Vulcanization process: a stochastic method for Identification and. Control problem. L. Autrique. Laboratoire d' Automatique de Nantes. Ecole Centrale de Nantes.
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Vulcanization process: a stochastic method for Identification and Control problem. J.E.Souza De Cursi Institut de Mecanique de Rouen 1.N.S.A de Rouen Place Emile Blonde1 BP 8 76131 Mont Saint Aignan cedex FRANCE Tel: (33) 35.52.83.70 FAX: (33) 35.52.84.21
L. Autrique Laboratoire d’Automatique de Nantes Ecole Centrale de Nantes 1 rue de la noe 44072 Nantes cedex 03 FRANCE Tel: (33) 40.37.16.00 (poste 1857) FAX: (33) 40.37.25.22 Abstract This paper deals with the coupling of deterministic and stochastic methods in order to solve identification and control problems arising in the industrial process of vulcanization (cf [I]). Elastomers and rubber composite materials are used in many high technology industries (cf [2] and [3]) and a severe control of their mech‘anical and thermal properties must be exerced. These properties are closely connected to the curing process, while an exothermal chemical reaction can occur, depending on the history of the temperature behaviour. This supplement of energy can, on the one hand, damage the material and, on the other hand, modify the final mechanical and thermal properties of the cured elastomer. A sharp control of this reaction is essential to avoid the destruction of the material and t o assure good performances in practical utilisation. A stochastic method is presented in order to solve these inverse problems.
1. A model for the i n d u s t r i a l process The following model has been introduced by [4] and [ 5 ] : the thermochemical behaviour of the elastomer is described by a set of two coupled equations: a heat equation for the thermal behaviour and an ordinary differential equation for the chemical one. These equations are coupled by their right members and involve a characteristic time called the induction ttme, before what a chemical reaction cannot occur. The symmetry of the real threedimensionnal process is used in order to reduce the model to a two-dimensionnal one. Thus, we consider a domain R: R = (0; L1) x (0; L z ) c RZ.
- The domain R and the purts of its boundary correspondzng t o Dirichlet and nleumunn data.
Fig.1
Its boundary a R is formed by Tn U r A - . corresponding to Dirichlet data (given or controlled temperature) and Neumann data (given or controlled heat flux) (see Fig.l). Let us denote by X = ( X I ; X z ) the space variables and t E [ O ; T ] the time. The temperature is 0 = 0 ( X ; t ) and the reticulation rate or curing degree is a = a (X; t) (a E [O; 13 is a measure of the chemical state of the material). These quantities satisfy the following equations, where Q = R x [ O ; q :
xErD,tE[O;T], E r N , tE [ o q ,
x
(6) Q (X; 0) = e o ( x ) x E 0. In this model, p denotes the density. c the thermal conductivity, Qm the enthalpy associated to the exothermal reaction. T h e function f is given by the Arrhenius model:
(6)
f ( 0 )= ko exp (7) where ko is a characteristic of the material, E is the energy of activation, R is the universal constant of the ideal gases. The induction time t t n d is the solution of the scalar equation: (8) h(trnd,@) = 0 , where: so, t , n d depends on the kzstory of the temperature In equation (9), E , n d and R are analogous to the correspondings ones of (7). 6 , n d is a constant, characteristic of the material. VVe observe that: t c h ( t ,0) is increasing. Thus, we introduce:
and equation (2) can be rewritten as: (2bis) = W ( h ( t ,e ) ) f ( 0 )g (a) . The function g : 10; 13 IR is called the reaction functzon. It must be determined from experimental data.
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0-7803-1872-2/94/$4.00 0 1994 IEEE -~ ~
Equation (1) is discretized as: 8 K + f - aA8K+i = s 2 8 K (13) for h’ = 0,.. . , Km,, - 1 where:
2. O p t i m i z a t i o n problems c o n n e c t e d to the
+
industrial Two optimization problems must be solved in order to ensure a good design:
(16)
5jc
= ~f(eK)g(QKjW(h(Li(,Oh.))
