design optimization school of engineering university of bradford 1 a discrete problem difficultiy in...
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1Design Optimization School of Engineering University of Bradford
A discrete problem
Difficultiy in the solution of a
discrete problem
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2Design Optimization School of Engineering University of Bradford
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
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3Design Optimization School of Engineering University of Bradford
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.
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4Design Optimization School of Engineering University of Bradford
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
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5Design Optimization School of Engineering University of Bradford
Genetic Algorithm
Evolutionary mechanism of the Genetic Algorithm
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6Design Optimization School of Engineering University of Bradford
Genetic Algorithm
A flowchart of a genetic algorithm
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7Design Optimization School of Engineering University of Bradford
Representation of a design by a binary string. Example.
Genetic Algorithm
Portal frame Chromosome of a design set using binary representation
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8Design Optimization School of Engineering University of Bradford
Genetic Algorithm - Encoded variables for UBs
Genetic Algorithm
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9Design Optimization School of Engineering University of Bradford
Genetic Algorithm - Single point crossover
Genetic Algorithm
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10Design Optimization School of Engineering University of Bradford
Genetic Algorithm - Arrangement of design variables
Genetic Algorithm
Five-bayfive-storey framework
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11Design Optimization School of Engineering University of Bradford
Genetic Algorithm - Solution for five-bay five-storey framework
Genetic Algorithm
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12Design Optimization School of Engineering University of Bradford
Genetic Algorithm
Genetic Algorithm - Five-bay five-storey framework (8 d.v.)
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13Design Optimization School of Engineering University of Bradford
Genetic Algorithm
Example.Three-bay by four-bay by four-storey structure
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14Design Optimization School of Engineering University of Bradford
Numerical optimization techniques
Genetic Algorithm - 3-bay by 4-bay by 4-storey structure
Genetic Algorithm
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15Design Optimization School of Engineering University of Bradford
Convergence history for 3-bay by 4-bay by 4-storey structure
Genetic Algorithm
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16Design Optimization School of Engineering University of Bradford
Optimization of front wing of J3 Jaguar Racing Formula 1 car
APPLICATION OF GENETIC ALGORITHM
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17Design Optimization School of Engineering University of Bradford
Optimization of front wing of J3 Jaguar Racing Formula 1 car
APPLICATION OF GENETIC ALGORITHM
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18Design Optimization School of Engineering University of Bradford
Genetic
Algorithm
APPLICATION OF GENETIC ALGORITHM
Front wing of J3 Jaguar Racing Formula 1 car
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19Design Optimization School of Engineering University of Bradford
Genetic
Algorithm
APPLICATION OF GENETIC ALGORITHM
Schematic layup of the composite structure of the
wing
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20Design Optimization School of Engineering University of Bradford
APPLICATION OF GENETIC ALGORITHM
GA convergence history
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Generations
Mas
s (K
g)
Optimization problem: minimize mass subject to displacement constraints (FIA and aerodynamics)Result of optimization:Design obtained by GA optimization: 4.95 KgBaseline design weight: 5.2 KgImprovement: 4.8%
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21Design Optimization School of Engineering University of Bradford
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)
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22Design Optimization School of Engineering University of Bradford
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. HoldenSowerby Research Centre, British Aerospace, UK
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23Design Optimization School of Engineering University of Bradford
Results (aerofoil, cont.)
EXAMPLES: SHAPE OPTIMIZATION
Results of MARS. Initial (dashed) and obtained (solid) configurations
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24Design Optimization School of Engineering University of Bradford
Results (aerofoil, cont.)
EXAMPLES: SHAPE OPTIMIZATION
Results of GA. Initial (dashed) and obtained (solid) configurations