A Meshless Discrete Galerkin Method Based on the Free Shape Parameter Radial Basis Functions for Solving Hammerstein Integral Equation

2018 ◽  
Vol 11 (3) ◽  
pp. 540-568 ◽  
Author(s):  
Pouria Assari and Mehdi Dehghan
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Radial Basis Functions ◽  
Shape Parameter ◽  
Basis Functions ◽  
Hammerstein Integral Equation ◽  
Radial Basis ◽  
Discrete Galerkin Method

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.

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.

Mixed-Mode Stress Intensity Factors by Mesh Free Galerkin Method

2009 ◽  
Vol 417-418 ◽  
pp. 957-960
Author(s):  
P.H. Wen ◽  
M.H. Aliabadi
Keyword(s):  
Stress Intensity ◽  
Galerkin Method ◽  
Radial Basis Functions ◽  
Stress Intensity Factors ◽  
Mixed Mode ◽  
Basis Functions ◽  
Intensity Factors ◽  
Mesh Free ◽  
Radial Basis ◽  
Mode Stress

An element-free Galerkin method is developed using radial basis interpolation functions to evaluate static and dynamic mixed-mode stress intensity factors. For dynamic problems, the Laplace transform technique is used to transform the time domain problem to frequency domain. The so-called enriched radial basis functions are introduced to accurately capture the singularity of stress at crack tip. The accuracy and convergence of mesh free Galerkin method with enriched radial basis functions for the two-dimensional static and dynamic fracture mechanics are demonstrated through several benchmark examples. Comparisons have been made with benchmarks and solutions obtained by the boundary element method.

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