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Slide 2 Genetic Algorithms Genetic algorithms imitate natural optimization process, natural selection in evolution. Developed by John Holland at the University of Michigan for machine learning in 1975. Similar algorithms developed in Europe in the 1970s under the name evolutionary strategies Main difference has been in the nature of the variables: Discrete vs. continuous Class is called evolutionary algorithms Will cover also differential evolution. Slide 3 Basic Scheme Coding: replace design variables with a continuous string of digits or genes Binary Integer Real Population: Create population of design points Selection: Select parents based on fitness Crossover: Create child designs Mutation: Mutate child designs Slide 4 Genetic operators Crossover: portions of strings of the two parents are exchanged Mutation: the value of one bit (gene) is changed at random Permutation: the order of a portion of the chromosome is reversed Addition/deletion: one gene is added to/removed from the chromosome Slide 5 Select parents Algorithm of standard GA Create initial population Calculate fitness 40 100 30 70 Create children Slide 6 Coding Integer variables are easily coded as they are or converted to binary digits Real variables require more care Key question is resolution or interval The number m of required digits found from Slide 7 Stacking sequence optimization For many practical problems angles limited to 0-deg, 45-deg, 90-deg. Ply thickness given by manufacturer Stacking sequence optimization a combinatorial problem Genetic algorithms effective and easy to implement, but do not deal well with constraints Slide 8 Coding - stacking sequence Natural coding: (0 0 =>1, 45 0 =>2, - 45 0 =>3, 90 0 =>4) (45/-45/90/0)s => (2/3/4/1) To satisfy balance condition, convenient to work with two-ply stacks (0 2 =>1, 45=>2, 90 2 =>3) or (45/-45/90 2 /0 2 ) s => (2/3/1) To allow variable thickness add empty stacks (2/3/1/E/E)=> (45/-45/90 2 /0 2 ) s Slide 9 Initial population Random number generator used Typical function call is rand(seed) Seed updated after call to avoid repeating the same number. See Matlab help on how to change seed (state). Need to transform random numbers to values of alleles. Slide 10 Fitness Augmented objective f*=f + pv-bm+sign(v). v = max violation m = min margin Repair may be more efficient than penalty Fitness is normalized objective or n s +1-rank What is the advantage of rank based objective? Slide 11 Roulette wheel selection Example fitnesses Slide 12 Single Point Crossover Parent designs [0 4 /45 2 /90 2 ] s and [45 4 /0 2 ] s Parent 1 [1/1/2/2/ 3] Parent 2 [2/2/2/2/ 1] One child [1/1/2/2/1] That is: [0 4 /45 2 /0 2 ] s Slide 13 Other kinds of crossover Multiple point crossover Hitchhiking problem Uniform crossover Random crossover for real numbers Multi-parent crossover Slide 14 Mutation and stack swap [1/1/2/2/3]=> [1/1/2/3/3] [0 4 /45 2 /90 2 ] s => [0 4 /45/90 4 ] s [1/1/2/2/3]=> [1/2/1/2/3] [0 4 /45 2 /90 2 ] s => [(0 2 /45) 2 /90 2 ] s Slide 15 Differential evolution (Wikipedia) Initialize m designs with n real number Repeat the following: Crossover: For each design x find three other random unique designs a,b,c to combine with y=y(x,a,b,c). For each design variable make a decision based on random number whether to leave alone or combine. Replacement: If y is better than x replace the x with y. Slide 16 Combination details Slide 17 Questions Global optimization balances exploration and exploitation. How is that reflected in genetic algorithms? What are all possible child designs of [0 2 /45/90] s and [45 2 /0] s that are balanced and symmetric? When we breed plants and animals we do not introduce randomness on purpose into the selection procedure. Why do we do that with GAs? Slide 18 Reliability Genetic algorithm is random search with