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Optimization of Composite Structuresby Estimation of Distribution Algorithms

Paris Research Center, October 5, 2004Laurent Grossetgrosset@emse.fr

Advisors:

Raphael T. Haftka, Department of Mechanical and Aerospace En-gineering, University of Florida

Rodolphe Le Riche, Département Mécanique et Matériaux, ÉcoleNationale Supérieure des Mines de Saint-Étienne

Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 1/50

OutlineRésumé des travaux (en français)

Introduction to composite laminate optimization

Introduction to Estimation of Distribution Algorithms (EDA)

Introduction to Estimation of Distribution Algorithms (EDA)Principles

Introduction to Estimation of Distribution Algorithms (EDA)PrinciplesImportance of the statistical model

Application of a simple EDA to laminate optimization

Introduction of variable dependencies through auxiliary variables andthe Double-Distribution Optimization Algorithm

Conclusions

Résumé des Travaux

Problématique:Optimisation de stratifiés composites

Stratifié composite : empilement de couchesde renforts (fibres) noyées dans la résine(polymère)

Exemples : aéronautique (ex. Boeing 7E7),construction navale, équipements sportifs,. . .

La réponse dépend de l’orientation des fibres

But de l’optimisation = déterminer l’orientation optimale de toutes lescouches pour une application particulière :

maximiser la résistance d’un élément de structure d’un avion

maximiser la résistance d’un élément de structure d’un avionminimiser le poids

maximiser la résistance d’un élément de structure d’un avionminimiser le poidsdéterminer la séquence d’empilement de la coque d’une voiturede course de manière à minimiser les vibrations

État de l’art, objectifs de la thèse

Méthodes traditionnelles basées sur les gradients mais problèmescontinus uniquement et recherche locale

Optimisation stochastique pour globalité (algorithmes génétiques,années 90)

La thèse fait suite à des travaux du groupe de Prof. Haftka:

La thèse fait suite à des travaux du groupe de Prof. Haftka:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)

La thèse fait suite à des travaux du groupe de Prof. Haftka:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)B. Liu: optimisation globale de structures composites paralgorithmes multi-niveaux (2001)

Objectifs de la thèse :

Maturation des méthodes évolutionnaires depuis 20 ans (ES, EDA)

Transférer ces nouvelles méthodes à l’optimisation de composites :utilisation plus simpleméthodes plus efficaces

Algorithmes à estimation de distribution

optimum

But : trouver le point de plus élevéd’une “fonction coût” F sur undomaine D

optimum

Difficulté : on ne dispose que d’unbudget limité de N évaluations de lafonction F (on ne peut pas calculertoutes les combinaisons pour choisirla meilleure!)

optimum

On représente notre croyance quel’optimum est dans une région deD par une densité de probabilitép(x)

optimum

p(x) est utilisée pour créer denouveaux points

optimum

p(x) est utilisée pour créer denouveaux pointsles nouveaux points sont utiliséspour mettre à jour p(x)

Enjeux, améliorations proposées et résultats

Éléments critiques de l’algorithme

Adaptation aux problèmes de composites (alphabet non binaire,nombre de variables réduit, évaluations coûteuses)

Représentation de la distribution p(x) : mauvais choix de la forme dep(x) ➫ gaspillage d’évaluations dans des régions médiocres

Solutions proposées

Amélioration du modèle statistique par injection de connaissance surla structure du problème

Mécanismes de contrôle de l’exploration

Résultats

Les EDAs sont facilement applicables à l’optimisation de stratifiés

Résultats

Les EDAs sont facilement applicables à l’optimisation de stratifiés

La stratégie proposée conduit à une amélioration de l’efficacité surles problèmes testés

Introduction to Composite LaminateOptimization

Composite Laminates

Composite laminate: structure made of layers(plies) of fibrous material embedded in amatrix

Composite Laminates

Composite laminate: structure made of layers(plies) of fibrous material embedded in a matrix

The fibers (graphite, glass,. . . ) provide themechanical properties, the matrix (polymer)hold the fibers together

Composite Laminates

Composite laminate: structure made of layers(plies) of fibrous material embedded in a matrix

The fibers (graphite, glass,. . . ) provide themechanical properties, the matrix (polymer)hold the fibers together

Applications:aerospace industry (rudder, wing box,flying control surfaces, helicopterblades,. . . )sporting goods (skis, sailing, tennis)wind turbinesmotorsports and performance cars

Optimization of Composite Laminates

Each layer is orthotropic: its mechanical properties depend on thedirection

Overall response F (stiffness, strength, natural frequency, bucklingload,. . . ) of the whole laminate depends on the stacking sequence[θ1, θ2, . . . , θn]:

