Parameter identification in thermoelastic problems for nonstationary thermal field
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SYSTEMS ANALYSIS PARAMETER IDENTIFICATION IN THERMOELASTIC PROBLEMS FOR NONSTATIONARY THERMAL FIELD I. V. Sergienkoa† and V. S. Deinekaa‡
UDC 519.6:539.3
Abstract. Explicit expressions of the gradients of residual functionals are obtained to use gradient methods to identify different parameters and thermoelastic states of compound bodies in a nonstationary thermal field. These gradients are found based on the theory of optimal control of the states of multicomponent distributed systems. Keywords: multicomponent bodies, thermal-stress state, parameter identification, gradient methods.
The studies [1, 2] are based on the theory of optimal control of states of multicomponent distributed systems [3, 4] and consider the construction of gradients of residual functionals to identify, by gradient methods, various parameters in problems of dynamic elastic and quasistationary thermoelastic deformation of bodies with inclusions, respectively. We will discuss deriving the explicit expressions for the gradients of residual functionals to identify, by gradient methods [5], various parameters and thermoelastic states of compound bodies in a nonstationary thermal field. 1. IDENTIFYING THE THERMAL AND THERMAL-STRESS STATES OF A BODY FROM SURFACE DISPLACEMENTS Following [6], for a long-term time independence of mechanical influences, i.e., in the absence of the inertial term r&&y ( ó is the displacement vector), the thermoelastic state of a body W Î R 3 can be described by the following initial–boundary-value problem. On a bounded connected strictly Lipschitz domain W Î R 3 , a thermoelastic equilibrium system
3
-å
k =1
¶s ik (T ; y ) ~ = f i ( x ) , i = 1, 3, t Î ( 0, T ) , ¶x k
is defined, where x = ( x1 , x 2 , x 3 ) , s ik = s ik (T ; y ) =
3
0 ( a, T )) , å ciklm ( e lm ( y ) - e lm
l , m =1
(1) s ik and e lm are the components
of the stress and strain tensors, respectively, and c iklm are elastic constants. If the material is elastic- and heat-isotropic, components of the strain tensor due to the difference T = T1 - T10 between the temperature T1 and its initial state T10 can be expressed as e 0lm = aTd lm , where a is the linear expansion coefficient and d lm is the Kronecker delta. Variations in the temperature Ò over the domain W satisfy the equation c where
3 ¶T ¶ æç ¶T k ij = å ç ¶t i , j = 1 ¶x i è ¶x j
ö ÷ + f , t Î ( 0, T ) , ÷ ø
(2)
ñ is the volumetric heat capacity and k ij are elements of the tensor of thermal conductivity coefficients. a
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 51–77, May–June 2011. Original article submitted January 27, 2010. †
1060-0396/11/4703-0375
©
2011 Springer Science+Business Media, Inc.
375
The following boundary conditions are specified on the boundary G = ¶W of the domain W: y = j, ( x, t ) Î G11T , 3
å
j =1
(3)
s ij ( y ) n j = g i , i = 1, 3, ( x, t ) Î G21T ,
(4)
T = j 0 , ( x, t ) Î G12T , 3
å
i
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