Modeling and optimization of the composite plates laminated by genetic algorithm
Safer /M
LPCCMCBM (physics laboratory of the Behavior and characterizations of composite materials for the biomedical one.)
USTOMB Oran, Algeria
E-mail :mal_rel85@yahoo.com
Belkheir /F
LPCCMCBM (physics laboratory of the Behavior and characterizations of composite materials for the biomedical one.)
USTOMB Oran, Algeria
Abstract
Our study made it possible to develop and test a program of optimization based on a technique of optimization based on the use of a genetic algorithm. This program moreover was applied to the resolution of several types of problems connected to the optimal design of laminated structures. Among these problems, orientations of the folds, the material and the number of folds as variable which were already solved . In this work, we have presented the principles of the genetic algorithm functioning , its application to the laminated composites and to the original technique of coding which allows giving an account of the industrial constraints of composites’ development.
The flexibility of the general structure of our software allows treatening the various types of problems like: calculation of the module of rigidity as well as the factors of buckling and rupture.
We develop the OpStrAG software to optimize the characteristics of rigidity of the laminates and also their constraints towards rupture and their deformations under a given loading.
Keywords— Composite materials ; laminated Plates; Modeling and optimizatio; genetic Algorithm.
I. INTRODUCTION
The use of composite materials is very widespread in various fields [1] such the aerospace one, aeronautics, automotive industry, nautical just as in sporting industry. These materials are snuffed because of their large rigidity compared to the weight and with resistance with the weight [2], in the same way for some their mechanical properties like the corrosion resistance. Moreover, the use of composite materials can be advantageous compared to traditional materials, because it allows the structural design the total properties which meet better the special needs of a situation. [3], [4]
The goal of our work is to conceive a reliable and practical software being able to be used for the varied resolution of a range of problems. The reliability of the software is given according to its capacity to obtain optimal solutions as often as possible.
Its practical character, is determined by its capacity to propose a family of quasi optimal designs rather than only one solution [5]. The idea here is to make it possible to the originator to choose the solution which it best suits him among a set of quasi optimal solutions. The approach used is to test the
software developed on a series of varied problems applied to the optimization of composite structures. The method of optimization recommended in this work is based on the genetic algorithms. we selected Borland C++ Builder6 like a programming language.
II. THEORETICAL AND EXPERIMENTAL APPROACH II.1 THEORETICAL REMINDERS
II.1.1 COMPOSITE MATERIAL
The key concept of the composites is the assembly of at least two nonmiscible different basic materials. we obtain a heterogeneous material [6]. A material alone does not have a good quality only in case it is linked with other materials that have excellent properties (united we stand, divided we fall).
The matrix and the fibres are components of composite material. [7].
Figu 1:The shape of composite materials.
Our system is a rectangular laminated plate presumedly balanced, symmetrical.
It supports on its four sides with biaxial loads of compression Nx and Ny applied to the average plan.
This plate laminated is made up of a number of folds Np, each fold is made of two homogenized materials and which are determined by the user with the precision his thickness “H” and orientation of fibres
“θ”.
Figu 2:The geometry of laminates used
II.1.2 STUDY OF A STANDARD GENETIC ALGORITHM
Each genetic algorithm is determined by its basic cycle, which is represented by the diagram follow [8]:
Figu 3 :Principal stages of the basic genetic algorithm.
II.2 EXPERIMENTAL APPROACHES
II.2.1 GENETIC CODING OF LAMINATED INDIVIDUAL Each individual represents a particular architecture of a laminate, and with the image of the world of the genetics, these characteristics are coded in the form of chromosomes. A specific constitution, named phenotype, corresponds to an arrangement of genes, or degrees of freedom. Each individual has a number of genes twice equal to the number of layers of the laminate, one gene corresponds to material and a second corresponds to the orientation of fibres in each neck. The gene regrouping connected to materials is called material chromosome and similarly, the whole of genes of angles is called chromosome of orientations of fibres. Each of them contains a number of genes .
Figu 4 :Structure génétique d’un individu stratifié
II.2.2 MECHANISMS OFAG EVALUATION OF THE INDIVIDUALS
Maximize :
Under the constraints : ,
F(x):performance value ; f(x): objective function;gi(x):constraints; λi: factors of penalization.
Maximize Under the constraints:
Ex:Young modulus;Gxy :modulus of shearing; Vxy :poisson’s coefficient
Objective Constraints
14.54 12.00 0.50
Tab1:Optimal solution.
INDIVIDUALS’ RANKING
After the evaluation stage, we classify the individuals in order waning according to the value of their function of evaluationF(x).
- INDIVIDUALS’ SELECTION - ELITIST SELECTION
It is the selection of N better individuals after having sorted them in a decreasing way according to the function of the
individuals’ adaptation.
