Fully Implicit Navier-Stokes Hypersonic Flow Softwareroystgnr/fins_talk.pdfFully Implicit...

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


Goals

Outline

1 Goals

2 FIN-S Motivation


4 Solver Techniques

5 Example Results

6 Upcoming Work


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


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


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


FIN-S Motivation

Outline

1 Goals

2 FIN-S Motivation


4 Solver Techniques

5 Example Results

6 Upcoming Work


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


FIN-S Motivation





I Development of adjoint-enhanced ablation code

I Exploration of adjoint-enhanced forward propagation of uncertaintyI Exploration of adjoint formulation of hypersonics flow codes


FIN-S Motivation





I Development of adjoint-enhanced ablation codeI Exploration of adjoint-enhanced forward propagation of uncertainty

I Exploration of adjoint formulation of hypersonics flow codes


FIN-S Motivation





I Development of adjoint-enhanced ablation codeI Exploration of adjoint-enhanced forward propagation of uncertaintyI Exploration of adjoint formulation of hypersonics flow codes


FIN-S Motivation


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


FIN-S Motivation


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


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


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


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


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.


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


Numeric Formulation

Outline

1 Goals

2 FIN-S Motivation


4 Solver Techniques

5 Example Results

6 Upcoming Work


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


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


Solver Techniques

Outline

1 Goals

2 FIN-S Motivation


4 Solver Techniques

5 Example Results

6 Upcoming Work


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


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


Example Results

Outline

1 Goals

2 FIN-S Motivation


4 Solver Techniques

5 Example Results

6 Upcoming Work


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


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


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


Upcoming Work

Outline

1 Goals

2 FIN-S Motivation


4 Solver Techniques

5 Example Results

6 Upcoming Work


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


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


Upcoming Work

Thank you!

Questions?


GoalsFIN-S MotivationNumeric FormulationSolver TechniquesExample ResultsUpcoming Work

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