Download - DESIGN OPTIMIZATION OF HIGH-LIFT CONFIGURATIONS USING …aero-comlab.stanford.edu/sanghok/reno2002-s.pdf · 2019-06-23 · Method by comparison with Gradients from Finite Diﬀerence

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DESIGN OPTIMIZATION OF HIGH-LIFT CONFIGURATIONSUSING A VISCOUS ADJOINT-BASED METHOD

Sangho Kim

Stanford University

Juan J. Alonso

Stanford University

Antony Jameson

Stanford University

40th AIAA Aerospace Sciences Meeting and Exhibit

Reno, NV

January, 2002

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Outline

• Introduction

• Objectives

• Description of Continuous Adjoint Method

• Viscous Aerodynamic Sensitivity Accuracy Study

• High-Lift System Design

• Conclusions and Future Work

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High–Lift System Configuration

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Secondary Flow

Viscous Shear Layer

Wake Line

Boundary Layer

SlatFlap

Main Airfoil

Angle of Attack

Lift Coefficient

Slats Extended

Slats Retracted

Flap angle 0 deg

Lift Coefficient

Angle of Attack

Increase Flap Angle

Flap angle 40 deg

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Introduction

• High-Lift System Configuration Design

– Wind tunnel testing

– CFD as an Analysis Tool

– Difficulties in Decision Making

• CFD as a Design Tool

– Automatic Design Procedure : CFD + Gradient Based Optimization

– Gradient Based Optimization

∗ Cost function to be Min/Maximized (Drag, L/D).

∗ Control function to be Optimized (Airfoil Shape).

∗ Design Variables (Mesh points, Sine Bump function)

∗ Constraint (Euler eq. NS eq)

∗ Gradients (Finite Difference, Complex, Adjoint)

∗ Optimization Algorithm (Steepest Descent)

– Control Theory Approach or Adjoint Method by Jameson 1988.

• Continuous Adjoint Design Method using the Navier-Stokes equations as a flow

model and the Gradient Accuracy study.

• Application to High-Lift System Design.

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Objectives

1. Viscous Aerodynamic Sensitivity Accuracy Study

• 2D Implementation of a Continuous Adjoint Design Method that uses the

Navier-Stokes equations as a flow model.

•Verification of the Accuracy and Efficiency of the present Continuous Adjoint

Method by comparison with Gradients from Finite Difference Method.

•Demonstration with Preliminary Examples.

2. High-Lift System Design

•Develop numerical optimization tools for design and development of high-lift system

configurations.

• Improve take-off and landing performance of high-lift systems using a Continuous

Adjoint Design Method.

– Cl, Cd, L/D and Clmax.

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Symbolic Description of Continuous Adjoint Method

Let I be the cost (or objective) function

I = I(w,F)

where

w = flow field variables

F = design variables

The first variation of the cost function is

δI =∂I

∂w

T

δw +∂I

∂FT

δFThe flow field equation and its first variation are

R(w,F) = 0

δR = 0 =

∂R

∂w

δw +

∂R

∂F δF

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Introducing a Lagrange Multiplier, ψ, and using the flow field equation as a

constraint

δI =∂I

∂w

T

δw +∂I

∂FT

δF − ψT

∂R

∂w

δw +

∂R

∂F δF

=

∂I

∂w

T

− ψT∂R

∂w

δw +

∂I

∂FT

− ψT∂R

∂F

δF

By choosing ψ such that it satisfies the adjoint equation∂R

∂w

T

ψ =∂I

∂w,

we have

δI =

∂I

∂FT

− ψT∂R

∂F

δF

= GTδFThis reduces the gradient(G) calculation for an arbitrarily large number of design

variables at a single design point to

One Flow Solution

+ One Adjoint Solution

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Navier–Stokes Inverse : Adjoint Gradient

0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5Comparison of Finite Difference and Adjoint Gradiensts on 512x64 Mesh

control parameter

grad

ient

Adjoint Finite Difference

1a: Finite Difference vs. Adjoint Gradient

0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5Continuous Adjoint Gradient for Different Flow Solver Convergence

control parameter

grad

ient

7(4) orders 6(3) orders 5(2) orders 4(1) orders 3(0) orders

1b: Continuous Adjoint on Flow

Convergence Levels

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Flowchart of the Design Process

