Trefftz finite element method and its special purpose elements

Abstract In this report, the hybrid Trefftz finite element method based on nonsingular T-complete functions is first briefly reviewed for the purpose of establishing notation; then various special purpose elements are developed to illustrat the major advantage of Trefftz methods. In HT-FEM, elements containing local defects are treated by simply replacing the regular T-complete functions with appropriate special purpose “trial” functions. Elements containing such special “trial functions” is known as special purpose elements. It should be mentioned that a common characteristic of such special trial functions is that it is not only the governing partial differential equations which are satisfied exactly, but also some prescribed boundary conditions at a particular portion of the element boundary. This enables various singularities to be specifically taken into consideration without troublesome mesh refinement. Formulations of special purpose hole elements for both heat conduction and plane stress/strain problems are presented. Finally, a numerical example involving contact problems with an elliptic hole, are considered. Numerical results from the proposed models are presented and compared with those from ABAQUS.