Download - Li Wang PhD Candidate Department of Mechanical Engineering University of Wyoming Laramie, WY
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Li Wang
PhD CandidateDepartment of Mechanical Engineering
University of WyomingLaramie, WY
April 21, 2009
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OutlineOutline
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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Computational Fluid Dynamics (CFD) Computational methods vs. Experimental methods
o Indispensible technologyo Inaccuracies and uncertainties
Improvement of numerical algorithmso High-order accurate methodso Sensitivity analysis techniqueso Adaptive mesh refinement (AMR)
IntroductionIntroduction
D. Mavriplis, DLR-F6 Wing-Body Configuration (2006)
M. Nemec, et. cl., Mach number contours around LAV (2008)
L. Wang, transonic flow over a NACA0012 airfoil with sub-grid
shock resolution (2008)
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Why Discontinuous Galerkin (DG) Methods? Finite difference methods
o Simple geometries
Finite volume methods
o Lower-order accurate discretizations
DG methods
o Solution Expansion
o Asymptotic accuracy properties:
o Compact element-based stencils
o Efficient performance in a parallel environment
o Easy implementation of h-p adaptivity
IntroductionIntroduction
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High-order Time-integration Schemes Explicit schemes (e.g. Explicit Runge-Kutta scheme)
o Easy to solve o Restricted time-step sizes :o Run a lot of time steps
Implicit schemeso No restriction by CFL stability limito Accuracy requiremento Accuracy o Computational cost
Efficient Solution Strategies Required for steady-state or time-implicit solvers p- or hp- nonlinear multigrid approach Element Jacobi smoothers
IntroductionIntroduction
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IntroductionIntroduction
Sensitivity Analysis Techniques Applications
o Shape optimization
o Output-based error estimation
o Adaptive mesh refinement
Adjoint Methods
o Linearization of the analysis problem + Transpose
o Discrete adjoint method
Reproduce exact sensitivities to the discrete system
Deliver Linear systems
o Simulation output : L(u), such as lift or drag
o Error in simulation output: e(L) ~ (Adjoint solution) • (Residual of the Analysis Problem)
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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
ObjectiveObjective
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Model ProblemModel Problem
Two-dimensional Compressible Euler Equations Conservative Formulation
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OutlineOutline
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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Triangulation Partition:
DG weak statement on each element, k
Integrating by parts
Solution Expansion
Steady-state system of equations
Discontinuous Galerkin DiscretizationsDiscontinuous Galerkin Discretizations
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Pressure contours using p=0 discretization and p=0 boundary elementsPressure contours using p=4 discretization and p=4 boundary elements
Compressible Channel Flow over a Gaussian BumpCompressible Channel Flow over a Gaussian Bump Free stream Mach number = 0.35 HLLC Riemann flux approximation Mesh size: 1248 elements
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Compressible Channel Flow over a Gaussian BumpCompressible 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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Element Jacobi Smoothers Single level method p-independent h-dependent
Compressible Channel Flow over a Gaussian BumpCompressible Channel Flow over a Gaussian Bump
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p- or hp-multigrid approach p-independent h-independent
Compressible Channel Flow over a Gaussian BumpCompressible Channel Flow over a Gaussian Bump
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OutlineOutline
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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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,
Output-based Spatial Error EstimationOutput-based Spatial Error Estimation
Goal: Find an approximation of without solving on the fine mesh
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Output-based Spatial Error EstimationOutput-based Spatial Error Estimation Goal: Find an approximation of without solving on the fine mesh
Taylor series expansion
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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
Output-based Spatial Error EstimationOutput-based Spatial Error Estimation
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o Set an error tolerance, ETOL
o Necessary refinement for an element if
is used to drive mesh adaptation Element-wise error indicator
Refinement CriteriaRefinement Criteria
Flag elements required for refinement
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hp-refinement◦ Local implementation of the h- or p-refinement individually
Mesh RefinementMesh Refinement h-refinement
◦ Local mesh subdivision
H
p
h
PP
P P p+1
H p-enrichment◦ Local variation of discretization orders
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Local smoothness indicator Element-based Resolution indicator [Persson, Peraire] Inter-element Jump indicator
Additional Criteria for Additional Criteria for hphp-refinement-refinement
For each flagged element: How to make a decision between h- and p-refinement?