Define Design Variables

Parameterize Configuration

Define Cost Function

Solve Flow Equation

Solve Adjoint Equations

Perturb Shape & Modify Grid

Evaluate Gradient

Adjoint

Perturb Shape & Modify Grid

Solve Flow Equations

Evaluate Gradient

Finite Difference

Optimization

New Shape

Optimum? No

Stop

Yes

Loop over Gradients

1 Loop

Loop over Gradients

0 �

20

40

60 �

80 �

100

120

Adjoint vs. Finite Difference

in Cost, N=100

Adjoint

Finite Difference

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Navier–Stokes Inverse : RAE2822 → NACA64A410

RAE2822 TO NACA64A410 MACH 0.750 ALPHA 0.000

CL 0.3196 CD 0.0029 CM -0.0952

GRID 513X65 NCYC 10 RES0.220E+00

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

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2a: Initial, Perror = 0.0504RAE2822 TO NACA64A410 MACH 0.750 ALPHA 0.000

CL 0.5650 CD 0.0111 CM -0.1445


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

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+00

-.8E

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2b: 100 Design Iterations, Perror = 0.0029

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Navier–Stokes Cd Min. : RAE2822 at Fixed Cl = .84

RAE DRAG MACH 0.730 ALPHA 2.739

CL 0.8411 CD 0.0169 CM -0.0975

GRID 513X65 NCYC 400 RES0.624E-03

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

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+01

-.2E

+01

-.2E

+01

Cp

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3a: Initial, CDt=0.0169

M = 0.73, α = 2.739, Cl = 0.8411

RAE DRAG MACH 0.730 ALPHA 2.949

CL 0.8406 CD 0.0097 CM -0.0824

GRID 513X65 NCYC 100 RES0.346E-02

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

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3b: 40 Design Iterations, Cd=0.0097

M = 0.73, α = 2.949, Cl = 0.8406

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Advantages of Viscous Adjoint Method Over Finite Differencing

• Computational Cost was Drastically Reduced.

• Required Level of Flow Solver Convergence is Lower than for Finite Differencing.

• Step Insensitive. (In Adjoint Method, Step is only used for Geometry Perturbation.)

• Required Level of Convergence of Adjoint is Minimal.

=⇒ More Efficient, More Robust Method for Sensitivity Analysis in Viscous Flows.

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High-Lift System Design : SYN103MB

• Design Variables

– Sine Bumps : Shape design for each element.

RAE AIRFOIL TO KORN MACH 0.730 ALPHA 2.790

CL 0.6465 CD 0.0415 CM -0.0884


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

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+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

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– Rigging Variables : Gaps, Overlaps and Deflections

– Angle of Attack (α) : Clmax maximization

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• Grid Topology : Multi-block structured, Overall C- and O- mesh topologies.

Figure 4: Viscous Mesh for 30P30N Multielement Airfoil

• Mesh Perturbation

New grids are generated by shifting the grid points along the radial coordinate lines

depending on the shape boundary modification.

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FLO103-MB PV3 CLIENT Z COORDINATE @ 1.5000 MACH NUMBER from 0.0000 to 0.3825

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High-Lift System Design : SYN103MB (cont.)

• FLO103-MB Methodology

– Runge-Kutta Explicit Time Stepping

– Cell Centered Spatial Discretization

– Jameson-Schmidt-Turkel(JST) Scheme

– Local Time Stepping

– Implicit Residual Smoothing

– Multigridding

• Turbulence

– Spalart-Allmaras One Equation Model solved by ADI Scheme for Multi-Element

Airfoil Designs. ( Dr. Creigh McNeil )

• ADJ103-MB : The same methodology.

• MPI (Message Passing Interface) parallel solution methodology.

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FLO103-MB CONVERGENCE HISTORY (SA Model)

0 200 400 600 800 1000 1200 1400 1600 1800 2000−2

0

2

4

6

8

Cl

Cl Convergence History

0 200 400 600 800 1000 1200 1400 1600 1800 200010

−2

100

102

104

Multigrid Cycles

Res

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FLO103MB Convergence History

4a: Convergence history for the Spalart-Allmaras model

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Cp Comparison : M=0.2, α = 8◦, Re = 9× 106, 30P30N

30P30N TEST MACH 0.200 ALPHA 8.010

CL 3.1514 CD 0.0651 CM -1.3693

GRID 118784 NCYC 2000 RES 0.950E+00

2.0

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4b: Experimental and computational Cp : o - Exp. +,x - Comp.

