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Rice University

Construction of Airfoil Performance Tables by the Fusion of Experimental and Numerical Data

by

Jose Navarrete

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE

Master of Science

APPROVED, THESIS COMMITTEE: _______________________________________ Andrew J. Meade, Jr., Committee Chair Professor of Mechanical Engineering &

Materials Science ___________________________________ Pol Spanos Professor of Mechanical Engineering &

Materials Science ___________________________________ Satish Nagarajaiah

Associate Professor of Civil Engineering & Mechanical Engineering & Materials Science

HOUSTON, TEXAS

AUGUST 2004

ii

ABSTRACT

Construction of Airfoil Performance Tables by the Fusion of Experimental and Numerical Data

by

Jose Navarrete

A method that combines experimental airfoil coefficient data with numerical data has been

developed to construct airfoil performance tables given limited data sets. This work

addresses the problem faced by engineers and aerodynamicists that currently rely on

incomplete performance tables when researching airfoil characteristics. The method

developed utilizes the Sequential Function Approximation (SFA) neural network tool and

employs a simple regularization scheme to fuse multi-dimensional experimental and

computational fluid dynamics (CFD) data efficiently. The method is considered an adaptive

and robust tool requiring relatively little computational demand and minimal user

dependence. An existing performance table for the NACA 0012 airfoil was used as a test

case to verify the feasibility of the SFA-fused network. A second test case assesses the

method’s viability for a more realistic and challenging problem using highly sparse and

scattered data sets for the SC1095 airfoil. Results from both studies realize the method’s

capability to make consistent approximations and smooth interpolations given only limited

experimental data. Comparisons are made with other scattered data approximation

techniques. The testing conditions, requirements, and limitations of this approach are

discussed and future applications and recommendations are made.

iii

Acknowledgments

I would like to thank my advisor Dr. Andrew J. Meade for making my graduate

studies a possibility. Special regards go to William Warmbrodt, Rotorcraft Branch Chief,

NASA Ames) for the amazing opportunity and generous support that allowed me to

complete this work at NASA Ames. Thanks to Wayne Johnson, Gloria Yamauchi and

everyone at the Rotorcraft branch for their enduring advice and insightful comments.

Their unwavering dedication, patience, and advice have made this experience

unforgettable and invaluable. Thanks to Carol Roland and Teresa Scherbing at the

Education Associate Program office at NASA Ames for their caring assistance in

acquiring a research grant. Acknowledgments go to Roger Strawn (Rotorcraft CFD

Group, NASA Ames), Eddie Mayda and Case van Dam (University of California at

Davis) for their efforts in generating the CFD table and granting access to the

experimental and computational data sets used in this report.

The completion of this work was also made possible through the kind assistance

and guidance of Judith Farhat and Kay McStay at the Mechanical Engineering and

Materials Science department graduate office. For their help I hold my deepest gratitude.

Of course, enthusiastic thanks go to Jorge “Sunshine” Gomez, Kalins Fotev, E.J.

Summerlin, Eric Cartman, Kyle Broflovski, Stan Marsh, Kenny McCormick, and God for

their unforgiving friendship and encouragement throughout the progress of this work.

Finally, thanks to my family for being a constant source of inspiration and my reason for

everything.

iv

Table of Contents

Chapter 1 Introduction ................................................................................................. 1

Chapter 2 Inverse Problem........................................................................................... 5

2.1 Ill-Posed Problem ............................................................................................ 6

2.2 Tikhonov Regularization.................................................................................. 7

Chapter 3 Multi-Dimensional Function Approximation.............................................. 10

3.1 Neural Networks............................................................................................ 10

3.2 Radial Basis Function Network......................................................................13

3.3 Sequential Function Approximation Network ................................................ 16

3.3.1 Method of Weighted Residuals .............................................................. 16

3.3.2 SFA Theory ........................................................................................... 18

3.3.3 Algorithm .............................................................................................. 21

3.4 Regularized Fusing Approach........................................................................ 22

Chapter 4 SFA Implementation as an Airfoil Approximation Tool............................. 25

4.1 NACA 0012 Airfoil Experimental and Numerical Data.................................. 26

4.1.1 C81 and CFD Data................................................................................. 26

4.1.2 Network Testing Conditions................................................................... 36

4.1.3 Data Size and Experimental Error Studies .............................................. 42

4.2 SC1095 Airfoil Experimental and Numerical Data......................................... 45

4.2.1 Experimental and CFD Data................................................................... 45

4.2.2 Network Testing Conditions................................................................... 53

4.2.3 SFA Network Approximated SC1095 Airfoil Performance Table........... 53

v

Chapter 5 Results and Analysis .................................................................................. 55

5.1 NACA 0012 Test Runs .................................................................................. 55

5.1.1 RMS Error ............................................................................................. 55

5.1.2 Coefficient Plots .................................................................................... 59

5.2 SC1095 Test Runs ......................................................................................... 89

Chapter 6 Neural Network Comparisons .................................................................... 97

6.1 Radial Basis Function Network......................................................................97

6.2 Generalized Regression Neural Network........................................................ 98

6.3 Spread Setting................................................................................................ 99

6.4 Network Properties ...................................................................................... 100

6.5 Reproducing Data ........................................................................................ 102

6.6 Interpolating Data ........................................................................................ 105

Chapter 7 Conclusions and Recommendations ......................................................... 113

Bibliography ............................................................................................................... 116

vi

List of Figures

Figure 3-1. MLP architecture ....................................................................................... 11

Figure 3-2. Schematic of the back-propagation training process .................................... 12

Figure 3-3. RBF network architecture........................................................................... 13

Figure 3-4. Example RBF network approximations ....................................................... 14

Figure 3-5. SFA algorithm ........................................................................................... 21

Figure 3-6. Fusing method flowchart............................................................................ 24

Figure 4-1. NACA 0012 airfoil..................................................................................... 26

Figure 4-2. Re and M distribution for the NACA 0012 data sets ................................... 27

Figure 4-3. Angle of attack locations available in the NACA 0012 experimental (EXP)

and CFD data sets.................................................................................................. 28

Figure 4-4. NACA 0012 Cl vs α comparison of C81 and CFD data at M = 0.30........... 30

Figure 4-5. NACA 0012 Cl vs α comparison of C81 and CFD data at M = 0.80............ 31

Figure 4-6. NACA 0012 Cd vs. α comparison of C81 and CFD data at M 0.3............ 32

Figure 4-7. NACA 0012 Cd vs. α comparison of C81 and CFD data at M 0.8............ 33

Figure 4-8. NACA 0012 Cm vs. α comparison of C81 and CFD data at M = 0.30......... 34

Figure 4-9. NACA 0012 Cm vs. α comparison of C81 and CFD data at M = 0.80......... 35

Figure 4-10. Cl vs. α for M = 0.7.................................................................................. 37

Figure 4-11. Error function vs. α for Cl data at M = 0.7................................................ 37

Figure 4-12. Error function vs. α for Cl data at M = 0.7, normalized............................. 38

Figure 4-13. Cl vs. α for M = 0.7, normalized .............................................................. 39

Figure 4-14. Optimum initial sigma ( o) setting............................................................ 41

vii

Figure 4-15. Optimum tolerance setting........................................................................ 42

Figure 4-16. SC1095 airfoil.......................................................................................... 45

Figure 4-17. Re and M number distribution for the SC1095.......................................... 47

Figure 4-18. α locations available in the SC1095 experimental (EXP) and CFD data sets

.............................................................................................................................. 48

Figure 4-19. SC1095 Cl vs α comparison of experimental and CFD data...................... 50

Figure 4-20. SC1095 Cd vs. α comparison of experimental and CFD data .................... 51

Figure 4-21. SC1095 Cm vs. α comparison of experimental and CFD data.................... 52

Figure 5-1. Error for Cl test case................................................................................... 57

Figure 5-2. Error for Cd test case .................................................................................. 58

Figure 5-3. Error for Cm test case.................................................................................. 58

Figure 5-4. Cl vs. α for NACA 0012 Test Run 1. Re = 3.72 x 106, M = 0.30, ε = 0, s =

390 and n = 390..................................................................................................... 61

Figure 5-5. Cl vs. α for NACA 0012 Test Run 22. Re = 8.68 x 106, M = 0.70, ε = ±

0.05, s = 390 and n = 168....................................................................................... 62

Figure 5-6. Cl vs. α for NACA 0012 at Re = 3.72 x 106, M = 0.30 ............................... 63



Figure 5-9. Cl vs. α for NACA 0012 at Re = 11.16 x 106, M = 0.90 ............................. 66

Figure 5-10. Cd vs. α for NACA 0012 Test Run 29 with ε = 0 ..................................... 70

Figure 5-11. Cd vs. α for NACA 0012 at Re = 3.47 x 106, M = 0.28............................. 71



viii

Figure 5-14. Cd vs. α for NACA 0012 at Re = 11.41 x 106, M = 0.92........................... 74

Figure 5-15. Cd vs. α for NACA 0012 at Re = 3.47 x 106, M = 0.28............................ 76



Figure 5-18. Cd vs. α for NACA 0012 at Re = 11.41 x 106, M = 0.92........................... 79

Figure 5-19. Cd vs. α for NACA 0012 for Test Run 49................................................. 80

Figure 5-20. Cm vs. α for NACA 0012 Test Run 57. Re = 8.68 x 106, M = 0.70, ε = 0, s

= 423 and n = 423.................................................................................................. 82

Figure 5-21. Cm vs. α for NACA 0012 Test Run 71. Re = 8.68 x 106, M = 0.70, ε = ±

0.01, s = 423 and n = 165....................................................................................... 83

Figure 5-22. Cm vs. α for NACA 0012 Test Run 78. Re = 8.68 x 106, M = 0.70, ε = ±

0.05, s = 423 and n = 59 ........................................................................................ 84

Figure 5-23. Cm vs. α for NACA 0012 at Re = 3.72 x 106, M = 0.3. ............................. 85

Figure 5-24. Cm vs. α for NACA 0012 at Re = 6.2 x 106, M = 0.5................................ 86

Figure 5-25. Cm vs. α for NACA 0012 at Re = 8.68 x 106, M = 0.7.............................. 87

Figure 5-26. Cm vs. α for NACA 0012 at Re = 11.16 x 106, M = 0.9 ............................ 88

Figure 5-27. Cl vs. α for SC1095. Re = 3.72 x 106, M = 0.30, ε = ± 0.05, s = 706 and n

= 240..................................................................................................................... 90


= 240..................................................................................................................... 90


= 240..................................................................................................................... 91

ix


= 240..................................................................................................................... 91

Figure 5-31. Cd vs. α for SC1095. Re = 3.72 x 106, M = 0.30, ε = ± 0.005, s = 571 and

n = 291.................................................................................................................. 92


n = 291.................................................................................................................. 92


n = 291.................................................................................................................. 93


n = 291.................................................................................................................. 93

Figure 5-35. Cm vs. α for SC1095. Re = 3.72 x 106, M = 0.30, ε = ± 0.01, s = 586 and n

= 137..................................................................................................................... 94


= 137..................................................................................................................... 94


= 137..................................................................................................................... 95

Figure 5-38. Cm vs. α for SC1095. Re = 11.16 x 106, M = 0.90, ε = ± 0.01, s = 586 and

n = 137.................................................................................................................. 95

Figure 6-1. Matlab RBF and GRNN spread setting..................................................... 100

Figure 6-2. Cl vs. α for NACA 0012 reproduction test comparisons. Re = 3.72 x 106,

M = 0.30, ε = 0 ................................................................................................... 103

Figure 6-3. Cl vs. α for NACA 0012 reproduction test comparisons. Re = 8.68 x 106, M

= 0.70, ε = 0 ....................................................................................................... 104

x

Figure 6-4. Cl vs. α for NACA 0012 interpolation test comparisons. Re = 3.72 x 106, M

= 0.30, ε = 0 ....................................................................................................... 109

Figure 6-5. Cl vs. α for NACA 0012 interpolation test comparisons. Re = 6.2 x 106, M =

0.50, ε = 0........................................................................................................... 110

Figure 6-6. Cl vs. α for NACA 0012 interpolation test comparisons. Re = 8.68 x 106, M

= 0.70, ε = 0 ....................................................................................................... 111

Figure 6-7. Cl vs. α for NACA 0012 interpolation test comparisons. Re = 11.16 x 106,

