S.Nagendra, D.Jestin, Z.Gurdal, R.T.Haftka and L.T.WatsonComputers & Structures, Vol. 58, No. 3, pp. 543-555, 1996.

Presented byVignesh Solai Rameshbabu

Introduction Simple programming technique which mimics natural genetics is known as

l hgenetic algorithm. It consists of reproduction, crossover, mutation, permutation, ply addition and

deletion Strengths: Laminate stacking sequence design Close to global minimumClose to global minimum

Weakness: High computational costs

Improved GA: Improved GA: Reduced computational costs Improve the reliability Lighter design – weight reduced by 4%

High Performance composites High tensile strength High tensile strength Impact resistant Thickness is less – Buckling criticalg Tuning of Flexural properties – Stacking sequence tailoring Stacking sequence optimization – GA Improved GA - tailoring the genetic operators and thereby improve the

reliability and reduce the computational cost. Composite Panels Layered composite laminate(skin) supported by stiffeners Composite Panels – Layered composite laminate(skin) supported by stiffeners

which is also made up of composite laminates(blade).

Problem Description•Length – 30 in ; Breadth – 32 inLength 30 in ; Breadth 32 in•Nx = 20,000 lb/in; Nxy = 5000 lb/in•Balanced and Symmetric –00 , ±450, 900

•Blade and Flange – Identical Laminates•Number of Identical Plies Adjacent ≤ 4 – to prevent matrix crackingprevent matrix cracking•No zero ply in skin laminate•Outer Plies - ±450 – To avoid damage due t i l dito compressive loading•Program used to run the genetic algorithm - PASCO

O F lOptimization FormulationFitness Function:

WhereWhere,W – Panel Weight

- - - Critical failure load factor.- Critical Buckling load factor = / ≥ 1 - Strength Failure load factor= / ≥ 1

Optimization Formulation (contd..)

q – Penalty parameterƐ – Bonus parameter

P Pl Pcont - Ply contiguity parameter =>where and are defined as the number of 00 or 900 stacks in excess to the constraint value in skin and blade laminates respthe constraint value in skin and blade laminates resp.

The objective function is penalized for designs that do not satisfy the failure constraints using penalty parameter.

A small fraction of the critical failure load factor is subtracted from the weight of the design when it satisfies the failure constraints using bonus parameter.

Genetic Code Design variables – Blade height , No. of plies, orientation Angle Geneticg g , p , Length of the string is fixed Upper bound for skin – 15; Upper bound for blade – 25;

Angle Geneticcode

Empty 0

0 1 Empty stacks - * ; Eg: 12-ply laminate- [*/*/*/902/±454/02] All zeros move towards outside when packed

L b d h d d f

0 1

±45 2

902 3 Lh – based on the desired precision of approximation Genetic code for Blade height includes 0s, 1s and 2s and the string is a nine digit

ternary numberternary number. Blade height is calculated from the design string as follows

G ti C d ( td )Genetic Code (contd..) For [Ylh

, . . . . . , Y2 , Y1] ;

where,X U b d 3 3 iXu – Upper bound = 3.3 inXl - Lower bound = 1.5 in

Substrings – Blade height(9), Skin laminates(15) and Blade laminates(25) Design string– concatenation of the substrings Design string Length – 15 + 25 + 9 = 49 No zero ply in skin laminateNo zero ply in skin laminate Outermost stack always has ±45 stack. Complete set of possible designs = 39 x (4-1)15-1 x 425-1

39 314 424= 39 x 314 x 424

= 2.65 x 1025

Implementation Generation of random initial population Translation of genetic string into PASCO

input Evaluation of mass, buckling loads,

strains and loads. Posting th lt Posting the results Evaluation of fitness, ranking Separation of elitist and creation of newSeparation of elitist and creation of new

population using various genetic operators.

Genetic Operators Processing of evaluated population to Basic Genetic Improved Genetic Processing of evaluated population to

create a new one. It combines most desirable characteristics

Algorithmp

Algorithm

Crossover Substring Crossover

of the older population It guarantees for the best design in the

Mutation Stack deletion, Stack Addition,

Orientationfinal population

Selection of two identical parents was not ll d i i d GA

Orientationmutation

Permutation Intralaminate and Interlaminateallowed in improved GA. Interlaminate

swap.

Crossover Trade the characteristics of their designs by exchanging parts of strings. Probability Pc is higher. (i.e) Pc ≥ 0.95.

Substring crossover:A li d t h b t i ti ki bl d d tiff h i ht Applied to each substring representing skin, blade and stiffener height separately.

Six point crossover instead of two point crossover.

M t tiMutation Small changes in children produced by crossover. Mutation Probability Pm is very low. (i.e) Pm ≤ 0.03.Mutation Probability Pm is very low. (i.e) Pm 0.03.

