Effect of Shape Parameterization on

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Effect of Shape Parameterization on Aerodynamic
Shape Optimization
Patrice Castonguay Undergraduate Student Siva
Nadarajah Assistant Professor Department of
Mechanical Engineering McGill University Quebec,
Canada
45th AIAA Aerospace Science Meeting and
Exhibit Reno, Nevada January 8-11, 2007
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Outline

• Motivations
• Objectives
• CFD as a Design Tool
• The Adjoint Method
• Optimization Procedure
• Shape Parameterization Techniques
• Results
• Conclusions
• Current and Future Work

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1. Motivations

• Only few works have studied the effect of shape
  parameterization on the aerodynamic optimization
  process
• Need to address the following questions
• Effect of smoothing the gradient
• Design space provided from Hicks-Henne bump
  functions
• Mesh-points method for industrial strength design
  ?
• B-Spline curves and PARSEC for inverse design and
  drag minimization ?

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2. Objectives

• Study the effect of shape parameterization on
  aerodynamic shape optimization by comparing four
  shape representation methods
• Mesh Points
• B-spline Curves
• Hicks-Henne Bump Functions
• PARSEC Method
• Compare them based on accuracy and impact on the
  computational cost in the context of 2D inverse
  design and drag minimization cases

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3. CFD as a Design Tool
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4. The Adjoint Method

• The aerodynamic properties that define the cost
  function I depends on the flow variables
• (?) and on the design variables (F) which
  represent the geometry. Then,

• and

• Now, suppose a governing equation R which
  expresses the dependence of the
• flow variables (?) and the design variables (F)
  is given by

• Then

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4. The Adjoint Method

• Because dR 0, it can be multiplied by a
  Lagrange multiplier ? and substracted from the
• variation dI. Thus, we get

• By choosing ? to satisfy the adjoint equation

• we find that

• where

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5. Optimization Procedure

• Let represent the design variable, and the
  gradient. An improvement can be made with a
• shape change

• The gradient can be replaced by a smoothed
  value in the descent process. This ensures
• that each new shape in the optimization sequence
  remains smooth and acts as a
• preconditioner which allows the use of larger
  steps. The smoothed gradient may be
• calculated from

• where is the smoothing parameter. The
  smoothing leads to a large reduction in the
  number
• of design iterations needed for convergence.

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6. Shape Parameterization Techniquesa) Mesh
Points

• Design variables are the x and y location of the
  mesh points representing the surface to
• be optimized
• Easy to implement
• Large number of design variables to represent 2D
  or 3D geometries
• Displacement of a single mesh point can lead to
  unsmooth shapes and cause the flow
• solver to become ill-conditioned

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6. Shape Parameterization Techniquesb)
Hicks-Henne Bump Functions
Geometry can be parameterized using the weighted
sum of a number of Hicks-Henne bump functions as
follows
where t1 locates the maximum point of the bump
and t2 controls the width of the bump. The
design variables are the weights aj multiplying
each Hicks-Henne bump functions.
A set of 16 Hicks-Henne bump functions with
parameter t2 10
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6. Shape Parameterization Techniquesc) B-Spline
Curves
Airfoil is initially represented as a B-spline
curve. Design is accomplished by directly moving
the control points representing the airfoil. The
design variables are therefore the x and y
locations of the control points. The position
vector along a curve is given by
where the Bi are the position vectors of the n1
control points, Ni,k(t) are the B-spline basis
functions of order k.
Airfoil represented from a set of 11 B-spline
control points
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6. Shape Parameterization Techniquesd) PARSEC
Method
PARSEC method uses 11 parameters to represent an
airfoil. They are

• Leading edge radius
• Upper crest position
• Upper crest curvature
• -Lower crest position
• -Lower crest curvature
• Trailing edge location
• Trailing edge angles

Two polynomials are used to define the upper and
lower surface of the airfoil
The coefficients an and bn can be found from the
11 PARSEC parameters.
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7. Results Inverse Design

