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A discrete problem

Difficultiy in the solution of a

discrete problem

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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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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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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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Genetic Algorithm - Encoded variables for UBs

Genetic Algorithm

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Genetic Algorithm - Single point crossover

Genetic Algorithm

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Genetic Algorithm - Arrangement of design variables

Genetic Algorithm

Five-bayfive-storey framework

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Genetic Algorithm - Solution for five-bay five-storey framework

Genetic Algorithm

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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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Numerical optimization techniques

Genetic Algorithm - 3-bay by 4-bay by 4-storey structure

Genetic Algorithm

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Convergence history for 3-bay by 4-bay by 4-storey structure

Genetic Algorithm

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Optimization of front wing of J3 Jaguar Racing Formula 1 car

APPLICATION OF GENETIC ALGORITHM

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Optimization of front wing of J3 Jaguar Racing Formula 1 car

APPLICATION OF GENETIC ALGORITHM

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Genetic

Algorithm

APPLICATION OF GENETIC ALGORITHM

Front wing of J3 Jaguar Racing Formula 1 car

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Genetic

Algorithm

APPLICATION OF GENETIC ALGORITHM

Schematic layup of the composite structure of the

wing

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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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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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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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Results (aerofoil, cont.)

EXAMPLES: SHAPE OPTIMIZATION

Results of MARS. Initial (dashed) and obtained (solid) configurations

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Results (aerofoil, cont.)

EXAMPLES: SHAPE OPTIMIZATION

Results of GA. Initial (dashed) and obtained (solid) configurations