Recent Advances in Radial Basis Function Collocation Methods

Chen, C.S.; Fu, Zhuojia; Chen, Wen

doi:10.1007/978-3-642-39572-7

Cited by 169 publications

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“…Many types of meshless RBF methods have been proposed in the literature [9,10]. Radial point interpolation method (RPIM) can be included among the methods that represent the functions of unknown field variables by series.…”

Section: Radial Point Interpolation Methodsmentioning

confidence: 99%

Inelastic damage using continuum damage mechanics in composite plate reinforced by unidirectional fibers

Žmindák

Dudinský

2018

MATEC Web Conf.

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Abstract.It is well that a finite element method is very popular simulation method to predict the physical behavior of systems and structures. In the last years an increase of interest in a new type of numerical methods known as meshless methods was observed. The paper deals with application of radial basis functions on modelling of inelastic damage using continuum damage mechanics of layered plate composite structures reinforced with long unidirectional fibers. For numerical simulations of elastic-plastic damage of layered composite plates own computational programs were implemented in MATLAB programming language. We will use the Newton-Raphson method to solve nonlinear systems of equations. Evaluation damage during plasticity has been solved using return mapping algorithm. The results of elastic-plastic damage analysis of composite plate with unsymmetrical laminate stacking sequence are presented.

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Section: Radial Point Interpolation Methodsmentioning

confidence: 99%

Inelastic damage using continuum damage mechanics in composite plate reinforced by unidirectional fibers

Žmindák

Dudinský

2018

MATEC Web Conf.

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“…Hence, in recent decades, such methods have attracted the attention of researchers in order to solve partial differential equations. See for example: Chen et al [6].…”

Section: Radial Basis Function Methodologymentioning

confidence: 99%

“…Hence, in recent decades, such methods have attracted the attention of researchers in order to solve partial differential equations. See for example: Chen et al [6].This type of technique approximates the solution by means of a linear combination of radial basis functions, which are globally defined and of exponential convergence, but they produce dense and badly conditioned interpolation matrices. On the other hand, some RBFs contain a shape parameter, a number that influences the precision of the numerical results [7,8].…”

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confidence: 99%

Applications of Radial Basis Function Schemes to Fractional Partial Differential Equations

Martínez¹,

Fuentes²

2017

Fractal Analysis - Applications in Physics, Engineering and Technology

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In modeling using diffusion equations, the relationship between fractal geometry and fractional calculus arises by modeling the conditions of the medium as a fractal whose fractional dimension determines the order of this equation. For this reason, it is very useful to have numerical methods that solve them and discretization of the domain is not determinant for the efficiency of the algorithm. In this work, it is proposed to show that meshless methods, in particular methods with radial basis functions (RBF), are an alternative to schemes in differences or structured meshes. We show that we can obtain numerical solutions to some fractional partial differential equations using collocation and RBF, over non equally distributed data.Keywords: fractional partial differential equations (FPDE), meshless methods, radial basis functions (RBF), Caputo derivative, Riemann-Liouville derivative, Riesz derivative, diffusion-convection 1. Background Radial basis function methodologyThe Hardy-based radial-based functions (RBF) methodology [1] arises from the need to apply multivariate interpolation to cartography problems, with randomly dispersed data (also known as collocation nodes). Micchelli [2], Powell et al. [3] gave it a great boost by proving non-singularity theorems. Later, Kansa [4,5] proposed to consider the analytical derivatives of the FBR to develop numerical schemes that deal with partial differential equations (PDE).Regarding PDE over spaces of dimension greater than one, we generally opt for finite element (FE) type discretizations on meshes, structured or not; also pseudo spectral (PS) methods through base functions such as Fourier or Chebyshev. The high degree of computational © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. efficiency in the procedure raises the cost in the regularity constraints on the form of the computational domain.The FE methods involve decomposition of the domain; for example, in two dimensions rectangles are constructed with curvilinear mappings that allow the refinement of the mesh in critical areas. However, this type of implementation is complex and very close nodes are needed, mainly at the domain boundaries, which impairs stability conditions in time.For this reason, we look for numerical techniques that do not depend to a great extent on data distribution. RBF collocation methods belong to the "meshfree" methods; that is, they only require scattered collocation nodes on the domain and the boundary. They are also an alternative for dealing with problems in larger dimensions and irregular domains. Hence, in recent decades, such methods have attracted the attention of researchers in order to solve partial differential equations. See for example: Chen et al. [6].This type of technique approximates the solution by means of a linear...

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“…One can conclude that although there are numerical examples showing that the matrix can be singular for arbitrary center locations [11], these cases are rare and many other works [12] indicate a successful application of the method.…”

Section: Kansa Methodsmentioning

confidence: 99%

The adaptation of the cross validation aproach for RBF-based collocation methods

Krowiak

Podgórski²

2017

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The adaptation of the cross validation aproach for rbf-based collocation methods Adaptacja podejscia krzyżowego sprawdzania do metod bazujących na funkcjach rbf Abstract The paper shows the adaptation of the cross validation approach, known from interpolation problems, for estimating the value of a shape parameter for radial basis functions. The latter are involved in two collocation techniques used on an unstructured grid to find approximate solution of differential equations. To obtain accurate results, the shape parameter should be chosen as a result of a trade-off between accuracy and conditioning of the system. The cross validation approach called "leave one out" takes these aspects into consideration. The numerical examples that summarize the investigations show the usefulness of the approach.Keywords: radial basis functions, shape parameter, cross-validation approach Streszczenie W artykule pokazano adaptację algorytmu krzyżowego sprawdzania, znanego z zagadnień statystyki i interpolacji, do wyznaczenia wartości współczynnika kształtu w radialnych funkcjach bazowych. Funkcje te użyto w dwóch typach technik kolokacyjnych stosowanych na nieregularnej siatce do przybliżonego rozwiązywania równań różniczkowych. Aby otrzymać rezultaty o odpowiedniej dokładności, współczynnik kształtu powinien być dobrany na bazie kompromisu pomiędzy dokładnością a uwarunkowaniem układu równań. Przedstawiony algorytm, zwany "leave one out", bierze te aspekty pod uwagę. Podsumowaniem artykułu są numeryczne testy, które pokazują użyteczność tego podejścia.Słowa kluczowe: radialne funkcje bazowe, współczynnik kształtu, algorytm cross-validation

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Recent Advances in Radial Basis Function Collocation Methods

Cited by 169 publications

References 0 publications

Inelastic damage using continuum damage mechanics in composite plate reinforced by unidirectional fibers

Inelastic damage using continuum damage mechanics in composite plate reinforced by unidirectional fibers

Applications of Radial Basis Function Schemes to Fractional Partial Differential Equations

The adaptation of the cross validation aproach for RBF-based collocation methods

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