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Stall Prediction : M = 0.2, Re = 9× 106

−5 0 5 10 15 20 25−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

angle of attack

Cl

plot Cl vs. alpha

Total

Main

Slat

Flap

−*− : Comp. o : Exp.

4c: Experimental and computational Cl vs. α

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Velocity Profiles : M = 0.2 α = 8◦ Re = 9× 106

4d: High-Lift Configuration

0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

u/ainf

d/c

velocity profile at x=0.45

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

u/ainf

d/c

velocity profile at x=0.89

Comp 9MExp 9M

4e: Comparison of velocity profiles

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High–Lift System Design Philosophy

• Neccessary in take-off and landing conditions

– Minimize runway length and Maximize payload.

• Typical Configurations

− High Maximum Lift Coefficient− High L/D (Climb)

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Take−Off Configuration Landing Configuration

− High Maximum Lift Coefficient− Less Strict Requirements on L/D

• Approaches

– Take–off : Maximize Cl while maintaining high L/D.

∗ Fix Cd and Maximize Cl (shape and α)

∗ Fix Cl and Minimize Cd (shape and α)

– Landing : Maximize Cl without worrying about Cd or L/D.

∗ Clmax maximization.

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Multi-Element Cl Max. : 30P30N with Fixed Cd=0.065


CL 4.0414 CD 0.0650 CM -1.4089

GRID 204800 NCYC 5000 RES 0.285E-02

2.0

00 0

.000

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-10.

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-12.

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-14.

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5a: Initial, Cl=4.0412

M = 0.2, α = 16.02◦, Re = 9× 106,

Cd = 0.0650


CL 4.1227 CD 0.0661 CM -1.4610

GRID 204800 NCYC 5000 RES 0.185E-01

2.0

00 0

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5b: 10 Design Iteration, Cl=4.1227

M = 0.2, α = 15.127◦, Re = 9× 106,

Cd = 0.0661

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RAE2822 Clmax Maximization.

0 2 4 6 8 10 12 14 16 180.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

angle of attack(α), degrees

Cl

Clmax

Maximization for RAE2822 Airfoil (Approah I)

O

A

B C

α at Clmax

: 11.1 %,Cl

max : 14 % improved.

Cl = 2π α (radian)

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RAE2822 Clmax Maximization.

RAE2822 TEST MACH 0.200 ALPHA 14.633

CL 1.6430 CD 0.0359 CM -0.4346

GRID 32768 NCYC 1000 RES 0.150E-02

2.0

00 0

.000

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-10.

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5c: Initial, Cl=1.6430

M = 0.2, α = 14.633◦, Re = 6.5M ,

Cd = 0.0359

RAE2822 TEST MACH 0.200 ALPHA 14.633

CL 1.8270 CD 0.0326 CM -0.4901

GRID 32768 NCYC 1000 RES 0.168E-03

2.0

00 0

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5d: 31 Design Iterations, Cl = 1.8270

M = 0.2, α = 14.633◦, Re = 9× 106,

Cd = 0.0326.

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RAE2822 Clmax Maximization Using α and Bump.

0 2 4 6 8 10 12 14 16 180.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Cl

Clmax

Maximization for RAE2822 Airfoil (Approach II)

α at Clmax

: 6.8 %,Cl

max : 13.3 % improved.

Cl = 2πα(radian)

Approach II Design Curve

Approach I Design Curve

O

A

B C

D

E

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30P30N Clmax Maximization.

19 20 21 22 23 24 25 264.2

4.25

4.3

4.35

4.4

4.45

4.5

4.55Maximum Cl Maximization for 30P30N


Cl

Design Curve

Baseline Cl vs. α Curve

α at Clmax

: 501 Cl counts,α

clmax : 0.1o increased.

S

Edesigned

Ebaseline

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Conclusions and Future Work

• Making use of the Large Computational Savings provided by the Adjoint

Method, a Numerical Optimization Procedure for Designs of

High-Dimensional Design Space has been developed.

• High-Lift System Configuration Design Examples have been performed in order

to validate the Sensitivity Calculation Procedure using our Continuous Viscous

Adjoint Method.

• Further Study of Adjoint Solver and Adjoint Boundary Conditions.

• Enhancement of Design Methods by Survey of Optimization Algorithm and Design

Parameterization.

• Extension to More Realistic Problems inculding 3D Design Problems and Multi

Point Designs

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