Identification of parameters in a dynamic problem of elasticity for a body with an inclusion
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SYSTEMS ANALYSIS IDENTIFICATION OF PARAMETERS IN A DYNAMIC PROBLEM OF ELASTICITY FOR A BODY WITH AN INCLUSION I. V. Sergienkoa† and V. S. Deinekaa‡
UDC 519.6
Explicit expressions for gradients of residual functionals are obtained for the identification of the parameters of elastic dynamic deformation of multicomponent bodies by gradient methods. The technique is based on the solutions of conjugate problems found using the theory of optimal control over states of multicomponent distributed systems that is developed by the authors. Keywords: multicomponent bodies, elastic dynamic deformation, parameter identification.
The paper [1] considers identification problems where gradient methods [2] are used to derive explicit expressions for gradients of residual functionals for different parameters of hyperbolic multicomponent systems. The technique proposed to derive explicit expressions of gradients of residual functionals is based on optimal control theory [3–5] and allows obtaining satisfactory numerical approximations of different heat-conduction characteristics of compound plates [6, 7]. In the paper, we will derive explicit expressions for the gradients of residual functionals to identify (using gradient methods) different parameters of the dynamic elastic deformation of multicomponent bodies. 1. IDENTIFICATION OF THE INITIAL STRESS–STRAIN STATE In practice, the dynamic stress–strain analysis of elastic bodies frequently requires their initial stress–strain state to be taken into account. Following [8], we have s = D( e - e 0 ) + s 0 , where s and e are stress and strain tensors; s 0 and e 0 are the initial stresses and strains, which are difficult to determine, and D is the stiffness matrix. Assume that the system of dynamic elastic equilibrium equations r is
defined
s ki = s ik ( y ) =
on 3
å
l , m =1
bounded
¶ 2 yi ¶t
connected
2
¶ s ik ~ + f i ( x, t ), i = 1, 3, t Î ( 0, T ), k =1 ¶ x k 3
=å
strictly
Lipschitz
domains
W1,
W2 ÎR3,
(1) where
x = ( x1 , x 2 , x 3 ) ,
c iklm ( e lm - e 0lm ) + s 0ik ; s ik and e lm ( s ik , e 0lm ) are the components of the (initial) stress and strain
¶y ö 1æ ¶y tensors, respectively, e lm = e lm ( y ) = çç l + m ÷÷ , y = ( y1 ( x ), y 2 ( x ), y 3 ( x )) is the displacement vector, y i ( x ) is its 2 è ¶ xm ¶ xl ø ~ ~ ~ ~ projection onto the ith axis of the Cartesian coordinate system, and f = ( f1 ( x ), f 2 ( x ), f 3 ( x )) is the vector of bulk forces. 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. 75–97, May–June 2009. Original article submitted December 26, 2008. †
1060-0396/09/4503-0397
©
2009 Springer Science+Business Media, Inc.
397
The elastic constants are symmetric: c iklm = c lmik = c kilm and satisfy the condition 3
å
c iklm e ik e lm ³ a 0
i , k , l , m =1
3
å
i , k =1
2 e ik , a 0 = const > 0 .
(1')
The displacements y=j
(2)
are specified at the boundary GT = G ´ ( 0, T )
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