Minimization of Sonic Boom on Supersonic Aircraft Using an Evolutionary Algorithm

Charles L. Karr

Rodney Bowersox

Vishnu Singh

Introduction

Things that we are going to cover.

• What is GA?

• The Problem at hand?

• How GA is used to solve it.

• How good is the solution.

• Conclusion.

• Evolutionary algorithms– search algorithms based on the mechanics of natural

selection

– growing in popularity (genetic algorithms, evolutionary strategies, evolutionary programming, etc.)

– effective in complex, nonlinear problems

– developed to the point of “cookbook” application?

– still require some expert tuning: “flavors” of crossover and mutation operators – dynamic genetic algorithm

The GA

–Initialize a Population of Strings

–Evaluate Each String’s Fitness–Value

–Select the Superior Strings for Reproduction

–Apply Crossover Methods 1….N

–Apply a Mutation Operator

–Termination–Criterion

–Stop

–Start

The Problem

• Minimization of Sonic Boom– The development of

supersonic transport vehicles will require much work in sonic boom mitigation

– Numerous approaches have been considered (pulse detonation, keel design, etc)

– Here, we are interested in designing a spike or keel that will mitigate the sonic boom

Changing the Area Distribution

• The Keel– Basically, adding a keel

will change the area distribution

– This change in area distribution will change the ground signature of the aircraft

Ground Signatures

In Real Life

Like With Any EA Application…

• There are two fundamental issues:– Coding

• How do I represent the problem as a string of characters so that the EA can operate on them

– Fitness Function

• How do I determine how good a proposed solution really is

Coding

• We have to represent an area distribution– Fit third or higher

order polynomials through five different sections

– Constraints• Continuous

through the second derivatives

– Twenty four (24) coefficients to be determined

Equations

44

314

2141444 54321

xCxCxCxCCY

43

313

2131333 54321

xCxCxCxCCY

42

312

2121222 54321

xCxCxCxCCY

41

311

2111111 54321

xCxCxCxCCY

310

2101000 4321

xCxCxCCY The five different sections are represented by the equations shown below

–We want to determine the coefficients

Coefficients

Now to find the Values of the first four coefficients we use the following method. Lets consider

30

20000 4321

XoCXoCXoCCY

20000 432

32 XoCXoCCY

30

20000 111

4321XCXCXCCY

20000 1312

432XCXCCY

The above equations can be written in a matrix form as

40

0

0

0

211

31

211

2

32

0

0

0

0

3

2

1

*

10

1

10

1

C

C

C

C

XX

XXX

XX

XXX

Y

Y

Y

Y

OO

OOO

Coefficients (continued)

• Now using matrix inversion method we can find the values of the coefficients

0

0

0

0

211

31

211

2

32

40

0

0

0

*

10

1

10

1

)(3

2

1

Y

Y

Y

Y

XX

XXX

XX

XXX

inv

C

C

C

C

OO

OOO

By using the other equations and applying the same method the

remaining coefficients can be found.

The Strings

C1 1.1349 1.1555 0.2367 … -1.2976 0.3356 4.3459-1.1369 C18

These strings are basically floating point arrays.

They represent the different values of X,Y and Y` which are needed to fully describe an area distribution.

They will be operated on by the evolutionary algorithm (dynamic GA)

Xo X1 X2 … Y1 Y2 Y 3 … …

Fitness Function

• How good is a potential solution– Given an area distribution, can we come up with a

representation of the effectiveness of the solution

– It turns out that this is a two-step process:

• Use a modified version of Whitham’s theory to generate the near field pressure signature

• Compute the ground signature using NFBOOM, an atmospheric propagation code

• Compute the magnitude of the sonic boom

Whitham’s Theory

2/ 0rr r xxr

0

0

( )

( )( 2 )

1 ( ) ( )

( )( 2 )

y

y

u f d

U y y r

v y r f d

U r y y r

4

20

2

0

0

( ) 2 2( 1)( )

2

( ) 2log ( )

( ) 2

( ) 2log ( )

( ) 2

yB

y

yB

B

y r y RMf d

y

y r yx r M f d

y r y

y R yf d y

y R y

2p p uM

p U

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0 10 20 30 40 50 60 70

x-br (cm)

p

/p

Theory

Experimental Data

Comparison of Quasi-Linear Theory with the Frontal Spike (r = 50.8 cm) Data of Swigart [11].

IT WORKS!!!

GA Particulars

• GA Parameters– Population size = 50

– Number of generations = 250

– Selection scheme = tournament selection

– Crossover schemes= ArithXover, Heuristic Xover, SimpleXover

– Mutation schemes = Uniform, Gaussian, Random

Feasibility Study

• Compare to a known solution– “Best” solution for this

aircraft was 106.3 dB

– The genetic algorithm found a solution of 104.0 for the given conditions

Conclusions

• This is a preliminary study

• The initial results are promising

• Next step is to develop a design tool