Vector Image Polygon (VIP) limiters in ALE Hydrodynamics

Transcription

1 New Models and Hydrocodes, Paris, France,24-28 May 2010 Vector Image Polygon (VIP) limiters in ALE Hydrodynamics Gabi Luttwak 1 and Joseph Falcovitz 2 1 Rafael, P.O. Box 2250, Haifa 31021, Israel 2 Institute of Mathematics, The Hebrew University of Jerusalem, Israel

2 VIP-Convex Hull based limiters What is VIP? The SMG/Q scheme VIP => for the Lagrangian phase of the SMG/Q scheme (Pavia 2009) VIP => for the advection phase of ALE/Euler SMG calculations Test Case - 2D Axial Symmetric Noh VIP - future perspectives

3 In Finite Volume Schemes The conservation laws solved over a control volume : (e.g. the computational zone-for zone centered variables) d dt V dv Fluxes evaluated on the control volume faces These require the face (and time step) centered values of the variables V ( u u g ) ds

4 Slope Limiters Second order Godunov and some other high resolution schemes use the gradients of the variables to get face-value: f c r r c In hyperbolic PDE discontinuities (shock or slip lines) are present, or can be generated, during the flow. At discontinuities the gradients diverge. Using them produces unphysical fluctuations in the solution Thus the gradients or the resulting fluxes must be limited to preserve a monotonic flow field f c

5 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: min( ) 0 lim r r0 max( ) 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

6 When is a vector monotonic relative to its neighbors? We seek a monotonicity preservation criterion for a vector, analogous to the scalar criterion: 1 max ( ) min 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

7 The VIP Monotonicity Criterion A point inside a convex hull can be expressed as: n n 2 v iv i ;0 i 1; i 1 i1 i1 In 1D (n=2) this reduces to: 3 v v1 (1 ) v2 ;0 1 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. 2 A 1 3

8 A Convex Hull (CH)

9 The VIP convex hull limiter Scalar Limiting V e V V r ) 0 c ( e r0 V e V 1 V 2 V 3 V 0 V c V 6 V 5 V 4 if V e is inside VIP => no limiting: V e unchanged If V 0 is outside, slopes are zeroed and V e =V 0 If V e is outside, VIP limiter is applied and V e => V c

10 The Staggered Mesh Godunov (SMG/Q) Scheme SMG schemes for 3D Lagrangian and ALE hydrodynamics were presented at Oxford 2005, Prague 2007, 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. p, The IRP are solved in the normal to shock direction, i.e. along the velocity difference ( u L u R ). This defines a uniaxial tensor pseudoviscosity.

11 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. This limiter also serves as a shock detector. The resulting scheme captures shocks with sharp monotonic profiles. It also has a strong inherent mesh stabilizing effect. u u u L u L Limited slope C u u u R R F Q F Q Original slope Velocity jump for RP

12 The Staggered Mesh Godunov (SMG/Q) Scheme Let p* be the RP solution pressure. We take q p* 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. p cell Some of these ideas are related to Christensen s split-q, and to the edge viscosity and compatible hydro-scheme by Caramana et al.

13 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 4 6 v ˆ 1,2 v2 v1 ; v1,2 v1,2 / v 1,

14 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: r 2 0.5( r ˆ ˆ 23r12) r12 The cell centered velocity gradient is used, taking its component along the velocity difference v : 1,2 v v i r 2 vˆ 12 v 12 2, v 2 2 ˆ e

15 SMG VIP limiter The value of chosen to keep 2 inside the VIP of e the edge neighbors (1,5,3,4) The same way chosen to keep v inside the VIP of the edge 1 1, e neighbors (2,6,7,8) The VIP limited slope along 12 generates a velocity jump between the corner zones : where: The above jump Lagrange phase v, red v12 v12 v12 v red 12 min ˆ v 12 1, 2 v ˆ 12 ( v r 12 ) v 12 serves as the data for the IRP in the

16 Momentum Advection on Staggered mesh Momentum integrated around corner zones around vertex O ( in the figure) d dt V st udv V u( u st u g ) ds O

17 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: m v f v here d,a are the donor and acceptor sides (1or 2) And the momentum flux => m d dav f m da d v red da

18 Slope Limiter - Flux Limiter Slope limiters are predictors Simple explicit Some Flux limiters, FCT - correctors can be implicit or more dissipative Slope Limiter + corrector step? Both with the VIP monotonicity criterion!

19 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 1/ 3 U S and density profile: 5/ 3, e 0 [ p,, ] [0,1, 1] u r 16 ; 0 r t / 3 ( r) 1 t / r ; t / 3 r t

20 2D LAG+VIP : Noh Lag+CL

21 2D Noh Lagrange-ALE both with VIP

22 2D Noh ALE+CL : ALE+VIP

23 2D NOH ALE+VIP More mesh smoothing Less mesh smoothing

24 Conclusions CH VIP limiter implemented both for Lag and ALE phase of SMG/Q In the 2D Noh test it improves the symmetry preservation

25 Future perspectives VIP in other high resolution schemes The concept of VIP CH based monotonicity can/should be applied to other schemes too TIP -Tensor image polyhedron e.g. stress

26 References G. Luttwak, J.Falcovitz, Slope Limiting for vectors: A Novel Vector Limiting Algorithm, Conf. on Numerical Methods For Multi- Material Flows, Pavia, Italy, Sept. 2009, to appear in Int.J.Num.Meth.Fluids,2010 M.Berger, M.J.Aftosmis, Analysis of slope limiters on Irregular Grids, NAS Technical Report NAS ,[2005] R. Loubere, On the effect of the different limiters for the tensor articial viscosity for the Compatible Lagrangian Hydrodynamics Scheme, LANL rep. LA-UR ,2005 W. J. Rider, D. B. Kothe, Constrained Minimization for Monotonic Reconstruction, AIAA , M. Hubbard, Multidimensional Slope Limiters for MUSCL-Type Finite Volume Schemes on Unstructured Grids, J.Comp. Phys., 155, 1999, pp

27 References Luttwak, G. Staggered Mesh Godunov (SMG) Schemes for Lagrangian Hydrodynamics, p , 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, wwwtroja.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, Luttwak, G., "Comparing Lagrangian Godunov and Pseudo-Viscosity Schemes for Multi-Dimensional Impact Simulations, p , Shock Compression of Condensed Matter-2001, Furnish M. D. et al Eds, (2002), AIP, CP620.

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