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Fully Implicit Navier-StokesHypersonic Flow Software
Roy H. Stogner
The University of Texas at Austin
May 4, 2009
Roy H. Stogner Hypersonic Flow May 4, 2009 1 / 30
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 2 / 30
Goals
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 3 / 30
Goals
PECOS Goals
Verification, Validation, Uncertainty Quantification• Advanced UQ research
I model validation/invalidationI parameter estimationI quantified parameter uncertaintyI quantified prediction uncertainty
• Demonstration on challenging multiphysics• Technology transfer to different physics
I QUESO software library
• Code Verification• Solution Verification
Roy H. Stogner Hypersonic Flow May 4, 2009 4 / 30
Goals
PECOS Goals
Numerics• Multiphysics Coupling
I Submodel testingI Adjoint Sensitivities
• Adaptive DiscretizationI Convergence testingI Error estimation
• Robustness
Software• Modularity
I Unit testingI Physics
independence• Extensibility
I Flexible submodelingI Operator verification
Physics• Complete model documentation
I Manufactured benchmarksI Exposed model parameters
Roy H. Stogner Hypersonic Flow May 4, 2009 5 / 30
Goals
PECOS Goals
Numerics• Multiphysics Coupling
I Submodel testingI Adjoint Sensitivities
• Adaptive DiscretizationI Convergence testingI Error estimation
• Robustness
Software• Modularity
I Unit testingI Physics
independence• Extensibility
I Flexible submodelingI Operator verification
Physics• Complete model documentation
I Manufactured benchmarksI Exposed model parameters
Roy H. Stogner Hypersonic Flow May 4, 2009 5 / 30
Goals
PECOS Goals
Numerics• Multiphysics Coupling
I Submodel testingI Adjoint Sensitivities
• Adaptive DiscretizationI Convergence testingI Error estimation
• Robustness
Software• Modularity
I Unit testingI Physics
independence• Extensibility
I Flexible submodelingI Operator verification
Physics• Complete model documentation
I Manufactured benchmarksI Exposed model parameters
Roy H. Stogner Hypersonic Flow May 4, 2009 5 / 30
Goals
PECOS Hypersonics Test Bed
FIN-S• Evolution from dissertation projects of Ben Kirk:• libMesh discretization library
I Parallel, adaptive unstructured meshesI Finite element, error estimation, time integration classesI Trilinos, PETSc linear algebra interfaces
• Fully Implicit Navier-Stokes applicationI Hypersonic flow formulation with stabilization, shock-capturing
MUTATION• MUlticomponent Transport And Thermodynamics of IONized gases.
I High enthalpy and plasma flowsI Thermal and chemical equilibrium/nonequilibrium modelsI Implementation independent of CFD code
Roy H. Stogner Hypersonic Flow May 4, 2009 6 / 30
FIN-S Motivation
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 7 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Local Sensitivity Information• Gives sensitivity estimates in one solve• Accelerates global sensitivity calculations• Reduces number of forward solves required by UQ
Adjoint-based Sensitivity Calculation• Builds on ICES experience with adjoint-based error analysis• Greatly accelerates calculation for problems with few quantities of
interest but many parameters• To be delivered in PECOS FY10:
I Development of adjoint-enhanced ablation codeI Exploration of adjoint-enhanced forward propagation of uncertaintyI Exploration of adjoint formulation of hypersonics flow codes
Roy H. Stogner Hypersonic Flow May 4, 2009 8 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Local Sensitivity Information• Gives sensitivity estimates in one solve• Accelerates global sensitivity calculations• Reduces number of forward solves required by UQ
Adjoint-based Sensitivity Calculation• Builds on ICES experience with adjoint-based error analysis• Greatly accelerates calculation for problems with few quantities of
interest but many parameters• To be delivered in PECOS FY10:
I Development of adjoint-enhanced ablation code
I Exploration of adjoint-enhanced forward propagation of uncertaintyI Exploration of adjoint formulation of hypersonics flow codes
Roy H. Stogner Hypersonic Flow May 4, 2009 8 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Local Sensitivity Information• Gives sensitivity estimates in one solve• Accelerates global sensitivity calculations• Reduces number of forward solves required by UQ
Adjoint-based Sensitivity Calculation• Builds on ICES experience with adjoint-based error analysis• Greatly accelerates calculation for problems with few quantities of
