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.