Error estimation of the two-dimensional heat problem simulation for thin plates
The problem of heat transmission in thin plates is analyzed in this paper. For the sake of computational resource economy it is proposed to make a two-dimensional simplification of the three-dimensional problem, if the size of a three-dimensional body is small enough along a certain vector. It is supposed that the plate thickness is sufficiently small, so one can treat the temperature function to be independent of thickness. This paper considers the Finite Element Method (FEM) as a method of heat problem simulation. To estimate an error of such an approach the estimation method, obtained in an empirical way, is used, and a short rationale of this estimation is also presented in this paper. A possibility of using this estimation as a sufficient condition for the thin plates theory is verified by a series of tests. In these tests the two-dimensional FEM implementation is compared numerically with the natural three-dimensional implementation for rectangular plates numerical errors in these tests being also taken into account. Exact solutions for some simplified two-dimensional cases are analyzed in detail in the book by Carslaw and Jaeger. However, no error estimation is presented in the book. Finding a temperature distribution in thin plates can not only save computational resources but also provides initial data for solving thermoelasticity problems of thin plates. The problem of plates bending is observed for example in the book by Zienkiewicz and Taylor. The temperature distribution can be used as initial data independently for the chosen hypotheses of the plate bending problem (like Kirchhoff's hypothesis or the hypothesis of thick plates). In particular, thin plate theory can be used to simulate thin heating elements or simulate heating of turbine blades etc., if the error estimation of thin plates is sufficiently small.