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Shape Optimization of Low Speed Airfoilsusing Matlab and Automatic Differentiation

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Shape Optimization of Low Speed Airfoils using MATLAB and Automatic Differentiation

Christian Wauquiez

Stockholm 2000 Licentiate’s Thesis

Royal Institute of Technology

Department of Numerical Analysis and Computing Science


The goal of the project is to develop an innovative tool to perform shape optimization of low speed airfoils. This tool is written in Matlab, and is constructed by coupling the Matlab Optimization Toolbox with a parametrised numerical aerodynamic solver.

The airfoil shape is expressed analytically as a function of some design parameters. The NACA 4 digits library is used with design parameters that control the camber and the thickness of the airfoil.

The solver has to provide fast and robust computation of the lift, pitching moment and drag of an airfoil placed in a low-speed viscous flow. A one-way coupled inviscid - boundary layer model is used.

The inviscid flow is computed with a linear vortex panel method, which provides the lift and moment coefficients. The boundary layer is computed using an integral formulation : the laminar part of the flow is computed with a two-equation formulation, and the turbulent part is solved with Head’s model. An e9-type amplification formulation is used to locate the transition area. Finally, the drag coefficient is computed using the Squire-Young formula.

In order to be used in optimization, the solver must provide derivatives of the objective function and limiting constraints with the solution for each set of parameters.

These derivatives are computed by automatic differentiation, a technique for augmenting computer programs with the computation of derivatives based on the chain rule of differ- ential calculus. The recent Matlab automatic differentiation toolbox ADMAT is used.

Finally as an application, sample optimization problems are solved using the Matlab Optimi- zation Toolbox, and the resulting optimal airfoils are analysed.

ISBN 91-7170-520-1 TRITA-NA-0004 ISSN 0348-2952 ISRN KTH/NA/R--00/04--SE


  1. The Aerodynamics Solver        7

1.1- Introduction - Overview of the Model 1.2- Airfoil and Flow Parameters

1.3- Inviscid Flow Model 1.4- Boundary Layer Model

  1. Automatic Differentiation        39

2.1- Introduction

2.2- Method Fundamentals

2.3- Computer Implementation

2.4- ADMAT, Automatic Differentiation Toolbox for Matlab 2.5- Application of ADMAT to the Aerodynamic Solver

  1. Airfoil Shape Optimization        51

3.1- Definition of the Optimization Problems 3.2- Solving the Optimization Problems

Conclusion        61



I would like to thank my supervisor, Associate Professor Jesper Oppelstrup, for his interest and support throughout this work, and Professor Arthur Rizzi for providing information and for many helpful discussions.

I would also like to thank, for advice and technical support, Associate Professor Ilan Kroo from Stanford University and Desktop Aeronautics, and Doctor Arun Verma from the Cornell University Theory Center.

Financial support from NUTEK through the Parallel Scientific Computing Institute, KTH, and TFR through the Center for Computational Mathematics and Mechanics, KTH, is gratefully acknowledged.


The performance of an airfoil can be characterized by three quantities : the lift, moment and drag coefficients : Cl, Cm and Cd respectively. They represent the aerodynamic loads applied to the airfoil. The actual loads are proportional to the coefficients times the square of the flow velocity.

  • Cl corresponds to the force acting on the airfoil in the direction orthogonal to the flow, which allows an aircraft to fly by compensating its weight.

  • Cm corresponds to the moment of the aerodynamic force with respect to the quarter of the airfoil chord length. For the equilibrium of an aircraft, the pitching moment of the main wing has to be compensated by the moment of a negative-lift tail. Cm should therefore not be too large.

  • Finally, Cd corresponds to the component of the force in the flow direction, which hinders aerodynamic performances and causes fuel consumption.

The present work focuses on shape optimization of airfoils in low speed viscous flows, based on the analysis of Cl, Cm and Cd.

The airfoils are chosen from the NACA 4 digits library [1], in which the shape is expressed analytically as a function of three parameters. The library is presented in section 1.2 of this report.

The formulation used to compute the aerodynamic coefficients is an inviscid - boundary layer model. The advantage of this kind of approach is that it provides a fast computation of the flow solution, the disadvantage being that cases with massive flow separation are impossible to handle. Famous codes based on this formulation include for 2D cases Desktop Aeronautics’ Panda [9], and Mark Drela’s ISES [10] and Xfoil [12], and for 3D cases Brian Maskew’s VSAERO [3]. The solver is presented in section 1.



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