ABSTRACT
The Trefftz finite element (FE) model, or T-element model for short, was originally developed in 1977 for the analysis of the effects of mesh distortion on thin plate elements [1]. During the subsequent three decades, the potential of T-elements for the solution of different types of applied science and engineering problems was recognised. Over the years, the Hybrid-Trefftz (HT) FE method has become increasingly popular as an efficient numerical tool in computational mechanics and has now become a highly efficient computational tool for the solution of complex boundary value problems. In contrast to conventional FE models, the class of FE associated with the Trefftz method is based on a hybrid method which includes the use of an auxiliary inter-element displacement or traction frame to link the internal displacement fields of the elements. Such internal fields, chosen so as to a priori satisfy the governing differential equations, have conveniently been represented as the sum of a particular integral of non-homogeneous equations and a suitably truncated T (Trefftz)-complete set of regular homogeneous solutions multiplied by undetermined coefficients. The mathematical fundamentals of T-complete sets were established mainly by Herrera and his co-workers [2 - 5], who named such a system a T-complete system. Following a suggestion by Zienkiewicz, Herrera changed this to a T-complete system of solutions, in honor of the originator of such non-singular solutions. As such, the terminology “TH-families” is usually used when referring to systems of functions which satisfy the criterion originated by Herrera [3]. Interelement continuity is enforced by using a modified variational principle together with an independent frame field defined on each element boundary. The element formulation, during which the internal parameters are eliminated at the element level, leads to a standard force-displacement relationship, with a symmetric positive definite stiffness matrix. Clearly, while the conventional FE formulation may be assimilated to a particular form of the Rayleigh-Ritz method, the HT FE approach has a close relationship with the Trefftz method [6]. As noted in Refs. [7, 8], the main advantages stemming from the HT FE model are: (a) The formulation calls for integration along the element boundaries only, which enables arbitrary polygonal or even curve-sided elements to be generated. As a result, it may be considered a special, symmetric,
which may be very laborious to build. (b) The HT FE model is likely to represent the optimal expansion bases for hybrid-type elements where inter-element continuity need not be satisfied, a priori, which is particularly important for generating a quasi-conforming plate bending element. And (c) The model offers the attractive possibility of developing accurate crack-tip, singular corner or perforated elements, simply by using appropriate known local solution functions as the trial functions of intra-element displacements.