interest but many parameters• To be delivered in PECOS FY10:
I Development of adjoint-enhanced ablation codeI Exploration of adjoint-enhanced forward propagation of uncertainty
I Exploration of adjoint formulation of hypersonics flow codes
Roy H. Stogner Hypersonic Flow May 4, 2009 8 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Local Sensitivity Information• Gives sensitivity estimates in one solve• Accelerates global sensitivity calculations• Reduces number of forward solves required by UQ
Adjoint-based Sensitivity Calculation• Builds on ICES experience with adjoint-based error analysis• Greatly accelerates calculation for problems with few quantities of
interest but many parameters• To be delivered in PECOS FY10:
I Development of adjoint-enhanced ablation codeI Exploration of adjoint-enhanced forward propagation of uncertaintyI Exploration of adjoint formulation of hypersonics flow codes
Roy H. Stogner Hypersonic Flow May 4, 2009 8 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Primal Problem
R(u(p), v; p) ≡ 0 ∀vdRdp
= 0
∂R∂p
+∂R∂u
∂u
∂p= 0
Adjoint Problem
q′ ≡ dQ(u; p)dp
∂R(u, φ(u, p); p)∂u
≡ ∂Q(u; p)∂u
Costs• Forward solution expensive: May be highly nonlinear• Adjoint solution efficient: Just one linear solve• Forward implementation simple: requires residual• Adjoint implementation complicated: requires full Jacobian
Roy H. Stogner Hypersonic Flow May 4, 2009 9 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Primal Problem
R(u(p), v; p) ≡ 0 ∀vdRdp
= 0
∂R∂p
+∂R∂u
∂u
∂p= 0
Adjoint Problem
q′ ≡ dQ(u; p)dp
∂R(u, φ(u, p); p)∂u
≡ ∂Q(u; p)∂u
Costs• Forward solution expensive: May be highly nonlinear• Adjoint solution efficient: Just one linear solve• Forward implementation simple: requires residual• Adjoint implementation complicated: requires full Jacobian
Roy H. Stogner Hypersonic Flow May 4, 2009 9 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Primal Problem
R(u(p), v; p) ≡ 0 ∀vdRdp
= 0
∂R∂p
+∂R∂u
∂u
∂p= 0
Adjoint Problem
q′ ≡ dQ(u; p)dp
∂R(u, φ(u, p); p)∂u
≡ ∂Q(u; p)∂u
Costs• Forward solution expensive: May be highly nonlinear• Adjoint solution efficient: Just one linear solve• Forward implementation simple: requires residual• Adjoint implementation complicated: requires full Jacobian
Roy H. Stogner Hypersonic Flow May 4, 2009 9 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Adjoint-based Sensitivity
q′ =∂Q
∂p+∂Q
∂u
∂u
∂p
=∂Q
∂p+∂R(u, φ(u, p))
∂u
∂u
∂p
=∂Q
∂p− ∂R(u, φ(u, p))
∂p
Costs• One adjoint solve per quantity of interest• Only one inner product per uncertain parameter
Roy H. Stogner Hypersonic Flow May 4, 2009 10 / 30
FIN-S Motivation
Adjoint-based Sensitivity Analysis
Adjoint-based Sensitivity
q′ =∂Q
∂p+∂Q
∂u
∂u
∂p
=∂Q
∂p+∂R(u, φ(u, p))
∂u
∂u
∂p
=∂Q
∂p− ∂R(u, φ(u, p))
∂p
Costs• One adjoint solve per quantity of interest• Only one inner product per uncertain parameter
Roy H. Stogner Hypersonic Flow May 4, 2009 10 / 30
FIN-S Motivation
Fully Coupled Multiphysics
Loose Coupling Experiences• Linear, often slow convergence• Limits timesteps in transient, pseudo-transient solves• Sensitivity analysis requires finite differencing, repeated expensive
forward solves
Full Coupling• Quadratic solver convergence• Robust with implicit timestepping• More natural direct sensitivity analysis• Requires intrusive development of flexible software• Can require per-subdomain physics
Roy H. Stogner Hypersonic Flow May 4, 2009 11 / 30
FIN-S Motivation
Fully Coupled Multiphysics
Example libMesh physics codes• Flow and transport
I Compressible, incompressible, non-Newtonian Navier-StokesI Depth-averaged surfactant-driven thin filmsI Double-diffusion in porous mediaI Electrically driven microfluidics
• Cahn-Hilliard Phase Decomposition• Laplace-Young Surface Tension• Cancer Angiogenesis• Bacterial Chemotaxis• SPn Radiation Transport
Roy H. Stogner Hypersonic Flow May 4, 2009 12 / 30
FIN-S Motivation
Error Estimation
Discretization error• Error estimates are an inextricable part of UQ
Adaptive Discretization• Adaptive timestepping is critical for robustness• Adaptive meshing is important for efficiency, particularly for
goal-oriented problems
Roy H. Stogner Hypersonic Flow May 4, 2009 13 / 30
FIN-S Motivation
PECOS Development
ITAR Status• DPLR is non-accessible to many PECOS staff• DPLR cannot be run or stored on most PECOS systems• DPLR cannot be run or stored on all TACC or DOE systems• FIN-S is open source; MUTATION is international.• Both are modular enough to separate out restricted components.
