high-order spatial and temporal methods for simulation and sensitivity analysis of high-speed flows...
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![Page 1: High-Order Spatial and Temporal Methods for Simulation and Sensitivity Analysis of High-Speed Flows PI Dimitri J. Mavriplis University of Wyoming Co-PI](https://reader036.vdocument.in/reader036/viewer/2022062408/56649eca5503460f94bd8615/html5/thumbnails/1.jpg)
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
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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
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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)
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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
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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
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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
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DG Navier-Stokes Solutions
• Mach =0.5, Re =5000• 2000 mesh elements• Non-symmetric grid
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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
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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
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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
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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
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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!
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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)
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Shock Capturing with Artificial Dissipation
• Sub-cell shock capturing resolution (p=4)
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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
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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,
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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
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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
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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
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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
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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
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• 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
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• 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
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• 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
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• 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
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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
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h-p Refinement for Mach 10 Case
• High-speed flow over a half circular-cylinder (M∞=10)
27
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H-p Refinement: Functional Convergence
28
M ∞=6, functional: integrated temperature M ∞=10, functional: drag
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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
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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