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PI Dimitri J. Mavriplis University of Wyoming Co-PI Luigi Martinelli Princeton University

High-Order Spatial and Temporal Methods for Simulation and Sensitivity Analysis of High-Speed Flows. PI Dimitri J. Mavriplis University of Wyoming Co-PI Luigi Martinelli Princeton University. Project Scope and Relevance.

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PI Dimitri J. Mavriplis University of Wyoming Co-PI Luigi Martinelli Princeton University

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  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 Luigi Martinelli Princeton University

  2. 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

  3. Advantages of DG Discretizations 2.5 million cell DG (h-p Multigrid) • Superior Asymptotic Properties • Smaller meshes • Easier to generate/manage • Superior Scalability: small meshes on many cores • Dense kernels, well suited for GPUs, Cell processors

  4. 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

  5. Overview of Current Work • Viscous discretizations and solvers for DG • ALE Formulation for moving meshes • BGK Flux flunction implementation/results • Shock capturing - Artificial dissipation - High-order filtering/limiting • Adjoint-based h-p refinement - Shocks captured with no limiting/added dissipation • Conclusions

  6. 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

  7. DG Navier-Stokes Solutions • Mach =0.5, Re =5000 • 2000 mesh elements • Non-symmetric grid

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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!

  13. 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)

  14. Shock Capturing with Artificial Dissipation • Sub-cell shock capturing resolution (p=4)

  15. 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

  16. Total Variation based nonlinear Filtering 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 • Formulation • Minimization

  17. Total Variation based nonlinear Filtering Controls amount of filtering 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 • Formulation • Minimization

  18. 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

  19. 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

  20. 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

  21. DG Filtering for Strong Shocks Lax-Friedrichs Van Leer • Shock resolution determined by convergence robustness • (not necessarily property of flux function) • Van Leer flux could be run with larger filter l value • Higher order solutions should deliver higher resolution shocks • Convergence issues remain above p=2

  22. ADJOINT-BASED ERROR ESTIMATION NOT DESIRED! 22 • 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

  23. ADJOINT-BASED ERROR ESTIMATION 23 • 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

  24. Combined h-p Refinement for Hypersonic Cases Target function of integrated temperature • hp-refinement • starting discretization order p = 0 (first-order accurate) 24 initial mesh: 17,072 elements High-speed flow over a half circular-cylinder (M∞=6)

  25. h-p Refinement for High-Speed Flows No shock refinement in regions not affecting surface temperature 25 High-speed flow over a half circular-cylinder (M∞=6) adapted mesh: 42,234 elements, discretization orders p=0~3

  26. h-p RefinementObjective=Surface TMach 6 Pressure Shock captured without limiting or dissipation Naturally remains at p=0 in shock region Mach Number 26

  27. h-p Refinement for Mach 10 Case Target function of drag 27 High-speed flow over a half circular-cylinder (M∞=10)

  28. H-p Refinement: Functional Convergence M ∞=6, functional: integrated temperature M ∞=10, functional: drag 28

  29. 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

  30. 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

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