This paper reviews the developments in the Trefftzian Methodology, which have been undertaken by the authors at the University of Sheffield during the past fifteen years and the application of these developments to engineering design analysis. Initially, in the late 197Os, this work concentrated on the Direct Boundary Element Method (DBEM) and the Indirect Boundary Element Method (IBEM). Unfort...

Based on the radial basis functions (RBF) and T-Trefftz solution, this paper presents a new meshless method for numerically solving various partial differential equation systems. First, the analog equation method (AEM) is used to convert the original patial differential equation to an equivalent Poisson’s equation. Then, the radial basis functions (RBF) are employed to approxiamate the inhomoge...

In this paper, a multiple-source-point boundary-collocation Trefftz method, with characteristic lengths being introduced in the basis functions, is proposed to solve the direct, as well as inverse Cauchy problems of the Laplace equation for a multiply connected domain. When a multiply connected domain with genus p (p>1) is considered, the conventional Trefftz method (T-Trefftz method) will fail...

Summary This work presents variational concepts associated with reduced Trefftz type approaches and discusses the interrelationship between various concepts of the displacement, hybrid and Trefftz methods. The basic concept of the displacement version of the reduced Trefftz method operates on the natural boundary conditions enforced in an integral form whereas the stress version of the reduced ...

A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339–363; Co...

In this paper, it is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation. In deriving the equivalence of the Trefftz method and the MFS for the Laplace and biharmonic problems, it is interesting to find that the comple...

In this article we present two numerical models for solving transient heat conduction problems. One is based on dual reciprocity method and Trefftz method (dubbed DRM-Trefftz), and the other is based on dual reciprocity method and fundamental solution (dubbed DRM-MFS). A time stepping method is used in handling the time variable to convert the problem into a set of inhomogeneous modified Helmho...

A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339–363; Co...

Following the success of the mathematical equivalence between the Trefftz method and the method of fundamental solutions for the annular Green’s function, we extend to solve the Green’s function of 3-D problems in this paper. The Green’s function of the concentric sphere is first derived by using the image method which can be seen as a special case of method of fundamental solutions. Fixed-fixe...

A new prediction technique, based on the indirect Trefftz method, has been developed for the steady– state dynamic analysis of coupled vibro–acoustic systems. In contrast with the finite element method, in which the dynamic field variables within each element are expanded in terms of local, non–exact shape functions, the dynamic field variables are expressed as global wave function expansions, ...