A Level-Set Method for Multimaterial Radiative Shock Hydrodynamics

MULTIMAT 2011, September 5-9, Arcachon, France

David Starinshak and Smadar Karni

Department of Mathematics, University of MichiganCenter for Radiative Shock Hydrodynamics (CRASH)

Funding: DoE NNSA-ASC grant DE-FC52-08NA28616, NSF Award DMS 0609766 and University of Michigan Rackham Travel Grant.

• Physics: Laser-driven HEDP experiments

• Mathematics: Model equations

• Numerics: Solver and Discretization Strategies

• Test Problems and Results

• Future Work and Extensions

1. Develop a 2D multimaterial rad-hydro code which employs state-of-the-art level set technology

2. Address concerns of species mass conservation and spurious evaporation in the context of HEDP and rad-hydro

OUTLINE:

Goals and Outline

The CRASH Experiment

Key Physical Features:

( ~ 3.8 kJ ) ( ~ 200 km/s )

1. Strong radiative shock front

2. High energy density regime

3. Strong radiation-hydro coupling

4. Multimaterial

Ma ~ 102-103, 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)

Wall Ablation & Entrainment Problem

Entrainment• Shear wave instability

driven by shock front• Interface dynamics drive

xenon entrainment

shockedberylliumplasma

hydroshock

wall shock

entrained xenon

dense shocked xenon ( ~ 10 ns )

Wall Ablation• Irradiated wall heats up and

ablates• Sends wall shock into Xe• Thick-thin Interface• Rad transport depends on

interface geometryplastic

unperturbedxenon

radiative precursor

hydroshock

wall shockinterface

heat front

( ~ 1 ns )

shockedberyllium

Simplifications and Amendments

Monoenergetic “Gray” Radiation

Zero Explicit Ionization

Zero Electron Heat Flux

Single Opacity

Reduced Model:

Multi-Group Flux-Limited Diffusion with Ionization

Model Equations

Model Equations

Conservation of mass, momentum, and (material + radiation) energy

Material Parameterization:Adiabatic index Ideal gas constant Opacity coefficient

System Closure:

Opacity Model:

Model Equations

Radiation Heat Flux

Source

Blackbody Emission Absorption

• Exchanges energy between matter and radiation

• Attempts to equilibrate system:

The Level Set Model for InterfacesKey AssumptionMaterials 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

Numerical Solver Summary

Operator Splitting:

Implicit : nonlinear root-finding + finite volume diffusion solver

• Newton-Raphson iteration on T and ER

• 2nd order Crank-Nicolson discretization in time

Explicit : 2nd order multimaterial HLL solver of MUSCL-Hancock type• Nonconservative products discretized to conserve total energy• Material designations determined from updated level set

Level Set Discretization StrategiesInitialize as signed distance function (satisfies Eikonal condition: )

Extension Velocity H-J

Extend normal velocity:

Efficient Implementation: Update only in a narrow band around interfaces

Many Attractive Methods Available1. Black box hydro solver [HLL, upwind, PPM, etc.]2. HOUC [Nourgaliev &

Theofanous JCP 2006]3. Particle LS [Enright et al

JCP 2002]4. LS-CIR / Semi-Lagrangian [Strain JCP 2000]5. Fast Marching [Adalsteinsson & Sethian

JCP 1999]6. ENO/WENO for H-J Equations [Osher & Shu SIAM 1991]

Advection Form

Reinitialize:

where

Hamilton-Jacobi Form

Reinitialize:

where

Interfaces and Material Terms

Strategy for Material Terms• Weighted by the computed volume fraction for each species

Sharp Volume Fractions• Integrate Heaviside function of interpolant

Sub-Zonal Interface reconstruction• bi-linear interpolation from nodal level set values

• Interface normal obtained from interpolant

Diffusion Solver DiscretizationNonlinear, material-dependent, flux-limited heat conduction:

Coefficient Flux limiter Radiative Knudson number

Discretization: Crank-Nicolson in time; Finite volume, centered-differencing in space

• 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

Diffusion coefficient constructed cell faces

Diffusion Solver DiscretizationNonlinear, material-dependent, flux-limited heat conduction:

Coefficient Flux limiter Radiative Knudson number

Discretization: Crank-Nicolson in time; Finite volume, centered-differencing in space

• 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

Diffusion coefficient constructed cell faces

Spurious Pressure OscillationsPressure 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

Single-Fluid Multimaterial Solver

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

Material 1 Material 2

Left State Right State

Interface

Resolve waves at cell boundary so that WL and WR “see” the same fluid on the other side

S Karni & R Abgrall, Ghost-Fluids for the Poor: A Single Fluid Algorithm for Multifluids. Proceedings ICHP. 2001.

