1. A Level-Set Method for Multimaterial Radiative Shock Hydrodynamics
David Starinshak and Smadar Karni
Department of Mathematics, University of Michigan
Center for Radiative Shock Hydrodynamics (CRASH)
MULTIMAT 2011, September 5-9, Arcachon, France
Funding: DoE NNSA-ASC grant DE-FC52-08NA28616, NSF Award DMS and University of Michigan Rackham Travel Grant.

2. Goals and Outline OUTLINE:
Develop a 2D multimaterial rad-hydro code which employs state-of-the-art level set technology
Address concerns of species mass conservation and spurious evaporation in the context of HEDP and rad-hydro
OUTLINE:
Physics: Laser-driven HEDP experiments
Mathematics: Model equations
Numerics: Solver and Discretization Strategies
Test Problems and Results
Future Work and Extensions
Material dependencies play a huge roll in multi-physics systems like radiation-hydrodynamics

3. The CRASH Experiment Key Physical Features:
( ~ 3.8 kJ )
( ~ 200 km/s )
Key Physical Features:
Strong radiative shock front
High energy density regime
Strong radiation-hydro coupling
Multimaterial
Ma ~ , radiation heats upstream media
ionization, > 1 g/cc, Tmat > 1 keV, Trad > 50 eV
prad ~ pmat , non-equilibrium
EOS and opacities vary across sharp interfaces
*X-ray Radiograph from CRASH experiment (FW Doss, 2011)

4. Wall Ablation & Entrainment Problem
Irradiated wall heats up and ablates
Sends wall shock into Xe
Thick-thin Interface
Rad transport depends on interface geometry
plastic
unperturbed
xenon
radiative precursor
hydro
shock
wall shock
interface
heat front
( ~ 1 ns )
shocked
beryllium
Entrainment
Shear wave instability driven by shock front
Interface dynamics drive xenon entrainment
shocked
beryllium
plasma
hydro
shock
wall shock
entrained xenon
dense shocked xenon
( ~ 10 ns )

5. Model Equations Multi-Group Flux-Limited Diffusion with Ionization
Simplifications and Amendments
Monoenergetic “Gray” Radiation
Zero Explicit Ionization
Zero Electron Heat Flux
Single Opacity
Reduced Model:

6. Model Equations
Conservation of mass, momentum, and (material + radiation) energy
System Closure:
Opacity Model:
Material Parameterization:
Adiabatic index
Ideal gas constant
Opacity coefficient

7. Model Equations Source Exchanges energy between matter and radiation
Blackbody Emission
Absorption
Exchanges energy between matter and radiation
Attempts to equilibrate system:
Radiation Heat Flux

8. The Level Set Model for Interfaces
Key Assumption
Materials are immiscible over timescales of interest
Sign of level curve determines material type
Level Curves advect at local flow velocity
Material-dependent quantities are discontinuous functions of the level curve

9. Numerical Solver Summary
Operator Splitting:
Explicit : 2nd order multimaterial HLL solver of MUSCL-Hancock type
Nonconservative products discretized to conserve total energy
Material designations determined from updated level set
Implicit : nonlinear root-finding + finite volume diffusion solver
Newton-Raphson iteration on T and ER
2nd order Crank-Nicolson discretization in time

10. Level Set Discretization Strategies
Initialize as signed distance function (satisfies Eikonal condition: )
Many Attractive Methods Available
Black box hydro solver [HLL, upwind, PPM, etc.]
HOUC [Nourgaliev & Theofanous JCP 2006]
Particle LS [Enright et al JCP 2002]
LS-CIR / Semi-Lagrangian [Strain JCP 2000]
Fast Marching [Adalsteinsson & Sethian JCP 1999]
ENO/WENO for H-J Equations [Osher & Shu SIAM 1991]
Advection Form
Reinitialize:
where
Hamilton-Jacobi Form
Reinitialize:
where
Extension Velocity H-J
Extend normal velocity:
Eikonal condition not the only way LS can be “well-behaved”
Lots of literature on new LS technology
Not a lot of literature on LS for compressible or rad-hydro
Efficient Implementation: Update only in a narrow band around interfaces

11. Interfaces and Material Terms
Sub-Zonal Interface reconstruction
bi-linear interpolation from nodal level set values
Interface normal obtained from interpolant
Sharp Volume Fractions
Integrate Heaviside function of interpolant
Fluids are immiscible HOWEVER numerically, it is oftentimes important to bring in physics below resolution of the cells
Strategy for Material Terms
Weighted by the computed volume fraction for each species