M = 0.90, ε = 0 ................................................................................................... 112

xi

List of Tables

Table 4-1. Data range for NACA 0012 airfoil................................................................ 27

Table 4-2. α matrix used for each test run ..................................................................... 43

Table 4-3. NACA 0012 test matrix................................................................................ 44

Table 4-4. Data range for SC1095 airfoil....................................................................... 46

Table 4-5. SC1095 airfoil test runs ............................................................................... 54

Table 6-1. Reproduction test results............................................................................ 102

Table 6-2. α locations used to train and test each network .......................................... 105

Table 6-3. Interpolation test results............................................................................. 107

xii

Nomenclature

( )

[ ]

sets data testingofnumber

sets data trainingofnumber

squaremean Root RMS

number ReynoldsRe

)(..., ),( vector,residual stage n

residualfunction theof stage n)(

bases ofnumber

functions trialMWR ofnumber

numberMach

function aldifferentilinear

dimensioninput

abledummy vari

)(..., ),(ector function v basis radial

function basis radial

magnitude maximum with vector component of valueg

function continuous

),(..., ),,(ector function verror

function,error

tcoefficien draglift -zero

deg,/ddC slope, curvelift

drag oft coefficien

lift oft coefficien

function mapping

1th

th

s1

n

EXPs

EXP11

1-l

0

t

s

rr

r

n

N

M

L

j

i

hh

h

f

ee

ee

C

C

C

C

A

snn

n

CFDs

CFD

EXPi

CFDii

d

l

d

l

ξξ

ξ

ξξ

ξξξξ

ξξ

α

n

1-n

r

h

r

e

−=

xiii

variablerandomt independen

ightnetwork weiteration n

function weight MWRW

function ingapproximatiteration n

functiontarget

th

i

th

x

w

u

u

n

an

Symbols

( )

productinner ,

parametertion regulariza , tolerancetraining

parameter RBF n

space coordinate real

domain problem

center RBF andinput sample

input sample

input sample ldimensiona

parametertion regulariza Tikhonov

functiontion regulariza

errort measuremen

accuracy data

M-1 ,correctionGlauert -Prandtl

deg attack, of angle

th

th*

th

2

⋅⋅

ℜ

Ω

−

τ

σ

ξ

ξ

ξ

λ

γ

ε

δ

β

α

λ

n

i

i

i

i

j

xiv

Subscripts

Superscripts

inverse1

nformualtio Tikhonov

dataion approximatnetwork SFA indicatesSFA

data alexperiment indicatesEXP

data CFD indicatesCFD

data testingindicatestest

data trainingindicatestrain

functionion approximat indicates

-

a

λ

conditionsboundary satisfies value,initialo

dimensionsinput

indexdummy

indexdummy

bases ofnumber

j

r

i

n

1

Chapter 1 Introduction

When approaching a problem an engineer will have the following tools available: 1)

theoretical/mathematical equations, 2) computational/numerical modeling, and 3)

experimental testing. These methods have their own unique benefits and drawbacks and

will typically be selected depending on the problem at hand. For engineers working in

the area of aerodynamics, a combination of the above tools is utilized at some point of

their analysis.

Theoretical calculations are commonly applied to approximate the general performance

characteristics of an aircraft, wing, or airfoil. Modern aerodynamic theory represented by

mathematical equations began taking shape in the early 1900’s and today this research

has set outstanding guidelines in the areas of airfoil theory, boundary layer, unsteady

aerodynamics, vortex wake, compressible flow, and many others. However advanced

this understanding is, engineers are still forced to make generalizations and assumptions

when applying theory to account for real-world complexities.

Today highly complex problems can be confronted with numerical methods and powerful

computers. Computational fluid dynamics (CFD) modeling is a common numerical tool

that involves the solution of a system of partial differential equations, such as the

Reynolds-Averaged Navier-Stokes (RANS) equations for example. Numerical methods

solving the RANS equations are broadly classified into finite difference, finite volume,

and finite element methods. These tools are very versatile and give highly detailed

2

results for any testing condition. However, CFD tools have limitations that influence the

reliability and fidelity of the predictions. For complex problems, CFD may require

generalizations in solving the partial differential equations. It is computationally very

expensive and may necessitate a reduction in grid resolution, and/or limit modeling to an

isolated form.

Experimental testing is universally regarded as the ultimate reference and is used to

validate theoretical and numerical models. In aerodynamics, experimental testing is

typically done in a wind tunnel where a scaled model is subjected to simulated conditions

representing the complexities of the actual test subject. Unfortunately, experimental

testing is generally very expensive and time consuming, usually resulting in a limited

scope of test runs. Further, data acquisition is susceptible to measurement noise that may

affect the precision of the data. Several numerical approximation methods such as

polynomial and least-squares approximation can sometimes be applied to fit and correct

for the scatter but not always successfully. Altogether, the results of experimental

methods remain the most credible and reliable data, although these data sets may be

sparse and scattered.

Today, increasing costs of wind tunnel testing and advances in the computational tools

have changed the focus of aerodynamic research. Industry is relying more on

computational tools while reducing the number of experimental tests. Even so,

experimental data, however limited, will remain the benchmark by which to compare and

validate theoretical and computational data. Recently, there have been novel attempts to

3

integrate both CFD and experimental data using regularization methods as a new type of

airfoil approximation tool [5]. However, reliable approximations are limited and

practical implementation of these methods requires high user interaction to fine tune and

set up the problem. Another approach has been the implementation of artificial neural

networks as an airfoil design tool [25]. The drawbacks to this approach include the

sensitivity and dependence on large sets of high quality experimental data, high

computational demands, and difficulties in determining optimal neural network

parameters.

The following work describes the Sequential Function Approximation (SFA) neural

network method as an alternate approach to building comprehensive airfoil aerodynamic

performance tables. The objective is to develop a method that approximates airfoil

section lift, drag, and moment coefficient data as functions of Reynolds number, Mach

number, and angle of attack given limited experimental and numerical data sets. To

achieve this purpose, the SFA neural network is combined with a regularization scheme

to fuse experimental and CFD data. The SFA-fused approach can be considered as a

method that 1) corrects the CFD data using high-fidelity experimental data, and 2) fits

noisy experimental data with the smooth curves of the CFD data. The approach is a self-

containing, highly adaptive and robust tool that requires relatively minimal user

interaction and computational demand in constructing airfoil performance tables.

Chapter 2 will describe the airfoil problem as an inverse problem and summarizes the

regularization methods previously applied. Chapter 3 gives a general introduction to

neural networks and presents the SFA neural network theory. The regularization

4

approach to fusing experimental and numerical data is described further. In Chapter 4 the

SFA-fused method is applied as an airfoil prediction tool using data sets for the NACA

0012 and SC1095 airfoils. Comparisons and validation of the approximations are made

in Chapter 5 to help assess the functionality and reliability of the SFA method. Chapter 6

evaluates and compares the results of the SFA approach to two conventional radial basis

methods. Finally, a summary of the findings and an evaluation of the potential and

limitations of the SFA-fused method are discussed in Chapter 7.

5

Chapter 2 Inverse Problem

Wind tunnel experimentation generally involves various channels of measurement to

acquire model aerodynamic forces and moments for different test conditions. Physical

theory and well-known equations can then take the measured experimental data, d, to

derive the desired quantities, q. Conventionally speaking, the collection of the quantities

q is referred to as the image, and the set of all images is called the image space. This

approach is called the forward problem [1],

)(qAd = (1)

where A is a linear or nonlinear mapping function modeling the experimental data and

where q is the image. Wind tunnel data will typically have inherent errors associated

with the instrumentation accuracy and the experimental process. The resulting errors are

referred to as noise, ε , and give the following equation form,

ε+= )(qAd (2)

The problem of reconstructing the original data, q, given the experimental data, d, the

noise, ε, and knowledge of the forward problem, A, is called the inverse problem.

This work details the inverse problem of extrapolating and interpolating airfoil lift, drag,

and moment coefficient data as a function of Reynolds number, Mach number, and angle-

of-attack using the knowledge of existing limited data sets. Typically, airfoil wind tunnel

models will be instrumented to measure the aerodynamic forces and moments for various

testing conditions. Experimentation is usually limited and the measuring process will

involve errors, resulting in sparse and scattered sets of data. For airfoils, these data sets

6

are constructed into what are known as C81 performance tables. These tables list lift,

drag, and moment coefficients with respect to the Mach number and angle-of-attack. The

problem arises when attempting to complete these C81 tables using limited data sets.

Application of well-known methods including polynomial fits, cubic splines

interpolation, and nonlinear least-squares approaches are all dependent on the size and

quality of the data sets and are frequently not applicable to ill-posed problems.

2.1 Ill-Posed Problem

The inverse problem of solving Equation (1) for q given d, is considered a well-posed

problem if:

1. A solution exists for any d within the given data space (Existence).

2. The solution is unique in the image space (Uniqueness).

3. The inverse mapping A-1 is continuous (Stability).

An ill-posed problem is one which does not satisfy all of these three conditions [2].

Conditions 1 and 2 state that the mapping function A has a well-defined inverse A-1

whose domain lies within the data space. In other words, an inverse does not exist if the

data lies outside of A or the solution is not unique because more than one image exists for

the same data. Condition 3 is a necessary but not sufficient condition for a stable

solution. Although it may be continuous, the inverse mapping may not be adequate in the

case where small fluctuations in the data results in unusually large changes in the image.

This is called an ill-conditioned problem. Ill-conditioning implies that standard methods

in linear algebra cannot be used to solve the problem in Equations (1) or (2). Instead,

regularization methods are used to solve these problems. Regularization imposes a priori

7

knowledge of the problem by way of a regularization parameter to condition the solution

within an acceptable tolerance.

2.2 Tikhonov Regularization

Approximating airfoil characteristics from wind tunnel data is an ill-posed inverse

problem. Ill-posedness is induced when the Stability Condition 3 is violated so that noise

of the initial data is significantly amplified by the inverse mapping process. The inverse

problem is to reconstruct a complete airfoil performance table given limited and noisy

experimental data.

There are many regularization methods available [3] for the solution of ill-posed

problems. Some are categorized as either direct methods, where the solutions are defined

by direct computation, or iterative methods that inherently require iterative solutions.

The most popular and widely used method is the Tikhonov direct regularization method

[4].

A novel approach to the airfoil problem was developed utilizing the Tikhonov

regularization method to merge experimental and CFD data [5]. That approach can be

viewed as an interpolating and extrapolating tool of the experimental data using a priori

information from the CFD models. The objective functional using the Tikhonov

formulation is given by

[ ] ( ) ( ) =

−+−=

i rr

CFDar

iaiCFDa dxdx

xfxfdxfxffff

1

0

2

2expexp

)()(

2

1)()(

2

1,, λγ λ

8

where ( )ijiii xxxx ,,2,1 ,...,,= and j denotes the dimensionality of the problem. From this

equation, it can be seen that for , fa(x) fexp(x), and the CFD data becomes less

relevant than the experimental data. For 0, the experimental data become less

relevant to the solution. The regularization parameter is determined by the following

tolerance (τ) constraint

0if,' ≠≤− ωτω

ωρ

where ( )

=−=

exp

1

2exp

exp

)()(1

's

iiai xfxf

sρ , sexp is the number of experimental data points, and

3

2δω = with as the data accuracy.

This Tikhonov regularized approach to fusing experimental and CFD data showed

varying degrees of success in the approximation of airfoil aerodynamics. Initial studies

for an airfoil problem showed promising results over polynomial and least-squares

approximations [5]. However, difficulties were observed in some regimes where large

CFD data inconsistencies overruled the more accurate but limited experimental data

points resulting in unreliable approximations. Additional drawbacks to this approach

included the high user interaction, problem set-up, and computational demands required

for every approximation test case.

This work develops an alternate method of fusing experimental and computational data to

approximate airfoil coefficients that improves on the drawbacks of the Tikhonov fused

method. The approach uses a neural network method as the inverse modeling tool and

9

applies a simplified Tikhonov-related regularization scheme to correct for the original

data error.

10

Chapter 3 Multi-Dimensional Function Approximation

This chapter covers a brief introduction to neural networks and discusses the SFA neural

network theory. A regularized fusing approach that merges experimental and numerical

data is also presented.

