Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept

Chen, Jeng‐Tzong; Lee, Ying‐Te; Yu, Shang-Ru; Shieh, S.C.

doi:10.1016/j.enganabound.2008.10.003

Cited by 55 publications

(43 citation statements)

References 28 publications

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“…where a is the radius of the sphere, ( , /2,0) It is interesting that the formulae for the location of image are the same as the 2-D case [6]. However, the magnitude of strength   / s R a is different from the 2-D case [6].…”

Section: The Image Solutionmentioning

confidence: 99%

“…Here, we also distribute singularities along the radial direction by using the method of image. Image solutions and Trefftz results for the annular Green's function were obtained [6]. Since a two-dimensional problem can be solved easily by using the complex variable, the image method can be seen as an alternative way to obtain the solution.…”

Section: Introductionmentioning

confidence: 99%

“…They found that the condition number of the MFS is much worst than that of the Trefftz method. Although the Trefftz method and the MFS individually have a long history, the link between the two methods was not discussed in the literature until Chen et al's papers [5,6]. Researchers have paid attention to construct the mathematical relationship between the Trefftz method and MFS since 2006.…”

Section: Introductionmentioning

confidence: 99%

“…They only proved the equivalence by demonstrating a simple circle with angular distribution of singularity to link the two methods. Following the success of deriving the annular Green's function [6], we plan to derive the Green's function of a concentric sphere. Here, we also distribute singularities along the radial direction by using the method of image.…”

Section: Introductionmentioning

confidence: 99%

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Equivalence between the Trefftz method and the method of fundamental solutions for Green’s function of concentric spheres using the addition theorem and image concept

Chen

Shieh

Tsai

et al. 2009

Mesh Reduction Methods

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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-fixed boundary conditions are considered. Also, the Trefftz method is employed to derive the analytical solution by using the T-complete sets. By employing the addition theorem, both solutions are found to be mathematically equivalent when the number of Trefftz bases and the number of image points are both infinite. In the successive image process, the final two images freeze at the origin and infinity, where their singularity strengths can be analytically and numerically determined in a consistent manner. The agreement among the three results, including two analytical solutions by using the Trefftz method and the image method, and one numerical solution by using the conventional MFS is observed.

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Section: The Image Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Equivalence between the Trefftz method and the method of fundamental solutions for Green’s function of concentric spheres using the addition theorem and image concept

Chen

Shieh

Tsai

et al. 2009

Mesh Reduction Methods

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“…In fact, the connection between the TCM and the MFS has recently been pointed out in [5] where it is shown that for the Laplace equation in a bounded, simply-connected domain the TCM is retrieved as a degenerate situation of the MFS with the source points placed at infinity. The corresponding result in annular and exterior domains may be found in [4] and [49], respectively.…”

Section: Introductionmentioning

confidence: 96%

Regularized collocation Trefftz method for void detection in two-dimensional steady-state heat conduction problems

Karageorghis

Lesnic

Marin

2013

Inverse Problems in Science and Engineering

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Abstract. We propose the use of the Trefftz collocation method for the solution of inverse geometric problems and, in particular, the determination of the boundary of a void. As was the case in the solution of such problems using the method of fundamental solutions, the algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The centre of this radial polar parametrization is considered to be unknown. The feasibility of this new method is illustrated by several numerical examples highlighting its advantages and shortcomings.

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Steam assisted biomass gasification—an overview

Kaushal¹,

Tyagi²

2011

Can J Chem Eng

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Biomass gasification technologies have matured over the years. In the past decade, numerous types of gasifiers have been developed and tested. This article reviews the status of steam assisted biomass gasifiers. It first discusses the challenges and the key operating parameters for steam‐biomass gasification and their effects on the gasifier performance. The article then summarises the progress and status of commercial or demonstration scale steam assisted biomass gasifiers available in the world.

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Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept

Cited by 55 publications

References 28 publications

Equivalence between the Trefftz method and the method of fundamental solutions for Green’s function of concentric spheres using the addition theorem and image concept

Equivalence between the Trefftz method and the method of fundamental solutions for Green’s function of concentric spheres using the addition theorem and image concept

Regularized collocation Trefftz method for void detection in two-dimensional steady-state heat conduction problems

Steam assisted biomass gasification—an overview

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