A discrete problem
Difficultiy in the
solution of a
discrete
problem
Design Optimization
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MATHEMATICAL OPTIMIZATION PROBLEM
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?
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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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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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Genetic Algorithm
Evolutionary mechanism of the Genetic Algorithm
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Genetic Algorithm
A flowchart of a genetic algorithm
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Genetic Algorithm
Representation of a design by a binary string. Example.
Portal frame
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Chromosome of a design set
using binary representation
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Genetic Algorithm
Genetic Algorithm - Encoded variables for UBs
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Genetic Algorithm
Genetic Algorithm - Single point crossover
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Genetic Algorithm
Genetic Algorithm - Arrangement of design variables
Five-bay
five-storey
framework
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Genetic Algorithm
Genetic Algorithm - Solution for five-bay five-storey framework
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Genetic Algorithm
Genetic Algorithm - Five-bay five-storey framework (8 d.v.)
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Genetic Algorithm
Example.
Three-bay by
four-bay by
four-storey
structure
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Genetic Algorithm
Numerical optimization techniques
Genetic Algorithm - 3-bay by 4-bay by 4-storey structure
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Genetic Algorithm
Convergence history for 3-bay by 4-bay by 4-storey structure
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APPLICATION OF GENETIC ALGORITHM
Optimization
of front wing
of J3 Jaguar
Racing
Formula 1
car
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APPLICATION OF GENETIC ALGORITHM
Optimization of front
wing of J3 Jaguar
Racing Formula 1 car
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APPLICATION OF GENETIC ALGORITHM
Genetic
Algorithm
Front wing of J3
Jaguar Racing
Formula 1 car
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APPLICATION OF GENETIC ALGORITHM
Genetic
Algorithm
Schematic layup
of the composite
structure of the
wing
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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%
GA convergence history
5.9
5.8
Mass (Kg)
5.7
5.6
5.5
5.4
5.3
5.2
5.1
5
4.9
Generations
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EXAMPLES: SHAPE OPTIMIZATION
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).
W.A. Wright, C.M.E. Holden, Sowerby Research Centre,
British Aerospace (1998)
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EXAMPLES: SHAPE OPTIMIZATION
Problem definition (aerofoil, cont.)
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
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EXAMPLES: SHAPE OPTIMIZATION
Results (aerofoil, cont.)
Results of MARS. Initial (dashed) and obtained (solid) configurations
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EXAMPLES: SHAPE OPTIMIZATION
Results (aerofoil, cont.)
Results of GA. Initial (dashed) and obtained (solid) configurations
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