3.1 Neural Networks

An artificial neural network (ANN) is a computational modeling tool that mimics the

ability of biological neural systems to capture and represent complex, multi-dimensional,

input/output linear and nonlinear relationships through a layered structure of units,

analogous to the brain’s neurons [6]. When used as a prediction and approximation tool,

ANN’s are divided into two categories:

1. Classification: The purpose is to output a particular class from a set of

independent input variables. Applications include credit rating determination

(good or bad), bomb detection (inspects for certain compounds), and optical

character/pattern recognition (e.g., reading handwriting).

2. Regression: The objective is to determine a single output value from a set of

continuous independent input variables. Examples include approximating

profits and forecasting stock, weather prediction, medical diagnosis

(identifying people with cancer risk), and estimating aerodynamic

performance of an airfoil.

11

There exists different types of neural network algorithms, all distinguished by the manner

in which they are trained, the number of layers and units, and the kind of operation

chosen to define the network units. The reader is referred to Bishop [7], Patterson [8],

and Carlin [9] for a comprehensive introduction of existing neural network algorithms.

There are also many derivatives of these classic neural network algorithms known as

Support Vector Machines (SVM) [10],[11]. The general approach to SVMs and their

abilities are similar to common neural networks. The most common neural network is

the supervised multilayer perceptron (MLP) network used for both classification and

regression problems [12]. The MLP is typically made up of one to two hidden layers of

units as shown in Figure 3-1.

Figure 3-1. MLP architecture.

Each MLP unit utilizes a sigmoid,)exp(1

1)(

xxh

−+= , or hyperbolic tangent as its

transfer function. The inputs are multiplied by interconnecting weights and fed forward

12

to the first hidden layer units. At the first hidden layer, within each unit, the inputs are

processed through a transfer function. Each unit output is then multiplied by the weights

and introduced to the second hidden layer where they are processed again. At the output

layer the weighted second layer outputs are summed to give a final value.

A supervised neural network requires a set of training data to construct the input/output

mapping. The training data set consists of multi-dimensional, independent inputs and one

dependent output. Training of an MLP network is achieved through a back-propagation

training algorithm in which the network weights are iteratively adjusted with respect to

the error until a desired error or tolerance is met. This is accomplished through

optimization by gradient descent along the error surface (Figure 3-2). One of the

obstacles in training ANN’s is determining the number of hidden layers and hidden units.

This involves a pruning algorithm, or trial-and-error testing, which may give sub-optimal

results.

Figure 3-2. Schematic of the back-propagation training process.

13

3.2 Radial Basis Function Network

As a modeling tool, ANN's have proven advantages over traditional approximation

methods, in their ability to learn and model complex nonlinear relationships. However,

ANN's, especially the MLP, are computationally demanding requiring iterative

optimization of the weights to minimize the error and also fine-tuning of the number of

units and layers in the network. A viable option to the MLP that bypasses some of the

training difficulties is the Radial Basis Function (RBF) neural network [13] as shown in

Figure 3-3.

Figure 3-3. RBF network architecture.

The RBF network in Figure 3-3 is comprised of a single hidden layer of transfer functions.

The RBF network is characterized by the use of a bell-shaped transfer function, e.g.,

−−= 2)

*(

2exp)( ξξσξh ,

where the parameter ξ∗ specifies the dimensional location of the transfer function center,

controls the width of the unit, and w is the network weighted value. The network

14

output consists of the weighted sum of the radial bases. Figure 3-4 (a) shows a sample

response from a RBF unit with two input variables. The Gaussian functions allow the

RBF network to model nonlinear multi-dimensional functions given sufficient radial units

as shown in Figure 3-4 (b) for the function,

).cos()sin( yxU an +=

(a) (b)

Figure 3-4. Example RBF network unit and response surface. (a) RBF transfer function with w =

1, ξ∗x,y=0, and σ = 1, (b) RBF network approximation of ).cos()sin( yxU a

n +=

Like the MLP, the RBF networks can, in theory, model any nonlinear function with a

sufficient number of transfer functions [13], [14]. Training of RBF networks involves the

same type of gradient descent optimization of the RBF network parameters as the MLP

but because of the RBF network’s single layer, training is accomplished many times

faster than the MLP. Further, as can be observed from Figure 3-4 (a), an input case

located far from the center will generate a zero since it acts as a local basis function. As a

result, RBF networks are not capable of extrapolation beyond what training data the

15

network has been exposed. This is generally regarded as a good trait since it may be

dangerous to predict a result beyond the known cases. An MLP network, on the other

hand, continues to predict even for extreme input cases since the transfer functions act as

global bases. Also, inherent to ANNs (specifically RBFs) are the difficulties experienced

in handling high dimensional inputs, commonly referred to as the “curse of

dimensionality.” Every input adds an additional dimension to the problem space where

the response surface is to be fitted. As the number of input dimensions increase so do the

number of free parameters that must be estimated. This results in a dramatic increase in

computational demand and a drop in the effectiveness of modeling the training data. In

contrast, for any RBF network that optimizes the amount of required units to fit a

response surface to the training data effectively, the number of network parameters that

must be determined is also minimized. As a result, these types of RBFs reduce or even

completely sidestep the curse of dimensionality.

Two novel approaches to determining the RBF network parameters (ξ∗, , and w) are

Orr’s Forward Selection training method [13] and Platt’s Resource Allocating Network

[15]. Both methods allocate only the necessary number of units required to model a

target function, thereby minimizing the computational demand. Specifically, Orr adapts a

sequential approximation algorithm with the form,

),,()()( *1 nnn

an

an hwuu ξσ

+= − (3)

In sequential approximation, the function parameters (wn, ξ∗n, and n) are optimized to

minimize the residual,

)()( ann uur −= .

16

This optimization is accomplished through forward selection. Forward selection takes a

subset of the training data and applies a simple nonlinear optimization algorithm that

searches for the basis function giving the smallest approximation error. Once found, that

function defines one network unit and the w, ξ∗, and σ are exclusive to that unit. This

repeats for the next iteration where one more unique basis function and unit are added to

the network, all while the error continues to decrease. The process continues until an

acceptable error or tolerance is met. With this approach the network automatically

determines the number of hidden units and their corresponding parameters while

bypassing a costly gradient descent backpropagation approach. It also helps to minimize

the effect of higher dimensionality and the need for large numbers of network units.

3.3 Sequential Function Approximation Network

A variation to Orr’s Forward Selection scheme that seeks to improve the computational

efficiency of an RBF network is the Sequential Function Approximation (SFA) method

developed by Meade [16]. This approach is based on the Method of Weighted Residuals

(MWR) [17] and minimizes the approximation function residual while accounting for the

number of network units.

3.3.1 Method of Weighted Residuals

MWR assumes that a solution can be approximated piecewise analytically. A partial

differential equation (PDE) solution can be determined as a superposition of a set of trial

functions with the form

17

+=

N

ii

a htwtutu )()(),(),( 0 (4)

where u0 satisfies the initial and boundary conditions. The functions are chosen to satisfy

a linear differential equation L(u) = 0, where u is the exact solution. Other than time

dependent problems, the wi’s are constants and Equation (4) reduces to a system of

algebraic equations resulting in a nonzero residual

)()()(),( 0 +==N

iii

ai hLwuLuLwr . (5)

The goal of the MWR is then to determine the coefficients wi that reduces r over the

problem domain . To do this the inner product of the weighted residual is set to zero:

=ΩΩ

drWi ,0, =iWr (6)

where Wi is a set of weighted functions (i = 1, .. ,N).

There are different approaches to the MWR using different weight functions. In the

Galerkin approach to the MWR used in this report, the weight functions are chosen to be

identical to the bases

)()( ii hW = . (7)

As a result, Equation (6) becomes 0, =ihr and can be written in matrix form for the

unknown coefficients wi. Substituting the resulting wi’s into Equation (4) gives the

approximate solution.

18

3.3.2 SFA Theory

Unlike Orr’s approach of choosing from a set of given basis functions to determine the

optimal values for cn and n, the SFA method fixes cn and uses the MWR to optimize the

approximating function residual with respect to n and wn. The residual takes the form

nnn

nnnan

annnn

hwr

hwuu

uur

−=

−−=

−=

−

−

1

*1

*

),()()(

)()(),,(

ξσ

ξσ

(8)

The objective in the SFA algorithm is to determine wn, n, and ξ∗n that minimize rn.

Using the Galerkin weighted residual method, the wn that makes rn orthogonal to hn is

sought at each forward selection iteration n such that,

0, ==Ω nnnn hrdhr . (9)

From Equation (8), n

nn w

rh

∂∂−= , and Equation (9) becomes

0,

21

, =∂

∂−=

∂∂−

n

nn

n

nn w

rr

w

rr . (10)

From Equation (8), the objective function becomes

nnnnnnnnnn hhwhrwrrrr ,,2,, 2111 +−= −−− (11)

and the first derivative with respect to wn gives

.,

,1

nn

nnn hh

hrw −= (12)

For our applications the input is made up of discrete samples. Our nonlinear

minimization is made with respect to n of the following function:

19

)()(2)( 2nnn1n1n1n hhhrrr ⋅+⋅−⋅ −−− nn gg (13)

where n is unconstrained. The value of gn is fixed at the value of the residual vector

component with the maximum magnitude. Using gn ensures that the residual is

continuously decreasing. The radial basis function center, *n

used to determine hn, has

the components of the input vector that satisfies rn-1( *n

) = max (|rn-1|) .

Finally, for our discrete samples, .)()(

nn

n1n

hhhr

⋅⋅= −

nw The network parameters (wn, n and

ξ∗n) account for one network unit at that nth iteration. The standard MATLAB nonlinear

unconstrained optimization routine called fminunc is used to minimize equation (13) with

the user choosing the initial sigma, 0, to begin the minimization process with respect to

n. The designation of 0 is discussed in Section 4.1.2.

MATLAB is used as the programming tool for developing and operating the SFA

network. To perform approximations, the user need only arrange the training data file,

train, specify the testing (approximating) data set, test, and indicate the network

parameters 0 and the tolerance . The algorithm begins training for n = 1 and determines

the parameters wn, n and ξ∗n for a new network unit and updates the residual vector rn.

At every training iteration, the parameter ξ∗n is allocated for the training data point with

the largest residual while the parameters wn and n are optimized by the network. The

iterative process continues until 1) gn reaches a pre-determined tolerance (|gn| ), or 2)

the number of network units is equal to the number of available input training sample

cases (n = s). In the end, the training process determines n number of network units and

20

their respective parameters (wn, n, and ξ∗n). The SFA network results are represented by

the following function,

=++=

n

iiinnnnn

an hwhwhwhwwu 2211

* ),,( σξ . (14)

Once training is complete, the network can be tested with a set of input test data, test.

The resulting output, ),,,( *nnn

testan wu σξ

, is the multi-dimensional approximation surface

of the SFA network. Figure 3-5 shows the SFA flowchart for the training and testing

process.

The computational demand for the whole training and testing process will usually take

less than one minute for every 100 derived radial basis function units using a Pentium 4©

PC (1.7 GHz computer and 512 RAM) running the SFA-fused code with the MATLAB

software.

21

3.3.3 Algorithm

SFA Network Algorithm

Run MATLAB unconstrained minimization routine to determine n.

yes no

end main training routine

Run testing routine

end testing subroutine

end SFA Network algorithm

Calculate basis function h n

Calculate

Calculate

Update residual,

Break main loop if |g n |= or if n=s .

Calculate weighted basis function, network unit,

Calculate basis function h n

Calculate summation of network units,

Load test data, test . Matrix size (t,j).

Load training data, train . Matrix size (s,j+1).

Calculate the weight wn =v n /zn.

Determine centers ξ*n.

Minimize

Main training routine (Every nth iteration determines a basis function)

Assign value for tolerance, .

Assign value for initial 0 for optimization routine.

Evaluate max residual g n =max(r n-1 )

with respect to n.

Dnnn hhz ,=

nnnntesta

n hwu =),,( * σξξ

=++=

n

iiinnnn

testan hwhwhwhwu 2211

* ),,( σξξ

),,( *nn

test σξξ

),,( *nn

train σξξ

nnnn hwrr −= −1

)()(2)( 2nnn1n1n1n hhhrrr ⋅+⋅−⋅ −−− nn gg

Dnnn hrv ,=

Figure 3-5. SFA algorithm.

22

3.4 Regularized Fusing Approach

The purpose of this chapter is to introduce a fusing approach using the SFA neural

network that maximizes the use of experimental data with the help of CFD data in

approximating a complete airfoil coefficient performance table. The fusing approach

attempts to correct the low-fidelity and high resolution CFD data with limited, yet

reliable, experimental data. It is also a method by which the noisy experimental data can

be conditioned with the smooth curves of the CFD data.