Pressure Defects in Pure HydroMultimaterial Sod Shock Tube:

• Pressure defect develops across interface• Single-fluid solver: defect removed with -0.17% error in energy conservation

Pressure Defects in Rad-HydroSource-Dominated Shock Tube:

Diffusion-Dominated Shock Tube:

Conservation Error < 0.12%

Conservation Error< 0.09%

1D Wall Ablation Problem

1D shock tube initial conditions:

Hot ionized gas

Cold dense wall

Interface

plasticRadiative precursor

xenon

Berylliumplasma

Radiation in equilibrium:

Temperature-dependent opacity:

- Radiating left boundary (T = 100 eV)

- Zero gradient at right boundary

Boundary conditions:

1D Wall Ablation ProblemSemi-analytic scaling*

*Used by permission (Drake et al, 2010, preprint)

Computed Solution

Wallshock

Interface

Heatfront

Denseshock

NOTE: System loses positivity if spurious pressure oscillations not addressed

1D Wall Ablation Problem

Species Mass ConservationInterfaces 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:

Species Mass Conservation ErrorsErrors manifest as mass exchange between fluid species

+ 28.8%

+ 20.7%

+ 14.3%

+ 2.78%

+ 0.42%

+ 0.18%

Reinitialize Level Curves

Summary

Future WorkImplementation

– 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

Accomplishments– 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

References

Smadar Karni, Eric Myra, Bruce Fryxell, Ken Powell, and Paul DrakeThe entire CRASH Team

University of Michigan, Texas A&M, and Simon Fraser contributorsU.S. Department of Energy’s NNSA-ASC Program, NSF, Rackham Graduate SchoolMULTIMAT 2011 organizing committee

1. R Abgrall & S Karni, Computations of Compressible Multifluids, JCP, 169, 594 (2001).2. R Drake et al, Behavior of Irradiated Low-Z Walls and Adjacent Plasma (preprint, 2010).3. 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).4. S Karni, Multicomponent Flow Calculations by a Consistent Primitive Algorithm, JCP, 112, 1 (1994).5. 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. 2001.6. C Levermore & G Pomraning, A Flux-Limited Diffusion Theory, Astrophys. J., 248: 321-334 (1981).7. R Lowrie, J Morel, & J Hittinger, The Coupling of Radiation and Hydrodynamics, Astrophys. J., 521: 432-450 (1999).8. R Lowrie, R Rauenzahn, Radiative Shocks in the Equilibrium Diffusion Limit, Shock Waves, 16: 445-453 (2007).9. R Lowrie, J Edwards, Radiative Shocks with Grey Nonequilibrium Diffusion, Shock Waves, 18: 129-143 (2008).10. Mihalas & Mihalas, Foundations of Radiation Hydrodynamics, 1983.11. W Mulder, S Osher, J Sethian, Computing Interface Motion in Compressible Gas Dynamics, JCP, 100, 2009 (1992).12. 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)

Acknowledgements

Implicit SolverSolve: 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:

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 thediscretized diffusion operator. This is doneat each iteration.Alternatives: preconditioned CG or GMRES

Pressure Evolution Solver

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.

Away from Interfaces Near InterfacesSolve 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

Spurious Pressure OscillationsPressure 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

Single-Fluid Multimaterial 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

Material 1 Material 2

Left State Right State

Interface

Waves updating WL

Start with primitive variablesForm WL and WR using EOS from Mat. 1

Waves updating WR

Start with primitive variables:Form WL and WR using EOS from Mat. 2

NOTE: Material and radiation energies not strictly conserved for this system; their sum is.