RP solution pressure. We take as a uni-axial tensor pseudo-viscosity acting along the shock direction.

Its impulse is imparted to the two neighboring vertices. Its work, which must be dissipative, is added to the zone

internal energy.– Some of these ideas are related to Christensen’s split-Q, and to

the edge viscosity and compatible hydro-scheme by Caramana et al.

cellppq *

When is a vector “monotonic” relative to its neighbors?

We seek a monotonicity preservation criterion for a vector, analogous to the scalar criterion:

We have defined VIP - Vector Image Polygon in 2D (Vector Image Polyhedron in 3D) as the convex hull of the vector-space points corresponding to the neighbor vectors.

If a slope-extrapolated vector lies inside the VIP, the slope is monotonicity preserving. Otherwise, slope limiting is required.

If the original vector lies entirely outside the VIP, we zero its slopes, in analogy to the scalar extremum case

)(maxmin

The VIP Monotonicity Criterion

A point inside a convex hull can be expressed as:

In 1D (n=2) this reduces to:

which is equivalent to the scalar criterion (1) A vector point lying on the line connecting two

neighbor points should be monotonicity complying. Since VIP is convex any segment connecting two vertices lies inside the VIP (or on boundary).

The VIP monotonicity criterion is a natural extension of (scalar) monotonicity to vectors.

1;10;11

n

ii

n

iiiivv

1

3

A

2

10;)1( 21 vvv

Falcovitz: Cold Noh Corner Test1000x1000 Eulerian GRP calculation