Charles L. Karr Rodney Bowersox Vishnu Singh 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 Start 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 Stop Criterion

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 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 The above equations can be written in a matrix form as

Coefficients (continued) Now using matrix inversion method we can find the values of the coefficients By using the other equations and applying the same method the remaining coefficients can be found.

The Strings These strings are basically floating point arrays. X1 X2 … Y1 Y2 Y 3 … Xo … 1.1349 1.1555 0.2367 -1.2976 0.3356 -1.1369 4.3459 C1 … 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)

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 IT WORKS!!! 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