Numerical Investigation of Electromagnetic Scattering Problems Based on the Compactly Supported Radial Basis Functions

2016 ◽  
Vol 71 (8) ◽  
pp. 677-690 ◽  
Author(s):  
Hadi Roohani Ghehsareh ◽  
Seyed Kamal Etesami ◽  
Maryam Hajisadeghi Esfahani
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Cross Section ◽  
Computational Cost ◽  
Basis Functions ◽  
Test Problems ◽  
Em Scattering ◽  
Compactly Supported ◽  
Radial Basis ◽  
Field Integral

AbstractIn the current work, the electromagnetic (EM) scattering from infinite perfectly conducting cylinders with arbitrary cross sections in both transverse magnetic (TM) and transverse electric (TE) modes is numerically investigated. The problems of TE and TM EM scattering can be mathematically modelled via the magnetic field integral equation (MFIE) and the electric field integral equation (EFIE), respectively. An efficient technique is performed to approximate the solution of these surface integral equations. In the proposed numerical method, compactly supported radial basis functions (RBFs) are employed as the basis functions. The radial and compactly supported properties of these basis functions substantially reduce the computational cost and improve the efficiency of the method. To show the accuracy of the proposed technique, it has been applied to solve three interesting test problems. Moreover, the method is well used to compute the electric current density and also the radar cross section (RCS) for some practical scatterers with different cross section geometries. The reported numerical results through the tables and figures demonstrate the efficiency and accuracy of the proposed technique.

scholarly journals Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 223
Author(s):  
Pedro González-Rodelas ◽  
Miguel Pasadas ◽  
Abdelouahed Kouibia ◽  
Basim Mustafa
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Volterra Integral Equation ◽  
Computational Cost ◽  
Basis Functions ◽  
Numerical Examples ◽  
Acceptable Accuracy ◽  
Convergence Results ◽  
Radial Basis ◽  
Low Computational Cost

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

scholarly journals A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 964 ◽  
Author(s):  
Zhiyong Liu ◽  
Qiuyan Xu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Radial Basis Functions ◽  
Computational Cost ◽  
Basis Functions ◽  
Partial Differential ◽  
Compactly Supported ◽  
Linear Partial Differential Equations ◽  
Numerical Experimentation ◽  
Radial Basis

In this paper, we derive and discuss the hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations. Similar to multilevel splitting of finite element spaces, the hierarchical radial basis functions are constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the compactly supported radial basis functions approximation and stationary multilevel approximation, the new method can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system. These observations are obtained by taking the direct approach of numerical experimentation.

A meshless local Galerkin integral equation method for solving a type of Darboux problems based on the radial basis functions

2021 ◽  
Vol 63 ◽  
pp. 469-492
Author(s):  
Pouria Assari ◽  
Fatemeh Asadi-Mehregan ◽  
Mehdi Dehghan
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Integral Equation Method ◽  
Basis Functions ◽  
Computer Memory ◽  
Shape Functions ◽  
Integration Rule ◽  
Radial Basis ◽  
Small Set ◽  
Discrete Galerkin Method

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions. doi:10.1017/S1446181121000377

A MESHLESS LOCAL GALERKIN INTEGRAL EQUATION METHOD FOR SOLVING A TYPE OF DARBOUX PROBLEMS BASED ON RADIAL BASIS FUNCTIONS

2021 ◽  
pp. 1-24
Author(s):  
P. ASSARI ◽  
F. ASADI-MEHREGAN ◽  
M. DEHGHAN
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Integral Equation Method ◽  
Basis Functions ◽  
Computer Memory ◽  
Shape Functions ◽  
Integration Rule ◽  
Radial Basis ◽  
Small Set ◽  
Discrete Galerkin Method

Abstract The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions.

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