Ru-feng Liu

National Taiwan Ocean University, Keelung, Taiwan, Taiwan

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Publications (2)2.06 Total impact

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    ABSTRACT: The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two-di-mensional domains. The radial basis function approach is used to find an approxi-mate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T-Trefftz method and F-Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill-posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L-curve concept are both employed. The Poisson equation of the type, ∇ 2 u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Nu-merical examples are provided to show the validity of the proposed numerical meth-ods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F-Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T-Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.
    Journal of the Chinese Institute of Engineers. 01/2006; 29:989-1006.
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    ABSTRACT: A symmetric indirect Trefftz 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 suffers 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. Copyright © 2003 John Wiley & Sons, Ltd.
    International Journal for Numerical Methods in Engineering 02/2003; 56(8):1175 - 1192. · 2.06 Impact Factor