The fusing approach first involves calculating the error function of the CFD and

experimental data defined by the following equation,

( ) ( ) ( )iEXP

iCFD

i uue ξξξ −= , (15)

for i = 1,…, s, where s is the number of training data sets. The error vector, e, is then

used to train the SFA network to a predetermined tolerance, . The resulting error

surface, e(ξ), will naturally involve some scatter directly related to the experimental data

noise. Training the network to the given tolerance allows the SFA to regulate the noisy

experimental data with a priori CFD information. Assuming the uCFD surface is known,

then the error surface approximation, eSFA, can be subtracted from the uCFD (ξ) data to

give the approximation surface,

SFACFDSFA euu −= . (16)

The τ value can be regarded as the regularization parameter and controls how well the

approximations fit the experimental or CFD data. A very high tolerance value allows the

training process to end prematurely with very few network units. As a result, the network

23

“under-learns” the training data and the majority of the approximations reach a value of

zero. For data points with an error value of zero, equation (16) shows that the

approximation value will reproduce the CFD data. On the other hand, a very small

tolerance value will force the network to use too many network units to reach the smallest

possible tolerance. In this case, the network “over-learns” the training data and will fit

even the experimental noise in the error surface. As a result, the approximations will

reproduce the experimental data. The user must carefully choose the tolerance value to

best fit the experimental data using the CFD information. Section 4.1.2 discusses the

suitable tolerance values used for each test case throughout this work.

Occasionally, our CFD data and experimental data will differ in the input ranges of

Reynolds number, Mach number, and angle of attack. To calculate the error vector e

directly, the CFD data must first be approximated for the same input conditions as the

experimental data. This is simply done by training the SFA network with the CFD data

and testing with the experimental data inputs to get a usable CFD data set, SFACFDu _ , with

inputs comparable to the experimental data. The fusing process continues as described

above and in the flowchart in Figure 3-6.

24

Train the network with the CFD data, CFDu . Test the network with exp. data, EXPu .

Re(10^6) M αααα Cl_CFD Re(10^6) M αααα Cl_EXP

1.24 0.1 -180 -0.001 2.48 0.20 -180.0 -0.0039 1.24 0.1 -178 0.1719 2.48 0.20 -172.5 0.7809 1.24 0.1 -176 0.28 2.48 0.20 -161.0 0.5970

The new CFD approximations, SFACFDu _ , can

then be compared with the EXP data points. Re(10^6) M αααα Cl_CFD_SFA

2.48 0.20 -180.0 0.00 2.48 0.20 -172.5 0.40 2.48 0.20 -161.0 0.70

Calculate the error vector: EXPSFACFD uu −= _e

Re(10^6) M αααα ERROR

2.48 0.20 -180 0.0042 2.48 0.20 -172.5 -0.3800 2.48 0.20 -161 0.1029

Train with the error vector to a given tolerance. Make approximations for the new

“regularized” error surface, SFAe . Re(10^6) M αααα ERROR_SFA

2.48 0.20 -180.0 0.01 2.48 0.20 -172.5 -0.38 2.48 0.20 -161.0 0.13

Calculate the approximation surface: SFACFD_SFASFA euu −=

Re(10^6) M αααα APPROX. 2.48 0.20 -180.0 -0.0084 2.48 0.20 -172.5 0.7765 2.48 0.20 -161.0 0.5695

Figure 3-6. Fusing method flowchart.

25

Chapter 4 SFA Implementation as an Airfoil

Approximation Tool

The objective of this work is to analyze the SFA fused approach in approximating

coefficient data in the construction of airfoil performance tables. Experimental and CFD

data sets for two different airfoils are used to test the SFA network method. The first

airfoil test case is the well-known NACA 0012 airfoil. This airfoil comes with a

complete aerodynamic table derived from over 40 different wind tunnel experiments.

Although a highly ideal case, this airfoil test case will serve to 1) verify that the SFA

network fusing approach is feasible, 2) determine the optimal SFA network parameters

and conditions, 3) validate the accuracy of the approximations for varying experimental

data sizes and random noise levels, and 4) demonstrate the ability to make smooth

interpolations. The second test case concerns the SC1095 airfoil and represents a real-

world application problem that is the ultimate objective of this work. This airfoil comes

with highly sparse and scattered experimental data sets of varying fidelity. The objective

of this test case is to 1) demonstrate that the SFA-fused approach can make reliable

approximations given highly sparse and scattered experimental data sets, and 2) evaluate

the capability of the SFA-fused approach to make smooth interpolations and transitions to

the CFD data.

26

4.1 NACA 0012 Airfoil Experimental and Numerical Data

The NACA 0012 airfoil (Figure 4-1) is a geometrically symmetrical configuration that is

one of the oldest and most tested. The experimental data used for the NACA 0012 airfoil

test case comes in the form of the so-called C81 aerodynamic performance table. This

complete table has been constructed from experimental data of over 40 wind tunnel

experiments and serves as an ideal test case to analyze the feasibility and accuracy of the

SFA-fused method. The NACA 0012 C81 table is one of the most complete airfoil data

sets. The C81 aerodynamic performance table is a comprehensive data set that lists lift,

drag, and moment coefficient data as functions of Reynolds number, Mach number, and

angle of attack. The angle of attack typically ranges from ± 180 degrees as Mach ranges

from 0 to 1.

Figure 4-1. NACA 0012 airfoil.

4.1.1 C81 and CFD Data

Both the C81 and CFD data sets share comparable spans in Mach number and angle of

attack. The reference Reynolds number for the C81 data is not well-defined and for the

purposes of this work it is assumed to have the same Mach-Reynolds proportionality

constant as the CFD data of 1:12.4 x 106. Table 4-1 shows the input data ranges and the

27

number of available data points for the NACA 0012 experimental and CFD data sets.

The Re, M, and α distributions are graphically represented in Figure 4-2 and Figure 4-3.

Table 4-1. Data range for NACA 0012 airfoil.

C81 "experimental" data CFD data Cl (390 data points) Cl (590 data points)

Re(106) M α Re(106) M α 2.48 0.2 -180 1.24 0.1 -180 12.4 1 180 12.4 1 180

Cd (650 data points) Cd (590 data points)

Re(106) M α Re(106) M α 2.232 0.18 -180 1.24 0.1 -180 12.4 1 180 12.4 1 180

Cm (423 data points) Cm (590 data points)

Re(106) M α Re(106) M α 2.48 0.2 -180 1.24 0.1 -180

11.16 0.9 180 12.4 1 180

Figure 4-2. Re and M distribution for the NACA 0012 data sets.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1 10 100

Re(10^6)

M

Cd EXP

Cl EXP

Cm EXP0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

Re(10^6)

M

Cd CFD

Cl CFD

Cm CFD

28

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 15 30 45 60 75 90 105 120 135 150 165 180α (deg.)

M Cl EXP

Cl CFD

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 15 30 45 60 75 90 105 120 135 150 165 180α (deg.)

M Cd EXP

Cd CFD

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 15 30 45 60 75 90 105 120 135 150 165 180α (deg.)

M Cm EXP

Cm CFD

(c)

Figure 4-3. Angle of attack locations available in the NACA 0012 experimental (EXP) and CFD

data sets. The locations are also representative of angles spanning from 0 to -180 degrees. (a)

Lift coefficient data set. (b) Drag coefficient data set. (c) Moment coefficient data set.

29

As discussed in Section 3.4, the SFA-fused approach works by regularizing the error

surface between the experimental and CFD data to make approximations. Since our

NACA 0012 experimental and CFD data sets vary in the α range for all airfoil

coefficients (and in Re and M for the Cd data), they cannot be directly compared. The

SFA must be used to train with the CFD data set and approximate the CFD data at the

identical Re, M, and α ranges of the experimental data.

The CFD data set was provided in collaboration with Mayda and van Dam at the

University of California at Davis from their work related to CFD-generated airfoil tables.

The CFD data set was constructed using the two-dimensional Reynolds-averaged Navier-

Stokes flow solver ARC2D [18] which also employs an automation technique that allows

for a largely “hands-off” approach to generating airfoil performance tables. Figure 4-4

through Figure 4-9 are sample plots showing CFD and C81 coefficient data with respect

to α. The CFD data show comparable lift-curve slopes and appear to capture the general

Cl sharp rise and decline although, as seen in Figure 4-4 (a), with lower accuracy as

α increases. Figure 4-6 shows the characteristic drag bucket and zero-lift drag coefficient

are well-defined by the CFD data and comparable to the C81 curves at Mach numbers

below M = 0.6 and at low α within ± 30 degrees. Figure 4-6 and Figure 4-8 show that

good comparisons are also observed at low Mach numbers and low α for the Cm data. At

high α, complex flow conditions take shape and the accuracy of the CFD degrades for all

coefficients, as seen in Figure 4-4 for example. The fidelity of the CFD data is further

degraded for all coefficients as Mach increases into the transonic regime (M = 0.7) where

the airfoil characteristics become nonlinear as shown in Figure 4-9.

30

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-200.0 -150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0

αααα (deg.)

Cl

Cl_CFD

Cl_EXP

(a)

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

Cl_EXP

(b)

Figure 4-4. NACA 0012 Cl vs α comparison of C81 and CFD data at M = 0.30. (a) Plot for α

range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

31

Re(10^6)= 9.92 , M= 0.80

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-200.0 -150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0

αααα (deg.)

Cl

Cl_CFD

Cl_EXP

(a)

Re(10^6)= 9.92 , M= 0.80

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

Cl_EXP

(b)

Figure 4-5. NACA 0012 Cl vs α comparison of C81 and CFD data at M = 0.80. (a) Plot for α


32

Re(10^6)= 3.47 , M= 0.28

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-200.0 -150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

(a)

Re(10^6)= 3.47 , M= 0.28

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

(b)

Figure 4-6. NACA 0012 Cd vs. α comparison of C81 and CFD data at M 0.3. (a) Plot for α


33

Re(10^6)= 10.17 , M= 0.82

0.00

0.50

1.00

1.50

2.00

2.50

-200.0 -150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

(a)

Re(10^6)= 10.17 , M= 0.82

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

(b)

Figure 4-7. NACA 0012 Cd vs. α comparison of C81 and CFD data at M 0.8. (a) Plot for α


34

Re(10^6)= 3.72 , M= 0.30

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-200.0 -150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0

αααα (deg.)

Cm

Cm_CFD

Cm_EXP

(a)

Re(10^6)= 3.72 , M= 0.30

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

Cm_EXP

(b)

Figure 4-8. NACA 0012 Cm vs. α comparison of C81 and CFD data at M = 0.30. (a) Plot for α


35

Re(10^6)= 9.92 , M= 0.80

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

-200.0 -150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0

αααα (deg.)

Cm

Cm_CFD

Cm_EXP

(a)

Re(10^6)= 9.92 , M= 0.80

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

Cm_EXP

(b)

Figure 4-9. NACA 0012 Cm vs. α comparison of C81 and CFD data at M = 0.80. (a) Plot for α


36

4.1.2 Network Testing Conditions

There are three testing conditions that govern the SFA network that must be suitably

established to ensure optimum results. The first is determining the proper scaling of the

multi-dimensional data, the second is finding an appropriate optimization network

parameter, and the third is designating the training tolerance value.

During the training process, the SFA network iteratively establishes a radial basis

function unit and sets the network parameters (ξ∗, , and w). The summation of these

units gives the final multi-dimensional approximation surface. The parameter ξ∗

allocates the dimensional location of the radial basis function center defined by the

discrete input dimensions (Re, M, and α) from the training data. Preliminary checkout

test runs confirmed the SFA’s ability to reproduce the training data, however, those tests

also showed that the SFA had difficulties in making interpolations, as seen in Figure 4-10

and Figure 4-11 for a Cl test case. This problem is prevalent in all coefficient test cases.

From the error function plot in Figure 4-11 the approximated (sfa) error curve confirms

the scatter that affects the approximations in Figure 4-10. This scatter occurs at locations

where data does not exist and where the SFA must interpolate. The SFA makes

interpolations that either 1) go to zero since no data exists at all, or 2) the interpolations

are over-sensitive to data at nearby Mach locations.

37

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_EXP

Figure 4-10. Cl vs. α for M = 0.7. Training data exist at = ± 21, 14, 12, 10, 6, and 2.