• - NACA 0012 to ONERA M6, M0.75, a 2
• Objective Function
• Design Procedure

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7. Results Inverse DesignB-Spline Control
Points with Smoothed Gradient
Final Airfoil Shapes for Various of B-Spline
Control Points
Convergence of Objective Function
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7. Results Inverse DesignB-Spline Control
Points with Smoothed Gradient
Final Pressure Distribution
Final Pressure Distribution Leading Edge
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7. Results Inverse DesignB-Spline Control
Points without Smoothed Gradient
Final Airfoil Shapes for Various of B-Spline
Control Points
Convergence of Objective Function
Final Pressure Distribution
Convergence of Objective Function
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7. Results Inverse DesignHicks-Henne Bump
Functions with Smoothed Gradient
Final Airfoil Shapes for Various of Bump
Functions
Convergence of Objective Function
Final Pressure Distribution - LE
Final Pressure Distribution
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7. Results Inverse DesignHicks-Henne Bump
Functions without Smoothed Gradient
Final Airfoil Shapes for Various of Bump
Functions
Convergence of Objective Function
Final Pressure Distribution
Final Pressure Distribution - LE
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7. Results Inverse DesignB-Spline Curves vs
Hicks-Henne Bump Functions
L2 Norm of Objective Function after 300 design
cycles for various cases
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7. Results Inverse DesignB-Spline Curves vs
Hicks-Henne Bump Functions
Number of design cycles for Objective Function
1.0E10-5
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7. Results Inverse DesignHicks-Henne Bump
Functions Effect of Bump Width
Bump width (controlled by parameter t2) affects
convergence of objective function
Optimum value of t2 depends on the number of bump
functions used
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7. Results Inverse DesignPARSEC Method
Target and Final Airfoil Shapes using PARSEC
method
Final Pressure Distribution
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7. Results Drag Minimization

• - NACA 0012, M0.75, a 2
• Objective Function
• Coefficients were chosen as 1/3
• 
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• - Airfoil thickness is constrained as well

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7. Results Drag MinimizationB-Spline Curves vs
Hicks-Henne Bump Functions
Final Airfoil Shapes
Final Pressure Distribution
Convergence of Objective Function
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7. Results Drag MinimizationB-Spline Curves vs
Hicks-Henne Bump Functions
Initial Lift and Drag cl 0.373
cd 0.0151
Final Lift and Drag for Various Cases
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8. Conclusions

• The smoothing of the gradient accelerates the
  convergence of the objective function even when
  using B-Spline curves as design variables
• Without the smoothing of the gradient, increasing
  the number of B-spline control points leads to a
  final shape that departs from the target shape
  due to unsmooth gradients
• The Hicks-Henne bump function are able to provide
  a design space where the target pressure is
  obtainable but the level of accuracy is lower
  than that achieved from the mesh points and
  B-spline curves
• The width of the Hicks-Henne bump functions
  affects the convergence of the objective function

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8. Conclusions

• The PARSEC method is unable to modify the leading
  edge shape of the NACA 0012 airfoil to reproduce
  the target shape (Onera M6 airfoil)
• Both Hicks-Henne bump functions and B-spline can
  sucessfully be used for drag minimization
• The Hicks-Henne bump function are able to modify
  the upper surface in the vicinity of the shock
  wave to eliminate the pressure discontinuity,
  however it was unable to provide the necessary
  shape modifications for the inverse design
  problem

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9. Current and Future Work

• Use an optimization algorithm that requires the
  Hessian matrix and a more accurate line search
  algorithm
• Study the effect of shape parameterization
  techniques for unsymmetrical airfoils
• Study the PARSEC method for drag minimization
• Extend the study for three dimensional shapes

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Questions ???

• Patrice Castonguay
• patrice.castonguay_at_mail.mcgill.ca
• Siva Nadarajah
• siva.nadarajah_at_mcgill.ca

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