F = F (θ1, θ2, . . . , θn)

Goal of the optimization: maximize F

F = F (θ1, θ2, . . . , θn)

Difficulty:

F = F (θ1, θ2, . . . , θn)

Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problems

F = F (θ1, θ2, . . . , θn)

Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problemsF non-convex in general ➫ many local optima

F = F (θ1, θ2, . . . , θn)

Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problemsF non-convex in general ➫ many local optima

➫ Require specific optimization methods

Laminate Optimization Methods

Gradient-based continuous optimization using ply thicknesses asdesign variables (Schmit and Farshi, 1977) or a penalty approach toforce discreteness of the ply angles (Shin et al., 1990)

➫ can use well-established methods, but no guaranty to find the globaloptimum

Stochastic optimization: Genetic Algorithms (Hajela and Lin, 1992;Le Riche and Haftka, 1993)

➫ stochastic search + discrete variables: can solve the problem directly,but

can be difficult to use (many problem-dependent parameters totune),

can be difficult to use (many problem-dependent parameters totune),and computationally expensive

Purpose of this work

Our goal is to

1. investigate the application of a new class of algorithms called“Estimation of Distribution Algorithms” (EDAs) to the field of laminateoptimization;

Our goal is to

2. propose improvements to EDAs in the context of laminateoptimization:use physics-based knowledge to improve efficiency

Our goal is to

2. propose improvements to EDAs in the context of laminateoptimization:use physics-based knowledge to improve efficiency

3. study the general behavior of EDAs and propose improvements toEDAs that can be used for other fields.

Introduction to Estimation of DistributionAlgorithms

Recall: Genetic Algorithms (GA)

Inspiration: Darwin’s theory of Evolution (“survival of the fittest”)

A species can adapt to an environment becausethose individuals that possess features that give them acompetitive advantage over other individuals are more likelyto have offspring, and therefore to pass on theiradvantageous traits to the next generation.

Application to function optimization:Idea: let a population of candidate solutions evolve to adapt to a taskto perform

Environment → Function F (x) to maximize (“fitness”)

Environment → Function F (x) to maximize (“fitness”)Population of individuals → population of points xi, i = 1, . . . , λ

Natural selection → Selection based on F

Reproduction → recombination (crossover) and perturbation(mutation) operators

GA Mechanism

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Initial population: uniform

GA Mechanism

0 0.2 0.4 0.6 0.8 10

Points obtained byselection ➫ parents

GA Mechanism

0 0.2 0.4 0.6 0.8 10

Creation of new points byrecombination(e.g. weighted average inR

n) ➫ children

GA Mechanism

0 0.2 0.4 0.6 0.8 10

n) ➫ children

Resulting new population

GA Mechanism

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Implicit pdf of children

n) ➫ children

Resulting new population

fitness function F

+ selection procedure+ variation operators

implicitprobability distribution p(x)

over the design space

Formalization of GAs ➫ Estimation of Distri-bution Algorithms

Goal: control the way the search distribution p(x) is constructed sothat it learns information about “promising regions”

Principle: express explicitly p(x) and use it to create new pointsin high-fitness areas

Potential advantages:

Potential advantages:Better understanding of the algorithm (based on statisticalprinciples, not nature imitation)➫ improve efficiency

Potential advantages:Better understanding of the algorithm (based on statisticalprinciples, not nature imitation)➫ improve efficiencyPotentially fewer parameters (avoid many ad hoc operators)➫ algorithm easier to use

The Estimation of Distribution Algorithm(EDA)

Initialize P(x)

Create l points

by sampling from P(x)

Select m good points

based on the fitness F

Estimate the distribution

P(x) of the selected points

Initialize P(x)

Create l points

Initialize P(x)

Create l points

Initialize P(x)

Create l points

1. Use created points to inferstatistical information aboutgood regions ➫ p(x)

Initialize P(x)

Create l points

Initialize P(x)

Create l points

1. Use created points to inferstatistical information aboutgood regions ➫ p(x)

2. Use p(x) to create new pointsin promising regions

Estimation of Distribution Algorithm: Illustra-tion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Initial population: uniformp(x)

0 0.2 0.4 0.6 0.8 10

Points obtained byselection

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gaussian kernels

Estimation of thedistribution of good pointsp(x)(continuous case, kernels,p(x) =const

∑Ni=1 exp(− (x−xi)

2σ2 ))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2σ2 ))

Estimated distribution ofgood points p(x)

0 0.2 0.4 0.6 0.8 10

2σ2 ))

Estimated distribution ofgood points p(x)

New population obtainedby sampling from p(x)

Estimating the distribution of good points

Task: given a sample of µ selected points, infer distribution p(x) of allthe points of comparable fitness (“promising regions”)

Procedure:

Procedure:1. choose a model p̂(x; m1, . . . , mr),

Procedure:1. choose a model p̂(x; m1, . . . , mr),2. find the value of the model parameters mj that maximizes the

likelihood of the good observed points.

