Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics Dissertation Defense

1. Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid DynamicsDissertation Defense Li Wang PhD Candidate Department of Mechanical Engineering University of Wyoming Laramie, WY April 21, 2009

2. Outline • Introduction • Objective • Steady Flow Problems • High-order Steady-State Discontinuous Galerkin Discretizations • Output-Based Spatial Error Estimation and Mesh Adaptation • Unsteady Flow Problems • High-order Implicit Temporal Discretizations • Output-Based Temporal Error Estimation and Time-step Adaptation • Conclusions and Future Work

3. Introduction • ComputationalFluid Dynamics (CFD) • Computational methods vs. Experimental methods • Indispensible technology • Inaccuracies and uncertainties • Improvement of numerical algorithms • High-order accurate methods • Sensitivity analysis techniques • Adaptive mesh refinement (AMR) L. Wang, transonic flow over a NACA0012 airfoil with sub-grid shock resolution (2008) M. Nemec, et. cl., Mach number contours around LAV (2008) D. Mavriplis, DLR-F6 Wing-Body Configuration (2006)

4. Introduction • Why Discontinuous Galerkin (DG) Methods? • Finite difference methods • Simple geometries • Finite volume methods • Lower-order accurate discretizations • DG methods • Solution Expansion • Asymptotic accuracy properties: • Compact element-based stencils • Efficient performance in a parallel environment • Easy implementation of h-p adaptivity

5. Introduction • High-order Time-integration Schemes • Explicit schemes (e.g. Explicit Runge-Kutta scheme) • Easy to solve • Restricted time-step sizes : • Run a lot of time steps • Implicit schemes • No restriction by CFL stability limit • Accuracy requirement • Accuracy • Computational cost • Efficient Solution Strategies • Required for steady-state or time-implicit solvers • p- or hp- nonlinear multigrid approach • Element Jacobi smoothers

6. Introduction • Sensitivity Analysis Techniques • Applications • Shape optimization • Output-based error estimation • Adaptive mesh refinement • Adjoint Methods • Linearization of the analysis problem + Transpose • Discrete adjoint method • Reproduce exact sensitivities to the discrete system • Deliver Linear systems • Simulation output : L(u), such as lift or drag • Error in simulation output: e(L) ~ (Adjoint solution) • (Residual of the Analysis Problem)

7. Objective • Development of Efficient Solution Strategies for Steady or Unsteady Flows • Development of Output-based Spatial Error Estimation and Mesh Adaptation • Investigation of Time-Implicit Schemes • Investigation of Output-based Temporal Error Estimation and Time-Step Adaptation

8. Model Problem • Two-dimensional Compressible Euler Equations • Conservative Formulation

9. Outline • Introduction • Objective • Steady Flow Problems • High-order Steady-State Discontinuous Galerkin Discretizations • Output-Based Spatial Error Estimation and Mesh Adaptation • Unsteady Flow Problems • High-order Implicit Temporal Discretizations • Output-Based Temporal Error Estimation and Time-step Adaptation • Conclusions and Future Work

10. Discontinuous Galerkin Discretizations • Triangulation Partition: • DG weak statement on each element, k • Integrating by parts • Solution Expansion • Steady-state system of equations

11. Compressible Channel Flow over a Gaussian Bump • Free stream Mach number = 0.35 • HLLC Riemann flux approximation • Mesh size: 1248 elements Pressure contours usingp=0discretization and p=0 boundary elements Pressure contours using p=4 discretization and p=4 boundary elements

12. Compressible Channel Flow over a Gaussian Bump • Spatial Accuracy and Efficiency for Various Discretization Orders Error convergence vs. Grid spacing Error convergence vs. Computational time

13. Compressible Channel Flow over a Gaussian Bump • Element Jacobi Smoothers • Single level method • p-independent • h-dependent

14. Compressible Channel Flow over a Gaussian Bump • p- or hp-multigrid approach • p-independent • h-independent

15. Outline • Introduction • Objective • Steady Flow Problems • High-order Steady-State Discontinuous Galerkin Discretizations • Output-Based Spatial Error Estimation and Mesh Adaptation • Unsteady Flow Problems • High-order Implicit Temporal Discretizations • Output-Based Temporal Error Estimation and Time-step Adaptation • Conclusions and Future Work