The reaction function g must be determined from experimental data. Since measures of (I are difficult to obtain, while these of 8 are easier, the natural solution is to determine g by minimizing the deviation between the expected 8,obtained from (1)-(10) and the measured 0.This leads to an identification problem.
anda=z P Q K and 8~ are the approximated values of Q (.; txj and 8 (.; t K ) respectively. Discretization in space is performed by introducing a grid made of N I 1 vertical lines V, and N Z 1 horizontal lines Hj.The intersection of K and H I is the point P,, of coor-
The curing must be controlled in order to obtain desired final mechanical and thermal properties of the elastomer and to avoid damages to the material. In this case, the main difficulty is the control of the evolution of the temperatures, since the final curing degree Q is always close of the unity and the damages are only caused by too large temperature. This leads to a control problem.
by a$ and (3; the approximated values of Q (P,,; t ~and) 8 (Pi,;t ~respectively. ) The discretized in time equations are discretized in space by a finite-element method (see ~ 1 , and [71)This leads .to a set of equations having the form: E; = 0 and F; = 0 (17)
+
dinates((i-l)hl,(j-
J =
1
l)hz),whereh, =
(0- q 2 d X d *
{
D i s c r e t i z a t i o n of J: The continuous cost-functionnal is discretized as. NI
J ( 9 ) 2:
Jh
NZ
h’ma,
(e;:.- 6,)’
(9) = Thi h 2 y K=O
,=I ,=I
In many practical situations, the discretization of the costfunctionnal J is implemented as follows:
Problem Let S E {g, U ,p} be the parameter of the optimization Find S such that: J ( S ) is a minimum of S c J (S) under constrants (1)-(10).
N Kma=
(18)
J(g)
= J,,( g j =
;;.E 1 (6;- - 6;c)2 r=l
h-=1
where: N is the number of sensors, 6;C is the temperature measured a t sensor i, and is the temperature interpolated at sensor i. This functionnal describes the case where we have: N < (NI 1) (N2 1). The corresponding algorithms are analogous. We shall present numerical results for this situation.
3. Discretization of the p r o b l e m Let us consider, for example, the problem of identification, where S = g. As we have observed, the control problem is analogous. The discretization is performed as follows:
-+
Discretization of S: For the identification problem, we consider a finite element discretization of g: NG
+
Thus, the problem is formulated as follows:
g(O)-gh(Q)=xgiSt(@)
,
a
,=I
where S, are the continuous piecewise linear function such that: , 3 = 1 , . . . , ,VC . S,verify: for each CY] =
Thus, our unknowns are (91,. , ~
Discretized problem Find S = (g,, . . . , g N G ) , such that S minimizes S Jh (S)under the constrints of (IT).
-
4. Descent methods and s t o c h a s t i c processes
The numerical solution of the discretized problem is performed by an iterative descent method which reads as follows: (19) Sm+> =z Q ( S m ) Here, lower subscripts refer to the iteration number. For example, the gradient method with a fixed parameter 7 > 0 involves:
X G )
Discretization of ( l ) - ( l O ) : Discretization in time is performed by finite differentiawhere A ‘,, is the tion involving parameter r =
+ mar
number of time interval. Equation (2bis) is discretized as for A- = 0, . . . , A-,,
Wedenote
K=0,.-.,Kmaz-l i = I , . . . . NI + I 3 = 2 , . . . , A T2 correspondings t o the discretization of (15) and (14) respectively. By reasons of limitation of room, we do not give here the expressions of E:. and Fi: (see [l]). for
under constraints (I)-(10). For the identification problem, we consider J as a function of g: J = J ( g ) , while for the control problem, we consider J = J(u) or J = J ( 9 ) Thus. both the situations can be summarized as:
(13)
k.