The choice of the model is a compromise between

The choice of the model is a compromise between1. the exploitation or accuracy of p, i.e. its ability to learn the

distribution of selected points, and

The choice of the model is a compromise between1. the exploitation or accuracy of p, i.e. its ability to learn the

distribution of selected points, and2. the generalization stability of p in unvisited regions, in particular

its ability to explore them.

Application of a Simple EDA to LaminateOptimization

Simple Model: Independent Variables

General principle in machine learning: in the absence of informationabout the distribution, use simple models (cf. “Occam’s razor”)

Simplest statistical model of selected points: independent variables:

p(x1, x2, . . . , xn) =

Only the marginal distributions of the variables appear in the model:discrete case = frequency of all possible values of each variable

p(x1, x2, . . . , xn) =

Proposed by S. Baluja (PBIL, 1994) and H. Mühlenbein (1996) in theUnivariate Marginal Distribution Algorithm (UMDA)

p(x1, x2, . . . , xn) =

Proposed by S. Baluja (PBIL, 1994) and H. Mühlenbein (1996) in theUnivariate Marginal Distribution Algorithm (UMDA)

Changes: non-binary alphabet, mutation to compensate forestimation error

Application to Laminates: Frequency problem

Constrained maximization of the first natural frequency of asimply-supported rectangular laminated plate:

maximize f1(θ1, . . . , θ15)

such that νl ≤ νeff ≤ νu

Poisson’s ratio = deformation observed in thetransverse direction when a unit deformation isapplied in the longitudinal direction

maximize f1(θ1, . . . , θ15)

θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}

maximize f1(θ1, . . . , θ15)

θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}

The constraints are enforced througha penalty approach

100−2000

−1000

maximize f1(θ1, . . . , θ15)

θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}

The constraints are enforced througha penalty approach

The constraints create a narrow ridgein the design space

100−2000

−1000

Results: reliability of the optimization

Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s

Compare UMDA to a GAand a hill-climbing algorithm(SHC)

Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s

UMDA and SHC: optimizedparameters, GA: same set-ting as UMDA

Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s

Reliability R: probability of finding the optimum ina given number of function evaluations. 0 2000 4000 6000 8000 10000

number of evaluations

SHCUMDAGA

Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s

Reliability R: probability of finding the optimum ina given number of function evaluations. 0 2000 4000 6000 8000 10000

SHCUMDAGA

➫ UMDA outperforms SHC because its distribution approach (global)allows it to handle narrow search spaces

➫ the performance of UMDA is substantially higher than that of GA forthis problem

Improvement of the Statistical Model ofSelected Points through Auxiliary Variables:

the Double-Distribution Optimization Algorithm

Limitations of simple models

Usual assumption: independent variables

p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)

Problem: does not work for problems with strong variable interactions

p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)

0 10 20 30 40 50 60 70 80 900

OPTIMUM

p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)

0 10 20 30 40 50 60 70 80 900

OPTIMUM

0 10 20 30 40 50 60 70 80 900

OPTIMUM

MAXIMUMPROBABILITY

p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)

0 10 20 30 40 50 60 70 80 900

OPTIMUM

0 10 20 30 40 50 60 70 80 900

OPTIMUM

MAXIMUMPROBABILITY

High-probability areas do not coincide with high-fitness regions

Representation of variable dependencies

Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)

X1 X2 X3

Disadvantages:

X1 X2 X3

Disadvantages:The number of parameters mj to estimate increases rapidly withthe model complexity

X1 X2 X3

Disadvantages:The number of parameters mj to estimate increases rapidly withthe model complexity

➫ the population size needed to estimate these parameters ensure(with a constant confidence) increases with the model complexitybecause flexible models do not generalize well the informationcontained in the sample to other regions

Representation of joint actions of the vari-ables via auxiliary variables

Observation : in many situations, a small number of high ordervariables V = (V1, V2, . . . , Vm) partially determine the objectivefunction

Examples :

Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,

Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,the geometry of a vehicle influences its aerodynamic behavior viathe drag coefficient CV ,

Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,the geometry of a vehicle influences its aerodynamic behavior viathe drag coefficient CV ,the locations of the holes/particles in a porous media affect theflow through the permeability.