M =0.7

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-25 -15 -5 5 15 25

AoA (deg)

erro

r

error, M=.7

error_s fa, M=.7

Figure 4-11. Error function vs. α for Cl data at M = 0.7.

38

The problem was found to be primarily the result of the large dimensional distance

between the data points along the α−coordinate where the data ranges from ± 180

degrees, while the Reynolds and Mach data ranges only from 1.2 to 12 and 0.1 to 1,

respectively. To address this issue the input dimension range was normalized by dividing

the α data by 180 degrees and the Re data by 12.4. This allowed the dimensional space

of the Re, M, and α data to be scaled down to 0.1 to 1, 0.1 to 1, and -1 to 1, respectively.

Performing the same test again gave much improved error function approximations as

seen in Figure 4-12. The final results in Figure 4-13 show smoother Cl approximation

curves and cleaner transitions to the CFD data when interpolating.

M =0.7

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

-25 -15 -5 5 15 25

AoA (deg)

erro

r

error, M=.7

error_s fa, M=.7

Figure 4-12. Error function vs. α for Cl data at M = 0.7, normalized.

39

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_EXP

Figure 4-13. Cl vs. α for M = 0.7, normalized. Training data exist at = ± 21, 14, 12, 10, 6, and

2.

Another factor that must be taken into account for each problem is normalization of the

input data. For example, normalizing the input dimension ranges for the Cd data to the

same scale as the Cl data gave poor results. This is because the Cd data have a more

complicated error surface to fit since there are significantly more data points to consider.

A better scale for the Re, M, and α data was found to be 0.15 to 1.5, 0.15 to 1.5, and -4 to

4, respectively. The same scale used in the Cl data was found to be appropriate for the Cm

data.

The SFA network utilizes a standard MATLAB nonlinear unconstrained optimization

routine fminunc to minimize the network parameter

n for every training iteration. The

parameter controls the width of the network radial basis function unit. For example,

considering a two-dimensional radial basis function, a very large value of gives a sharp

40

and narrow basis function. As a result a large number of these units may be required to

fit a smooth function. A very small results in a wide and shallow basis function and

gives a smoother function approximation but the network may require many of these

units to fit a fast changing function. During the training process, the SFA optimizes the

value for each network unit using a MATLAB minimization routine. This routine is

initialized by a user-defined

o setting. Ideally, the MATLAB routine should determine

the global minima regardless of

o. However, the routine will occasionally end with a

local minimum resulting in suboptimal values for

n that may affect the final

approximations. This problem can be avoided as long as the

o is set to an appropriate

value.

A simple trial-and-error method was used to help determine the optimum o value for

each coefficient test case. Six preliminary SFA network test runs where performed for

each coefficient at varying

o settings. The network was trained to a tolerance of zero

with the CFD data and tested against the C81 data as a benchmark. The maximum

difference in the resulting error, EXPCFD_SFA uue −= , is plotted against

o in Figure 4-14.

This plot shows that

o = 1 is a suitable setting for all coefficient test cases.

41

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.01 0.1 1 10 100

initial sigma

max

. di

ffere

nce

Cl

Cd

Cm

Figure 4-14. Optimum initial sigma ( o) setting.

The tolerance value, , is the regularization parameter that serves the purpose of ending

the training process as it stabilizes the SFA approximations between the CFD and

experimental data. A large causes the approximations to lean towards the CFD data,

while a small closely fits the experimental data. A suitable tolerance value is the

magnitude of the measurement error present in the experimental data, . With = | | the

SFA network is allowed to train within an acceptable error window while conditioning

with the CFD information as seen in the sample plot of Figure 4-15.

42

Re(10^6)= 3.47 , M= 0.28

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

+/- .005 tolerance

SFA

Figure 4-15. Optimum tolerance setting. Using = to train within the acceptable error

window.

4.1.3 Data Size and Experimental Error Studies

The NACA 0012 airfoil C81 and CFD data sets serve as an ideal test case to analyze the

SFA-fused approach. In controlled tests that mimic real-world problems, these complete

data sets allow us to study the accuracy of the approximations as fewer experimental data

points are available. At the same time, since the C81 table is in the form of clean and

noiseless data, the effect of varying levels of random experimental error on the

approximations can be studied. Altogether, the objective is to verify the feasibility and

accuracy of the SFA-fused approach as the size of the experimental data sets and the

magnitude of the experimental noise varies.

43

For our test we start with seven test runs performed with different experimental data set

sizes using the “clean” C81 data for every coefficient. Data size is reduced by

eliminating data points at α locations for each of the Re and M regimes. In many real-

world airfoil experiments, aerodynamicists and engineers are particularly interested in

collecting data within α of ± 30 degrees. Therefore, a similar α range is considered for

reducing data. Table 4-2 shows the α matrix for each test run. The data outside ± 30

degrees is still used in our simulations to demonstrate the SFA’s ability to build a

complete C81 table.

Table 4-2. α matrix used for each test run. The highlighted areas represent the α’s that are

eliminated for that test. First column represents the available C81 data.

390 360 340 320 300 280 260 650 600 560 520 480 440 400 423 387 351 315 279 252 198α α α α (deg.) α α α α (deg.) α α α α (deg.)

-21.0 -21 -30-16.5 -16 -23-15.0 -15 -16-14.0 -14 -15-13.0 -13 -14-12.0 -12 -13-11.0 -11 -12-10.0 -10 -11-8.0 -9 -10-6.0 -8 -9-4.0 -7 -8-2.0 -6 -70.0 -5 -62.0 -4 -44.0 -3 -36.0 -2 -28.0 -1 -110.0 0 011.0 1 112.0 2 213.0 3 314.0 4 415.0 5 616.5 6 721.0 7 8

8 99 1010 1111 1212 1313 1414 1515 1616 2321 30

C l Exp. Data Size C d Exp. Data Size C m Exp. Data Size

44

The seven test runs are repeated for conditions where an experimental error is added to

the C81 data using the MATLAB rand function to simulate noisy experimental data.

This error also helps determine the training tolerance value, hence τ=|ε|. The tests are

repeated for three different settings resulting in a total of 28 test runs for each

coefficient test case. The three settings are chosen from the error range observed in

actual wind tunnel experiment data [19]. The complete test matrix for the NACA 0012

test case is shown in Table 4-3.

Table 4-3. NACA 0012 test matrix.

σσσσ 1 390 360 340 320 300 280 260

0.00 TEST RUN 1 TEST RUN 2 TEST RUN 3 TEST RUN 4 TEST RUN 5 TEST RUN 6 TEST RUN 7




σσσσ 1 650 600 560 520 480 440 4000.000 TEST RUN 29 TEST RUN 30 TEST RUN 31 TEST RUN 32 TEST RUN 33 TEST RUN 34 TEST RUN 35




σσσσ 1 423 387 351 315 279 252 198





C m Exp. Data Size

C l Exp. Data Size

C d Exp. Data Size

Exp. Error,

εεεε

Exp. Error,

εεεε

Exp. Error,

εεεε

For every test run, the SFA-fused approach is used to approximate a full C81 table with

the same input dimension as the C81 data for direct comparison. A simple way to

measure the accuracy of the SFA-fused approximations is to take into account the root-

mean-square (RMS) error for all n samples,

45

RMS = ( )

τξξ

=−

n

n

i

Ci

SFAi

281

. (17)

This process results in an ideal tolerance limit to help measure accuracy.

4.2 SC1095 Airfoil Experimental and Numerical Data

The SC1095 (Figure 4-16) is one of two important airfoils currently used in the UH-60A

helicopter main rotor blade. Unlike the NACA 0012 airfoil, the SC1095 is a semi-

symmetrical airfoil that has been investigated in only a handful of wind tunnels. The

following studies involve highly sparse and scattered experimental data sets from six

experiments. The SC1095 test case represents a real-world application problem that is

the ultimate objective of the SFA-fused method.

Figure 4-16. SC1095 airfoil.

4.2.1 Experimental and CFD Data

The available airfoil SC1095 experimental data sets are highly sparse and scattered and

have varying degrees of fidelity. The objective of this test case is to 1) demonstrate that

the SFA-fused approach can correct the CFD data and make consistent approximations

46

given limited experimental data sets, and 2) evaluate the capability of the SFA-fused

approach to make smooth interpolations and transitions to the CFD data.

The experimental data sets are limited in the Reynolds number and α range as noted in

Table 4-4. Re and M ranges vary drastically between the available CFD data sets and

experimental, as seen in Figure 4-17. Ideally we would like the CFD data input ranges to

be comparable to the experimental data inputs. To achieve these tests it is assumed that

the results are not dependent on the Reynolds number. Mach number and α are the only

dependent input parameters for this test. This is a fair assumption for airfoil research

performed in standard atmospheric conditions.

Table 4-4. Data range for SC1095 airfoil.

Experimental data CFD data Cl (706 data points) Cl (590 data points)

Re(106) M α Re(106) M α 0.9 0.11 -6.24 1.24 0.1 -180 6.7 1.075 25 12.4 1 180

Cd (571 data points) Cd (590 data points)

Re(106) M α Re(106) M α 0.9 0.199 -6.24 1.24 0.1 -180 6.7 1.071 19.94 12.4 1 180

Cm (586 data points) Cm (590 data points)

Re(106) M α Re(106) M α 0.9 0.11 -6.24 1.24 0.1 -180 6.7 1.071 25 12.4 1 180

47

0

0.2

0.4

0.6

0.8

1

1.2

0.1 1 10 100

Re(106)

M

Cl EXP

Cd EXP

Cm EXP

Cl, Cd, Cm CFD

Figure 4-17. Re and M number distribution for the SC1095. Experimental (EXP) and CFD data.

The α range of the SC1095 data is limited to around -6 to +25 degrees for all coefficients

as seen in Figure 4-18. Beyond this range the SFA-fused method will depend fully on the

CFD data to complete the C81 aerodynamic table. Altogether, the SC1095 test case will

assess the method’s ability to make clean and consistent interpolations for highly sparse

and scattered data sets as well as test its ability to make smooth transitions to CFD

dominated regimes.

48

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

-25 -20 -15 -10 -5 0 5 10 15 20 25

α (deg.)

M

Cl EXP

Cl CFD

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

-25 -20 -15 -10 -5 0 5 10 15 20 25

α (deg.)

M

Cd EXP

Cd CFD

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

-25 -20 -15 -10 -5 0 5 10 15 20 25

α (deg.)

M

Cm EXP

Cm CFD

(c)

Figure 4-18. α locations available in the SC1095 experimental (EXP) and CFD data sets. (a) Cl

data, (b) Cd data, (c) Cm data.

49

The CFD-generated airfoil data was provided by Mayda and van Dam [18] using the two-

dimensional Reynolds-averaged Navier-Stokes flow solver, ARC2D. The CFD data has

a consistently higher lift-curve slope (αl

C ) at all Mach numbers when compared to the

majority of the experimental data as seen in Figure 4-19. These plots show that the

scatter of the experimental data increases with angle of attack. This is also apparent in

drag and moment coefficient data. The CFD data has a slightly higher maximum lift

coefficient (Cl,max) for M < 0.6. With fewer experimental points at higher Mach numbers

the αl

C and Cl,max of both data sets become more difficult to compare. Figure 4-20 shows

that the CFD generally captures the characteristic drag bucket very well with similar

zero-lift drag coefficient values (0dC ). At higher Mach numbers (M>0.6) the bucket

shape and 0dC values of the CFD begin to diverge from the experimental data. Similar

findings are seen in the moment coefficient data, as shown in Figure 4-21.

50

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

C l

Cl_CFD

Cl_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

Cl_EXP

(b)

Re(10^6)= 11.16 , M= 0.90

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

Cl_EXP

(c)

Figure 4-19. SC1095 Cl vs α comparison of experimental and CFD data. (a) M = 0.30, (b) M =

0.70, (c) M = 0.90.

51

Re(10^6)= 3.72 , M= 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

Cd_EXP

(b)

Re(10^6)= 11.16 , M= 0.90

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (de g.)

Cd

Cd_CFD

Cd_EXP

(c)

Figure 4-20. SC1095 Cd vs. α comparison of experimental and CFD data. (a) M = 0.30, (b) M =

0.70, (c) M = 0.90.

52

Re(10^6)= 3.72 , M= 0.30

-0.2

-0.2

-0.1

-0.1

0.0

0.1

0.1

0.2

0.2

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (de g.)

