On Certain Problems of Identification of Thermal Density of Temperature State of the Hollow Cylinder Shell

Abstract

Introduction. In conditions of the active use of composite materials, as well as in the tasks of extending the service life of existing structures, there are problems of recovering unknown parameters of their components in the presence of data on their surface. In [1-4], to solve the problems of identification of parameters of a wide range, it is proposed to construct explicit expressions of the gradients of residual functionals by means of the corresponding conjugate problems obtained from the theory of optimal control of the states of multicomponent distributed systems, which is the development of the corresponding researches of Zh. Lyons. In [5-7], this technology is extended to the problem of thermoelastic deformation of multicomponent bodies. In this article some problems of optimal control of the temperature state of a cylindrical body with a cavity are considered. The purpose of the paper is to show the algorithm for identifying the parameters of a cylindrical hollow shell, based on the theory of optimal control and using the gradient methods of Alifanov. Results. Based on the theory of optimal control, the temperature control of a cylindrical shell is studied. To solve the problem of identifying the parameters of a hollow cylindrical shell, namely, find-ing the heat flux powers on its surfaces, based on [1,2,5-7], a direct and conjugate problem and gradients of non-viscous functionals are constructed. Discretization by the finite element method using piecewise quadratic functions is carried out and accuracy estimates for it are presented. The initial problem in the model examples presented is solved using gradient methods, where at each step of determining the (n + 1) th approximation of the solution, the direct and adjoint problems are solved using the finite element method using piecewise quadratic functions by minimizing the corresponding energy functional. A number of model examples have been resolved.

Keywords

• temperature state
• gradient methods
• cylindrical bodies