Shape RepresentationWahid Ghaly

Mechanical and Industrial Engineering

NATO RTO AVT-167 Lecture SeriesOctober 26-27, 2009Montreal, Canada

NATO RTO AVT-1672

Outline

• Objectives and context• Shape representation/parameterization options• Compressor and turbine airfoil representation• Turbine stage representation in 3D flow• Summary

NATO RTO AVT-1673

Shape Representation

• Accurate, flexible and robust shape representation• Most suitable representation for a given shape• Least number of shape parameters that are directly

related to the design parameters and are used as optimization variables

• Preferably a CAD-native parameterization• Can the geometric representation make the

optimization approach more efficient?• Can it reduce the design problem complexity?

NATO RTO AVT-1674

Intended applications

• Component level optimization, e.g. turbine or compressor

• Single and multiple blade rows, disciplines, objectives, single and multipoint

• Airfoils (2D) and blades (3D) profiles • Global - low fidelity - representation • Local/Global high fidelity representation

NATO RTO AVT-1675

Low and high fidelity representations

• Global - low fidelity – representation– Shape is represented by a few low order polynomials– Change in any point on the curve affects the shape

globally

• Local/Global high fidelity representation– Shape is represented by a continuous curve with e.g.

NURBS, B-splines, Bezier curves, …(Note that the 2nd and 3rd representations are subsets of NURBS)

NATO RTO AVT-1676

Global – low fidelity - representation

AXIAL CHORD

TANGENTIALCHORD

INLETBLADEANGLE

LEADINGEDGERADIUS INLET 1/2

WEDGEANGLE

UNGUIDEDTURNING

EXITBLADEANGLE

TRAILINGEDGERADIUS

THICKNESS

Turbine airfoil is represented by 5 Conic sections

NATO RTO AVT-1677

VKI

Global – low fidelity - model

DFVLRE/TU-4

NATO RTO AVT-1678

Global/Local high fidelity rep., NURBS

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jjpj

n

iiipi

wuN

PwuN

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)(

)(

)(

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if1 10,

iii

uuuuN

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uuuN

uu

uuuN pi

ipi

pipi

ipi

ipi 1,1

11

11,,

}1,1,1...,,0,0,0{1

p

U

– C NURBS curve– Pi Control points

– wi Weights

– Ni,p Basis function

– p degree of polynomial,

(p=2 in this work)

– U Knot vector

NATO RTO AVT-1679

Examples of C2 Continuity Curves

DFVLR

VKI

ETU-4

NATO RTO AVT-16710

Shape optimization methodology

– Shape representation:• Low order - global - representation• High order representation, e.g. NURBS, B-Splines, Bezier

– Optimization method:• Direct: GA, SA• Indirect: Gradient/Newton-based, Control Theory-based

– Choice and computation of objective function:• High fidelity simulations (CFD solver of your choice)• Low fidelity using a surrogate model (ANN, RBF, wavelets)

NATO RTO AVT-16711

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

OriginalOptimal

Compressor airfoils in 2D flow

Inviscid transonic case viscous subsonic cases

x

y

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

OptimalOriginal

NATO RTO AVT-16712

Geometric description and parameterization

• The airfoil shape is described by a camber line and a thickness distribution

• Camber line overall flow turning• Thickness structural constraints• They are parameterized using a high

fidelity NURBS function with 11 control points for camber line, f(x), and 9 for thickness distribution, T(x).

• Y-coordinates of the control points are used as the design variables (17 points)

NATO RTO AVT-16713

NACA Transonic compressor redesign

mass flow rate

adiabatic

efficiency

0.3 0.32 0.34 0.36

0.7

0.75

0.8

0.85

0.9

0.95

OriginalImproved

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

OriginalOptimal

OriginalOptimal

Performance map shows Dh ~ 1.7% Original and redesigned compressor airfoils

NATO RTO AVT-16714

NACA 65 subsonic compressor redesign

mass flow rate

adiabatic

efficiency

0.3 0.35 0.4 0.45

0.7

0.75

0.8

0.85

0.9

0.95

1 OriginalImproved

x

y

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

OptimalOriginal

Performance map shows Dh ~ 7%x

y

0 0.2 0.4 0.6 0.8 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ReferenceLowerUpper

Range of airfoil profilesexplored in the design space

Original and redesigned compressor airfoils

NATO RTO AVT-16715

A turbine airfoil profile in 2D flow

• Optimization is done successively on two geometric parameterizations:– Starting from a global shape representation of the airfoil using

the design parameters, optimization is carried out– The resulting profile is used as input to a high fidelity shape

representation so as to refine the profile locally

NATO RTO AVT-16716

The original turbine airfoil

• Total pressure loss coeff. = 17.26 %• Adiabatic efficiency = 91.67 %• Pressure ratio (inlet/outlet) =1.518• Inlet flow angle = 57.4o

• Exit flow angle = -65.82o

• Corrected mass flow rate = 0.191

• Note that this is a low subsonic turbine airfoil with over 91% adiabatic efficiency

NATO RTO AVT-16717

Airfoil shape: global representation, MRATD

MRATD model: Feature-based representation.By construction, it eliminates infeasible turbine airfoil shapes

