Slide1: A discrete problem Difficultiy in the solution of a discrete problem
Slide2: Example of a discrete problem
Optimization of a composite structure where individual parts of it are described by 10 design variables. Each design variable represents a ply angle varying from 0 to 45 degrees with an increment of 5 degrees, i.e. 10 possible angles.
One full FE analysis of each design takes 1 sec. on a computer.
Question: how much time would it take to check all the combinations of the angles in order to guarantee the optimum solution? MATHEMATICAL OPTIMIZATION PROBLEM
Slide3: Genetic Algorithm stochastic, directed and highly parallel search technique based on principles of population genetics
Difference with traditional search techniques:
Coding of the design variables as opposed to the design variables themselves, allowing both discrete and continuous variables
Works with population of designs as opposed to single design, thus reducing the risk of getting stuck at local minima
Only requires the objective function value, not the derivatives. This aspect makes GAs domain-independent
GA is a probabilistic search method, not deterministic, making the search highly exploitative.
Slide4: Genetic Algorithm Representation scheme: finite-length binary alphabet of ones and zeros
The fitness function defines how well each solution solves the problem objective.
Darwin's principle of survival of the fittest: evolution is performed by genetically breeding the population of individuals over a number of generations
crossover combines good information from the parents
mutation prevents premature convergence
Slide5: Genetic Algorithm Evolutionary mechanism of the Genetic Algorithm
Slide6: Genetic Algorithm A flowchart of a genetic algorithm
Slide7: Representation of a design by a binary string. Example. Genetic Algorithm Portal frame Chromosome of a design set using binary representation
Slide8: Genetic Algorithm - Encoded variables for UBs Genetic Algorithm
Slide9: Genetic Algorithm - Single point crossover Genetic Algorithm
Slide10: Genetic Algorithm - Arrangement of design variables Genetic Algorithm Five-bay
five-storey framework
Slide11: Genetic Algorithm - Solution for five-bay five-storey framework Genetic Algorithm
Slide12: Genetic Algorithm Genetic Algorithm - Five-bay five-storey framework (8 d.v.)
Slide13: Genetic Algorithm Example.
Three-bay by four-bay by four-storey structure
Numerical optimization techniques: Numerical optimization techniques Genetic Algorithm - 3-bay by 4-bay by 4-storey structure Genetic Algorithm
Slide15: Convergence history for 3-bay by 4-bay by 4-storey structure Genetic Algorithm
Optimization of front wing of J3 Jaguar Racing Formula 1 car: Optimization of front wing of J3 Jaguar Racing Formula 1 car APPLICATION OF GENETIC ALGORITHM
Optimization of front wing of J3 Jaguar Racing Formula 1 car: Optimization of front wing of J3 Jaguar Racing Formula 1 car APPLICATION OF GENETIC ALGORITHM
Slide18: Genetic
Algorithm APPLICATION OF GENETIC ALGORITHM Front wing of J3 Jaguar Racing Formula 1 car
Slide19: Genetic
Algorithm APPLICATION OF GENETIC ALGORITHM Schematic layup
of the composite structure of the wing
APPLICATION OF GENETIC ALGORITHM: APPLICATION OF GENETIC ALGORITHM Optimization problem: minimize mass subject to displacement constraints (FIA and aerodynamics)
Result of optimization:
Design obtained by GA optimization: 4.95 Kg
Baseline design weight: 5.2 Kg
Improvement: 4.8%
Optimization of an aerofoil: Optimization of an aerofoil B-spline representation of the NACA 0012 aerofoil. The B-spline poles are numbered from 1 to 25. Design variables: x and y coordinates of 22 B-spline poles (N = 44). EXAMPLES: SHAPE OPTIMIZATION W.A. Wright, C.M.E. Holden, Sowerby Research Centre, British Aerospace (1998)
Problem definition (aerofoil, cont.): Problem definition (aerofoil, cont.) EXAMPLES: SHAPE OPTIMIZATION Problem formulation:
Objective function (to be minimized): drag coefficient at Mach 0.73 and Mach 0.76:
F0 (x) = 2.0 Cd total (M=0.73) + 1.0 Cd total (M=0.76)
Constraints: on lift and other operational requirements (sufficient space for holding fuel, etc.)
Techniques used:
Powell’s Direct Search (PDS)
Genetic Algorithm (GA)
MARS
Carren M.E. Holden Sowerby Research Centre, British Aerospace, UK
Results (aerofoil, cont.): Results (aerofoil, cont.) EXAMPLES: SHAPE OPTIMIZATION Results of MARS. Initial (dashed) and obtained (solid) configurations
Results (aerofoil, cont.): Results (aerofoil, cont.) EXAMPLES: SHAPE OPTIMIZATION Results of GA. Initial (dashed) and obtained (solid) configurations