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

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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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slide1

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
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
introduction
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)

introduction4
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
introduction5
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
introduction6
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)
objective
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
model problem
Model Problem
  • Two-dimensional Compressible Euler Equations
    • Conservative Formulation
outline9
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
discontinuous galerkin discretizations
Discontinuous Galerkin Discretizations
  • Triangulation Partition:
  • DG weak statement on each element, k
  • Integrating by parts
  • Solution Expansion
  • Steady-state system of equations
compressible channel flow over a gaussian bump
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

compressible channel flow over a gaussian bump12
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

compressible channel flow over a gaussian bump13
Compressible Channel Flow over a Gaussian Bump
  • Element Jacobi Smoothers
    • Single level method
    • p-independent
    • h-dependent
compressible channel flow over a gaussian bump14
Compressible Channel Flow over a Gaussian Bump
  • p- or hp-multigrid approach
  • p-independent
  • h-independent
outline15
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
output based spatial error estimation
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
output based spatial error estimation17
Output-based Spatial Error Estimation
  • Taylor series expansion
  • Goal: Find an approximation of without solving on the fine mesh
slide18

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
refinement criteria
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
mesh refinement

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
additional criteria for hp refinement
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],

subsonic flow over a four element airfoil
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)

subsonic flow over a four element airfoil23
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

h refinement for target functional of lift
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

h refinement for target functional of lift25
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)

p enrichment for target functional of drag
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

p enrichment for target functional of drag27
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

hypersonic flow over a half circular cylinder
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

hypersonic flow over a half circular cylinder29
Hypersonic Flow over a half-circular Cylinder
  • Final pressure and Mach number solutions
hypersonic flow over a half circular cylinder30
Hypersonic Flow over a half-circular Cylinder
  • Convergence of the objective functional
outline31
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
implicit time integration schemes
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)
convection of an isentropic vortex
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)

convection of an isentropic vortex34
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

shedding flow over a triangular wedge
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

shedding flow over a triangular wedge36
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

shedding flow over a triangular wedge37
Shedding Flow over a Triangular Wedge
  • t = 100
  • Various spatial discretizations and temporal schemes

p = 1 and BDF2

p = 1 and IRK4

shedding flow over a triangular wedge38
Shedding Flow over a Triangular Wedge
  • t = 100
  • Various spatial discretizations and temporal schemes

p = 1 and BDF2

p = 3 and IRK4

outline39
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
output based temporal error estimation
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

output based temporal error estimation41
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
shedding flow over a triangular wedge42
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)

shedding flow over a triangular wedge43
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

outline44
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
conclusions
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
future work
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
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