Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

Hematiyan, M. R.; Arezou, M.; Dezfouli, N. Koochak; Khoshroo, M.

doi:10.32604/cmes.2019.08275

Cited by 7 publications

(9 citation statements)

References 31 publications

(36 reference statements)

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“…This same result is found as number of source points increases (Golberg & Chen, 1998, Liu, 2012. Some strategies to circumvent ill-conditioning were considered in Chen et al (2006), Young et al (2008), Chen et al (2016) and Hematiyan et al (2016).…”

Section: /15supporting

confidence: 61%

Meshless method using fundamental solution applied to computational simulation of groundwater flow of real aquifers: study case (Guariroba’s APA and Juazeiro do Norte)

Neto

Queiroz

Barbosa

et al. 2020

RBRH

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We investigated the influence of fictitious boundary distance, a parameter of MFS, to determine piezometric levels of two unconfined sedimentary aquifers assuming Dupuit-Forchheimer and steady-state flow hypothesis. Two study areas were modelled: Guariroba’s Environmental Protection Area, in Mato Grosso do Sul State, Brazil, and Juazeiro do Norte City, in Ceará State, Brazil. It was observed that in order to use the MFS as a numerical method in modeling groundwater flow, it is necessary to determine the best distance value of the fictitious boundary. This value can be chosen from the use of field data within the analyzed domain, where the relative error is a parameter to be minimized. Applying this methodology and comparing with the results of the MODFLOW application for the same set of initial data, we concluded that the MSF allows to estimate the piezometric level values within the analyzed domains and that the results of the statistical comparison between them point to the need to investigate the representativeness of both methods to determine which one is most appropriate for modelling the groundwater flow in each region.

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Section: /15supporting

confidence: 61%

Meshless method using fundamental solution applied to computational simulation of groundwater flow of real aquifers: study case (Guariroba’s APA and Juazeiro do Norte)

Neto

Queiroz

Barbosa

et al. 2020

RBRH

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“…The exact solution for radial displacement and stresses are uniform over the boundary of the disk; however, the solutions obtained by the MFS have some oscillations over the boundary in the cases with smaller values of K. It is observed that by increasing the value of K, the accuracy of the solutions is enhanced. The MFS results for isotropic elastostatic problems [40] show that K = 0.85 yields sufficiently accurate results; however, as it is seen in Fig. 5, the MFS results with K = 0.9 for the present example, which has a moderate degree of anisotropy, are not sufficiently accurate.…”

Section: A Circular Domainmentioning

confidence: 53%

“…Therefore, a parameter which describes the distance between main and pseudo boundaries relative to the number of soure points should be considered in the next numerical studies. Hematiyan et al [34,40] defined the source point location parameter for this purpose. The source point location parameter can be defined using Fig.…”

Section: Tablementioning

confidence: 99%

“…There are many studies on the location of source points and collocation points in the MFS. Among them, one can refere to the works on Laplace and Helmholtz equations [1,[31][32][33][34][35], biharmonic equation [36,37], torsion problem [38], transient heat conduction [39], and isotropic elasticity [40][41][42]. Yet, no research has been carried out on the configuration of source points for anisotopic elasticity.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Recently, in [4,5] was introduced the approach that dispenses with the use of the RBF method and we are going to derive it for the time dependent boundary value problem for the hyperbolic elasticity equation. There are a few studies for the elastodynamic problems that are based on the MFS-RBF, see [14,15,17] or on the boundary integral equations method [8,10]. This is due to the important application of the problem, for example, in ISSN 20785097.…”

Section: Introductionmentioning

confidence: 99%

An Application of the Method of Fundamental Solutions for the Elastodynamic Problem

Borachok

2022

VAM

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A method of fundamental solutions (MFS) is employed for the numerical solution of the initial boundary value problem for the elastodynamic equation in annular planar domains. Using the Laguerre transformation in the time, the non-stationary problem is semi discretized to a sequence of stationary Dirichlet problems with an inhomogeneous equation, for which the sequence of fundamental solutions is known. The solutions of stationary problems are found by the MFS, when the unknown functions are approximated by a linear combination of narrowing of the elements from the fundamental sequence, and the source points are placed uniformly on articial boundaries, located at xed distances from the boundaries of the domain. The unknown coecients in the MFS-approximations are found using the collocation method, taking into account the Dirichlet conditions on the boundaries of the domain. As a result, we obtain a sequence of recurrent SLAEs with the same matrix and recurrent right-side parts, that depend on the solutions from previous iterations. In general, for the numerical solution of a problem with an inhomogeneous equation by the method of fundamental solutions, it is necessary to nd a partial solution of the inhomogeneous equation, for example, by the method of radial basis functions, however, according to our approach, this is not necessary. A step-by-step algorithm for the numerical solution of the given problem is described and the algorithm for the distribution of collocation points and source points is shown. The results of numerical experiments for dierent domain congurations are presented, which conrm the applicability and eectiveness of the proposed approach.

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Some Remarks on the Method of Fundamental Solutions for Two-Dimensional Elasticity

Cited by 7 publications

References 31 publications

Meshless method using fundamental solution applied to computational simulation of groundwater flow of real aquifers: study case (Guariroba’s APA and Juazeiro do Norte)

Meshless method using fundamental solution applied to computational simulation of groundwater flow of real aquifers: study case (Guariroba’s APA and Juazeiro do Norte)

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

An Application of the Method of Fundamental Solutions for the Elastodynamic Problem

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