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. • Applied to laminate design beginning in the 1990s.
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
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
40 100 30 70 Create initialpopulation Calculatefitness Select parents Algorithm of standard GA Create children
Coding integer variables • Done directly or via binary representation. • For example, in fun optimization problem we had four jobs, with design variables ranging in • The optimum choice of tasks could be coded as [2/0/50/8] or as [000010,00000,110010,01000] (but with no commas).
Coding Real variables • Real variables require more care • Key question is resolution or interval • The number m of required digits found from • If every x varied in [0,1], what are the increments?
Differential evolution (Wikipedia) • Initialize all designs in n dimensional space with random positions. • Repeat the following: • Crossover: For each design, • find three other random unique designs to combine with. • For each design variable make a decision based on random number whether to leave alone or combine. • Replacement: If new design is better than old design replace the old with the new.
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
Coding - stacking sequence • Binary coding common. Natural coding works better. (00=>1, 450=>2, - 450=>3, 900=>4) (45/-45/90/0)s => (2/3/4/1) • To satisfy balance condition, convenient to work with two-ply stacks (02=>1, 45=>2, 902=>3) or (45/-45/902/02)s => (2/3/1) • To allow variable thickness add empty stacks (2/3/1/E/E)=> (45/-45/902/02)s • Permutation coding may be used when number of plies is given
Coding - dimensions • Binary coding most common. Real number coding possible but requires special treatement. Trinary coding used in one example. • Genetic algorithm not effective for getting high precision. It is better to go for coarse grid of real values. With n binary digits get 2nvalues. • Segregate stacking sequence and geometry chromosomes.
Initial population • Random number generator used • Typical function call is rand(seed) • In Matlab rand(n) generates nxn matrix of uniformly distributed (0,1) random numbers • 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.
Fitness • When defining fitness we want • Low objective function (e.g. mass) • Penalty for violating constraints. • Bonus for margin in constraint satisfaction. • 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 ns-1-rank
Selection • Roulette wheel and tournament based selection • In tournament selection • Select randomly two designs • Take the better one as first parent • Repeat to select second parent • With roulette wheel bias towards better designs is implemented by larger portion of roulette wheel. • No twin rule.
Roulette wheel • Example fitnesses[0.62,0.60,0.65,0.61,0.57,0.64]
Single Point Crossover • Parent designs [04/±452/902]s and [±454/02]s • Parent 1 [1/1/2/2/3] • Parent 2 [2/2/2/2/1] • One child [1/1/2/2/1] • That is: [04/±452/02]s
Other kinds of crossover • Multiple point crossover • Uniform crossover and hitchhiking problem • Bell-curve crossover for real numbers • Multi-parent crossover • Complex crossovers for permutation coding
Mutation and stack swap • [1/1/2/2/3]=> [1/1/2/3/3] • [04/±452/902]s =>[04/±45/904]s • [1/1/2/2/3]=> [1/2/1/2/3] • [04/±452/902]s =>[(02/±45)2/902]s
Questions • Global optimization balances exploration and exploitation. How is that reflected in genetic algorithms? • What are all possible child designs of [02/±45/90]s and [±452/0]s that are balanced and symmetric? • When we breed plants and animals we do not introduce on purpose into the selection procedure. Why do we do that with GAs?
Reliability • Genetic algorithm is random search with random outcome. • Reliability is defined as the fraction of runs that arrived at the global optimum with some predetermined tolerance. • It can be estimated from multiple runs for similar problems with known solutions. • Variance of reliability, r, from n runs
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
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-offcurve between weight and cost (set of Pareto-optimal designs)
Graphite-epoxy Longitudinal modulus, E1: 20.01 106 psi Transverse modulus, E2: 1.30 106 psi Shear modulus, G12: 1.03 106 psi Poisson’s ratio, 12: 0.3 Ply thickness, t: 0.005 in Density, : 5.8 10-2 lb/in3 Ultimate shear strain, ult: 1.5 10-2 Cost index: $8/lb Glass-epoxy Longitudinal modulus, E1: 6.30 106 psi Transverse modulus, E2: 1.29 106 psi Shear modulus, G12: 6.60 105 psi Poisson’s ratio, 12: 0.27 Ply thickness, t: 0.005 in Density, : 7.2 10-2 lb/in3 Ultimate shear strain, ult: 2.5 10-2 Cost index: $1/lb Material properties
Material properties Source: http://composite.about.com for the stiffnesses, Poisson's ratios and densities
Method for constructing the Pareto trade-off curve • Simple method: weighting method. A composite function is constructed by combining the 2 objectives:W: weight C: cost: weighting parameter (01) • A succession of optimizations with varying from 0 to 1 is solved. The set of optimum designs builds up the Pareto trade-off curve
Multi-material Genetic Algorithm • Two variables for each ply: • Fiber orientation • Material • Each laminate is represented by 2 strings: • Orientation string • Material string • Example:[45/0/30/0/90] is represented by: • Orientation: 45-0-30-0-90 • Material: 2-2-1-2-1 • GA maximizes fitness: Fitness = -F 1: graphite-epoxy2: glass-epoxy
Simple vibrating plate problem • Minimize the weight (W) and cost (C) of a 36”x30” rectangular laminated plate • 19 possible ply angles from 0 to 90 in 5-degree step • Constraints: • Balanced laminate (for each + ply, there must be a - ply in the laminate) • first natural frequency > 25 Hz Frequency calculated using Classical Lamination Theory
How constraints are enforced • GAs do not permit constrained optimization • Balance constraint hard coded in the strings: stacks of ± are usedExample: (45-0-30-25-90) represents [±45/0/±30/±25/90]s • Other constraints (frequency, maximum strain…) are incorporated into the objective function by a penalty, which is proportional to the constraint violation >0: penalty parameter, g: constraint
Pareto Trade-off curve A (16.3,16.3) point C • 64% lighter than A; 17% more expensive • 53% cheaper than B; 25% heavier (lb) C B (5.9,55.1) ($)
Optimum laminates Intermediate optimum laminates:sandwich-type laminates • Cost minimization: [±5010/0]s, cost = 16.33, weight = 16.33 • Weight minimization: [±505/0]s, cost = 55.12, weight = 6.89 • Intermediate design: [±502/±505]s,cost = 27.82, weight = 10.28 Graphite-epoxy as outer pliesfor a high frequency Midplane Glass-epoxy in the core layersto increase the thickness