global optimization problems in various industrial fields

UP Baguio Seminar, 02/02/2007

1. Global optimization problems in various industrial fields1.1 Car shape optimization in automotive industry

(in collaboration with PSA Peugeot Citroën)

1.2 Optical fiber optimization in telecommunication industry

(in collaboration with Alcatel)

2. Existing optimization methods

3. Presentation and validation of a new hybrid method

4. Application to the previous industrial problems

5. Conclusions

Laurent Dumas Laboratoire Jacques-Louis Lions, UPMC, Paris, [email protected]

Hybrid optimization and application to various industrial problems


1.1.1 1.1.1 Car shapeCar shape optimization for fuel consumption reduction optimization for fuel consumption reduction

(joint work with PSA Peugeot Citroën)(joint work with PSA Peugeot Citroën)

Possibility of action to reduce consumption :Possibility of action to reduce consumption : - Motors evolution

- Weight reduction

- Car shape optimization

Aerodynamic drag

74%

Others

26%

Fuel consumption repartition for a car at 120 km/h:Fuel consumption repartition for a car at 120 km/h:


Fx

Fz

•Aerodynamic drag force:FFF P

S

SSS

P

dSvCfVF

dSPdSPdSCpVF

//

2

2

2

1

2

1

• Drag coefficient : Cx1

SCxVFx 2

2

1.1.2 Definition of the drag coefficient1.1.2 Definition of the drag coefficient


Lower exterior shape

CoolingWheels15%

Upper exterior shape

40%25 to 30 %

10 to 15%

Others5%

65%to 70 % of Cx dépends on the exterior shape

Aerodynamic flow at the rear of Peugeot 206 (DRIA)

1.1.3 Origins of the drag coefficient1.1.3 Origins of the drag coefficient


fastSolving

constrained problems

robust

Parametrization of the geometry and definition of

a cost function

1.1.4 General formulation of the optimization problem1.1.4 General formulation of the optimization problem

Détermination of N control parameters

Optimisation method of J : RN R


1.2.1 Optical fiber optimization for the construction of a mono/multi channel wavelength filter

> Such filters can be obtained by using an optical fiber called FBG (Fiber Bragg Grating) having a fast periodic modulation of its refractive index in the core:

> The index variation can be optimized in order to give the desired reflectivity spectrum: inverse problem

m

(reflectivity spectrum)

(joint work with Alcatel)(joint work with Alcatel)


1.2.2 Mathematical modelization of a FBG

• The refractive index of a FBG is expressed through a quasi-sinusoïdal function in the longitudinal direction z:

n(z)=n0+n(z) cos(2z/0) z [0, L]

with the following notations:

n0 : index refraction of the core

0: nominal period of the FBG

n(z): slowly varying amplitude (also called apodisation)

• The inverse-type optimization problem will consist in finding the ‘best’ apodisation function leading to the desired reflectivity spectrum.


1.2.3 Computation of the reflectivity spectrum of a FBG

•The reflectivity spectrum is a function R() =| r() |2 where

r() = bB(0,) / bF(0,)

• In the above expression, the enveloppes of the forward and backward propagating waves are obtained by the resolution of the following system of coupled ODE’s:

where , and


1.2.4 Examples of reflectivity spectra

1.5494 1.5496 1.5498 1.55 1.5502 1.5504 1.5506 1.5508

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

lambda

|r|2

reflectivité

1.5494 1.5496 1.5498 1.55 1.5502 1.5504 1.5506 1.5508

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

lambda

|r|2

reflectivité

1.5494 1.5496 1.5498 1.55 1.5502 1.5504 1.5506 1.5508

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

lambda

|r|2

reflectivité

1.5494 1.5496 1.5498 1.55 1.5502 1.5504 1.5506 1.5508

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

lambda

|r|2

reflectivité

0 0.005 0.011.45

1.4501

1.4502

1.4503

1.4504

z

dn(z

)

enveloppe de l'indice

0 0.005 0.011.4502

1.4502

1.4502

1.4503

1.4503

1.4503

1.4503

1.4503

1.4504

1.4504

1.4504

z

dn(z

)


