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Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.


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Journal ArticleDOI
TL;DR: This paper presents different types of RBF methods to solve PDEs, including Kansa method, Hermite symmetric approach, localized and hybrid methods, and discussed the preference of using meshfree methods like RBF over the mesh based methods.
Abstract: Background/Objectives: The approximation using radial basis function (RBF) is an extremely powerful method to solve partial differential equations (PDEs). This paper presents different types of RBF methods to solve PDEs. Methods/ Statistical Analysis: Due to their meshfree nature, ease of implementation and independence of dimension, RBF methods are popular to solve PDEs. In this paper we examine different generalized RBF methods, including Kansa method, Hermite symmetric approach, localized and hybrid methods. We also discussed the preference of using meshfree methods like RBF over the mesh based methods. Findings: This paper presents a state-of-the-art review of the RBF methods. Some recent development of RBF approximation in solving PDEs is also discussed. The mathematical formulation of different RBF methods are discussed for better understanding. RBF methods have been actively developed over the years from global to local approximation and then to hybrid methods. Hybrid RBF methods help in reduction of computational cost and become very effective in solving large scale problems. Application/Improvements: RBF methods have been applied to various diverse fields like image processing, geo-modeling, pricing option and neural network etc.

15 citations

Proceedings ArticleDOI
01 Jan 2004
TL;DR: A domain- decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue is developed.
Abstract: Mesh reduction methods such as the boundary element methods, method of fundamental solutions or the so-called meshless methods all lead to fully populated matrices. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of non-symmetric equations. Researchers have developed several approaches to address this issue including the class of fast-multipole techniques, use of wavelet transforms, and matrix decomposition. In this paper, we develop a domain- decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue. The meshless method we employ is based on expansions using radial basis functions (RBFs). An efficient physically-based procedure provides an effective initial guess of the temperatures along the sub-domain interfaces. The iteration process converges very efficiently, offers substantial savings in memory, and features superior computational efficiency. The meshless iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation under MPI standards on a small Windows XP PC cluster. Numerical results reveal the domain decomposition meshless methods produce accurate temperature predictions while requiring a much- reduced effort in problem preparation in comparison to other traditional numerical methods.

15 citations

Posted Content
TL;DR: In this article, mesh-free finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian are defined on unstructured point clouds, allowing for very complicated domains and a non-uniform distribution of discretisation points.
Abstract: We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which allows for very complicated domains and a non-uniform distribution of discretisation points. The schemes are monotone, which ensures that they converge to the viscosity solution of the underlying PDE as long as the equation has a comparison principle. Numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds, complex domains, degenerate equations, and singular solutions.

15 citations

Journal ArticleDOI
TL;DR: A meshless element-free Galerkin method is applied for the first time to solve pulsed eddy-current problems and is validated against analytic solutions for two canonical cases.
Abstract: Meshless methods have attracted great attention due to their advantage in geometric representation. In this paper, a meshless element-free Galerkin method is applied for the first time to solve pulsed eddy-current problems. Detailed mathematical derivations and the numerical implementation are discussed. The model is validated against analytic solutions for two canonical cases.

15 citations

Journal ArticleDOI
TL;DR: In this paper, a new hybrid numerical method, the hM-DOR method, which is based on an order-reduction technique for partial differential equations, combines the true-meshless collocation technique with a fixed reproducing kernel approximation.

15 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202355
2022112
2021102
202092
201996
201897