minimization of sonic boom on supersonic aircraft using an evolutionary algorithm charles l. karr...
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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