FBG with Gaussian apodisation FBG with raised-cosine apodisation

FBG with weak constant apodisation ( n=1E-4) FBG with strong constant apodistion (n=4E-4)

•Four examples of reflectivity spectra are displayed below corresponding to four different FBG (L=20cm, n0=1.45, B=1550nm):


fastSolving

constrained problems

robust

Parametrization of the geometry and definition of

a cost function

1.2.5 General formulation of the optimization problem1.2.5 General formulation of the optimization problem

Détermination of N control parameters

Optimisation method of J : RN R

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-4

z

dn(z

)


dn(premiere generation)dn(derniere generation)


2. Existing optimization methods

• Numerous global or local optimization methods exist to find the minimum of a given function J : O RN R . Among them, the main two classes are the following:

Deterministic gradient-based methods (steepest descent, quasi Newton, etc…) where the gradient of the cost function is needed.

Stochastic methods (simulated annealing, genetic algorithms, evolution strategies, … ) seeking for a global optimum.

•Both methods will be applied and compared on the classical Rastrigin function with n parameters which exhibits many local minima but only one global:

Rastrigin function with 2 parameters

nx2cosxXFn

1i

i2iRastrigin


2.1 Principle of gradient-based methods

• Gradient based methods use the gradient of the cost function J in order to construct a sequence of points (xk), with decreasing values by J.

• At each iteration, once a descent direction is chosen, a linesearch principle is used to ensure a sufficient decrease in this direction.

• In a gradient-based method such as the steepest descent or the BFGS method, the main difficulty is to compute the gradient of the cost function,x J(x). In some

cases, the gradient is even not achievable and can only be approximated by finite differences.


Initialisation: random selection of a population of Np ‘individuals’ associated to different values of the parameters xO Rn.

Evaluation: to each individual is associated a ‘fitness’ value inversely proportional to the cost function J to minimize.

The population is evolving at each generation through three ‘Darwinian’ principles of selection, crossover and mutation (detailed below).

After Ng generations, the average and the best fitness value of individuals have improved.

2.2 Principle of genetic algorithms (GA)

Initialisation

Fitness evaluation of the population

Darwinian principles :Selection – Crossover- Mutation

Convergencetest

Ngen=Ngen+1

No


2.3 Examples of Darwinian principles

•Selection: the individuals are selected through a non-uniform random wheel:

•Crossover: (with probability pc): starting from two individuals x and y (, two new individuals are randomly generated from a barycentric combination of each component of x and y.

•Mutation: (with probability pm): starting from an individual x, a new individual can be randomly created with a normal law centered in x

• In general, a one-elitism strategy is included in order to keep the current best individual at the next generation.


Initialisation: same as in GA (Np individuals randomly chosen)

The Darwinian principles consist only of crossover and mutation.

The number of generated offspring is higher than Np. The next generation of Np individuals is then obtained after a deterministic selection among parents and offsprings (called plus selection) or among offsprings (comma selection).

2.4 Principle of evolution strategies (ES)

Initialisation

Darwinian principles :Crossover- Mutation

Convergencetest

Ngen=Ngen+1

No

Evaluation of the population

Deterministic selection


Convergence speed

Global minimization No regularity conditions on J Robustness Multi-objective Parallélisable

Local minimization Gradient evaluation Not multi-objective

Convergence speed

drawbacksadvantagesdrawbacksadvantages

Stochastic methods (GA, ES)Gradient-based methods

2.5 Comparison of the two types of methods


3.1 Description of hybrid methods

Example of convergence history

Cost function evaluation number

Co

st f

un

ctio

n

gradient

• Idea : couple a stochastic optimization method (GA or ES) with a deterministic one in order to improve its convergence speed

•Principle: application of a descent type method to one or a few well chosen individuals at well chosen moments

GA or ES


3.2 Description of the coupling principle

Final local search

END

BEGIN

G2L

Stopping criterion

Global search (AG) Local search (gradient)

Stopping criterion

L2G

• The coupling principle between the global search and the local search is done on an adaptative way with respect to two coefficients G2L and L2G.