12. Diffusion Solver Discretization
Nonlinear, material-dependent, flux-limited heat conduction:
Coefficient
Flux limiter
Radiative Knudson number
Discretization: Crank-Nicolson in time; Finite volume, centered-differencing in space
Diffusion coefficient constructed cell faces
Arithmetic average of ER
Single-material faces:
Harmonic avg of opacity
(Arithmetic of DR)
Two-material faces:
Arithmetic avg of opacity
(Harmonic of DR)
“Mixed” cells treated as separate material

13. Diffusion Solver Discretization
Nonlinear, material-dependent, flux-limited heat conduction:
Coefficient
Flux limiter
Radiative Knudson number
Discretization: Crank-Nicolson in time; Finite volume, centered-differencing in space
Diffusion coefficient constructed cell faces
Arithmetic average of ER
Single-material faces:
Harmonic avg of opacity
(Arithmetic of DR)
Two-material faces:
Arithmetic avg of opacity
(Harmonic of DR)
“Mixed” cells treated as separate material

14. Spurious Pressure Oscillations
Pressure equilibrium not respected across diffused material fronts
Well-understood phenomenon for multi-fluid systems
Consequence of prescribing a mixed-cell EOS across (numerically) diffused interfaces
Does not occur in single-fluid systems
Oscillations
Occur at every time level
Occur in 1st order solvers
Not removed by mesh refinement
Not removed by high-order solvers
Better control over pressure values is required near interfaces
Remedies: Ghost Fluids Pressure Evolution Single-fluid Solver
R Fedkiw et al, JCP, 1999
S Karni, JCP, 1994
S Karni & R Abgrall, Proceedings, 2001

15. Single-Fluid Multimaterial Solver
Interface
Left State
Right State
Material 1
Material 2
Resolve waves at cell boundary so that WL and WR “see” the same fluid on the other side
Properties
Total mass and momentum perfectly conserved
(Material + radiation) energy essentially conserved
Conserved almost everywhere
Errors are small, do not accumulate, and tend to zero with mesh refinement
NOTE: Material and radiation energies not individually conserved
S Karni & R Abgrall, Ghost-Fluids for the Poor: A Single Fluid Algorithm for Multifluids. Proceedings ICHP

16. Pressure Defects in Pure Hydro
Multimaterial
Sod Shock Tube:
Pressure defect develops across interface
Single-fluid solver: defect removed with -0.17% error in energy conservation

17. Pressure Defects in Rad-Hydro
Source-Dominated Shock Tube:
Conservation Error
< 0.12%
Diffusion-Dominated Shock Tube:
Conservation Error
< 0.09%

18. 1D Wall Ablation Problem
Hot ionized gas
plastic
Radiative precursor
Beryllium
plasma
Interface
xenon
Cold dense wall
1D shock tube initial conditions:
Radiation in equilibrium:
Temperature-dependent opacity:
Boundary conditions:
- Radiating left boundary (T = 100 eV)
- Zero gradient at right boundary

19. 1D Wall Ablation Problem
Semi-analytic scaling*
Computed Solution
Dense
shock
Heat
front
Wall
shock
Interface
*Used by permission (Drake et al, 2010, preprint)
NOTE: System loses positivity if spurious pressure oscillations not addressed

20. 1D Wall Ablation Problem

21. Species Mass Conservation
Interfaces are sharp, but material fronts diffuse numerically
Conservation of individual species masses not guaranteed
1. Density can vary by orders of magnitude across interfaces
Small changes in interface position large species mass errors
2. Opacity and EOS sensitive to density changes
Radiative transfer rates
Overflowing bounds of EOS / opacity table
3. Spurious evaporation of entrained fluid species
Complications to Rad-Hydro Models:

22. Species Mass Conservation Errors
Errors manifest as mass exchange between fluid species
+ 28.8%
+ 20.7%
+ 14.3%
+ 2.78%
+ 0.42%
+ 0.18%
Reinitialize Level Curves
Errors do not accumulate for isolated fronts
Errors tend to zero with grid refinement (1st order)
Can accelerate convergence by steepening contact

23. Summary Accomplishments Future Work Implementation
Verified code for 1D, 2-material rad-hydro
Implemented and begun testing 2D, N-material hydro with “modern” LS solvers
Characterized species mass errors in 1D
Future Work
Implementation
2D extensions of source and diffusion solvers
Generalize LS methods to 3+ materials
Sub-Grid Material Resolution
Mixed-cell diffusion solver (tensor?)
Multimaterial source (!)
LS-informed adaptive grids
Species Mass Conservation Errors
Consistency conditions: VOF and LS
Modify reinitialization methods

