The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates

Alves, Carlos J.S.; Antunes, Pedro R. S.

doi:10.1002/nme.2404

Cited by 35 publications

(27 citation statements)

References 20 publications

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“…9(b), but a lump singularity in one radial direction as shown in Fig. 9(c) as mentioned by Alves and Antunes [35]. In this paper, our image location in the MFS only lumps on the radial direction which agrees with the optimal location in [34,35].…”

Section: Methods Of Fundamental Solutions (Image Method)supporting

confidence: 83%

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

Lee

et al. 2009

Engineering Analysis with Boundary Elements

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Section: Methods Of Fundamental Solutions (Image Method)supporting

confidence: 83%

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

Lee

et al. 2009

Engineering Analysis with Boundary Elements

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“…The MFS is a simple and efficient scheme and has been widely applied to various engineering and science problems, such as heat conduction [17][18][19], acoustics [20,21], diffusion-reaction [22,23], axisymmetric elasticity [24], stokes problem [25][26][27][28], and free vibration [29][30][31], just to mention a few. Furthermore, Smyrlis and Karageorghis [32][33][34][35][36], Drombosky [37], Marin [38], and Lin et al [39] presented some fundamental theoretical analysis about the MFS.…”

Section: The Methods Of Fundamental Solutionsmentioning

confidence: 99%

Boundary-Type RBF Collocation Methods

Chen

2013

Recent Advances in Radial Basis Function Collocation Methods

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The mesh generation in the standard BEM is still not trivial as one may imagine, especially for high-dimensional moving boundary problems. To overcome this difficulty, the boundary-type RBF collocation methods have been proposed and endured a fast development in the recent decade thanks to being integration-free, spectral convergence, easy-to-use, and inherently truly meshless. First, this chapter introduces the basic concepts of the method of fundamental solutions (MFS). Then a few recent boundary-type RBF collocation schemes are presented to tackle the issue of the fictitious boundary in the MFS, such as boundary knot method (BKM), regularized meshless method, and singular boundary method. Following this, an improved multiple reciprocity method (MRM), the recursive composite MRM (RC-MRM), is introduced to establish a boundary-only discretization of nonhomogeneous problems. Finally, numerical demonstrations show the convergence rate and stability of these boundary-type RBF collocation methods for several benchmark examples.Keywords Meshless Á Integration-free Á Collocation Á Fundamental solutions Á Singularity Á Method of fundamental solutions Á Boundary knot method Á Regularized meshless method Á Singular boundary method Á Boundary particle method During the past two decades we have witnessed a research boom on the boundarytype meshless techniques since the construction of a mesh in the standard BEM is not trivial. Among the typical techniques are the boundary node method, the local boundary integral equation method, the boundary cloud method, the boundary point method, the boundary point interpolation method (BPIM), and the method of fundamental solutions (MFS). The essence of all these techniques, excluding the MFS and BPIM, is basically a combination of the moving least square (MLS) technique with various boundary element schemes, whereas the MFS is a boundary-type RBF collocation scheme.Such MLS-based methods involve singular integration and are mathematically complicated, and their low-order approximations also depress computational efficiency. The numerical integration requires a background mesh, and thus the

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“…The coefficients α j and β j will be calculated by fitting the boundary conditions. As in [2] or [3] we define m collocation points x 1 , . .…”

Section: Brief Description Of the Numerical Methodsmentioning

confidence: 99%

Convex shape optimization for the least biharmonic Steklov eigenvalue

Antunes

Gazzola

2013

ESAIM: COCV

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Abstract. The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L 2 -norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.

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The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates

Cited by 35 publications

References 20 publications

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

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

Boundary-Type RBF Collocation Methods

Convex shape optimization for the least biharmonic Steklov eigenvalue

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