Application of symmetric indirect Trefftz method to free vibration problems in 2D

Abstract

SUMMARYA symmetric indirect Tre tz method is developed to solve the free vibration problem of a 2D membrane. It is proved that in this approach the spurious eigensolution exists, and an auxiliary matrix is constructed to help extraction of the spurious solution using the generalized singular-value decomposition. In addition to the spurious eigensolution, this regular formulation su ers from its ill-posed nature, i.e. the numerical instability. In order to deal with the numerical instability, the Tikhonov's regularization method, in conjunction with the generalized singular-value decomposition, is suggested. The proposed approach has some merits when compared with other regular boundary element formulations reported so far; namely the capacity of representing eigenmodes and the ability to deal with a multiply connected domain of genus 1. Several numerical examples are demonstrated to show the validity of the current approach.

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

“…In this regard, the T-Trefftz method is sometimes viewed as the eigenfunction expansion method since the T-complete functions are the eigenfunctions. Due to the limited use of one source point (Trefftz origin), the T-Trefftz method cannot deal with a multiply connected domain with genus p(p > 1) as mentioned in Chang's paper (Chang et al, 2003). By adopting the domain partition technique, the multiply connected Downloaded by [Universidad de Sevilla] at 02:12 01 December 2014 domain with genus p(p > 1) can be divided into several simply connected domains and consequently the above-mentioned difficulty can be overcome (Leitão, 1998).…”

“…The most serious one is the ill-posed behavior, i.e., the numerical instability. In previous publications (Chang et al, 2002(Chang et al, , 2003(Chang et al, , 2004, we found that the T-Trefftz method encounters numerical instability when solving the Helmholtz equation. Besides, it was also reported that (Goldberg and Chen, 1996;Chen et al, 1998;Chen, 1995;Christiansen and Sararen, 1996;Christiansen and Hansen, 1994) the condition number for the influence matrix of the F-Trefftz method is very large.…”

“…It was originated by Trefftz (1926). Since then, it has been studied by Cheung et al (1989), Jin et al (1990), Herrera (1995), Jrousek and Leon (1977), Kamiya and Wu (1994), Piltner (1992), Zielinski and Ziekiewicz (1985), Chang et al (2002Chang et al ( , 2003Chang et al ( , 2004 and so on. To date, two kinds of Trefftz methods, the T-Trefftz method and the FTrefftz method, have been developed.…”

Effectiveness of nonsingular solutions of the boundary problems based on Trefftz methods

“…The latter group, [17][18][19][20], is derived from either classical formulation or original variational one. In this case, the solution is approximated by the series of the T-functions with unknown coefficients, which are determined so that the approximate solution satisfies boundary conditions and it also leads to BIE.…”

Adaptive error estimation of the Trefftz method for solving the Cauchy problem

“…direct formulations [8] and indirect formulations [7], have been developed. The unknown coefficients are determined by satisfying the boundary condition with the approximate solution.…”