High-Order Spatial and Temporal Methods for Simulation and

Sensitivity Analysis of High-Speed Flows

PIDimitri J. Mavriplis

University of WyomingCo-PI

Luigi MartinelliPrinceton University

Project Scope and Relevance

• Develop novel approaches for improving simulation capabilities for high-speed flows

– Emerging consensus about higher-order methods• May be only way to get desired accuracy

– Asymptotic arguments• Superior scalability

– Sensitivity analysis and adjoint methods• Now seen as indispensible component of new emerging class

of simulation tools• Automated (adaptive) solution process with certifiable accuracy

– Other novel approaches: BGK methods

Advantages of DG Discretizations

• Superior Asymptotic Properties• Smaller meshes

– Easier to generate/manage

• Superior Scalability: small meshes on many cores

•Dense kernels, well suited for GPUs, Cell processors

2.5 million cell DG (h-p Multigrid) 2.5 million cell DG (h-p Multigrid)

Disadvantages of DG Discretizations

• High-Risk, Revolutionary– Still no production level DG code for subsonics

• Relies on smooth solution behavior to achieve favorable asymptotic accuracy– Difficulties for strong shocks– Robustness issues

Overview of Current Work

1. Viscous discretizations and solvers for DG

2. ALE Formulation for moving meshes

3. BGK Flux flunction implementation/results

4. Shock capturing- Artificial dissipation

- High-order filtering/limiting

5. Adjoint-based h-p refinement- Shocks captured with no limiting/added dissipation

6. Conclusions

Extension to Viscous Flows• DG methods developed initially for hyperbolic

problems– Diffusion terms for DG non-trivial

• Interior Penalty (IP) method– Simplest approach, compact stencil– Explicit expression for penalty parameter derived (JCP)

• IP method derived and implemented for compressible Navier-Stokes formulation up to p=5– Studied symmetric and non-symmetric forms for IP– h and p independent convergence observed for Poisson and Navier-

Stokes problems

DG Navier-Stokes Solutions

• Mach =0.5, Re =5000• 2000 mesh elements• Non-symmetric grid

DG Navier-Stokes Solutions

• h-p multigrid convergence maintained (50 – 80 cycles)• Accuracy validated by comparison with high-resolution finite-volume results

– Separation location ~ 81% chord (p=3)

p=1: second-order accuracy p=3: fourth-order accuracy

Solution of DG Discretization for NS Equations

• h-p multigrid solver: h and p independent convergence rates• Used as preconditioner to GMRES for further efficiency improvements

Kinetic Based Flux Formulations (BGK)L. Martinelli

Princeton University

• Alternative for extension to Navier-Stokes: – It is not necessary to compute the rate of strain tensor in order

to calculate viscous fluxes

• Automatic upwinding via the kinetic model.• Satisfy Entropy Condition (H-Theorem) at the discrete

level.• Implemented in 2D Unstructured Finite-Volume code by

Martinelli • Extension to 2D DG code under development

BGK Finite Volume SolverMach 10 Cylinder

• Robust 2nd order accurate solution• BGK –DG solutions obtained for low speed flows

– BGK-DG cases with strong shocks initiated

Treatment of Shock Waves

• High-order (DG) methods based on smooth solution behavior

• 3 approaches investigated for high-order shock wave simulation– Smoothing out shock: Artificial viscosity

• Use IP method discussed previously• Sub-cell shock resolution possible

– Limiting or Filtering High Order Solution• Remove spurious oscillations• Sub-cell shock resolution possible

– h-p adaption• Start with p=0 (1st order) solution• Raise p (order) only were solution is smooth• Refine mesh (h) where solution is non-smooth (shock)• No limiting required!