24. Acknowledgements References
Smadar Karni, Eric Myra, Bruce Fryxell, Ken Powell, and Paul Drake
The entire CRASH Team
University of Michigan, Texas A&M, and Simon Fraser contributors
U.S. Department of Energy’s NNSA-ASC Program, NSF, Rackham Graduate School
MULTIMAT 2011 organizing committee
References
R Abgrall & S Karni, Computations of Compressible Multifluids, JCP, 169, 594 (2001).
R Drake et al, Behavior of Irradiated Low-Z Walls and Adjacent Plasma (preprint, 2010).
R Fedkiw, T Aslam, B Merriman, & S Osher, A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method), JCP, 152, 457 (1999).
S Karni, Multicomponent Flow Calculations by a Consistent Primitive Algorithm, JCP, 112, 1 (1994).
S Karni & R Abgrall, Ghost-Fluids for the Poor: A Single Fluid Algorithm for Multifluids. Proceedings of the 10th International Conference on Hyperbolic Problems, Theory and Numerics
C Levermore & G Pomraning, A Flux-Limited Diffusion Theory, Astrophys. J., 248: (1981).
R Lowrie, J Morel, & J Hittinger, The Coupling of Radiation and Hydrodynamics, Astrophys. J., 521: (1999).
R Lowrie, R Rauenzahn, Radiative Shocks in the Equilibrium Diffusion Limit, Shock Waves, 16: (2007).
R Lowrie, J Edwards, Radiative Shocks with Grey Nonequilibrium Diffusion, Shock Waves, 18: (2008).
Mihalas & Mihalas, Foundations of Radiation Hydrodynamics, 1983.
W Mulder, S Osher, J Sethian, Computing Interface Motion in Compressible Gas Dynamics, JCP, 100, 2009 (1992).
B van der Holst, G Toth, I Sokolov, K Powell, J Holloway, E Myra, Q Stout, M Adams, J Morel, S Karni, R Drake, CRASH: A Block-Adaptive-Mesh Code for Radiative Shock Hydrodynamics (preprint, 2011)

26. Implicit Solver Solve: Use EOS to transform E to T:
Kinetic terms do not vary in time, and material terms are frozen at * time level:
Implicit Backward Euler discretization in time:
FV Spatial Discretization
Tridiagonal matrix in 1D
Banded tridiagonal in 2D
Arithmetic averaging gives DR*
at cell boundaries
Vectorize:

27. Implicit Solver (Cont)
Solve the nonlinear vector-operator equation:
Equivalently, perform nonlinear root-finding:
where
Newton-Raphson Iteration:
Initial iterate:
NOTE:
Inverting the Jacobian matrix
amounts to inverting the matrix from the
discretized diffusion operator. This is done
at each iteration.
Alternatives: preconditioned CG or GMRES

28. Pressure Evolution Solver
Away from Interfaces
Near Interfaces
Solve material energy equation:
Solve material pressure equation:
Form pressure using EOS:
Form material energy using EOS:
Solving for pressure directly ensures its continuity across material fronts
Solver Properties
Perfectly conserves total mass and momentum
Essentially conserves energy
Conserved almost everywhere in computational domain
Conservation errors are small and do not accumulate
Errors decrease with mesh refinement
NOTE: Material and radiation energies not strictly conserved for this system; their sum is.

29. Spurious Pressure Oscillations
Pressure equilibrium not respected across material fronts
Pressure computed from diffused hydro quantities using the EOS
Two EOS exist across interfaces: a mixed-cell EOS is needed
As interface moves: mixed-fluid EOS becomes inconsistent with hydro variables
A pressure defect develops, sending signals into neighboring cells
Oscillations
Occur at every time level
Occur in 1st order solvers
Not removed by mesh refinement
Not removed by high-order solvers
Better control over pressure values is required near interfaces
Remedies: Ghost Fluids Pressure Evolution Single-fluid Solvers
R Fedkiw et al, JCP, 1999
S Karni, JCP, 1994
S Karni & R Abgrall, Proceedings, 2001

30. Single-Fluid Multimaterial Solver
Interface
Left State
Right State
Material 1
Material 2
Waves updating WL
Start with primitive variables
Form WL and WR using EOS from Mat. 1
Waves updating WR
Start with primitive variables:
Form WL and WR using EOS from Mat. 2
Properties
Perfectly conserves total mass and momentum
Essentially conserves energy
Conserved almost everywhere in computational domain
Conservation errors are small and do not accumulate
Errors decrease with mesh refinement
NOTE: Material and radiation energies not strictly conserved for this system; their sum is.