Cm

Cm_CFD

Cm_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-0.2

-0.2

-0.1

-0.1

0.0

0.1

0.1

0.2

0.2

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cm

Cm_CFD

Cm_EXP

(b)

Re(10^6)= 11.16 , M= 0.90

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cm

Cm_CFD

Cm_EXP

(c)

Figure 4-21. SC1095 Cm vs. α comparison of experimental and CFD data. (a) M = 0.30, (b) M

= 0.70, (c) M = 0.90.

53

4.2.2 Network Testing Conditions

The network condition results of the NACA0012 airfoil can be directly applied to the

SC1095 test case. The one major difference between these test cases is that the SC1095

only has two independent inputs dimensions (M and α ).

The Cl, Cd, and Cm CFD and experimental data were scaled in a manner similar to the

NACA 0012 test case so that the M and α ranges are normalized by dividing by 1 and

180, respectively. The optimum optimization parameter of o = 1 determined with the

NACA0012 data is also used for the SC1095 test case.

A suitable tolerance value is the magnitude of the random error present in the

experimental data. Determining the actual experimental error of wind tunnel data is

complex and involves random and bias errors. For the SC1095 experimental data, which

is composed of several wind tunnel experiments performed at different facilities and for

different airfoil models, the experimental error can only be estimated. A suitable

tolerance value can be derived from the values used in the NACA0012 studies. The τ

used for the following Cl, Cd, and Cm SC1095 test cases are 0.05, 0.005, and 0.01,

respectively.

4.2.3 SFA Network Approximated SC1095 Airfoil Performance Table

A total of three test runs were performed with the test conditions shown in Table 4-5 for

the SC1095 test case. The SFA-fused method was used to construct a complete C81

54

aerodynamic table with the same M and α dimension array as the CFD data. The CFD

M-Re proportionality of 1:12.4 x 106 is assumed for the approximations to estimate the Re

data.

Table 4-5. SC1095 airfoil test runs

Cl Data Cd Data Cm Data TEST RUN 1 TEST RUN 2 TEST RUN 3

DATA SIZE 706 571 586

σσσσ 1 1 1

ττττ 0.05 0.005 0.010

55

Chapter 5 Results and Analysis

5.1 NACA 0012 Test Runs

The NACA 0012 airfoil test case is used to verify the feasibility and accuracy of the

SFA-fused approach. Altogether, each coefficient has 28 test runs with different levels of

measurement noise. Every test run uses the SFA-fused approach to create a full

aerodynamic C81 performance table. The SFA results are directly compared to the actual

C81 data to derive the RMS error as a measurement of accuracy. The plots of

coefficients versus α at different Re and M regimes graphically demonstrate the ability of

the SFA-fused approach to make smooth interpolations.

5.1.1 RMS Error

The studies of the RMS errors are graphically represented in Figure 5-1, Figure 5-2, and

Figure 5-3 for the Cl, Cd, and Cm test cases, respectively. The RMS error is plotted

against training tolerance, which is equal to the amount of random error or noise present

in the experimental data. These plots help determine the effect on the SFA-fused

approach as available experimental data points and random noise vary. The following

observations are made from these plots:

• Approximation errors increase as the percentage of available data points falls.

• As the tolerance increases the SFA-fused approximations are more likely to

fall within the tolerance limit.

56

• All test cases using 100% of available data points closely follow the original

experimental data.

• As expected, for all data with ε = 0 (no noise) and a network training tolerance

of τ = 0, the SFA-fused approximations will always have an error associated

with the limitations of the method. This error increases as fewer data points

are used.

• As tolerance increases, the RMS error decreases below the tolerance limit line

in all plots. As can be seen for the Cm test case (Figure 5-3) for τ = 0.05, the

SFA-fused scheme efficiently conditions the approximations with the CFD

information, therefore minimizing the total error and improving the accuracy

of the approximations over the experimental data.

Altogether, the accuracy of the SFA-fused approximations diminishes as the number of

available experimental data points decrease. However, it is not the number of data points

that affect the accuracy of the approximations as much as choosing the most favorable

data points represented in Table 4-2. We have shown that the SFA-fused network will

give optimal results given CFD and experimental data sets that consistently span similar

Re, M, and α ranges. By choosing strategic points (α locations) that define the

coefficient curves well, and assuming comparable CFD and experimental data sets, the

SFA-network will give acceptable approximations with small errors. It can be seen that

the majority of those test runs using 70% to 90% of the available experimental data for

the Cl and Cd test cases, and 60% to 90% for the Cm test case, the RMS errors are

generally in close proximity to each other. In those test runs the number of experimental

57

data points is not as significant as the location of the chosen data points. Once the

experimental data size becomes very small, and the prime locations diminish, the RMS

error rises considerably.

Accuracy is also dependent on the amount of random noise, ε, in the experimental data as

well as the tolerance value, τ, used during the network training process. Generally, as the

ε (or τ) increases, the RMS error for all test runs converges below the tolerance limit.

For the extreme case in Figure 5-3 where τ = 0.05, the SFA-fused approximations show

that the RMS error of the approximations eventually goes below that of the experimental

data.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.01 0.02 0.03 0.04 0.05

Tolerance

RM

S E

rror

Exp. Data Only

390 (100%)

360 (92%)

340 (87%)

320 (82%)

300 (77%)

280 (72%)

260 (67%)

Tolerance Limit

Figure 5-1. Error for Cl test case.

58

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 0.002 0.004 0.006 0.008 0.01

Tolerance

RM

S E

rror

Exp. Data Only

650 (100%)

600 (92%)

560 (86%)

520 (80%)

480 (74%)

440 (68%)

400 (62%)

Tolerance Limit

Figure 5-2. Error for Cd test case.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.01 0.02 0.03 0.04 0.05

Tolerance

RM

S E

rror

Exp. Data Only

423 (100%)

387 (91%)

351 (83%)

315 (74%)

279 (66%)

252 (60%)

198 (47%)

Tolerance Limit

Figure 5-3. Error for Cm test case.

59

5.1.2 Coefficient Plots

The RMS analysis gives important but inconclusive information about the SFA’s ability

to approximate and interpolate. We must consider the coefficient plots for a good

evaluation of the SFA-fused approach. There are a total of 84 test runs performed, as

described in the NACA0012 test matrix in Table 4-3 and a complete C81 performance

table is approximated for every test run. The approximations are compared in plots

showing coefficient data versus α, spanning the given Re and M regimes. The CFD,

actual C81 data, the fabricated experimental data (EXP), and the SFA-fused

approximations (approx.) are all compared in these plots.

The RMS error analysis shows that the accuracy of the SFA-fused approximations is

optimal for test cases with more available experimental data. Figure 5-4 and Figure 5-5

are both representative for the test cases using 100% of the available experimental data.

From these plots we verify that the SFA-fused network can make appropriate

approximations that follow the available experimental data. For Figure 5-5, where the

experimental data has a random noise of ε = ± 0.05, we also observe the SFA-fused

network regularizing the approximations with the CFD data while making the

approximations committed to the experimental data to within the tolerance, τ. The result

is an attempt to approximate a curve that fits the noisy experimental data with the help of

the CFD data.

As available experimental data decreases, the accuracy of the SFA-fused approximations

falls. To demonstrate the extent of this deviation we consider the test cases using the

60

smallest experimental data sets. The plots shown below for these test cases compare the

coefficient data for a α-range of ± 30 degrees, for ε = 0 and ε = ± 0.05. This is the α

regime we are most interested in and where most of the data is concentrated and also

reduced, as previously shown in Table 4-2. For extreme α below α = -30 and above α =

+30 degrees, the approximations compare well with the C81 data for all test runs. Figure

5-4 (a) and Figure 5-5 (a) are sufficient in representing all test runs for the extreme α

range.

61

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(a)

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(b)

Figure 5-4. Cl vs. α for NACA 0012 Test Run 1. Re = 3.72 x 106, M = 0.30, ε = 0, s = 390 and

n = 390. (a) Plot for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

62

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(b)

Figure 5-5. Cl vs. α for NACA 0012 Test Run 22. Re = 8.68 x 106, M = 0.70, ε = ± 0.05, s =

390 and n = 168. (a) Plot for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

63

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(a)

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(b)

Figure 5-6. Cl vs. α for NACA 0012 at Re = 3.72 x 106, M = 0.30. (a) Test Run 7, ε = 0, s = 260

and n = 260. (b) Test Run 28, ε = ± 0.05, s = 260 and n = 152.

64

Re(10^6)= 6.20 , M= 0.50

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(a)

Re(10^6)= 6.20 , M= 0.50

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(b)



65

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(b)



66

Re(10^6)= 11.16 , M= 0.90

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(a)

Re(10^6)= 11.16 , M= 0.90

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_C81

Cl_EXP

(b)

Figure 5-9. Cl vs. α for NACA 0012 at Re = 11.16 x 106, M = 0.90. (a) Test Run 7, ε = 0, s =

260 and n = 260. (b) Test Run 28, ε = ± 0.05, s = 260 and n = 152.

67

From the lift coefficient plots we can better understand the finding of the RMS error

analysis. Comparing Figure 5-4 (b) and Figure 5-5 (b) using 100% of the available

experimental data with Figure 5-6 and Figure 5-8 using 67% of the available

experimental data, respectively, we see that the SFA-fused network makes smooth

interpolations for α locations where no experimental data exists. These interpolations

tend towards the CFD data although the regularization technique keeps the

approximations committed to the existing experimental data points. How smooth the

interpolations are depends greatly on how well the CFD and experimental data sets agree.

For the subsonic regime of M < 0.6, the CFD and experimental data compare well for a

range of ± 30 degrees. The amount of random noise does not significantly affect the

approximations, as shown for the (b) subplots with ε = 0.05, as long as M < 0.6. The

interpolations made for this regime are consistent and generally smooth, as seen in Figure

5-6 and Figure 5-7. The SFA-fused approach does a good job of approximating the

characteristic lift-curve slope and the rise and drop that reveal the maximum lift

coefficient.

However, for the transonic regime of M > 0.6 where the airfoil characteristics become

nonlinear and flow and shock wave complexities begin to take shape, the CFD and

experimental data begin to diverge. The main differences occur within α = 5 to α = 20

degrees where there is no defined maximum lift coefficient in this regime. This variance

results in a fast changing error surface, especially for M = 0.7 to M = 0.8, that must be

conditioned during the fusing process. As a result, the approximation curves in this

regime are more sensitive to ε and the number of available data points, as can be seen in

68

Figure 5-8. As the airfoil nears supersonic speeds, the flow complexities are diminished

and so are the fluctuations in the error curve allowing for smoother approximations, as

observed in Figure 5-9 for M = 0.9.

For the lift coefficient test runs 1, 8, 15, and 22 that use 100% of all the available

experimental data points, the effect of random noise on the approximations is minimal

even in the transonic regime. However, the effect of ε on the drag coefficient

approximations is more prevalent, especially at α-range within ± 20 degrees where the

characteristic airfoil drag bucket takes shape. The following plots are primarily focused

in this α regime where experimental data is reduced. As in the lift coefficient test runs,

the drag coefficient approximations compare well with the experimental data for extreme

α. Τhe sample plots shown in Figure 5-10 are sufficient to represent all test runs for the

extreme α range below α = -20 and above α = +20 degrees.

Within the α range of interest, Figure 5-11 through Figure 5-14 compare the results for

Test Run 29 and Test Run 50 that use 100% of the available experimental data with ε = 0

and ε = ± 0.01, respectively. For subplots (a) with ε = 0, the SFA-fused drag

approximations follow the experimental data closely and smoothly as in the lift

coefficient approximations. For subplots (b) with ε = ± 0.01, the approximations are

very sensitive to the random noise resulting in a less defined curve influencing the

minimum drag range and the zero-lift drag coefficient. This sensitivity to ε is not as

prominent in the lift coefficient approximations. A random error of ε = ± 0.01 is

unusually high and actual drag data with this much error will typically be disregarded.

69

Yet, it is important to highlight the reaction of the SFA-fused approximations to this

extreme case. Overall, the other test runs with experimental data having smaller random

errors show approximation curves exhibiting a similar sensitivity to ε though at a much

smaller scale.

70

Re(10^6)= 3.47 , M= 0.28

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 8.93 , M= 0.72

0.00

0.50

1.00

1.50

2.00

2.50

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)

Figure 5-10. Cd vs. α for NACA 0012 Test Run 29 with ε = 0. (a) Re = 3.47 x 106, M = 0.28, s

= 650 and n = 650. (b) Re = 8.93 x 106, M = 0.72, s = 650 and n = 650.

