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
n
jjpj
n
iiipi
wuN
PwuN
uC
0,
0,
)(
)(
)(
otherwise0
if1 10,
iii
uuuuN
uNuu
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
cum
fere
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
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