random outcome. Reliability r can be estimated from multiple runs for similar problems with known solution Variance of reliability, r, from n runs Slide 19 Reliability curves Slide 20 Multi-material laminate Materials: one material = 1 lamina ( matrix or fiber materials) E.g.: glass-epoxy, graphite-epoxy, Kevlar-epoxy Use two materials in order to combine high efficiency (stiffness) and low cost Graphite-epoxy: very stiff but expensive; glass-epoxy: less stiff, less expensive Objective: use graphite-epoxy only where most efficient, use glass-epoxy for the remaining plies Slide 21 Multi-criterion optimization Two competing objective functions: WEIGHT and COST Design variables: number of plies ply orientations ply materials No single design minimizes weight and cost simultaneously: A design is Pareto-optimal if there is no design for which both Weight and Cost are lower Goal: construct the trade-off curve between weight and cost (set of Pareto-optimal designs) Slide 22 Material properties Graphite-epoxy Longitudinal modulus, E 1 : 20.01 10 6 psi Transverse modulus, E 2 : 1.30 10 6 psi Shear modulus, G 12 : 1.03 10 6 psi Poisson s ratio, 12 : 0.3 Ply thickness, t: 0.005 in Density, : 5.8 10 -2 lb/in 3 Ultimate shear strain, ult : 1.5 10 -2 Cost index: $8/lb Glass-epoxy Longitudinal modulus, E 1 : 6.30 10 6 psi Transverse modulus, E 2 : 1.29 10 6 psi Shear modulus, G 12 : 6.60 10 5 psi Poisson s ratio, 12 : 0.27 Ply thickness, t: 0.005 in Density, : 7.2 10 -2 lb/in 3 Ultimate shear strain, ult : 2.5 10 -2 Cost index: $1/lb Slide 23 Material properties Carbon-epoxyGlass-epoxy E 1 (psi)20.01 x 10 6 5.7 x 10 6 E 2 (psi)1.30 x 10 6 1.24 x 10 6 G 12 (psi)1.03 x 10 6 0.54 x 10 6 12 0.30.28 (lb/in 3 ) 0.0580.076 Cost (lb -1 )81-2 Thickness (in)0.005 1 lim 0.010.02 2 lim 0.010.02 12 lim 0.0150.025 Source: http://composite.about.com for the stiffnesses, Poisson's ratios and densitieshttp://composite.about.com Slide 24 Method for constructing the Pareto trade-off curve Simple method: weighting method. A composite function is constructed by combining the 2 objectives: W: weight C: cost : weighting parameter (0 1) A succession of optimizations with varying from 0 to 1 is solved. The set of optimum designs builds up the Pareto trade-off curve Slide 25 Multi-material Genetic Algorithm Two variables for each ply: Fiber orientation Material Each laminate is represented by 2 strings: Orientation string Material string Example: [45/0/30/0/90] is represented by: Orientation:45-0-30-0-90 Material: 2-2-1-2-1 GA maximizes fitness: Fitness = -F 1: graphite- epoxy 2: glass-epoxy Slide 26 Simple vibrating plate problem Minimize the weight (W) and cost (C) of a 36x30 rectangular laminated plate 19 possible ply angles from 0 to 90 in 5- degree step Constraints: Balanced laminate (for each + ply, there must be a - ply in the laminate) first natural frequency > 25 Hz Frequency calculated using Classical Lamination Theory Slide 27 How constraints are enforced GAs do not permit constrained optimization Balance constraint hard coded in the strings: stacks of are used Example: (45-0-30-25-90) represents [45/0/30/25/90] s Other constraints (frequency, maximum strain) are incorporated into the objective function by a penalty, which is proportional to the constraint violation >0: penalty parameter, g: constraint Slide 28 Pareto Trade-off curve ($) (lb) A (16.3,16.3) B (5.9,55.1) C point C 64% lighter than A; 17% more expensive 53% cheaper than B; 25% heavier Slide 29 Optimum laminates Cost minimization: [50 10 /0] s, cost = 16.33, weight = 16.33 Weight minimization: [50 5 /0] s, cost = 55.12, weight = 6.89 Intermediate design: [50 2 /50 5 ] s, cost = 27.82, weight = 10.28 Glass-epoxy in the core layers to increase the thickness Graphite-epoxy as outer plies for a high frequency Midplane Intermediate optimum laminates: sandwich-type laminates