Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics
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Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics Dissertation Defense Li Wang PhD Candidate Department of Mechanical Engineering University of Wyoming Laramie, WY April 21, 2009 Outline Introduction Objective Steady Flow Problems

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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


Outline l.jpg
Outline Discretizations in Fluid Dynamics

  • 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


Introduction l.jpg
Introduction Discretizations in Fluid Dynamics

  • 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)


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Introduction Discretizations in Fluid Dynamics

  • 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


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Introduction Discretizations in Fluid Dynamics

  • 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


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Introduction Discretizations in Fluid Dynamics

  • 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)


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Objective Discretizations in Fluid Dynamics

  • 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


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Model Problem Discretizations in Fluid Dynamics

  • Two-dimensional Compressible Euler Equations

    • Conservative Formulation


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Outline Discretizations in Fluid Dynamics

  • 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


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Discontinuous Galerkin Discretizations Discretizations in Fluid Dynamics

  • Triangulation Partition:

  • DG weak statement on each element, k

  • Integrating by parts

  • Solution Expansion

  • Steady-state system of equations


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Compressible Channel Flow over a Gaussian Bump Discretizations in Fluid Dynamics

  • 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


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Compressible Channel Flow over a Gaussian Bump Discretizations in Fluid Dynamics

  • Spatial Accuracy and Efficiency for Various Discretization Orders

Error convergence vs. Grid spacing

Error convergence vs. Computational time


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Compressible Channel Flow over a Gaussian Bump Discretizations in Fluid Dynamics

  • Element Jacobi Smoothers

    • Single level method

    • p-independent

    • h-dependent


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Compressible Channel Flow over a Gaussian Bump Discretizations in Fluid Dynamics

  • p- or hp-multigrid approach

  • p-independent

  • h-independent


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Outline Discretizations in Fluid Dynamics

  • 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


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Output-based Spatial Error Estimation Discretizations in Fluid Dynamics

  • 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


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Output-based Spatial Error Estimation Discretizations in Fluid Dynamics

  • Taylor series expansion

  • Goal: Find an approximation of without solving on the fine mesh


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Output-based Spatial Error Estimation Discretizations in Fluid Dynamics

  • 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


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    Refinement Criteria Discretizations in Fluid Dynamics

    • 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


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    P Discretizations in Fluid Dynamics

    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


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    Additional Criteria for Discretizations in Fluid Dynamicshp-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],


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    Subsonic Flow over a Four-Element Airfoil Discretizations in Fluid Dynamics

    • 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)


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    Subsonic Flow over a Four-Element Airfoil Discretizations in Fluid Dynamics

    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


    H refinement for target functional of lift l.jpg
    h Discretizations in Fluid Dynamics-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


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    h Discretizations in Fluid Dynamics-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)


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    p Discretizations in Fluid Dynamics-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


    P enrichment for target functional of drag27 l.jpg
    p Discretizations in Fluid Dynamics-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


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    Hypersonic Discretizations in Fluid Dynamics 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


    Hypersonic flow over a half circular cylinder29 l.jpg
    Hypersonic Discretizations in Fluid Dynamics Flow over a half-circular Cylinder

    • Final pressure and Mach number solutions


    Hypersonic flow over a half circular cylinder30 l.jpg
    Hypersonic Discretizations in Fluid Dynamics Flow over a half-circular Cylinder

    • Convergence of the objective functional


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    Outline Discretizations in Fluid Dynamics

    • 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


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    Implicit Time-integration Schemes Discretizations in Fluid Dynamics

    • 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)


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    Convection of an Isentropic Vortex Discretizations in Fluid Dynamics

    • 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)


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    Convection of an Isentropic Vortex Discretizations in Fluid Dynamics

    • Temporal accuracy and efficiency study for various temporal schemes

    Error convergence vs. time-step sizes

    Error convergence vs. Computational time


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    Shedding Flow over a Triangular Wedge Discretizations in Fluid Dynamics

    • Free-stream Mach number = 0.2

    • Unstructured mesh with 10836 elements

    • Various spatial discretizations and temporal schemes

    Unstructured computational mesh with 10836 elements


    Shedding flow over a triangular wedge36 l.jpg
    Shedding Flow over a Triangular Wedge Discretizations in Fluid Dynamics

    • 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


    Shedding flow over a triangular wedge37 l.jpg
    Shedding Flow over a Triangular Wedge Discretizations in Fluid Dynamics

    • t = 100

    • Various spatial discretizations and temporal schemes

    p = 1 and BDF2

    p = 1 and IRK4


    Shedding flow over a triangular wedge38 l.jpg
    Shedding Flow over a Triangular Wedge Discretizations in Fluid Dynamics

    • t = 100

    • Various spatial discretizations and temporal schemes

    p = 1 and BDF2

    p = 3 and IRK4


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    Outline Discretizations in Fluid Dynamics

    • 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


    Output based temporal error estimation l.jpg
    Output-based Temporal Error Estimation Discretizations in Fluid Dynamics

    • 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


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    Output-based Temporal Error Estimation Discretizations in Fluid Dynamics

    • 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


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    Shedding Flow over a Triangular Wedge Discretizations in Fluid Dynamics

    • 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)


    Shedding flow over a triangular wedge43 l.jpg
    Shedding Flow over a Triangular Wedge Discretizations in Fluid Dynamics

    • 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


    Outline44 l.jpg
    Outline Discretizations in Fluid Dynamics

    • 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


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    Conclusions Discretizations in Fluid Dynamics

    • 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


    Future work l.jpg
    Future Work Discretizations in Fluid Dynamics

    • 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


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