Adaptive RBF-FD method for elliptic problems with point singularities in 2D

Oanh, Đặng Thị; Davydov, Oleg; Phú, Hoàng Xuân

doi:10.1016/j.amc.2017.06.006

Cited by 40 publications

(44 citation statements)

References 9 publications

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“…Note that similar estimates involving general growth functions ρ q,D (z, X, · ) can be obtained for the error of the kernel-based numerical differentiation, generalizing the results in [13]. Indeed, (21) and (25) can be applied to the bound given in [13,Lemma 7].…”

Section: Minimal Formulas and Growth Functionsmentioning

confidence: 71%

“…• Growth functions ρ q,D (z, X, 1, q) and ρ q,D (z, X, 2, q) that appear in the estimates (29), (43), (44) can be computed as ρ q,D (z, X, 1, q) = w 1,q 1,q and ρ q,D (z, X, 2, q) = w 2,q 2,q , respectively, and used to assess the accuracy of numerical differentiation on a given set X, which may be used to improve the selection of sets of influence in the generalized finite difference methods. The ability to select good sets of influence is essential for the design of competitive adaptive algorithms [21]. Moreover, it may be possible to generate discretization nodes for a given domain Ω in such a way that the local growth functions for a given degree are small on these nodes, which guarantees small consistency error.…”

Section: Possible Applicationsmentioning

confidence: 99%

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Minimal numerical differentiation formulas

Davydov

Schaback

2018

Numer. Math.

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We investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector. Error bounds are given in terms of a growth function that carries the information about the geometry of the centers. Specific forms of weighted ℓ 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments. The results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods.

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Section: Minimal Formulas and Growth Functionsmentioning

confidence: 71%

Section: Possible Applicationsmentioning

confidence: 99%

Minimal numerical differentiation formulas

Davydov

Schaback

2018

Numer. Math.

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“…Details of particle distributions adopted in this work will be provided in Section 4.2. For a discussion of how generalized finite difference-type methods depend upon particle anisotropy and their relation to adaptive finite element methods, we refer to recent works [20,21]. The basis for the moving least squares method is to seek such an approximation as the solution of an optimization problem u h (x) = p * (x), where p * is the minimizer of…”

Section: Classical Mlsmentioning

confidence: 99%

Compact moving least squares: An optimization framework for generating high-order compact meshless discretizations

Trask

Maxey

2016

Journal of Computational Physics

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A generalization of the optimization framework typically used in moving least squares is presented that provides high-order approximation while maintaining compact stencils and a consistent treatment of boundaries. The approach, which we refer to as compact moving least squares, resembles the capabilities of compact finite differences but requires no structure in the underlying set of nodes. An efficient collocation scheme is used to demonstrate the capabilities of the method to solve elliptic boundary value problems in strong form stably without the need for an expensive weak form. The flexibility of the approach is demonstrated by using the same framework to both solve a variety of elliptic problems and to generate implicit approximations to derivatives. Finally, an efficient preconditioner is presented for the steady Stokes equations, and the approach's efficiency and high order of accuracy is demonstrated for domains with curvi-linear boundaries.

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“…This approach has been later successfully extended to the meshless solutions of elasticity problems, both weak form, using meshless finite volume method, node‐based smoothed point interpolation method, and strong form, using FPM . In RBF‐FD context, a ZZ type of error indicator has been discussed in a solution of Laplace equation in the work of Oanh et al An alternative class of error indicators is available for commonly used least squares–based meshless methods, which relies on the least squares approximation residual . The residual‐based error indicator has been successfully used in a meshless solution of an elasticity problem with a discrete least squares meshless method .…”

Section: Introductionmentioning

confidence: 99%

“…Since the algorithm does not guarantee smooth distributions, required by RBF-FD, 17 authors had to use special stencil selection algorithms to improve stability. 22 Similar h-refinement algorithm achieving a ratio of 2 17 between the densest and coarsest parts of the discretisation has been recently applied to elasticity problems. 8 No special stencil selection algorithms were needed in this case; however, a few steps of regularisation were needed to sufficiently improve the distribution.…”

Section: Introductionmentioning

confidence: 99%

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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Adaptive RBF-FD method for elliptic problems with point singularities in 2D

Cited by 40 publications

References 9 publications

Minimal numerical differentiation formulas

Minimal numerical differentiation formulas

Compact moving least squares: An optimization framework for generating high-order compact meshless discretizations

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

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