Shock Capturing with Artificial Dissipation (p=4)

• IP Method used for artificial viscosity terms (Laplacian)• Artificial Viscosity scales as ~ h/p• An alternative to limiting or reducing accuracy in vicinity of non-smooth solutions (Persson and Peraire

2006)

Shock Capturing with Artificial Dissipation

• Sub-cell shock capturing resolution (p=4)

Mach 6 Flow over Cylinder

•Third order accurate (p=2)

•Relatively coarse grid

•Sub-cell shock resolution captured with artificial dissipation

•Principal issue: Convergence/Robustness

Euler-Lagrange equation (1st variation)

Nonlinear partial differential equations (PDE) based

Pseudo-time stepping (Rudin, Osher and Fatemi 1992)

Solved locally in each element

Total Variation based nonlinear FilteringTotal Variation based nonlinear Filtering

Formulation Minimization

where,

Euler-Lagrange equation (1st variation)

Nonlinear partial differential equations (PDE) based

Pseudo-time stepping (Rudin, Osher and Fatemi 1992)

Solved locally in each element

Total Variation based nonlinear FilteringTotal Variation based nonlinear Filtering

Formulation Minimization

where,

Controls amount of filtering

Shock Capturing with Filteringp=3 (4th order accuracy)

• Weak (transonic) shock captured with sub-cell resolution using filtering/limiting

• Enables highest order polynomial without oscillations

DG Filtering for High Speed Flows

• Mach 6 flow over cylinder at p=2 (3rd order)– Lax Friedrichs flux

Relatively robust

Shock spread over more than one element

DG Filtering for High Speed Flows

• Mach 6 flow over cylinder at p=2 (3rd order)– Van-Leer Flux

Relatively robust

Thinner Shock spread over approximately one element

DG Filtering for Strong Shocks

• Shock resolution determined by convergence robustness – (not necessarily property of flux function)– Van Leer flux could be run with larger filter value– Higher order solutions should deliver higher resolution shocks

• Convergence issues remain above p=2

Lax-Friedrichs Van Leer

• Formulation– Key objective functionals with engineering applications

• Surface integrals of the flow-field variables• Lift, drag, integrated temperature, surface heat flux• A single objective, expressed as

– Current mesh (coarse mesh, H)• Coarse flow solution, • Objective on the coarse mesh,

– Globally refined mesh (fine mesh, h)• Fine flow solution, • Objective on the fine mesh,

• Goal : find an approximate for without solving on the fine mesh

ADJOINT-BASED ERROR ESTIMATION

22

)~(uJ

Hu~

)~( HHJ u

hu~

)~( huhJNOT DESIRED!

hJ

• Formulation– Coarse grid solution projected onto fine grid gives non-zero residual– Change in objective calculated on fine grid:

= inner product of residual with adjoint

• Procedure– Compute coarse grid solution and adjoint– Project solution and adjoint to fine grid– Form inner product of residual and adjoint on fine grid

• Global Error estimate of objective• Local error estimate (in each cell)

– Use to drive adaptive refinement– Smoothness indicator used to choose between h and p refinement– Naturally maintains p=0 in shock region

ADJOINT-BASED ERROR ESTIMATION

23

)()~( HHh JJ uhu

• High-speed flow over a half circular-cylinder (M∞=6)

Combined h-p Refinement for Hypersonic Cases

Target function of integrated temperature

• hp-refinement• starting discretization order p = 0 (first-order accurate)

dSTJw

24initial mesh: 17,072 elements

• High-speed flow over a half circular-cylinder (M∞=6)

h-p Refinement for High-Speed Flows

25adapted mesh: 42,234 elements,

discretization orders p=0~3

No shock refinement in regions not affecting surface temperature

h-p RefinementObjective=Surface T

Mach 6

26

Pressure

Mach Number

Shock captured without limiting or dissipation

Naturally remains at p=0 in shock region

h-p Refinement for Mach 10 Case

• High-speed flow over a half circular-cylinder (M∞=10)

27

H-p Refinement: Functional Convergence

28

M ∞=6, functional: integrated temperature M ∞=10, functional: drag

Conclusions and Future Work

• DG methods hold promise for advancing state-of-the-art for difficult problems such as Hypersonics

• Recent advances in:– Viscous discretizations– Flux functions (BGK)– ALE formulations– Solver technology (h-p multigrid)– Shock capturing

• Extend into 3D DG parallel code– Diffusion terms– Shock capturing– h-p adaptivity (adjoint based)

• Real gas effects– 5 species, 2 temperature model for DG code

Remaining Difficulties

• DG Methods need to be robust– Often requires accuracy reduction (limiting)

• Shock capturing with artificial viscosity becomes very non-linear/difficult to converge for high p and high Mach

• Limiting is very robust initially, but convergence to machine zero stalls– Other limiter formulations are possible

• Adjoint h-p refinement is promising but will likely require use with limiter for necessary robustness– Linearization of limiter/filter