Li Wang PhD Candidate Department of Mechanical Engineering University of Wyoming Laramie, WY
description
Transcript of Li Wang PhD Candidate Department of Mechanical Engineering University of Wyoming Laramie, WY
Li Wang
PhD CandidateDepartment of Mechanical Engineering
University of WyomingLaramie, WY
April 21, 2009
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
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)
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
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
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)
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
Model ProblemModel Problem
Two-dimensional Compressible Euler Equations Conservative Formulation
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
Triangulation Partition:
DG weak statement on each element, k
Integrating by parts
Solution Expansion
Steady-state system of equations
Discontinuous Galerkin DiscretizationsDiscontinuous Galerkin Discretizations
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
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
Element Jacobi Smoothers Single level method p-independent h-dependent
Compressible Channel Flow over a Gaussian BumpCompressible Channel Flow over a Gaussian Bump
p- or hp-multigrid approach p-independent h-independent
Compressible Channel Flow over a Gaussian BumpCompressible Channel Flow over a Gaussian Bump
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
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
Output-based Spatial Error EstimationOutput-based Spatial Error Estimation Goal: Find an approximation of without solving on the fine mesh
Taylor series expansion
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
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
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
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],
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
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
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
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
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
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
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
Final pressure and Mach number solutionsHypersonic Flow over a half-circular CylinderHypersonic Flow over a half-circular Cylinder
Convergence of the objective functional
Hypersonic Flow over a half-circular CylinderHypersonic Flow over a half-circular Cylinder
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
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
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)
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
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
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
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
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
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
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
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
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
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:
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
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
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