Techniques for Highorder Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics Dissertat

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

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Introduction

• Computational Fluid 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

• 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

• 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

• 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

• 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

• Two-dimensional Compressible Euler Equations
• Conservative Formulation

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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

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Discontinuous Galerkin Discretizations

• 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

• Free stream Mach number 0.35
• HLLC Riemann flux approximation
• Mesh size 1248 elements

Pressure contours using p0 discretization and
p0 boundary elements
Pressure contours using p4 discretization and
p4 boundary elements
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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
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Compressible Channel Flow over a Gaussian Bump

• Element Jacobi Smoothers
• Single level method
• p-independent
• h-dependent

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Compressible Channel Flow over a Gaussian Bump

• p- or hp-multigrid approach
• p-independent
• h-independent

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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

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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,

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Output-based Spatial Error Estimation

• Taylor series expansion

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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

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Refinement Criteria

• is used to drive mesh adaptation
• Element-wise error indicator

• Flag elements required for refinement

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Mesh Refinement

• h-refinement
• Local mesh subdivision

• p-enrichment
• Local variation of discretization orders

• hp-refinement
• Local implementation of the h- or p-refinement
  individually

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Additional Criteria for hp-refinement

• For each flagged element
• How to make a decision between h- and
  p-refinement?

• 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

• 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
Mach number contours
Flow and adjoint problems target functional of
lift
Adjoint solution, ?(2)
Comparisons on hp-Multigrid convergence for the
flow and adjoint solutions
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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
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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)
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p-Enrichment for Target Functional of Drag

• Fixed underlying grids (1508 elements)

Final p-adapted mesh discretization orders p14
Spatial error distribution for the objective
functional of drag
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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
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Hypersonic Flow over a half-circular Cylinder

• Free-stream Mach number of 6
• Objective functional surface integrated
  temperature,
• hp-refinement
• Starting discretization order of p0 (first-order
  accurate)
• hp-adapted meshes

Final hp-adapted mesh 42,234 elements.
Discretization orders p03
Initial mesh 17,072 elements
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Hypersonic Flow over a half-circular Cylinder

• Final pressure and Mach number solutions

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Hypersonic Flow over a half-circular Cylinder

• Convergence of the objective functional

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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

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Implicit Time-integration Schemes

• Time-Implicit System
• First-order accurate 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

• 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

• 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

• Free-stream Mach number 0.2
• Unstructured mesh with 10836 elements
• Various spatial discretizations and temporal
  schemes

Unstructured computational mesh with 10836
elements
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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
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Shedding Flow over a Triangular Wedge

• t 100
• Various spatial discretizations and temporal
  schemes

p 1 and BDF2
p 1 and IRK4
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Shedding Flow over a Triangular Wedge

• t 100
• Various spatial discretizations and temporal
  schemes

p 1 and BDF2
p 3 and IRK4
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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

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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
• Unsteady Flow solution
• Unsteady adjoint solution
• Linearization of the unsteady flow equations
• Transpose operation results in a backward
  time-integration

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Output-based Temporal Error Estimation

• Two successively refined time-resolution levels
• H coarse level functional
• h fine level functional

• Approximation of fine level functional

• Localized functional error (for each time step i)

• Local time-step subdivision if

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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)
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Shedding Flow over a Triangular Wedge

• Adaptive time-step refinement approach vs.
  Uniform time-step refinement approach
• Objective functional

Error convergence vs. computational cost
Error convergence vs. time steps (i.e. DOF)
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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

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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

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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