16. Output-based Spatial Error Estimation • Some key functional outputs in flow simulations • Lift, Drag, Integrated surface temperature, etc. • Surface integrals of the flow-field variables • Single objective functional, L • Coarse affordable mesh, H • Coarse level flow solution, • Coarse level functional, • Fine (Globally refined) mesh, h • Fine level flow solution, • Fine level functional, • Goal: Find an approximation of without solving on the fine mesh

17. Output-based Spatial Error Estimation • Taylor series expansion • Goal: Find an approximation of without solving on the fine mesh

18. Output-based Spatial Error Estimation • Discrete adjoint problem (H) • Transpose of Jacobian matrix • Delivers similar convergence rate as the flow solver • Reconstruction of coarse level adjoint • : Estimates functional error • : Indicates error distribution and drives mesh adaptation • Approximated fine level functional

19. Refinement Criteria • is used to drive mesh adaptation • Element-wise error indicator • Set an error tolerance, ETOL • Necessary refinement for an element if • Flag elements required for refinement

20. P P p p+1 P P h H H Mesh Refinement • h-refinement • Local mesh subdivision • p-enrichment • Local variation of discretization orders • hp-refinement • Local implementation of the h- orp-refinement individually

21. Additional Criteria for hp-refinement • For each flagged element: • How to make a decision between h- and p-refinement? Smoothness indicator • Local smoothness indicator • Element-based Resolution indicator[Persson, Peraire] • Inter-element Jump indicator [Krivodonova,Xin,Chevaugeon,Flaherty],

22. Subsonic Flow over a Four-Element Airfoil • Free-stream Mach number = 0.2 • Various adaptation algorithms • h-refinement • p-enrichment • Objective functional: drag or lift (angle of attack = 0 degree) • Starting interpolation order of p = 1 • HLLC Riemann solver • hp-Multigrid accelerator Initial mesh (1508 elements)

23. Subsonic Flow over a Four-Element Airfoil Mach number contours Flow and adjoint problems target functional of lift Adjoint solution, Λ(2) Comparisons on hp-Multigrid convergence for the flow and adjoint solutions

24. h-Refinement for Target Functional of Lift • Fixed discretization order of p = 1 Final h-adapted mesh (8387 elements) Close-up view of the final h-adapted mesh

25. h-Refinement for Target Functional of Lift • Comparison between h-refinement and uniform mesh refinement Error convergence history vs. degrees of freedom Error convergence history vs. CPU time (sec)

26. p-Enrichment for Target Functional of Drag • Fixed underlying grids (1508 elements) Spatial error distribution for the objective functional of drag Final p-adapted mesh discretization orders: p=1~4

27. p-Enrichment for Target Functional of Drag • Comparison between p-enrichment and uniform order refinement Error convergence history vs. CPU time (sec) Error convergence history vs. degrees of freedom

28. Hypersonic Flow over a half-circular Cylinder • Free-stream Mach number of 6 • Objective functional: surface integrated temperature, • hp-refinement • Starting discretization order of p=0 (first-order accurate) • hp-adapted meshes Final hp-adapted mesh: 42,234 elements. Discretization orders: p=0~3 Initial mesh: 17,072 elements

29. Hypersonic Flow over a half-circular Cylinder • Final pressure and Mach number solutions

30. Hypersonic Flow over a half-circular Cylinder • Convergence of the objective functional

31. Outline • Introduction • Objective • Steady Flow Problems • High-order Steady-State Discontinuous Galerkin Discretizations • Output-Based Spatial Error Estimation and Mesh Adaptation • Unsteady Flow Problems • High-order Implicit Temporal Discretizations • Output-Based Temporal Error Estimation and Time-step Adaptation • Conclusions and Future Work

32. Implicit Time-integration Schemes • Time-Implicit System • First-orderaccurate backwards difference scheme (BDF1) • Second-order accurate multistep backwards difference scheme (BDF2) • Second-order Crank Nicholson scheme (CN2) • Fourth-order implicit Runge-Kutta scheme (IRK4)

