An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

Cheng, Alexander H.‐D.; Hong, Yongxing

doi:10.1016/j.enganabound.2020.08.013

Cited by 110 publications

(30 citation statements)

References 298 publications

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“…These methods appeared long before the ones suggested by Nowak and Hall [44,45], but in the acoustical and electromagnetic wave contexts. As explained in the excellent review articles [4,8,72] (see also [55,53,41,58,19,20,64,42,18,5,63,21]), all these methods are based on employing linear combinations of the extended boundary condition, first-kind integral equation and second-kind integral equations so as to result, after discretization, in a matrix equation whose matrix is not singular at any frequency. In particular, this was the procedure adopted by Brakhage and Werner [14], Schenck [52], Bolomey and Tabbara [11,12], Burton and Miller [15], Mautz and Harrington [38,39], just to name a few.…”

Section: Review Of the Methods Of Cure Prior To Nowak And Hallmentioning

confidence: 99%

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The spurious resonance disease and how to cure it: application to the seismic response of a canyon

Wirgin

2020

Preprint

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Three types of boundary integral equation (BIE) methods are employed to obtain closed-form solutions of a wave-scattering problem which are compared to the exact, closed-form (reference), solution deriving from the separation-of-variables technique. The problem involves either Dirichlet (D) or Neumann (N) boundary conditions (BC) for a scatterer that is a circular cylinder submitted to one or two incident waves. The three BIE methods lead to different expressions for the traction (for D-BC) or boundary displacement (for N-BC) by which numerous resonances are predicted whose frequency of occurrence differs from one method to another. This is interpreted as being the sign that the three methods are generally-defective and the resonances are 'spurious'. This 'disease' is cured by combining two BIE into one in such a way that the resulting BIE gives rise to a closed-form solution identical to the exact reference solution devoid of spurious resonances.

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Section: Review Of the Methods Of Cure Prior To Nowak And Hallmentioning

confidence: 99%

“…The solution for {C n } is quite obvious (recall that {A n } is known via (17) and from the fact that r s , θ s are known), but I wish to bring to the fore a feature that will be useful further on. Thus, I choose to project (21) as follows:…”

Section: The Relation Of U I To Smentioning

confidence: 99%

The spurious resonance disease and how to cure it: application to the seismic response of a canyon

Wirgin

2020

Preprint

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“…If the number of collocation points is greater than the number of source points, i.e., M > N, an overdetermined system of equations is generated, that can be solved by a linear least squares algorithm. As suggested by Cheng et al [44], the MFS with M = N and M > N are called the standard MFS and the least squares MFS, respectively.…”

Section: The Mfs Formulation For Two-dimensional Anisotropic Elasticitymentioning

confidence: 99%

“…In the second type, which is more practical, the boundary conditions are of mixed type and are prescribed by functions which do not satisfy the governing equations. As mentioned in reference [44], most examples studied in the previous works are of the first type. In this work, we take into account both types of problems; however, the main focus is maintained on the second type, which is more important.…”

Section: Tablementioning

confidence: 99%

The Method of Fundamental Solutions for Two-Dimensional Elastostatic Problems with Stress Concentration and Highly Anisotropic Materials

Hematiyan¹,

Jamshidi²,

Mohammadi³

2022

Computer Modeling in Engineering &Amp; Sciences

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The method of fundamental solutions (MFS) is a boundary-type and truly meshfree method, which is recognized as an efficient numerical tool for solving boundary value problems. The geometrical shape, boundary conditions, and applied loads can be easily modeled in the MFS. This capability makes the MFS particularly suitable for shape optimization, moving load, and inverse problems. However, it is observed that the standard MFS lead to inaccurate solutions for some elastostatic problems with stress concentration and/or highly anisotropic materials. In this work, by a numerical study, the important parameters, which have significant influence on the accuracy of the MFS for the analysis of two-dimensional anisotropic elastostatic problems, are investigated. The studied parameters are the degree of anisotropy of the problem, the ratio of the number of collocation points to the number of source points, and the distance between main and pseudo boundaries. It is observed that as the anisotropy of the material increases, there will be more errors in the results. It is also observed that for simple problems, increasing the distance between main and pseudo boundaries enhances the accuracy of the results; however, it is not the case for complicated problems. Moreover, it is concluded that more collocation points than source points can significantly improve the accuracy of the results.

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“…source points and the ill conditioning. The location of the source points has been widely addressed in the literature ( [1,2,3,13,19,23,24,26,27,33]) and several choices have been advocated to be effective. Some works have proposed techniques to alleviate the ill conditioning if the MFS [9,10,17,37], but none of these approaches seem to completely solve the problem of ill conditioning.…”

Section: Introductionmentioning

confidence: 99%

A well conditioned Method of Fundamental Solutions

Antunes¹

2021

Preprint

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The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved typically are illconditioned and this may prevent to achieve high accuracy.In this work, we propose a new algorithm to remove the ill conditioning of the classical MFS in the context of Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization we define well conditioned basis functions spanning the same functional space as the MFS's. Several numerical examples show that this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.

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An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

Cited by 110 publications

References 298 publications

The spurious resonance disease and how to cure it: application to the seismic response of a canyon

The spurious resonance disease and how to cure it: application to the seismic response of a canyon

The Method of Fundamental Solutions for Two-Dimensional Elastostatic Problems with Stress Concentration and Highly Anisotropic Materials

A well conditioned Method of Fundamental Solutions

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