Solving Direct and Inverse Thermoelasticity Problems by Means of Trefftz Base Functions for Finite Element Method

Abstract

The Trefftz functions method has been developed very quickly. The paper presents the application of this method to solving direct and inverse problems of elasticity and thermoelasticity. The system of equations for displacements is reduced to a system of wave equations. Then the wave polynomials (Trefftz functions for wave equation) as base functions for several variants of Finite Element Method are used. In the paper, continuous FEMT and substructuring are considered. In the case of thermoelasticity, the temperature field occurs as inhomogeneity in one of the wave equations. It is shown how to get the particular solution in 2D and 3D. When using FEMT, the difference of solutions between the elements has to be minimized. The mechanical energy of the body depends on the velocity of the displacements. Therefore, the difference of the velocities between the elements is also minimized – it is a kind of physical regularization. The quality of the approximate solutions of direct and inverse problem was verified on the test example.

Keywords:

• FEM
• Inverse problems
• Substructuring
• Thermoelasticity
• Trefftz functions

Acknowledgments

This work was carried out in the framework of the research project No. N513 003 32/0541, which was financed by the resources for the development of science in the years 2007–2009.