33. Convection of an Isentropic Vortex • Initial condition • Isentropic vortex perturbation; Periodic boundary conditions • HLLC Flux approximation • p = 4 spatial discretization • ∆ t = 0.2 BDF1 (First-order accurate) IRK4 (Fourth-order accurate)

34. Convection of an Isentropic Vortex • Temporal accuracy and efficiency study for various temporal schemes Error convergence vs. time-step sizes Error convergence vs. Computational time

35. Shedding Flow over a Triangular Wedge • Free-stream Mach number = 0.2 • Unstructured mesh with 10836 elements • Various spatial discretizations and temporal schemes Unstructured computational mesh with 10836 elements

36. Shedding Flow over a Triangular Wedge • Free-stream Mach number = 0.2 • Unstructured mesh with 10836 elements • Various spatial discretizations and temporal schemes Density solution using p = 1 discretization and BDF2 scheme

37. Shedding Flow over a Triangular Wedge • t = 100 • Various spatial discretizations and temporal schemes p = 1 and BDF2 p = 1 and IRK4

38. Shedding Flow over a Triangular Wedge • t = 100 • Various spatial discretizations and temporal schemes p = 1 and BDF2 p = 3 and IRK4

39. Outline • Introduction • Objective • Steady Flow Problems • High-order Steady-State Discontinuous Galerkin Discretizations • Output-Based Spatial Error Estimation and Mesh Adaptation • Unsteady Flow Problems • High-order Implicit Temporal Discretizations • Output-Based Temporal Error Estimation and Time-step Adaptation • Conclusions and Future Work

40. Output-based Temporal Error Estimation • Same methodology can be applied in time • Global temporal error estimation and time-step adaptation • Implementation to BDF1 and IRK4 schemes • Time-integrated objective functional: • UnsteadyFlow solution • Unsteady adjoint solution • Linearization of the unsteady flow equations • Transpose operation results in a backward time-integration Forward time-integration Current Backward time-integration

41. Output-based Temporal Error Estimation • Two successively refined time-resolution levels • H: coarse level functional • h: fine level functional • Approximation of fine level functional BDF1: • Localized functional error (for each time step i) IRK4: • Local time-step subdivision if

42. Shedding Flow over a Triangular Wedge • Implementation for BDF1 scheme ( p = 2) • Validation of adjoint-based error correction • Objective function: Drag at t = 5 • Error prediction for two time-resolution levels Refined time-resolution levels Computed functional error (Reconstructed adjoint) • (Unsteady residual)

43. Shedding Flow over a Triangular Wedge • Adaptive time-step refinement approach vs. Uniform time-step refinement approach • Objective functional: Error convergence vs. time steps (i.e. DOF) Error convergence vs. computational cost

44. Outline • Introduction • Objective • Steady Flow Problems • High-order Steady-State Discontinuous Galerkin Discretizations • Output-Based Spatial Error Estimation and Mesh Adaptation • Unsteady Flow Problems • High-order Implicit Temporal Discretizations • Output-Based Temporal Error Estimation and Time-step Adaptation • Conclusions and Future Work

45. Conclusions • High-order DG and Implicit-Time Methods • Optimal error convergence rates are attained for the DG discretizations • Perform more efficiently than lower-order methods • Both h- and p-independent convergence rates • An attempt to balance spatial and temporal error • Perform more efficiently than lower-order implicit temporal schemes • h-independent convergence rates and slightly dependent on time-step sizes • Discrete Adjoint based Sensitivity Analysis • Formulation of discrete adjoint sensitivity for DG discretizations • Accurate error estimate in a simulation output • Superior efficiency over uniform mesh or order refinement approach • hp-adaptation shows good capturing of strong shocks without limiters • Extension to temporal schemes • Superior efficiency over uniform time-step refinement approach

46. Future Work • Dynamic Mesh Motion Problems • Discretely conservative high-order DG • Both high-order temporal and spatial accuracy • Unsteady shape optimization problems with mesh motion • Robustness of the hp-adaptive refinement strategy • Incorporation of a shock limiter • Investigation of smoothness indicators • Combination of spatial and temporal error estimation • Quantification of dominated error source • More effective adaptation strategies • Extension to other sets of equations • Compressible Navier-Stokes equations (IP method) • Three-dimensional problems