The Hybrid-Trefftz Finite Element Method For Elastic Contact Problems

Contact problems arise frequently in industrial processes and engineering applications. The non-linearities appearing in these problems are due to the undetermined size of contact zone and to the irreversible nature of frictional effects. Even though a number of analytical solutions are available, the vast majority of practical problems lie beyond their applicability. Hence a large family of numerical approaches, mainly based on the conventional finite element method (FEM) or boundary element method (BEM), has been developed for treating these problems. As a highly efficient and well established tool, the hybrid-Trefftz (HT) FEM, initiated about three decades ago, has now become more and more popular. There are two main advantages in HT FEM: firstly, the formulation only calls for integration along the element boundary which enables arbitrary polygonal or even curve-sided elements to be generated. As a result, it may be considered as a special, symmetric, substructure-oriented boundary solution approach and thus possesses the advantages of the conventional BEM. In contrast, however, HT FEM avoids the introduction of singular integral equations and, secondly, HT FEM, compared with the conventional FEM, can accurately treat problems with various local effects due to loading and/or geometry (for example, corners, holes, cracks, inclusions, concentrated loads, etc.) without troublesome mesh refinement. Up to now, HT FEM has been thoroughly explored in a number of fields in engineering. However, there are no contributions by HT FEM for contact problems. The main content of this paper is twofold: some research on HT FEM itself and, development of contact solution algorithm based on it. Starting from the fundamental analytical solutions to plane elastic problems, an alternative way to construct Trefftz functions for regular element is presented. Additionally, a novel Trefftz hole element, based on the work of Piltner, is developed with the introduction of rotated mapping function and with the help of Muskhelishvili complex function approach. This element can be easily and conveniently employed to analyze problems with elliptical holes opened in any direction. And thus the shortage of the Piltner hole element has been covered.This paper initially applies HT FEM, in conjunction with the direct constraint technique, to elastic contact problems with or without friction. The static condensation is employed to condense a larger model down to a smaller one which involves discrete...