[Krivodonova,Xin,Chevaugeon,Flaherty],
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Subsonic Flow over a Four-Element AirfoilSubsonic Flow over a Four-Element Airfoil
Initial mesh (1508 elements)
• 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
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Comparisons on hp-Multigrid convergence for the flow and adjoint solutions
Flow and adjoint problemstarget functional of lift
Subsonic Flow over a Four-Element AirfoilSubsonic Flow over a Four-Element Airfoil
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hh-Refinement for Target Functional of Lift-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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Comparison between h-refinement and uniform mesh refinement
Error convergence history vs. degrees of freedom
Error convergence history vs. CPU time (sec)
hh-Refinement for Target Functional of Lift-Refinement for Target Functional of Lift
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Fixed underlying grids (1508 elements)
Final p-adapted meshdiscretization orders: p=1~4
Spatial error distribution for the objective functional of drag
pp-Enrichment for Target Functional of Drag-Enrichment for Target Functional of Drag
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Error convergence history vs. degrees of freedom
Error convergence history vs. CPU time (sec)
pp-Enrichment for Target Functional of Drag-Enrichment for Target Functional of Drag Comparison between p-enrichment and uniform order refinement
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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
Initial mesh: 17,072 elements
Hypersonic Flow over a half-circular CylinderHypersonic Flow over a half-circular Cylinder
Final hp-adapted mesh: 42,234 elements. Discretization orders: p=0~3
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Final pressure and Mach number solutionsHypersonic Flow over a half-circular CylinderHypersonic Flow over a half-circular Cylinder
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Convergence of the objective functional
Hypersonic Flow over a half-circular CylinderHypersonic Flow over a half-circular Cylinder
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OutlineOutline
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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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)
Implicit Time-integration SchemesImplicit Time-integration Schemes
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Initial condition Isentropic vortex perturbation; Periodic boundary conditions HLLC Flux approximation p = 4 spatial discretization ∆ t = 0.2
Convection of an Isentropic VortexConvection of an Isentropic Vortex
BDF1 (First-order accurate)
IRK4 (Fourth-order accurate)
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Temporal accuracy and efficiency study for various temporal schemes
Convection of an Isentropic VortexConvection of an Isentropic Vortex
Error convergence vs. time-step sizes Error convergence vs. Computational time
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Shedding Flow over a Triangular WedgeShedding Flow over a Triangular Wedge
Unstructured computational mesh with 10836 elements
Free-stream Mach number = 0.2 Unstructured mesh with 10836 elements Various spatial discretizations and temporal schemes
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Shedding Flow over a Triangular WedgeShedding Flow over a Triangular Wedge
Density solution using p = 1 discretization and BDF2 scheme
Free-stream Mach number = 0.2 Unstructured mesh with 10836 elements Various spatial discretizations and temporal schemes
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t = 100 Various spatial discretizations and temporal schemes
Shedding Flow over a Triangular WedgeShedding Flow over a Triangular Wedge
p = 1 and BDF2
p = 1 and IRK4
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t = 100 Various spatial discretizations and temporal schemes
Shedding Flow over a Triangular WedgeShedding Flow over a Triangular Wedge
p = 1 and BDF2
p = 3 and IRK4
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OutlineOutline
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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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
o Linearization of the unsteady flow equations
o Transpose operation results in a backward time-integration
Output-based Temporal Error EstimationOutput-based Temporal Error Estimation
Forward time-integration
Backward time-integration
Current
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Two successively refined time-resolution levels H: coarse level functional h: fine level functional
Approximation of fine level functional
Output-based Temporal Error EstimationOutput-based Temporal Error Estimation
Localized functional error (for each time step i)
BDF1:
IRK4:
Local time-step subdivision if
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Implementation for BDF1 scheme ( p = 2) Validation of adjoint-based error correction Objective function: Drag at t = 5
Shedding Flow over a Triangular WedgeShedding Flow over a Triangular Wedge
Error prediction for two time-resolution levels
Computed functional error
(Reconstructed adjoint) • (Unsteady residual)
Refined time-resolution levels
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Shedding Flow over a Triangular WedgeShedding Flow over a Triangular Wedge
Error convergence vs. computational costError convergence vs. time steps (i.e. DOF)
Adaptive time-step refinement approach vs. Uniform time-step refinement approach
Objective functional:
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OutlineOutline
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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ConclusionsConclusions 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 WorkFuture 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