Before mutation - Lm 2 2 2 3 2 2 3 3 0 0 0 0 0 0 0After mutation - Lm 2 2 2 3 2 0 3 3 0 0 0 0 0 0 0

New Mutation: Stack deletion: (Pdel)

Stack closest to the mid plane is deleted and packed.Before stack deletion - Lm 2 3 2 2 3 3 2 2 2 2 0 0 0 0 0 After stack deletion - Lm 0 3 2 2 3 3 2 2 2 2 0 0 0 0 0After string packing - Lm 3 2 2 3 3 2 2 2 2 0 0 0 0 0 0

Stack addition:(Padd)Add d t th id l d ll th t k hift d ith t h i th d Added at the mid-plane and all other stacks are shifted without changing the order.Before stack deletion - Lm 2 3 2 2 3 3 2 2 2 2 0 0 0 0 0

After stack deletion - Lm 3 2 3 2 2 3 3 2 2 2 2 0 0 0 0 Orientation Mutation:(Pom)

Mutation occurs at random for orientation change.

PPermutation Inversion of the order of genes between two randomly assigned points.

Before permutation – Lm 2 2 2 1 3 1 2 2 0 0 0 0 0 After permutation – Lm 2 1 3 1 2 2 2 2 0 0 0 0 0

Intralaminate swap: (P l ) Intralaminate swap: (Pils) Swapping within the laminate, either in skin or in blade(never in both)

Before intralaminar swap – Lm 2 3 2 2 3 3 2 2 2 2 0 0 0 0 0 p m

After intralaminar swap – Lm 2 2 2 2 3 3 2 3 2 2 0 0 0 0 0

Interlaminate swap: (Pilsw) Exchanges stack between skin and blade laminates.

Before interlaminate swap After interlaminate swap

SKIN L 2 3 2 2 3 3 2 2 2 2 0 0 0 0 0 SKIN L 2 2 2 2 3 3 2 2 2 2 0 0 0 0 0SKIN – Lm 2 3 2 2 3 3 2 2 2 2 0 0 0 0 0BLADE – Lm2 2 1 1 3 3 2 3 3 2 2 1 1 2 2 0 0

0 0 0 0 0 0 0

SKIN – Lm 2 2 2 2 3 3 2 2 2 2 0 0 0 0 0BLADE – Lm2 2 1 1 3 3 2 3 3 3 2 1 1 2 2 0 0

0 0 0 0 0 0 0

P l DPopulation Diversity Diverse population adapts quickly to the changes and allows to search for the

d ti i hproductive niches. Avoid premature convergence by

increasing the design space.g g p Heterogeneity measure

where,Sij – No of genes that are different between designs i and j Sij No. of genes that are different between designs i and j nd - population sizeL’ – string length

Tendency to become homogeneous increases with the increase in the number of trials.

Al h P f d TAlgorithm Performance and Tuning

Probabilities of the gene operators (P P P )were tuned to reduce the Probabilities of the gene operators (Pc , Pm , Pp)were tuned to reduce the computational costs.

Using basic GA, by fixing Hb = 3.2121 in and blade laminate set to g y g b[±45/(±452/04)2/(±452/04)2/±45/02]s and varying only the skin laminate, the design space reduced from 2.65 x 1025 to approximately 4.78 x 106 and the mass got reduced from 25 5 lb to 22 68 lb but the result was not reliablegot reduced from 25.5 lb to 22.68 lb, but the result was not reliable.

Hence the probabilities of the new genetic operators( Pdel , Padd , Pils , Pilsw ) were tuned to obtain the optimal performance of the algorithm.

T f d GATuning of improved GA Step 1: Tuning Pils , Pilsw ,Pdel

Increasing Pdel causes decrease in reliability because of deletion of largenumber of plies which causesppremature convergence.

Step2: Tuning Hb.Th b t d i h d f 22 77 lb The best design had a mass of 22.77 lbfor Hb = 3.25 in. (1.18% increase)

Full problem tests: (based on 150 generations and population of 20)Method 1% Reliability 2% Reliability

Basic GA – crossover 10 30

Improved GA-crossover 35 63

Basic GA – Mutation 10 40

Improved GA – Mutation 35 63

C lConclusion

Weight was reduced from 25.5 lb to 22.68 lb.

It has given the near optimal design Reliability of the improved GA is higher.

References ” Design and Optimization of Laminated Composite Materials ” Gurdal Zafer Design and Optimization of Laminated Composite Materials. Gurdal, Zafer,

Raphael T. Haftka and Prabhat Hajela , New York: John Wiley & Sons, Inc., 1999. “ An Introduction to Genetic Algorithms” Melanie Michelle , A Bradford book. The g

MIT press. S.Nagendra, R.T.Haftka, Z.Gurdal and J,H,Starnes Jr, Design of a blade stiffened

it l ith h l C it St t 18 195 219(1991)composite panels with a hole. Composite Structures. 18, 195-219(1991)