AXIAL CHORD

TANGENTIALCHORD

INLETBLADEANGLE

LEADINGEDGERADIUS INLET 1/2

WEDGEANGLE

UNGUIDEDTURNING

EXITBLADEANGLE

TRAILINGEDGERADIUS

THICKNESS

NATO RTO AVT-16718

Global-Shape Aerodynamic Optimization

• Objectives– Improve efficiency– Maintain or increase pressure

ratio

• Constraints: Keep the same operating point– Same rotor speed, inlet Pt, Tt,

and exit Ps (CFD)

– Fixed corrected mass flow rate and flow angles (penalty terms added to the objective function)

• Design variables– All parameters affecting the

airfoil SS (6 in all)

Original airfoil: ETU turbine profile

MRATD (design) parameters1. Number of blades = 30

2. Radius = 0.162 m

3. Axial chord C = 0.0396472 m

4. Tangential chord = 78.19%

5. Throat = 33.54%

6. Unguided turning = 12o

7. TE radius = 0.55%

8. Inlet metal angle = 39.4o

9. Exit metal angle = -66.0o

10. SS Inlet wedge angle =15o

11. PS Inlet wedge angle = 30o

12. PS Outlet wedge angle =2.5o

13. Maximum thickness = 26.86%

14. Axial location of maximum thickness = 35%

15. LE ellipse major diameter = 12.61%

16. LE ellipse minor diameter =5.04%

NATO RTO AVT-16719

Global-Shape Optimal profile (MRATD)

• 6 design variables• Dh = 0.4%

Same pressure ratio, reduced mass flow rate and flow angles

NATO RTO AVT-16720

Original vs. Optimal MRATD parameters

MRATD Design parameters

Original Optimal

Tangential chord 0.031m 0.02609m

Throat 0.01330m 0.01312m

Unguided turning 12° 9.95°

SS inlet wedge angle 15° 14.83°

Maximum thickness 0.01065m 0.0122m

LE ellipse minor diameter 0.002m 0.001828m

NATO RTO AVT-16721

ReferenceCandidates

Airfoil shape: local refinement, NURBS

• A close look at the curvature and pressure distributions helps to pinpoint regions where improvements can be made.

x

p

0 0.5 1

0.7

0.8

0.9

1OriginalOptimal

NATO RTO AVT-16722

NURBS optimal vs. MRATD optimal profile

MRATD initialOptimal

MRATD initialOptimal

Efficiency improved by an additional 0.165%, for the same pressure ratio, reduced mass flow rate and flow angles, using 6 NURBS control points.

NATO RTO AVT-16723

Turbine blade profiles in 3D flow

• Geometry representation: 2D Airfoils: MRATD, B-splines and NURBS Hub-to-tip: stacking line going through the 2D airfoils 3D blade shape: obtained by skinning the stacked 2D

airfoils, using compatible B-splines

NATO RTO AVT-16724

CATIA-CFD integration

• NURBS and B-splines are CAD-native parameterizations can be directly integrated into the CAD system

• All blade features are extracted and updated into solid model during the optimization process using:– CAD neutral packages, e.g. CARPI from MIT– CATIA Application Program Interface (API)

Note: MRATD can be integrated into CAD using e.g. CATIA-API

NATO RTO AVT-16725

The Stacking Curve (or line)

NATO RTO AVT-16726

0P

1P

2P

Quadratic Rational Bezier Curve (QRBC)

22

102

222

11002

)1(2)1(

)1(2)1()(

wuwuuwu

PwuPwuuPwuuC

]1,0[u

NATO RTO AVT-16727

0P

1P

2P

QRBC as Stacking Curve

NATO RTO AVT-16728

Leaning the Stacking Curve

Circumferential Direction

NATO RTO AVT-16729

Sweeping the Stacking Curve

Axial Direction

NATO RTO AVT-16730

Bowing the Stacking Curve

Circumferential Direction

NATO RTO AVT-16731

0P

1P

2P

2P

1P

0P

0P

2P

1P

Q

Q

Q

Q

0P

1P

2P

2P

1P

0P

Cir

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ntia

l Pla

ne

Q

QRBCin 1w

Meridional P

lane

2P Q

1P

0P

0P

1P

2PQ

Design Variables

NATO RTO AVT-16732

Design Variable QRBC Parameter Symbol

1. Sweep angle Axial coordinate of P2b

2. Lean angle Circumferential coordinate of P2a

3. Bowing shape in radial direction

Radial coordinate of P1g

4. Bowing shape in circumferential direction

Circumferential coordinate of P1q

5. Bowing intensity Weight of P1 w1

Design Variables

NATO RTO AVT-16733

Stator solidity 1.56Aspect ratio 0.57

Rotor solidity 1.5Aspect ratio 0.904

Single Stage Turbine (E/TU3)

• Low speed subsonic turbine 7800 (rpm)

• Flow coefficient 0.74• Stage loading 1.93• Stage P.R. = 2• Reaction 31%• Reav = 2 Millions

NATO RTO AVT-16734

Stage Optimization

Dh = 1.2% with 5 design variables

Stator Rotor

aso bs

o a ro br

o wrhtt

Min. -30 -15 -5 -10 0 -

Max. 10 5 20 15 3 -

Original 0 0 0 0 0 87.50

Optimum -29.5 -9.4 2.2 -9.7 0.05

88.56

NATO RTO AVT-16735

Summary

• Geometric representation can improve the efficiency of the optimization approach

• It can also reduce the design problem complexity by:– reducing the number of design variables– Eliminating infeasible blade profiles

• It is critical to pick the ‘right’ representation and the ‘right’ parameterization for a given shape

NATO RTO AVT-16736

Thank You