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Numerical analysis and weighted residuals
Published in Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee, Numerical Methods in Mechanics of Materials, 2017
Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee
In contrast to the Ritz method, the Trefftz method (Trefftz, 1927) is based on choosing trial functions ϕ¯ such that the differential equation (Equation 2.62) is satisfied over R, and the arbitrary constants in the trial functions are chosen to minimize the integral, over the cross-section, of the error gradient. That is, the method is a boundary method. With this method, the approximate magnitude of the twisting moment is larger than its exact value, in contrast to the Ritz method (Equation 2.80). Hence, by using the Ritz method in conjunction with the Trefftz method, the exact value of the twisting moment M (hence the twist β) may be bounded from above and below.
Variational and Related Methods
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
There are a number of good books on the variational method, including Mura and Koya (1992), Washizu (1982), Kantorovich and Krylov (1964), Reiss (1965), and Reddy (2002). There are also more specialized methods similar to the Galerkin method, such as the Trefftz method. In this method, the approximate solution is selected such that the governing equation is exactly satisfied and the boundary conditions are satisfied approximately in a variational sense. However, it is not easy to find the approximation that satisfies the governing equation, and thus the Trefftz method is not discussed in the present chapter.
Application of the Trefftz Method for Pool Boiling Heat Transfer on Open Microchannel Surfaces
Published in Heat Transfer Engineering, 2021
Sylwia Hożejowska, Robert Kaniowski, Robert Pastuszko
In order to find approximate solutions of heat transfer problems given by system of Equations (5)–(8) and Equations (9)–(13), the Trefftz method was used [6, 14, 15]. The idea of the Trefftz method is to approximate unknown solutions of a partial differential equation by a linear combination of the functions (called Trefftz functions or briefly T-functions) that exactly satisfy given differential equation. Since we deal with heat transfer problems in polar coordinates (Equations (5)–(8)) and in Cartesian coordinates (Equations (9)–(13)), then two sets of T- functions are required for the Trefftz method to be applied to Equations (5) and (9), respectively. T-functions for Equation (5) are the functions given by [14]: where the symbol [] denotes the floor function.
A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition
Published in Applicable Analysis, 2020
Shelvean Kapita, Peter Monk, Virginia Selgas
The Trefftz method, in which a linear combination of simple solutions of the underlying partial differential equation on the whole solution domain are used to approximate the solution of the desired problem, dates back to the 1926 paper of Trefftz [1]. A historical discussion in relation to Ritz and Galerkin methods can be found in [2]. From our point of view, a key paper in this area is that of Cessenat and Déspres [3] who analyzed the use of a local Trefftz space on a finite element grid to approximate the solution of the Helmholtz equation [3]. This was later shown to be a special case of the Trefftz Discontinuous Galerkin (TDG) method [4,5] which opened the way for a more general error analysis. For more recent work in which boundary integral operators are used to construct the Trefftz space, see, for example [6,7]. The aforementioned work all concerns the standard pressure field formulation of acoustics which results in a scalar Helmholtz equation. Indeed, TDG methods are well developed for the Helmholtz, Maxwell and Navier equations with standard boundary conditions and a recent survey can be found in [8]. For the displacement form, a TDG method has been proposed by Gabard [9] also using simple boundary conditions.
Numerical analysis of heat transfer in arbitrary plane domains using a novel Trefftz energy method
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Fajie Wang, Chein-Shan Liu, Wenzhen Qu
forms the T-complete functions, and the numerical solution in terms of these bases is known as the Trefftz method [101112]. The Trefftz method is designed to satisfy the governing equation and leaves the unknown coefficients determined by satisfying the boundary conditions with the help of the collocation, the least square or the Galerkin method, etc. [13, 14]. Even the Trefftz method can reduce the dimensionality on the boundary as a boundary-type method, it is notoriously ill-conditioned.