O

O

O

• elements of the population O center of mass of clusters

3.3 Choice of the elements for local search

• A clustering method is used. It consists to divide the population into N regularly distributed sub-population. The local search process is then applied to every best element of each cluster.


3.4 Another way to improve GA: approximate evaluations

Initialisation

Exact evaluations

Darwinian principles :Selection – Crossover- Mutation

Convergencetest

Ngen=Ngen+1

No

Approximated evaluation of each

individual

Search for the best individuals

Exact evaluation for:- the ‘best’ individuals- a randomly chosen individual

Ngen>1


3.5 Validation of the hybrid method on the Rastrigin function with 3 parameters

•GA +gradient (coupling1): average gain of a factor 2 in time

• GA + gradient (coupling 2) : average gain of a factor 10 in time

• GA +approximate evaluation(RBF) : average gain of a factor 4 in time

(population of 30 individuals)


4.1 Industrial application 1: simplified car shape optimization (L.D, V. Herbert, F. Muyl, Computers and Fluids, 2004)

• Aim : minimization of the drag coefficient Cx of a simplified monospace shape with respect to three rear angles:

- back-light angle

- boat-tail angle

- ramp angle

• State of the art : Morel 1978, Ahmed et al 1985 experimental results on bluff body.

Han et al 1992 numerical and experimental results on the same bluff body.


4.2 Description of the global optimization procedure

Aerodynamic simulation

Algorithme hybride

3D mesh : 380000 cells

Navier Stokes with k - model

CFD code: FLUENT


4.3 Results

• GA +gradient: no significant improvement because of the lack of precision of the gradient evaluation

•GA+approximate evaluations (RBF): average gain of a factor 7 in time


4.4 Aerodynamic interpretation of the results

Cx_initial=0.21949

iso-surfaces of pressure

Cx_final=0.11725

(flow lines coloured by the longitudinal speed)


• The first treated example has been to find a FBG with the following characteristics:

R() > –3dB in a 0.3nm band

R() < –20dB outside a 0.4nm band

|Dispx) |< 50ps/nm in a 0.112nm

• The obtained results after optimization fulfill all the conditions as it can be seen:

optimized apodistion (blue) reflectivity spectrum (dB) dispersion

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-4

z

dn(z

)


dn(premiere generation)dn(derniere generation)

1.55 1.55 1.55 1.55 1.55 1.55 1.5501

x 10-6

-100

-50

0

50

lambda

Dp(

ps/n

m)

dispersion

Disp(premiere generation)Disp(derniere generation)

1.55 1.5501 1.5501 1.5501 1.5502 1.5503 1.5503

x 10-6

-30

-25

-20

-15

-10

-5

0

lambda

|r| (d

B)

reflectivité(decibels)

|r|2(premiere generation)|r|2(derniere generation)

4.5 Application 2: design of a monochannel FBG filter

(L.D, O. Durand, B. Ivorra, B. Mohammadi, IJCSE, 2006)


5. Conclusions

• Various global optimization methods have been developped, all consisting in a convergence acceleration of stochastic methods (GA or ES) by incorporating a new ‘intelligent’ mutation principle (namely a gradient-based method). The observed acceleration convergence speed ranges from a factor 2 to 10.

• Different industrial problems have been solved more efficiently with the help of these new methods, either direct (drag minimization, etc…) or inverse problems (filter design).

… presentation and Scilab scripts available next week at:

http://www.ann.jussieu.fr/~dumas/UP-Baguio.html

http://www.ann.jussieu.fr/~dumas/UP-Baguio.html

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