Roy H. Stogner Hypersonic Flow May 4, 2009 14 / 30
FIN-S Motivation
PECOS Development
In-house expertise• ICES is a center of finite element research• PECOS paid staff includes core libMesh/FIN-S and MUTATION
developers
Collaborations• libMesh is under continuing development, internationally and locally
at ICES and TACC
• libMesh adjoint functionality is under development with collaboratorsat ICES and Idaho National Laboratory
• FIN-S formulations are under development with Ben Kirk at NASAand Steve Bova at Sandia National Laboratories
Roy H. Stogner Hypersonic Flow May 4, 2009 15 / 30
Numeric Formulation
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 16 / 30
Numeric Formulation
Current FIN-S ModelGoverning Equations
∂ρ
∂t+∇ · (ρ~u) = 0
∂ρ~u
∂t+∇ · (ρ~u~u) = −∇P +∇ · τ
∂ρE
∂t+∇ · (ρE~u) = −∇ · ~q −∇ · (P~u) +∇ · (τ~u)
Notation
∂~U
∂t+∂ ~Fi∂xi
=∂ ~Gi∂xi
∂~U
∂t+ Ai
∂~Ui∂xi
=∂
∂xi
(Kij
∂~U
∂xj
)Roy H. Stogner Hypersonic Flow May 4, 2009 17 / 30
Numeric Formulation
FIN-S Formulation
Weak FormulationWeighted Residuals with Streamline Upwind Petrov-Galerkin stabilizationand Consistent Artificial Diffusion shock capturing
∫Ω
[~W ·
(∂~U
∂t+∂ ~Fi∂xi
)+∂ ~W
∂xi·
(Kij
∂~U
∂xj
)]dΩ−
∫Γ
~W · ~gdΓ
+nel∑e=1
∫Ωe
τSUPG∂ ~W
∂xk·Ak
[∂~U
∂t+∂ ~Fi∂xi− ∂∂xi
(Kij
∂~U
∂xj
)]dΩ
+nel∑e=1
∫Ωe
δ
(∂ ~W
∂xi·AH
∂~U
∂xi
)dΩ = 0
Roy H. Stogner Hypersonic Flow May 4, 2009 18 / 30
Numeric Formulation
Ongoing Formulation Research
Stabilization• Impact of τSUPG on boundary fluxes• Stabilization with turbulence
Shock Capturing• Smoothing, differentiability of shock capturing operator δ• Shock capturing with chemistry
Enthalpy Preservation• Higher order adiabatic Fi,h
Roy H. Stogner Hypersonic Flow May 4, 2009 19 / 30
Numeric Formulation
Ongoing Formulation Research
Stabilization• Impact of τSUPG on boundary fluxes• Stabilization with turbulence
Shock Capturing• Smoothing, differentiability of shock capturing operator δ• Shock capturing with chemistry
Enthalpy Preservation• Higher order adiabatic Fi,h
Roy H. Stogner Hypersonic Flow May 4, 2009 19 / 30
Numeric Formulation
Ongoing Formulation Research
Stabilization• Impact of τSUPG on boundary fluxes• Stabilization with turbulence
Shock Capturing• Smoothing, differentiability of shock capturing operator δ• Shock capturing with chemistry
Enthalpy Preservation• Higher order adiabatic Fi,h
Roy H. Stogner Hypersonic Flow May 4, 2009 19 / 30
Solver Techniques
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 20 / 30
Solver Techniques
Time Integration
Current FIN-S Adaptive Time Stepping• First or second order accurate
backward finite difference schemes withnon-uniform time steps
• Automatic smooth time step controlbased on solution rate of change
Time StepNor
mal
ized
Uns
tead
yR
esid
ual,∆
U/∆
t∞,a
ndT
ime
Ste
p,∆t
Non
dim
ensi
onal
Tim
e
0 50 100 150 200 250 30010-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
0
50
100
150
200
250Time Step Size
Unsteady Residual
Tim
e
libMesh Adaptive Time Stepping• Backward finite difference, trapezoidal rule, Crank-Nicholson
schemes or space-time finite element integration
• Automatic smooth time step control based on time discretization errorestimates, backtracking on solver failure
Roy H. Stogner Hypersonic Flow May 4, 2009 21 / 30
Solver Techniques
Time Integration
Current FIN-S Adaptive Time Stepping• First or second order accurate
backward finite difference schemes withnon-uniform time steps
• Automatic smooth time step controlbased on solution rate of change
Time StepNor
mal
ized
Uns
tead
yR
esid
ual,∆
U/∆
t∞,a
ndT
ime
Ste
p,∆t
Non
dim
ensi
onal
Tim
e
0 50 100 150 200 250 30010-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
0
50
100
150
200
250Time Step Size
Unsteady Residual
Tim
e
libMesh Adaptive Time Stepping• Backward finite difference, trapezoidal rule, Crank-Nicholson
schemes or space-time finite element integration
• Automatic smooth time step control based on time discretization errorestimates, backtracking on solver failure
Roy H. Stogner Hypersonic Flow May 4, 2009 21 / 30
Solver Techniques
Nonlinear Algebraic Solver
