The method of fundamental solutions for annular shaped domains

Li, Zi‐Cai

doi:10.1016/j.cam.2008.09.027

Cited by 43 publications

(19 citation statements)

References 17 publications

(37 reference statements)

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“…The error bounds of the MFS are derived in Li [15] for annular shaped domains without numerical examples. The purposes of the numerical experiments in this section are twofold: (1) to support the analysis in [15] and (2) to compare MFS and MPS.…”

Section: Annular Shaped Domainsmentioning

confidence: 99%

“…For Laplace's equation on bounded domains, in the next section the algorithms of MFS and MPS are described, and in Section 3 the error and stability analysis is briefly provided. In Sections 4 and 5, two numerical experiments are reported to support the analysis in [15], and to make comparisons of MFS and MPS. In the last section, a few remarks are made.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Comparisons of fundamental solutions and particular solutions for Trefftz methods

Li¹,

Young²,

Huang³

et al. 2010

Engineering Analysis with Boundary Elements

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Section: Annular Shaped Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Comparisons of fundamental solutions and particular solutions for Trefftz methods

Li¹,

Young²,

Huang³

et al. 2010

Engineering Analysis with Boundary Elements

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“…From Theorem 6.2, the optimal choice of L and p is found by p = O(L). i.e., c 0 L ≤ p ≤ CL, where c 0 and C are two constants independent of L and p. For error analysis of CTM and HTM using FS, we may follow Li [25].…”

Section: Proofmentioning

confidence: 99%

“…Let ∂S be divided into uniform small sections j with length h = 1 M , and M is the division number along BC. On the middle nodes (x j , y j ) of j , we obtain the collocation equations from (7.3) and (7.4) 25) where ω = 1 N , and u N and v N are given in (7.5) and (7.6), respectively. In computation we also choose M(= 50) along AB in Fig.…”

Section: Numerical Examplesmentioning

confidence: 99%

Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

2011

Numerical Methods Partial

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For linear elastostatics, the Lagrange multiplier to couple the displacement (i.e., Dirichlet) condition is well known in mathematics community, but the Lagrange multiplier to couple the traction (i.e., Neumann) condition is popular for elasticity problems by the Trefftz method in engineering community, which is called the Hybrid Trefftz method (HTM). However, there has not been any analysis for these Lagrange multipliers to couple the traction condition so far. New error analysis of the HTM for elasticity problems is explored in this paper, to derive error bounds with the optimal convergence rates. Numerical experiments are reported to support this analysis. The error analysis of the HTM for linear elastostatics is the main aim of this paper. In this paper, the collocation Trefftz method (CTM) without a multiplier is also introduced, accompanied with error analysis. Numerical comparisons are made for HTM and CTM using fundamental solutions (FS) and particular solutions (PS). The error analysis and numerical computations show that the accuracy of the HTM is equivalent to that of the CTM, but the stability of the CTM is good. For elasticity and other complicated problems, the simplicity of algorithms and programming grants the CTM a remarkable advantage. More numerical comparisons show that using PS is more efficient than using FS in both HTM and CTM. However, since the optimal convergence rates are the most important criterion in evaluation of numerical methods, the global performance of the HTM is as good as that of the CTM. The comparisons of HTM and CTM using FS and PS are the next aim of this article.

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“…This is why people prefer collocation on boundary even though there are some singularity issues to solve. Li proposed two other kinds of semianalytical approaches, one can combine with FEM and FDM, and the other use the method of fundamental solutions as given in the Li's book [14] and paper [15], respectively. Nonsingular formulation of the Trefftz method was studied by Li et al [16].…”

Section: Introductionmentioning

confidence: 99%

Scattering of sound from point sources by multiple circular cylinders using addition theorem and superposition technique

Chen

Lee

Lin

et al. 2011

Numer. Methods Partial Differential Eq.

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In this study, we use the addition theorem and superposition technique to solve the scattering problem with multiple circular cylinders arising from point sound sources. Using the superposition technique, the problem can be decomposed into two individual parts. One is the free-space fundamental solution. The other is a typical boundary value problem (BVP) with specified boundary conditions derived from the addition theorem by translating the fundamental solution. Following the success of null-field boundary integral formulation to solve the typical BVP of the Helmholtz equation with Fourier densities, the second-part solution is easily obtained after collocating the observation point exactly on the real boundary and matching the boundary condition. The total solution is obtained by superimposing the two parts which are the fundamental solution and the semianalytical solution of the Helmholtz problem. An example was demonstrated to validate the present approach. The parameter study of size and spacing between cylinders are addressed. The results are well compared with the available theoretical solutions and experimental data.

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The method of fundamental solutions for annular shaped domains

Cited by 43 publications

References 17 publications

Comparisons of fundamental solutions and particular solutions for Trefftz methods

Comparisons of fundamental solutions and particular solutions for Trefftz methods

Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

Scattering of sound from point sources by multiple circular cylinders using addition theorem and superposition technique

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