These meaningful quantities V reflect joint actions of the variables x

and can be used as auxiliary variables to capture such interactions

Algorithm Principle

The distribution p(x) is represented by a simple model (UMDA)

A simple distribution f(V) is used to introduce variable dependencies

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)

Two simple models ⇒ Complex model

Algorithm Principle

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)

Required attributes of theV ’s:

Algorithm Principle

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)

Required attributes of theV ’s:1. m < n

Algorithm Principle

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)

2. m does not grow with n

Algorithm Principle

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)

2. m does not grow with n

3. inexpensive tocompute

The Double-Distribution Optimization Algo-rithm

Initialize P(xi) and f(V)

(Apply mutation)

based on the fitness

Estimate the distributions

P(xi) and f(V) of the selected points

Create l target points

by sampling from f(V)

Create n candidate points

by sampling from P(xi)

Keep l candidate points

closest to the targets points

Initialize P(xi) and f(V)

(Apply mutation)

based on the fitness

Estimate the distributions

P(xi) and f(V) of the selected points

Keep l candidate points

closest to the targets points

Goal: create points whose distribution that reflects both p(x) and f(V):

p(xi) provides a pool of points that have correct marginaldistributions.

f(V) is used as a filter that favors promising regions.

The relative influence of the 2 distributions is adjusted through ν/λ

Application to Laminate Optimization Problems

Case of composite laminates:Auxiliary variables = “Lamination Parameters”

V ≡ Lamination parameters = geometric contribution of the plies tothe stiffness.

E.g. in-plane problem:

U1 U2 U3

U1 −U2 U3

U5 0 −U3

U4 0 −U3

V ∗1

V ∗3

h: total laminate thickness, Ui’s: material invariants

E.g. in-plane problem:

U1 U2 U3

U1 −U2 U3

U5 0 −U3

U4 0 −U3

V ∗1

V ∗3

h: total laminate thickness, Ui’s: material invariants

Symmetric balanced laminates [±θ1,±θ2, . . . ,±θn]s:

V ∗{1,3} =

∫ h/2

{cos 2θ, cos 4θ}dz =1

{cos 2θk, cos 4θk}

Representation of the probability distributions

Distribution in the θ-domain

x ≡ θ

Discrete univariate model➫ estimate marginal frequen-cies

x ≡ θ

Distribution in the V-domain

x ≡ θ

Continuous variables

x ≡ θ

Continuous variables

Use kernel density estimate:

f(V) =1

K (V − Vi)

In this work, we usedGaussian kernels:

K(u) =1

(2π)d/2σdexp

Estimated Distributions: UMDA and DDOAUMDA: probability concentrated

around (55, 55)Fitness function and selected points

0 10 20 30 40 50 60 70 80 900

OPTIMUM

around (55, 55)

DDOA: the mass is distributedalong the “tunnel”

0 10 20 30 40 50 60 70 80 900

around (55, 55)

DDOA: the mass is distributedalong the “tunnel”

0 10 20 30 40 50 60 70 80 900

θ 2Sampling from p(θ1), . . . , p(θn) leads to an inaccurate distribution

V-based selection improves the distribution by introducing variabledependencies

Application I: Design of “zero-CTE” laminates

Objective: minimize the longitudinal coefficient of thermal expansion(CTE) subject to a constraint on the first natural vibration frequency,for a 50 in × 15 in plate

minimize |αx|

such that f1 ≥ fmin

minimize |αx|

Constraint enforced through a penalty approach

minimize |αx|

Possible values of the angles: θk ∈ {0◦, 22.5◦, 45◦, 67.5◦, 90◦}

minimize |αx|

Case n = 12: optimum = [904/ ± 67.5/06/ ± 22.56]s

minimize |αx|

Case n = 12: optimum = [904/ ± 67.5/06/ ± 22.56]s

Particularity: response is a function of the lamination parameters only➫ V ’s provide reliable information about the optimum

Comparison with UMDA and GA

0 500 1000 15000

GAUMDADDOA

0 500 1000 15000

GAUMDADDOA

GA and UMDA’s progress falls off after 600 analyses

0 500 1000 15000

GAUMDADDOA

DDOA reaches high a reliability (90%)

0 500 1000 15000

GAUMDADDOA

DDOA reaches high a reliability (90%)

➫ For this problem, DDOA benefits from the use of auxiliary variables(incorporation of physics-based information improves accuracy ofp(x))

Application II: Strength Maximization

Strength maximization problem:

maximize λs =n

mink=1

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

maximize λs =n

mink=1

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

Characteristics of this problem:

maximize λs =n

mink=1

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

Characteristics of this problem:the V ’s do not capture allthe response:λs = λs(V1, V3, θ1, . . . , θn)

maximize λs =n

mink=1

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

Characteristics of this problem:the V ’s do not capture allthe response:λs = λs(V1, V3, θ1, . . . , θn)

many local optima 0

Case n = 12