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