Numerical Solution of Inverse Problems of Thermoelasticity for a Composite Cylinder
- PDF / 142,251 Bytes
- 8 Pages / 594 x 792 pts Page_size
- 15 Downloads / 197 Views
NUMERICAL SOLUTION OF INVERSE PROBLEMS OF THERMOELASTICITY FOR A COMPOSITE CYLINDER UDC 519.6:539.3
A. A. Aralova
Abstract. The thermoelastic state of a composite cylinder is analyzed. Classical generalized problems defined on classes of discontinuous functions are presented. Explicit expressions are obtained for residual gradients (using the solution of direct and adjoint problems) for the implementation of Alifanov’s gradient methods; functions of the finite element method are used to construct highly accurate computation schemes for the numerical discretization of direct and adjoint problems. Keywords: mathematical modeling, thermoelastic state, cylindrical composite body, finite element method, discontinuous solution, gradient method of identification. INTRODUCTION In [1], based on the results of the theory of optimal control of states in different multicomponent distributed systems [2, 3], a technology of constructing explicit expressions of gradients of residual functionals was proposed for the identification of different parameters of multicomponent distributed systems by gradient methods [4]. In [5–8], this technology was used for the identification of parameters of problems of axisymmetric, thermal, and thermoelastic deformation of a long hollow cylinder. This article considers methods for solving inverse thermoelasticity problems for a composite hollow cylinder with the help of gradient methods. Results of solution of some model inverse boundary-value problems are presented. 1. MATHEMATICAL MODEL OF THE THERMOELASTIC STATE OF A COMPOSITE CYLINDRICAL SHELL We consider a long composite hollow isotropic cylinder. Proceeding from [2, 9] and taking into account its symmetry, we describe its thermoelastic state with the help of the equation ds r s r - s j (1) + = 0, r Î W . dr r Here, W = W 1 È W 2 , W 1 = ( r1 , x ) , W 2 = ( x , r2 ) , 0 < r1 < x < r2 < ¥ , W = [ r1 , r2 ] , and r1 and r2 are radiuses of the internal and external circular surfaces, respectively; r is a radial coordinate of a cylindrical coordinate system; s r = s r ( y, T ) and s j = s j ( y, T ) are components of the stress tensor. Assume that s r = ( l + 2m ) e r + le j - ( 3l + 2m ) aT , s j = le r + ( l + 2m ) e j - ( 3l + 2m ) aT , (2) dy y er = , ej = , dr r V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 164–172, September–October, 2014. Original article submitted March 27, 2014. 1060-0396/14/5005-0797
©
2014 Springer Science+Business Media New York
797
where y = y( r ) is a radial displacement of the point with a coordinate r; l and m are the Lame constants; a is the coefficient of linear thermal expansion; e r and e j are components of the deformation tensor; Ò is a change in the temperature. With allowance made for expressions (2), Eq. (1) assumes the form y d æ dy ö dT ü ì - í ( l + 2m ) ç r ÷ - ( l + 2m ) - ( 3l + 2m ) ar ý = 0, r Î W . dr dr þ dr r ø è î
(3)
A change in the temperature Ò satisfies
Data Loading...
