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Global optimization problems in various industrial fields

Hybrid optimization and application to various industrial problems. Laurent Dumas Laboratoire Jacques-Louis Lions, UPMC, Paris, dumas@ann.jussieu.fr. Global optimization problems in various industrial fields 1.1 Car shape optimization in automotive industry

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Global optimization problems in various industrial fields

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  1. Hybrid optimization and application to various industrial problems Laurent DumasLaboratoire Jacques-Louis Lions, UPMC, Paris, dumas@ann.jussieu.fr • Global optimization problems in various industrial fields • 1.1 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 UP Baguio Seminar, 02/02/2007

  2. Possibility of action to reduce consumption : Fuel consumption repartition for a car at 120 km/h: - Motors evolution - Weight reduction - Car shape optimization 1.1.1 Car shape optimization for fuel consumption reduction (joint work with PSA Peugeot Citroën) Others 26% Aerodynamic drag 74% UP Baguio Seminar, 02/02/2007

  3. Fz • Aerodynamic drag force: 1 = r 2 Fx V SCx 2 Fx 1.1.2 Definition of the drag coefficient r 1 òò òò òò = r n = n - n F V Cp dS P dS P dS P 2 ^ ^ ¥ ^ 2 S S S r 1 r òò = r F V Cf v dS n 2 // 2 S • Drag coefficient : Cx UP Baguio Seminar, 02/02/2007

  4. Others Wheels Cooling 5% 15% 10 to 15% Lower exterior shape Upper exterior shape 25 to 30 % 40% Aerodynamic flow at the rear of Peugeot 206 (DRIA) 65%to 70 % of Cx dépends on the exterior shape 1.1.3 Origins of the drag coefficient UP Baguio Seminar, 02/02/2007

  5.   1.1.4 General formulation of the optimization problem Parametrization of the geometryand definition of a cost function Optimisation method of J : RN R fast Solving constrained problems Détermination of N control parameters robust UP Baguio Seminar, 02/02/2007

  6. L = 0.5 mm 1.2.1 Optical fiber optimization for the construction of a mono/multi channel wavelength filter (joint work with Alcatel) • 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 (reflectivity spectrum) UP Baguio Seminar, 02/02/2007

  7. 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+dn(z) cos(2pz/L0) z [0, L] • with the following notations: • n0 : index refraction of the core • L0:nominal period of the FBG • dn(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. UP Baguio Seminar, 02/02/2007

  8. 1.2.3 Computation of the reflectivity spectrum of a FBG • The reflectivity spectrum is a function l R(l) =| r(l) |2 where • r(l) = bB(0,l) / bF(0,l) • 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 UP Baguio Seminar, 02/02/2007

  9. 1.2.4 Examples of reflectivity spectra • Four examples of reflectivity spectra are displayed below corresponding to four different FBG (L=20cm, n0=1.45, lB=1550nm): FBG with weak constant apodisation ( dn=1E-4)FBG with strong constant apodistion (dn=4E-4) • FBG with Gaussian apodisation FBG with raised-cosine apodisation UP Baguio Seminar, 02/02/2007

  10. 1.2.5 General formulation of the optimization problem Parametrization of the geometryand definition of a cost function Optimisation method of J : RN R fast Solving constrained problems robust Détermination of N control parameters UP Baguio Seminar, 02/02/2007

  11. Rastrigin function with 2 parameters 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: UP Baguio Seminar, 02/02/2007

  12. 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,Ñx J(x). In some cases, the gradient is even not achievable and can only be approximated by finite differences. UP Baguio Seminar, 02/02/2007

  13. 2.2 Principle of genetic algorithms (GA) Initialisation Initialisation: random selection of a population of Np ‘individuals’associated to different values of the parametersxÎO Ì 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 Nggenerations, the average and the best fitness value of individuals have improved. Fitness evaluation of the population Darwinian principles : Selection – Crossover- Mutation Convergence test Ngen=Ngen+1 No UP Baguio Seminar, 02/02/2007

  14. 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. UP Baguio Seminar, 02/02/2007

  15. 2.4 Principle of evolution strategies (ES) Initialisation 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). Darwinian principles : Crossover- Mutation Evaluation of the population Deterministic selection Convergence test Ngen=Ngen+1 No UP Baguio Seminar, 02/02/2007

  16. 2.5 Comparison of the two types of methods Gradient-based methods Stochastic methods (GA, ES) advantages drawbacks advantages drawbacks • Convergence speed • Global minimization • No regularity conditions onJ • Robustness • Multi-objective • Parallélisable • Convergence speed • Local minimization • Gradient evaluation • Not multi-objective UP Baguio Seminar, 02/02/2007

  17. 3.1 Description of hybrid methods • 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 gradient GA or ES Cost function Cost function evaluation number Example of convergence history UP Baguio Seminar, 02/02/2007

  18. 3.2 Description of the coupling principle • 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. BEGIN G2L Global search (AG) Local search (gradient) L2G Stopping criterion Stopping criterion Final local search END UP Baguio Seminar, 02/02/2007

  19. 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. • elements of the population O center of mass of clusters UP Baguio Seminar, 02/02/2007

  20. 3.4 Another way to improve GA: approximate evaluations Initialisation Approximated evaluation of each individual Ngen>1 Exact evaluations Search for the best individuals Darwinian principles : Selection – Crossover- Mutation • Exact evaluation for: • the ‘best’ individuals • a randomly chosen individual Convergence test Ngen=Ngen+1 No UP Baguio Seminar, 02/02/2007

  21. 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) UP Baguio Seminar, 02/02/2007

  22.   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. UP Baguio Seminar, 02/02/2007

  23. Algorithme hybride Aerodynamic simulation 4.2 Description of the global optimization procedure 3D mesh : 380000 cells Navier Stokes with k - model CFD code: FLUENT UP Baguio Seminar, 02/02/2007

  24. 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 UP Baguio Seminar, 02/02/2007

  25. Cx_initial=0.21949 Cx_final=0.11725 (flow lines coloured by the longitudinal speed) iso-surfaces of pressure 4.4 Aerodynamic interpretation of the results UP Baguio Seminar, 02/02/2007

  26. 4.5 Application 2: design of a monochannel FBG filter (L.D, O. Durand, B. Ivorra, B. Mohammadi, IJCSE, 2006) • The first treated example has been to find a FBG with the following characteristics: • R(l) > –3dB in a 0.3nm band • R(l) < –20dB outside a 0.4nm band • |Dispxl) |< 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 UP Baguio Seminar, 02/02/2007

  27. 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 UP Baguio Seminar, 02/02/2007

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