71

Re(10^6)= 3.47 , M= 0.28

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 3.47 , M= 0.28

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)

Figure 5-11. Cd vs. α for NACA 0012 at Re = 3.47 x 106, M = 0.28. (a) Test Run 29, ε = 0, s =


72

Re(10^6)= 5.95 , M= 0.48

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 5.95 , M= 0.48

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



73

Re(10^6)= 8.93 , M= 0.72

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 8.93 , M= 0.72

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



74

Re(10^6)= 11.41 , M= 0.92

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 11.41 , M= 0.92

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



75

From the test runs previously discussed we noted the significance of random noise on the

drag coefficient approximations. We now can look at how well the SFA-fused network

makes approximations for the drag coefficient as experimental data is reduced. Figure

5-15 through Figure 5-18 compare the drag coefficient for Test Runs 35 and 56 having

ε = 0 and ε = ± 0.01, respectively, and using just 62% of the available experimental data.

Again, all of the Test Run 35 plots (subplots (a)) for ε = 0 show approximation curves

having generally good correlation with the available experimental data at all M. Overall,

smooth and consistent interpolations are made with only slight inconsistencies in

symmetry. For the Test Run 56 plots (subplots (b)) having ε = ± 0.01, the approximation

curves are again sensitive to the random noise and show more inconsistencies in

symmetry. The plots show however that the SFA-fused network attempts to regularize

the approximations with help from the CFD curves allowing the network to make

relatively smooth interpolations.

For the test runs with a smaller random error, the approximations are more realistic as

seen in Figure 5-19 for Test Run 49 with ε = ± 0.005. Compared to Figure 5-15 (b) for

Test Run 56 having ε = ± 0.01, Figure 5-19 (b) shows that there are much fewer

irregularities in the approximation curves for experimental data with a smaller ε.

76

Re(10^6)= 3.47 , M= 0.28

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 3.47 , M= 0.28

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



77

Re(10^6)= 5.95 , M= 0.48

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 5.95 , M= 0.48

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



78

Re(10^6)= 8.93 , M= 0.72

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 8.93 , M= 0.72

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



79

Re(10^6)= 11.41 , M= 0.92

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 11.41 , M= 0.92

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)



80

Re(10^6)= 3.47 , M= 0.28

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(a)

Re(10^6)= 8.93 , M= 0.72

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_C81

Cd_EXP

(b)

Figure 5-19. Cd vs. α for NACA 0012 for Test Run 49. (a) Re = 3.47 x 106, M = 0.28,

ε = ± 0.005, s = 400 and n = 364. (b) Re = 8.93 x 106, M = 0.72, ε = ± 0.005, s = 400 and n =

364.

81

The following moment coefficient plots confirm the findings of the lift and drag studies.

As in the lift and drag test cases where 100% of the available experimental data is used

during the SFA-fused network training process, the moment coefficient approximations

are also good and follow the C81 data well when using 100% of the experimental data, as

seen in Figure 5-20. When a small random error is added to the experimental data as in

Test Run 71 where ε = ± 0.01, the approximations become sensitive to ε, but the

discrepancies remain small when compared to the C81 data, as seen in Figure 5-21. The

SFA-fused network attempts to regularize the approximations with the CFD data while

remaining committed to the experimental data to within the training tolerance. However,

when a very large random error is introduced this tolerance is high, and the

approximations will rely more on the CFD data, as in Figure 5-22 for Test Run 78 for ε =

± 0.05.

Figure 5-23 through Figure 5-26 show the effect of using only 47% of the available

experimental data and compare the results for Test Runs 63, 77, and 84 having a random

error of ε = 0, ε = ± 0.01, and ε = ± 0.05, respectively. As in the lift and drag test cases,

these plots also demonstrate the SFA-fused network’s ability to make smooth and

consistent interpolations given only limited data. The Test Run 84 plots (subplots (c))

further show how the approximations will ultimately depend exclusively on the CFD data

as the random error present in the experimental data becomes very large.

82

Re(10^6)= 8.68 , M= 0.70

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Figure 5-20. Cm vs. α for NACA 0012 Test Run 57. Re = 8.68 x 106, M = 0.70, ε = 0, s = 423

and n = 423. (a) Plot for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

83

Re(10^6)= 8.68 , M= 0.70

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Figure 5-21. Cm vs. α for NACA 0012 Test Run 71. Re = 8.68 x 106, M = 0.70, ε = ± 0.01, s =


84

Re(10^6)= 8.68 , M= 0.70

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

-180.0 -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Figure 5-22. Cm vs. α for NACA 0012 Test Run 78. Re = 8.68 x 106, M = 0.70, ε = ± 0.05, s =


85

Re(10^6)= 3.72 , M= 0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 3.72 , M= 0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Re(10^6)= 3.72 , M= 0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(c)

Figure 5-23. Cm vs. α for NACA 0012 at Re = 3.72 x 106, M = 0.3. (a) Test Run 63, ε = 0, s = 423 and n = 423. (b) Test Run 77, ε = ± 0.01, s = 423 and n = 165. (c) Test Run 84, ε = ± 0.05, s = 423 and n = 59.

86

Re(10^6)= 6.20 , M= 0.50

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 6.20 , M= 0.50

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Re(10^6)= 6.20 , M= 0.50

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(c)


87

Re(10^6)= 8.68 , M= 0.70

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 8.68 , M= 0.70

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Re(10^6)= 8.68 , M= 0.70

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(c)


88

Re(10^6)= 11.16 , M= 0.90

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(a)

Re(10^6)= 11.16 , M= 0.90

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(b)

Re(10^6)= 11.16 , M= 0.90

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_C81

Cm_EXP

(c)


89

5.2 SC1095 Test Runs

The SC1095 airfoil test case consists of highly sparse and scattered experimental data

sets that will benefit from the SFA-fused network approach to complete a full airfoil

table. The following work graphically represents the results of a complete SC1095 airfoil

C81 performance table constructed using the SFA-fused approach. This test case reflects

the type of realistic problem the SFA-fused network will face in constructing airfoil

performance tables.

The following coefficient plots compare the available experimental and CFD coefficient

data with the SFA-fused approximations. From the NACA 0012 airfoil test case analysis,

we have a better understanding of how the SFA-fused network deals with limited

experimental data sets and random noise and can apply that knowledge to the SC1095

airfoil results.

90

Re(10^6)= 3.72 , M= 0.30

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_EXP

Figure 5-27. Cl vs. α for SC1095. Re = 3.72 x 106, M = 0.30, ε = ± 0.05, s = 706 and n = 240.

Re(10^6)= 6.20 , M= 0.50

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_EXP


91

Re(10^6)= 8.68 , M= 0.70

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_EXP


Re(10^6)= 11.16 , M= 0.90

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cl

Cl_CFD

approx.

Cl_EXP


92

Re(10^6)= 3.72 , M= 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_EXP

Figure 5-31. Cd vs. α for SC1095. Re = 3.72 x 106, M = 0.30, ε = ± 0.005, s = 571 and n = 291.

Re(10^6)= 6.20 , M= 0.50

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_EXP


93

Re(10^6)= 8.68 , M= 0.70

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_EXP


Re(10^6)= 11.16 , M= 0.90

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cd

Cd_CFD

approx.

Cd_EXP

Figure 5-34. Cd vs. α for SC1095. Re = 11.16 x 106, M = 0.90, ε = ± 0.005, s = 571 and n =

291.

94

Re(10^6)= 3.72 , M= 0.30

-0.2

-0.2

-0.1

-0.1

0.0

0.1

0.1

0.2

0.2

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_EXP

Figure 5-35. Cm vs. α for SC1095. Re = 3.72 x 106, M = 0.30, ε = ± 0.01, s = 586 and n = 137.

Re(10^6)= 6.20 , M= 0.50

-0.2

-0.2

-0.1

-0.1

0.0

0.1

0.1

0.2

0.2

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_EXP


95

Re(10^6)= 8.68 , M= 0.70

-0.2

-0.2

-0.1

-0.1

0.0

0.1

0.1

0.2

0.2

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_EXP


Re(10^6)= 11.16 , M= 0.90

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

-25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0

αααα (deg.)

Cm

Cm_CFD

approx.

Cm_EXP


96

As expected from the NACA 0012 test case, the SFA-fused network makes suitable

approximations that fit the available experimental data while utilizing the CFD data to

make smooth and consistent interpolations as shown in all the plots below. The

approximation curves smoothly manage through the highly condensed data points and

efficiently control the large scatter of the experimental data. Further, for regimes where

there are no experimental data, the approximations show a smooth transition to the CFD

data and eventually rely entirely on the CFD curves. Except for the regimes where the

approximations are conditioned with experimental data, the SFA-fused-generated C81

table overwhelmingly reflects the CFD data.

97

Chapter 6 Neural Network Comparisons

The application of the SFA-fused approach as an airfoil approximation tool has been

studied and verified in the last chapters. In this chapter the SFA network is compared

with a standard RBF network and a Generalized Regression Neural Network (GRNN)

found in the Matlab Neural Network Toolbox Version 3.0.1. The following will compare

the required network properties and evaluate the ability of these networks to 1) reproduce

the training data set and to 2) make interpolations for a given Cl test case for the NACA

0012 airfoil. The test case for the following analysis involves the NACA 0012 Cl data

provided in the C81 table.

6.1 Radial Basis Function Network

The Matlab radial basis function network NEWRB is a two layer network (a hidden layer

of radial units and an output layer of linear units) that can be used to approximate

functions. Initially, the RBF network begins with no network units and will add units to

the hidden layer until a desired mean squared error is met. The network uses a gradient

descent backpropagation algorithm that updates weights and biases (defining the position

and width of the radial basis function) in order to minimize the error. The RBF network

trains in the same amount of time as the SFA and only requires the input of the training

data sets, the desired mean squared error, and the spread of the RBF. A trial-and-error

process must be used to determine the optimum spread value for each problem. Finally,

the RBF network is trained to some goal or mean squared error setting (similar to the

tolerance setting of the SFA), or until the number of network units (which include the

98

RBF transfer function centers and their respective weights) in the first layer equals the

number of available training data points. The number of network units in the second

layer always equals the number of available training data points.

6.2 Generalized Regression Neural Network

The GRNN is a type of RBF network that is also known as a Bayesian network [20].

Generally, Bayesian statistics are applied to estimate the probability density of a model’s

parameters given a set of available data. To minimize error, the model parameters that

maximize this probability density function (PDF) are selected. An approach to

estimating the PDF is a kernel-based approximation technique [21]. In kernel-based

estimation, simple functions (typically Gaussian functions) are located at each available

case and added together to estimate the overall PDF. A good approximation to the true

PDF is derived given sufficient training points. This is the general groundwork of the

GRNN that is used in the Matlab NEWGRNN function. The GRNN is a two layer

network with the first layer containing radial units and weights equal to the vector

distance of the input training data. The second layer contains units that estimate the

weighted average of the training data outputs. The obvious advantage to the GRNN is

that it requires no optimization algorithm and as a result trains immediately. The

NEWGRNN is set up similarly as NEWRB but also requires the trial-and-error designation

of the RBF spread for optimum results. The two layers of the GRNN have network units

equal to the number of available training data points. As a result, the GRNN will tend to

99

be memory intensive and slow to execute for large scale problems with large training data

sets.

6.3 Spread Setting

The spread setting in the Matlab versions of the RBF network and the GRNN is

analogous to the σn in the SFA network and represent the width of the radial basis

functions. For the SFA, σn values are unique to every network unit and are derived using

a standard MATLAB nonlinear unconstrained minimization routine in the training

algorithm. As discussed in Chapter 4.2.2, this minimization routine is prone to

suboptimal results when initiated with a poor optimization parameter value (σ0). To

ensure the best results, a trial and error approach was used to determine the most suitable

σ0 applicable for all coefficient test cases.

For the Matlab RBF network and GRNN, the spread setting is the same for all the

network units and is determined by the user. The larger the spread setting the smoother

the function approximation, although the network may require more units to fit a fast

changing function. Too small a spread and the network may need many units to fit a

smooth function. A trial and error process must be used to determine the best spread

setting. This first involves training the networks with all of the NACA 0012 Cl data and

testing with the same training input conditions (Re, M, α) to determine how well the

network reproduces the training data. The networks are trained using varying spread

settings and the RMS error is calculated for each test. The results of this analysis show

100

that a spread of σ0 = 0.01 gives the best results for both the RBF network and GRNN, as

shown in Figure 6-1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3

Spread

RM

S

RBF

GRNN

Figure 6-1. Matlab RBF and GRNN spread setting.