- UNIVERSAL STOCHASTIC SELECTION(SUS)
This operation is complementary to the ranking of the individuals; each individual occupies a segment Sj
with - INDIVIDUALCROSSING
Un individu du stratifié
materials
matk={ 1, 2, 3, ..., K , np-1}
k = 0…Nmat-1
m a t 1
m a t 2
… θ
k Orientation of the folds
θk={0, 1, 2, 3, ..., K , np-1}
k=1…Nplis-1
θ
1
θ
2
θ
3
…θ
k k Chromosome of
orientation Material
chromosome
thee code génétique
Lamiated 1
Lamiated K
Lamiated N
Lamiated K
Matérial
… mat2 mat1
Angle
90°
±45°
0°
The following interface in general presents all the possible procedures and treatments to carry out an example of optimization of a laminate.
II.2.3 EXAMPLE TO DETERMINE THE MODE OF THE RUPTURE We take an initial laminated plate of symmetrical folds, each one of these folds is an Carbon-Epoxy composite and the properties of this material are given to the table below
Propreties Carbon-epoxy
Longitudinal module, E1
Transverse module, E2
Shearing modulus in the plan, G12
Poisson's Coefficient in the plan, V12
Maximum principal deformation Maximum transverse deformation Maximum shearing
Folds thickness, h
127.59 GPa 13.03 GPa 6.41 GPa 0.300.008 0.029 0.015 0.127 mm Tab2: Properties of Carbon-Epoxy.
•
Folds number is: 5 folds,•
The length “a” and the width “B” of the plate is respectively : 2m and 1,5m,•
The loads Nx, Ny are respectively (-175 and -43,75 N/m),L The possible folds orientation is limited to the four values distinct from 0, +45, -45 and 90 degrees.
Figu 5: the histogram of the initial Population of the rupture mode.
The Figure above represents the histogram rupture factors by deformation and the critical invoices of bucklings of the Five individuals of the initial population who are classified minimum towards the maximum, it is noticed here that the factor of buckling is lower than the ruptures’ factors.
Figu 6:Factor of rupture and buckling of the minimal individual of final population.
This histogram (Figu 6) represents the rupture factor by deformation and the critical buckling factor of the individual optimal after the generation of the final population. There is the buckling factor lower than the rupture factor , therefore the mode of the rupture in this example is
III . CONCLUSION
To solve this problem, we used the genetic algorithms. The latter proved its effectiveness. It converges quickly towards a set of good solutions It is possible more effectively to improve it by combining it with a deterministic algorithm. That could make it possible to find one more a large number of good solutions among the final population. Also, as it is not necessary to twice evaluate the same individual, it could be interesting 'to study the possibility of establishing a control mechanism on the individuals already evaluated in order to avoid the useless repetition of the individual evaluation. That would make it possible to decrease the number of times that one calls on the function of evaluation (optimization of computing time). However, this way of making can take a storage capacity and an increased time computing. .This is why a methodology considered could be to store a fixed number of results, a hundred for example. It would then be necessary to establish a criterion which would make it possible to determine when one adds a result in the database and which must be removed once the limit of storage reached.
IV.REFERENCES
[1] BooK «MATERIAUX COMPOSITES, Comportement mécanique et analyse des structures ». J-M. BERTHELOT.
[1] Book “MATERIALS COMPOSITE, mechanical Behavior and analyzes structures”. J-M. BERTHELOT.
[2]«The 02/12/2003 »Etude théorique et expérimentale du comportement mécanique en statique et en fatigue des matériaux composites stratifiés et sandwiches en flexion 3- points ; BEZAZI Abderrezak
[3] «2009 »Optimisation de plaques stratifiées en représentation polaire A. Jibawy1,2,4, C. Julien1,2,4, B.
Desmorat1,2,3, A. Vincenti1,2, F. Lene1,2 ( *1 UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert 4 place Jussieu, Case 161, F-75252 CEDEX Paris France
ali.jibawy@etu.upmc.fr, cedric.julien@etu.upmc.fr, francoise.lene@upmc.fr; *2CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, 4 place Jussieu, Case 161, F-75252 CEDEX Paris, France angela.vincenti@upmc.fr; *3 Univ Paris-Sud 11 F-91405 Orsay, Franceboris.desmorat@upmc.fr
; *4 Segula Technologies. 5 rue Albert Durant, F-31700 Blagnac, France.)
[4]DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE spécialité : Génie Mécanique, Mécanique des matériaux
« Stratégies de calcul pour l'optimisation multiobjectif des structures composites. », soutenue par François-Xavier IRISARRI, Le 23 Janvier 2009
[5] « 2006» Optimisation multicritère de stratifié par algorithmes génétiques Laurent-Blanchard ; Nathalie- Blaszka ; Stéphane-Lantéri ; Jean-Antoine Désidéri (Alcatel
Space Industries ; Institut National de recherche en Informatique et en Automatique).
[6] L.Guillaumat-F. Dau-A. Alzina. Clermont-Ferrand, le 28 janvier 2008. « Matériaux composites et fiabilité. »
[7] Prof Rafic Younes. “Composite materials. ”
[8] Michel Van Caneghem « Le voyageur de commerce, Algorithme “branch and bound”, Algorithme Glouton, Méthode de recherche locale. » Décembre 2002.