Current FIN-S Solver Options• Trilinos, PETSc interfaces• Preconditioned Jacobian-Free Newton-Krylov• Complex perturbation method element Jacobians
New libMesh Solver Options• Custom preconditioned Newton-Krylov with line search• Finite differenced element Jacobians with automatic analytic Jacobian
verification
Roy H. Stogner Hypersonic Flow May 4, 2009 22 / 30
Solver Techniques
Nonlinear Algebraic Solver
Current FIN-S Solver Options• Trilinos, PETSc interfaces• Preconditioned Jacobian-Free Newton-Krylov• Complex perturbation method element Jacobians
New libMesh Solver Options• Custom preconditioned Newton-Krylov with line search• Finite differenced element Jacobians with automatic analytic Jacobian
verification
Roy H. Stogner Hypersonic Flow May 4, 2009 22 / 30
Example Results
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 23 / 30
Example Results
Simulation: Mach 20 Cylinder
Solution Verification• Cylinder in Mach 20 Perfect Gas flow• Predicted shock standoff distance
within both codes’ smoothed shockregion
Code-to-Code Verification• DPLR result and libMesh with “nu”
shock capturing provide near-matchingshock layers on identical meshes
Roy H. Stogner Hypersonic Flow May 4, 2009 24 / 30
Example Results
Simulation: Double Cone
Nondimensional Distance from Cone Apex (x/L)
Pre
ssur
eC
oeff
icie
nt
Hea
tTra
nsfe
rC
oeff
icie
nt
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
1
2
3
4
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16Pressure CoefficientHeat Transfer Coefficient
Roy H. Stogner Hypersonic Flow May 4, 2009 25 / 30
Example Results
Simulation: Hollow Cylinder
Nondimensional Distance from Cone Apex (x/L)
Pre
ssur
eC
oeff
icie
nt
Hea
tTra
nsfe
rC
oeff
icie
nt
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
1
2
3
4
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16Pressure CoefficientHeat Transfer Coefficient
Roy H. Stogner Hypersonic Flow May 4, 2009 26 / 30
Upcoming Work
Outline
1 Goals
2 FIN-S Motivation
3 Numeric Formulation
4 Solver Techniques
5 Example Results
6 Upcoming Work
Roy H. Stogner Hypersonic Flow May 4, 2009 27 / 30
Upcoming Work
Adjoint Analysis
Tasks• Completion of linear system assembly refactoring• Addition of missing Jacobian terms• Verification of Newton convergence• Test against numeric Jacobians• Integration with libMesh adjoint solve()• Verification of adjoint against finite differenced sensitivities• Verification against DAKOTA+FIN-S results
Roy H. Stogner Hypersonic Flow May 4, 2009 28 / 30
Upcoming Work
New Physics Coupling
Chemistry• Successful with N /N2 at Sandia Labs• MUTATION interface under development
Turbulence• Algebraic models initially
Ablation• Implementable as nonlinear fully implicit boundary condition
Radiation• Loose coupling to 1D tangent slab code• libMesh collaborators have SPn code experience
Roy H. Stogner Hypersonic Flow May 4, 2009 29 / 30
Upcoming Work
New Physics Coupling
Chemistry• Successful with N /N2 at Sandia Labs• MUTATION interface under development
Turbulence• Algebraic models initially
Ablation• Implementable as nonlinear fully implicit boundary condition
Radiation• Loose coupling to 1D tangent slab code• libMesh collaborators have SPn code experience
Roy H. Stogner Hypersonic Flow May 4, 2009 29 / 30
Upcoming Work
New Physics Coupling
Chemistry• Successful with N /N2 at Sandia Labs• MUTATION interface under development
Turbulence• Algebraic models initially
Ablation• Implementable as nonlinear fully implicit boundary condition
Radiation• Loose coupling to 1D tangent slab code• libMesh collaborators have SPn code experience
Roy H. Stogner Hypersonic Flow May 4, 2009 29 / 30
Upcoming Work
New Physics Coupling
Chemistry• Successful with N /N2 at Sandia Labs• MUTATION interface under development
Turbulence• Algebraic models initially
Ablation• Implementable as nonlinear fully implicit boundary condition
Radiation• Loose coupling to 1D tangent slab code• libMesh collaborators have SPn code experience
Roy H. Stogner Hypersonic Flow May 4, 2009 29 / 30
Upcoming Work
Thank you!
Questions?
Roy H. Stogner Hypersonic Flow May 4, 2009 30 / 30
GoalsFIN-S MotivationNumeric FormulationSolver TechniquesExample ResultsUpcoming Work