6.4 Network Properties

The focus of the following tests is to measure and compare the networks abilities to

reproduce the training data set and to make concise interpolations. These tests also reveal

the required network sizes (number of network units) and the computational time to

complete a training/testing process. Since all the networks are trained to a tolerance (or

“goal” for the RBF network and GRNN) of zero, the SFA and the RBF network always

train until the number of network units equals the number of available data points, s. For

the SFA this means s network units, while the two-layered RBF network requires 2s

network units. The nature of the GRNN is always to use all the available data points as

network units for both its layers. It therefore also has 2s network units.

101

Recall that each network unit consists of a (or spread for the RBF network and GRNN),

center, and weight. The SFA optimizes the values for and the weights for each unit,

while the centers are iteratively determined. For the RBF network, the user specifies the

spread, the centers are iteratively calculated, and the weights are optimized. The GRNN

also requires a user defined spread but the centers and the weights are predetermined.

The SFA’s algorithm is dependent on optimization although it possesses fewer network

units. The RBF network requires more network units but its algorithm is not as

demanding as the SFA. The GRNN also uses more network units than the SFA and

requires no optimization. As a result, from the reproduction tests using s = 390 training

data samples, the RBF network and SFA each required about a full minute to complete

the training and testing process, while the GRNN was trained and tested within seconds.

For these interpolation tests using s = 280 training data samples, the RBF network and

the SFA were trained/tested within one minute while the GRNN again required only

seconds.

These results offer insight into the required network properties of each network for a

small scale test such as the airfoil problem discussed here. Larger scale comparisons of

the networks are beyond the scope of the work here and are recommended in future work.

102

6.5 Reproducing Data

A way to compare the RBF network, GRNN, and the SFA is to evaluate how well the

networks reproduce the training data. In the following tests, all the networks were trained

and tested with the same C81 data set consisting of 390 data points. The Cl

approximations from all the networks were then compared to the actual C81 data to

calculate the RMS error, as shown in Table 6-1.

The results of these tests proved that all of the networks make efficient and adequate

reproductions of the training data. Figure 6-2 and Figure 6-3 compare the results at M =

0.3 and at M = 0.7, respectively, and are representative for all tests. The GRNN does a

good job at fitting the data with smooth curves although not exactly through every point,

resulting in an error RMS = 0.0219. The RBF network and the SFA network fit every

data point and share almost identical results with errors of RMS = 0.0170 and RMS =

0.0166, respectively.

Table 6-1. Reproduction test results.

RMS error s (number of training samples)

n (number of units)

RBF 0.0170 390 780 GRNN 0.0219 390 780 SFA 0.0167 390 390

103

Re(10^6)= 3.72 , M= 0.3

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

αααα

C l

C81

RBFN (s=.01)

GRNN (s =.01)

SFA

(a)

Re(10^6)= 3.72 , M= 0.3

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-30 -20 -10 0 10 20 30

αααα

C l

C81

RBFN (s=.01)

GRNN (s =.01)

SFA

(b)

Figure 6-2. Cl vs. α for NACA 0012 reproduction test comparisons. Re = 3.72 x 106, M =

0.30, ε = 0. RBF: s = 390 and n = 780. GRNN: s = 390 and n = 780. SFA: s = 390 and n = 390.

(a) Plot for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

104

Re(10^6)= 8.68 , M= 0.7

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

αααα

C l

C81

RBFN (s=.01)

GRNN (s =.01)

SFA

(a)

Re(10^6)= 8.68 , M= 0.7

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-30 -20 -10 0 10 20 30

αααα

C l

C81

RBFN (s=.01)

GRNN (s =.01)

SFA

(b)

Figure 6-3. Cl vs. α for NACA 0012 reproduction test comparisons. Re = 8.68 x 106, M = 0.70,

ε = 0. RBF: s = 390 and n = 780. GRNN: s = 391 and n = 782. SFA: s = 390 and n = 390. (a) Plot

for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

105

6.6 Interpolating Data

The ability to reproduce the training data is just one way to make comparisons between

the different networks. Another important aspect to assess is the network’s ability to

make interpolations. For these tests, data points are reduced from the original training

data set of 390 data points to 280 data points. Table 6-2 shows the locations of the points

that are eliminated. All of the networks were then trained with this new sparse data set

and tested for inputs covering the range where no training data points exist.

Table 6-2. α locations used to train and test each network. The highlighted areas represent the α

values that were eliminated.

± α α α α (deg.) 390 280 ±±±± α α α α (deg.)180.0 180172.5 178161.0 176147.0 174129.0 17249.0 17039.0 16821.0 16616.5 16415.0 16214.0 16013.0 15812.0 13011.0 11010.0 908.0 606.0 454.0 302.0 220.0 20

181614121086420

Training Data Testing Data

106

Where all the networks show similar aptitude in reproducing the training data in the

previous section, the results from the interpolation tests reveal major differences between

the networks. Figure 6-7 through Figure 6-7 illustrate the comparisons of the networks

with a sparse set of NACA 0012 lift coefficient C81 data at different Mach numbers.

From these plots it becomes apparent that the RBF network has diminished capabilities

for making smooth and consistent interpolations. Using a larger spread with the RBF

network will give smoother curves but with a poorer fit of the data. The GRNN and the

SFA give much better interpolations and their results are comparable for smaller α (-30 <

α < 30). The GRNN will approximate smoother curves at smaller α although its

interpolations lack consistency at high α where the approximations transition into

locations where very little data exists. The SFA will adapt its smooth curves to fit closely

all the available sparse data sets, resulting in less stable approximations at small α (where

more data points exists) yet still making smooth interpolations and transitions regardless

of limited data points. The trained networks were tested for the same input range as the

C81 data to allow calculation of the RMS error. As shown in he approximations for the

RBF.

107

Table 6-3, the SFA again has the smaller error of the three networks. The SFA curves at

low α are not as clean as the GRNN curves although results can be much improved if a

tolerance is applied to the training process. A tolerance will allow the SFA to end

training much sooner requiring less network units to better fit the low α range. The same

approach would greatly improve the approximations for the RBF.

108

Table 6-3. Interpolation Test Results.

RMS error s (number of training samples)

n (number of units)

RBF 0.0860 280 560 GRNN 0.0171 280 560 SFA 0.0133 280 280

Each network tool tested has its distinct strengths and weaknesses. The SFA exhibits the

best overall characteristics. The RBF network and the SFA share similar capabilities in

reproducing the training data although the RBF network suffers from its difficulties in

making interpolations. The GRNN and the SFA make good interpolations with the

GRNN approximating smoother curves and the SFA making better fits to the available

data. However, the SFA makes better interpolations for regimes where fewer data points

are available. In terms of computational demands, the GRNN has the advantage of

training/testing immediately while the RBF network and the SFA have similar

computational needs. The GRNN’s advantage is outweighed by the time consuming

trial-and-error approach required to determine the optimum spread setting for each test

problem. The SFA required a similar approach to determine its initialization parameter

only once for all tests.

109

Re(10^6)= 3.72 , M= 0.3

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(a)

Re(10^6)= 3.72 , M= 0.3

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-30 -20 -10 0 10 20 30

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(b)

Figure 6-4. Cl vs. α for NACA 0012 interpolation test comparisons. Re = 3.72 x 106, M = 0.30,

ε = 0. RBF: s = 280 and n = 560. GRNN: s = 280 and n = 560. SFA: s = 280 and n = 280. (a)

Plot for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

110

Re(10^6)= 6.2 , M= 0.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(a)

Re(10^6)= 6.2 , M= 0.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30 -20 -10 0 10 20 30

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(b)




111

Re(10^6)= 8.68 , M= 0.7

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(a)

Re(10^6)= 8.68 , M= 0.7

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30 -20 -10 0 10 20 30

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(b)




112

Re(10^6)= 11.16 , M= 0.9

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(a)

Re(10^6)= 11.16 , M= 0.9

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-30 -20 -10 0 10 20 30

αααα

Cl

C81 (sparse)

RBFN (s=.01)

GRNN (s=.01)

SFA

(b)

Figure 6-7. Cl vs. α for NACA 0012 interpolation test comparisons. Re = 11.16 x 106, M =

0.90, ε = 0. RBF: s = 280 and n = 560. GRNN: s = 280 and n = 560. SFA: s = 280 and n = 280.

(a) Plot for α range of ± 180 degrees. (b) Plot for α range of ± 30 degrees.

113

Chapter 7 Conclusions and Recommendations

A method that combines experimental airfoil coefficient data with numerical data has

been developed to address the issue of constructing full airfoil performance tables given

limited data sets. The method utilizes the Sequential Function Approximation neural

network and a regularization scheme to fuse multi-dimensional experimental and CFD

data efficiently. The SFA-fused network was proven to be an adaptive and robust tool

requiring relatively little computational demand. The approach was found to require

more user interaction than just the SFA method alone, depending on the user to determine

1) the optimal network parameter, σ0, and 2) the scaling factor of the data set for each test

case. These parameters were determined through a trial-and-error process although

future work will look into employing an optimization code to determine these values

automatically.

The SFA-fused network was first tested and analyzed for the NACA 0012 airfoil test

case. This ideal test subject consisted of a complete C81 airfoil performance table with

matching CFD data. A total of 28 test runs were conducted for each lift, drag, and

moment coefficient test case using different experimental data sizes and varying random

noise levels. The RMS error of the approximations was calculated with respect to the

actual C81 data and represents the accuracy of the approximations. Those studies

demonstrate that the accuracy of the approximation falls only slightly as fewer

experimental data points are used, and becomes significantly larger when there are an

insufficient number of data points to clearly define the experimental curve. The

coefficient plots reveal that the SFA-fused approach makes consistent interpolations that

114

closely fit the available experimental data points while smoothly transitioning to the CFD

data when no experimental data exists. The approximation curves become less smooth

for regimes where the error surface, e, deviates significantly over a small dimensional

space. The SFA-fused approximations are also more sensitive as experimental random

error increases, specifically at the same time that available data sizes become smaller,

resulting in less smooth interpolations. The accuracy of the approximations eventually

improves over the accuracy of the experimental data having a very large random error by

conditioning the approximations towards the CFD data. Overall, the NACA 0012 test

case verifies the feasibility of the SFA-fused approach as an airfoil performance table

approximation tool and reveals the strengths and limitations of the method.

For a realistic test case, a complete performance table was constructed for the SC1095

airfoil. This test case is representative of existing airfoils having incomplete performance

tables. The SC1095 comes with highly sparse and scattered experimental data sets

composed from six different wind tunnel experiments. Consequently, the Reynolds

number range for the experimental data differs significantly from the CFD data and

exposes a limitation of the SFA-fused method. Without comparable experimental and

CFD data sets, we cannot calculate the error surface, e, used to fuse both data sets. For

this test case we were able to disregard the Reynolds number data assuming that it plays

little significance. The results for this test case showed smooth approximation curves

fitting the scattered experimental data and consistent transitions to the CFD data.

115

Finally, to verify the efficiency of the SFA network, comparison tests with two other

common neural networks were performed. These tests compared the networks’ abilities

to reproduce the training data and to make interpolations. Final evaluations showed that

the SFA exhibited better overall characteristics when applied to an airfoil prediction

problem. The SFA reproduced the training data with the smallest error and made smooth

and consistent interpolations with less dependence on the number of available data.

Moreover, the SFA was shown to involve less user interaction to optimize the results and

required less computational space than the other standard networks.

Altogether, the SFA-fused approach has been established as an efficient method to

process and combine sparse and scattered experimental and numerical data in the

construction of airfoil performance tables. Future work will add to the usefulness of this

approach by developing a more reliable optimization routine for the determination of n

during the SFA training process. Following the developments of this work, further airfoil

test cases should be considered to fully realize the application scope of this method. It is

hoped that the SFA-fused approach can be used to tackle a higher-dimensional problem

that, for example, considers the geometric measurements (chord, thickness ratio, etc.) of

an airfoil as independent parameters. The SFA-fused method can then move from being

a fusing/approximation/interpolation tool to a research design tool.

116

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