PI
Both these situations lead to optimization problems, where we must minimize a functionnal: (11)
+
-1
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7. References
Howewer, the convergence to the global minimum cannot be ensured without convexity assumptions which are not verified for the given functionnal J . Recent develop ments have considered the perturbation: (21) S m + l = Q(sm) Amzm where A, is a real number and 2, is a random variable. They have shown that, for suitable A, and Z,, this perturbation ensures, on the one hand, convergence to a global minimum and, on the other hand, improves the convergence (see [SI).
[I] L.Autrique and J.E.Souza De Cursi, “Resolution numirique d’un problhme d’identification comprenant des Cquations couplCes B l’aide de processus stochastiques”, Rapport interne 93.12, Laboratoire d’Automatique de Nantes, Ecole Centrale de Nantes, France, 1993.
+
[2] F.Golay and O.DCbordes, “ModClisation par ddments finis de la polymirisation de structures composites Applications industrielles” , Revue europe‘enne des e‘liments finis, Vol 1, nb 1, p p 51-73, 1992.
5. N u m e r i c a l e x p e r i m e n t s
[3] G.J.W.Hou, T.H.Hou and J.S.Sheen, “Optimal Cure Cyde Design for the Fiber Reinforced Composite Lamination Processing”, International Polymer Processrnq, Vol 5, n02, pp 88-99, 1990.
Numerical experiments have been performed in the case where:
LI = 0,2m N I = 16 N2 = 11
Kmaz= 295 E I n d = 81040 J.mole-I R = 8,314 J.mole-’.K-’ Qoo= 2 , 177.10i J.m-3 p = 2, 0422.106 J.m-3.h’-1 c = 0,271 W . m - ’ . K - ’
L2 = 0 , 0 4 m hl = 0,0125m h2 = 0,00364m T = 30s 6ind = 2, 5 2 . 1 0 - 6 ~ E = 81040 J.mole-’ ko = j.106s-’
[4] B.Ga.rnier, D.Delaunay and F.Danes, “Coupling between Heat Transfer and Rubber Vulcanization”, IXth International Heat Transfer Conference, Jerusalem, Israel, 1990.
[5] J.Burger, P.Chauvet, J.S.Le Brizaut, “Identification d’une fonction de reaction dans un processus de vulcanisation”, RAIRO A.P.I.I., Vol 25, n“5,pp 463-476. 1990.
00= 85°C 0 0
=0
v(X1;X2)=X2(X2-L2)
[6] G.Beauquet, M.Pogu, “Programmation des dements finis PI,ZD”,Cepadues Editions, Toulouse, France, 1967.
and the controlled temDerature is: G + &j - if t 5 3600 u(1) = if t > 3600 We compare the results obtained by ( 2 0 ) and (21). The
{
evaluation of the gradient methods:
3
[7] E.Zeidler, “Non linear functional analysis and its applications”, Tome II. B: Probleme d’e‘uolution, Springer Verlag, 1990.
is performed by three
[8] M.Pogu, J.E.Souza De Cursi, “Global optimization by random perturbation of the gradient method with a fixed parameter”, Journal of Global optimization, Kluwer Academic publishers, accepted nb 18793, 1994.
(MI):Reduction (differentiation of the discretized equations corresponding to the continuous equations (1) and (2bis). This leads to a linear system giving
aQ‘J
the values of -=&. These values are used in the differentiation of (18)); 0
(M2): Discretized Lagrangian method (penalty method for taking the discretized equations (17) into account); ( M 3 ) : Continuous Lagrangian method (penalty method before discretization)
In each case, we have shown the interest of the addition of random perturbations on the descent method. The convergence speed has been increased and the value of the discretized cost-function J h is close to 0. 6 . Concluding remarks
The interest of the coupling of deterministic and stochastic methods has been shown on the optimization problem. Since the problem presentation is sufficiently general, random perturbations can improve the control problem resolution. Differents numerical experiments are actually implemented in order to proove that important result. Other possible developments are the introduction of genetic-like strategics which have shown to be effective to calculate in mechanical engineering problems.
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