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 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

Select parents Algorithm of standard GA Create initial population Calculate fitness 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. (0 0 =>1, 45 0 =>2, =>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 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 2 n values. 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 n s -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 [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

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] [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

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 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

Reliability curves

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-off curve between weight and cost (set of Pareto-optimal designs)

Material properties Graphite-epoxy –Longitudinal modulus, E 1 : psi –Transverse modulus, E 2 : psi –Shear modulus, G 12 : psi –Poisson ’ s ratio, 12 : 0.3 –Ply thickness, t: in –Density,  : lb/in 3 –Ultimate shear strain,  ult : –Cost index: $8/lb Glass-epoxy –Longitudinal modulus, E 1 : psi –Transverse modulus, E 2 : psi –Shear modulus, G 12 : psi –Poisson ’ s ratio, 12 : 0.27 –Ply thickness, t: in –Density,  : lb/in 3 –Ultimate shear strain,  ult : –Cost index: $1/lb

Material properties Carbon-epoxyGlass-epoxy E 1 (psi)20.01 x x 10 6 E 2 (psi)1.30 x x 10 6 G 12 (psi)1.03 x x  (lb/in 3 ) Cost (lb -1 )81-2 Thickness (in)0.005  1 lim  2 lim  12 lim Source: for the stiffnesses, Poisson's ratios and densitieshttp://composite.about.com

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

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: –Material: GA maximizes fitness: Fitness = -F 1: graphite- epoxy 2: 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 used Example: ( ) 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 ($) (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

Optimum laminates Cost minimization: [±50 10 /0] s, cost = 16.33, weight = Weight minimization: [±50 5 /0] s, cost = 55.12, weight = 6.89 Intermediate design: [±50 2 /±50 5 ] s, cost = 27.82, weight = 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