A multiresolution prewavelet-based adaptive refinement scheme for RBF approximations of nearly singular problems

Libre, Nicolas Ali; Emdadi, Arezoo; Kansa, Edward J.; Shekarchi, Mohammad; Rahimian, M.

doi:10.1016/j.enganabound.2009.02.007

Cited by 29 publications

(20 citation statements)

References 47 publications

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“…There are three types of meshless techniques: meshless techniques based on weak forms such as the element free Galerkin method , meshless techniques based on collocation techniques (strong forms) such as the meshless collocation technique based on radial basis functions (RBFs) , and meshless techniques based on the combination of weak forms and collocation technique. Because of the ill‐conditioning of the resultant linear systems in RBF‐collocation method, various approaches are proposed to overcome this difficulty; References being among them. The weak forms are used to derive a set of algebraic equations through a numerical integration process using a set of quadrature domain that might be constructed globally or locally in the domain of the problem.…”

Section: Preliminariesmentioning

confidence: 99%

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

Shivanian

2015

Numerical Meth Engineering

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SUMMARYIn this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form u t t C˛u t Cˇu D u xx C u yy C u´´C f .x; y;´; t/; x T D .x; y;´/ 2 Â R d ; .d D 2; 3/; t > 0 subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3-D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two-step time discretization method is employed to evaluate the time derivatives. This suggests Crank-Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones.

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Section: Preliminariesmentioning

confidence: 99%

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

Shivanian

2015

Numerical Meth Engineering

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“…Due to the ill-conditioning of the resultant linear systems in the RBFcollocation method, various approaches are proposed to circumvent this problem, Ling et al (2006), Ling and Schaback (2008), Libre et al (2008), Libre et al (2009) being among them. The weak forms are used to derive a set of algebraic equations through a numerical integration process using a set of quadrature domain that may be constructed globally or locally in the domain of the problem.…”

Section: Introductionmentioning

confidence: 99%

Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations

Shivanian

2014

Ocean Engineering

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“…Due to the ill-conditioning of the resultant linear systems in RBF-collocation method, various approaches are proposed to circumvent this problem, Refs. [7][8][9][10][11][12] being among them. The weak forms are used to derive a set of algebraic equations through a numerical integration process using a set of quadrature domain that may be constructed globally or locally in the domain of the problem.…”

Section: Introductionmentioning

confidence: 99%

Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem

Shivanian

Khodabandehlo

2016

Ain Shams Engineering Journal

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In this paper, the meshless local radial point interpolation (MLRPI) method is applied to one-dimensional inverse heat conduction problems. The meshless LRPIM is one of the truly meshless methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. Numerical experiments are carried out and compared with implicit Finite difference method. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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A multiresolution prewavelet-based adaptive refinement scheme for RBF approximations of nearly singular problems

Cited by 29 publications

References 47 publications

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations

Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem

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