Energy based approach for shape parameter selection in radial basis functions collocation method

Iurlaro, Luigi; Gherlone, Marco; Sciuva, Marco Di

doi:10.1016/j.compstruct.2013.07.041

Cited by 18 publications

(7 citation statements)

References 27 publications

(49 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…MQ-RBF's includes the user-defined shape parameter c, which has an influence on accuracy and stability of the solution [37,38,39]. Some of the most popular RBFs function are listed in table 1.…”

Section: Radial Basis Function (Rbf)mentioning

confidence: 99%

The numerical study on the boundary layer flow and heat transfer over a vertical slender cylinder using RBF collocation method

Parand¹,

Baharlouei²

2018

Communications in Numerical Analysis

View full text Add to dashboard Cite

In this study, the effects of blowing/injection and suction on the steady mixed convection boundary layer flows over a vertical slender cylinder have been considered. It is assumed that the mainstream velocity and the wall surface temperature of the cylinder are proportional to the axial distance along the surface. The governing equations of motion, energy, and nanoparticles are simplified by applying appropriate boundary layer approximation. These partial differential equations are reduced to highly non-linear ordinarily differential equations, by using similarity transformation. The resulting differential system is solved by employing the meshfree method for the first time. The convergence of the obtained series solution is carefully checked and comparison with other algorithms shows an excellent agreement. The physical behavior of the different parameters involved in these equations is discussed in details and presented graphically.

show abstract

Section: Radial Basis Function (Rbf)mentioning

confidence: 99%

The numerical study on the boundary layer flow and heat transfer over a vertical slender cylinder using RBF collocation method

Parand¹,

Baharlouei²

2018

Communications in Numerical Analysis

View full text Add to dashboard Cite

show abstract

“…As stated above, the choice of a suitable value of the shape parameter in the infinitely smooth RBFs is very important and has been an ongoing challenge problem. Up to now, no mathematical theory has been developed to rule the selection of an adequate value but only several approaches are available in the open literature [7][8][9][10][11]. In the early works, many researchers tried to present some empirical formulas based on the number and distribution of data points to select a good value for the shape parameter; see [12][13][14] for examples.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, some improvements of the LOOCV algorithm have been deeply studied; see [16][17][18][19][20] and references therein. Other techniques for finding the optimal shape parameter include the energy-based method [7,9], the sample solution approach [10], the interval method [21], and so on. Another type of technique [22,23], i.e., bypassing the problem of the optimal shape parameter selection by reducing its influence on the stability of the method, is also of great interest.…”

Section: Introductionmentioning

confidence: 99%

A Coupled RBF Method for the Solution of Elastostatic Problems

Chen

Cao

2021

Mathematical Problems in Engineering

View full text Add to dashboard Cite

Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.

show abstract

“…Bayona (2011, 2012) [ 40 , 41 ] has attempted to find optimal constant and variable shape parameters for the multiquadric approximation method combined with the finite difference method. Tsai et al (2010) [ 42 ] proposed the golden section algorithm, Esmaeilbeigi and Hosseini (2014) [ 43 ] the genetic algorithm, Iurlaro et al (2014) [ 44 ] the energy-based approach, and Chen et al (2010) [ 14 ] the sample solution approach. Despite the very intensive development of methods for finding a good value of the shape parameter in the recent years, the choice of the shape parameter for multiply-connected, complex-shaped domain problems remains an open problem [ 14 ].…”

Section: Introductionmentioning

confidence: 99%

Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter

Popczyk

Dziatkiewicz

2021

Materials

View full text Add to dashboard Cite

New engineering materials exhibit a complex internal structure that determines their properties. For thermal metamaterials, it is essential to shape their thermophysical parameters’ spatial variability to ensure unique properties of heat flux control. Modeling heterogeneous materials such as thermal metamaterials is a current research problem, and meshless methods are currently quite popular for simulation. The main problem when using new modeling methods is the selection of their optimal parameters. The Kansa method is currently a well-established method of solving problems described by partial differential equations. However, one unsolved problem associated with this method that hinders its popularization is choosing the optimal shape parameter value of the radial basis functions. The algorithm proposed by Fasshauer and Zhang is, as of today, one of the most popular and the best-established algorithms for finding a good shape parameter value for the Kansa method. However, it turns out that it is not suitable for all classes of computational problems, e.g., for modeling the 1D heat conduction in non-homogeneous materials, as in the present paper. The work proposes two new algorithms for finding a good shape parameter value, one based on the analysis of the condition number of the matrix obtained by performing specific operations on interpolation matrix and the other being a modification of the Fasshauer algorithm. According to the error measures used in work, the proposed algorithms for the considered class of problem provide shape parameter values that lead to better results than the classic Fasshauer algorithm.

show abstract

Energy based approach for shape parameter selection in radial basis functions collocation method

Cited by 18 publications

References 27 publications

The numerical study on the boundary layer flow and heat transfer over a vertical slender cylinder using RBF collocation method

The numerical study on the boundary layer flow and heat transfer over a vertical slender cylinder using RBF collocation method

A Coupled RBF Method for the Solution of Elastostatic Problems

Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter

Contact Info

Product

Resources

About