On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs

Bayona, Víctor; Flyer, Natasha; Fornberg, Bengt; Barnett, G. A.

doi:10.1016/j.jcp.2016.12.008

Cited by 221 publications

(252 citation statements)

References 23 publications

(58 reference statements)

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“…effectively adding an additional constraint ∑ n i i=1 w i = 0. The effects of adding polynomial constraints to RBF-FD approximations have been studied recently by Bayona et al 7 In cases discussed in this paper, additional consistency constraints were not needed to obtain satisfying results. Another possible modification to the RBF-FD above procedure is to include more stencil points then basis functions, putting basis functions only on the closest m i nodes.…”

Section: Rbf-generated Finite Differencesmentioning

confidence: 98%

“…A popular variant among many local meshless methods is the radial basis function–generated finite differences (RBF‐FD) method, which uses finite difference‐like collocation weights on an unstructured set of nodes. The method has been successfully used in several problems and is still actively researched . In the field of solid mechanics, where problems are traditionally tackled with the finite element method (FEM), meshless methods surfaced as a response to the cumbersome meshing of realistic three‐dimensional (3D) domains required by FEM .…”

Section: Introductionmentioning

confidence: 99%

“…The method has been successfully used in several problems and is still actively researched. [6][7][8] In the field of solid mechanics, where problems are traditionally tackled with the finite element method (FEM), 9 meshless methods surfaced as a response to the cumbersome meshing of realistic three-dimensional (3D) domains required by FEM. 8,[10][11][12] The development began with weak form methods such as the element-free Galerkin method 13 and meshless local Petrov-Galerkin method, 14 followed by strong form collocation approach that emerged with finite point method (FPM) 15 and further developed with similar local strong form methods.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive radial basis function–generated finite differences method for contact problems

Slak

Kosec

2019

Numerical Meth Engineering

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Summary This paper proposes an original adaptive refinement framework using radial basis function–generated finite differences method. Node distributions are generated with a Poisson disc sampling–based algorithm from a given continuous density function, which is altered during the refinement process based on the error indicator. All elements of the proposed adaptive strategy rely only on meshless concepts, which leads to great flexibility and generality of the solution procedure. The proposed framework is tested on four gradually more complex contact problems. First, a disc under pressure is considered and the computed stress field is compared to the closed‐form solution of the problem to assess the behaviour of the algorithm and the influence of free parameters. Second, a Hertzian contact problem is studied to analyse the proposed algorithm with an ad hoc error indicator and to test both refinement and derefinement. A contact problem, typical for fretting fatigue, with no known closed‐form solution is considered and solved next. It is demonstrated that the proposed methodology produces results comparable with finite element method without the need for manual refinement or any human intervention. In the last case, generality of the proposed approach is demonstrated by solving a three‐dimensional Boussinesq's problem.

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Section: Rbf-generated Finite Differencesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive radial basis function–generated finite differences method for contact problems

Slak

Kosec

2019

Numerical Meth Engineering

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“…There are workarounds for both of these problems. In the stationary interpolation case, it is possible to recover high-order convergence by adding suitable polynomial terms to the interpolant [14,3]. On a surface, however, the polynomials may themselves introduce ill-conditioning, especially if it is an algebraic surface as polynomial unisolvency becomes an issue.…”

Section: Parameter Studiesmentioning

confidence: 99%

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

Lehto¹,

Shankar²,

Wright³

2017

SIAM J. Sci. Comput.

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Abstract. We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in R d . The novelty of the method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds.

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“…The current direction in the research on RBF approximation methods for PDEs is towards the use of localized RBF approximation methods. The main categories are stencil-based methods (RBF-FD) [2,11] and partition of unity methods (RBF-PUM) [23,32]. The (FP-RBF) technique should carry over in both cases, with minor differences in the implementation, whereas the (RS-RBF) method should be applicable to RBF-PUM, but not as easily to RBF-FD.…”

Section: Introductionmentioning

confidence: 99%

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

2017

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Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.

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On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs

Cited by 221 publications

References 23 publications

Adaptive radial basis function–generated finite differences method for contact problems

Adaptive radial basis function–generated finite differences method for contact problems

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

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