Point interpolation method based on local residual formulation using radial basis functions

2002 ◽  
Vol 14 (6) ◽  
pp. 713-732 ◽  
Author(s):  
G.R. Liu ◽  
L. Yan ◽  
J.G. Wang ◽  
Y.T. Gu
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
Basis Functions ◽  
Radial Basis ◽  
Point Interpolation Method ◽  
Point Interpolation

scholarly journals On the Factors Affecting the Accuracy and Robustness of Smoothed-Radial Point Interpolation Method

2016 ◽  
Vol 9 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Abderrachid Hamrani ◽  
Idir Belaidi ◽  
Eric Monteiro ◽  
Philippe Lorong
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
Basis Functions ◽  
Integration Scheme ◽  
Shape Parameters ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Radial Basis ◽  
Point Interpolation Method ◽  
Point Interpolation

AbstractIn order to overcome the possible singularity associated with the Point Interpolation Method (PIM), the Radial Point Interpolation Method (RPIM) was proposed by G. R. Liu. Radial basis functions (RBF) was used in RPIM as basis functions for interpolation. All these radial basis functions include shape parameters. The choice of these shape parameters has been and stays a problematic theme in RBF approximation and interpolation theory. The object of this study is to contribute to the analysis of how these shape parameters affect the accuracy of the radial PIM. The RPIM is studied based on the global Galerkin weak form performed using two integration technics: classical Gaussian integration and the strain smoothing integration scheme. The numerical performance of this method is tested on their behavior on curve fitting, and on three elastic mechanical problems with regular or irregular nodes distributions. A range of recommended shape parameters is obtained from the analysis of different error indexes and also the condition number of the matrix system. All resulting RPIM methods perform very well in term of numerical computation. The Smoothed Radial Point Interpolation Method (SRPIM) shows a higher accuracy, especially in a situation of distorted node scheme.

scholarly journals A computational homogenization analysis of materials using the stabilized mesh-free method based on the radial basis functions

2020 ◽  
Vol 14 (1) ◽  
pp. 65-76
Author(s):  
Ho Le Huy Phuc ◽  
Le Van Canh ◽  
Phan Duc Hung
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
Computational Homogenization ◽  
Basis Functions ◽  
Computational Aspect ◽  
Mesh Free ◽  
Kronecker Delta ◽  
Radial Basis ◽  
Point Interpolation Method ◽  
Mesh Free Method

This study presents a novel application of mesh-free method using the smoothed-radial basis functions for the computational homogenization analysis of materials. The displacement field corresponding to the scattered nodes within the representative volume element (RVE) is split into two parts including mean term and fluctuation term, and then the fluctuation one is approximated using the integrated radial basis function (iRBF) method. Due to the use of the stabilized conforming nodal integration (SCNI) technique, the strain rate is smoothed at discreted nodes; therefore, all constrains in resulting problems are enforced at nodes directly. Taking advantage of the shape function satisfies Kronecker-delta property, the periodic boundary conditions well-known as the most appropriate procedure for RVE are similarly imposed as in the finite element method. Several numerical examples are investigated to observe the computational aspect of iRBF procedure. The good agreement of the results in comparison with those reported in other studies demonstrates the accuracy and reliability of proposed approach. Keywords: homogenization analysis; mesh-free method; radial point interpolation method; SCNI scheme.

Time fractional modified anomalous sub-diffusion equation with a nonlinear source term through locally applied meshless radial point interpolation

2018 ◽  
Vol 32 (22) ◽  
pp. 1850251 ◽  
Author(s):  
Elyas Shivanian ◽  
Ahmad Jafarabadi
Keyword(s):  
Diffusion Equation ◽  
Radial Basis Functions ◽  
Source Term ◽  
Interpolation Method ◽  
Two Dimensions ◽  
Basis Functions ◽  
Nonlinear Source ◽  
Radial Point Interpolation ◽  
Radial Basis ◽  
Point Interpolation

In this paper, an alternative approach of spectral meshless radial point interpolation (SMRPI) is applied to the modified anomalous fractional sub-diffusion equation with a nonlinear source term in one and two dimensions. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on a combination of meshless methods and the spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of the SMRPI. It is proved that the scheme is unconditionally stable with respect to the time variable in [Formula: see text] and convergent with the order of convergence [Formula: see text], [Formula: see text]. In this work, the thin plate splines (TPS) are used as the radial basis functions. In order to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is used. The results of numerical experiments are compared to the analytical solutions in order to confirm the accuracy and the efficiency of the presented scheme.

Fundamental Solutions of PDEs as Radial Basis Functions in Multivariate Interpolation

2007 ◽  
Vol 7 (4) ◽  
pp. 321-340
Author(s):  
A. Masjukov
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
A Priori ◽  
Fundamental Solutions ◽  
Basis Functions ◽  
Scale Parameters ◽  
Radial Basis ◽  
Undecidable Problem ◽  
The Right ◽  
Smooth Approximations

AbstractFor bivariate and trivariate interpolation we propose in this paper a set of integrable radial basis functions (RBFs). These RBFs are found as fundamental solutions of appropriate PDEs and they are optimal in a special sense. The condition number of the interpolation matrices as well as the order of convergence of the inter- polation are estimated. Moreover, the proposed RBFs provide smooth approximations and approximate fulfillment of the interpolation conditions. This property allows us to avoid the undecidable problem of choosing the right scale parameter for the RBFs. Instead we propose an iterative procedure in which a sequence of improving approx- imations is obtained by means of a decreasing sequence of scale parameters in an a priori given range. The paper provides a few clear examples of the advantage of the proposed interpolation method.

scholarly journals Three-Dimensional Swirl Flow Velocity-Field Reconstruction Using a Neural Network With Radial Basis Functions

2001 ◽  
Vol 123 (4) ◽  
pp. 920-927 ◽  
Author(s):  
J. Pruvost ◽  
J. Legrand ◽  
P. Legentilhomme
Keyword(s):  
Neural Network ◽  
Velocity Field ◽  
Radial Basis Functions ◽  
Swirling Flow ◽  
Interpolation Method ◽  
Measurement Techniques ◽  
Three Dimensional ◽  
Basis Functions ◽  
Hydrodynamic Characteristics ◽  
Radial Basis

For many studies, knowledge of continuous evolution of hydrodynamic characteristics is useful but generally measurement techniques provide only discrete information. In the case of complex flows, usual numerical interpolating methods appear to be not adapted, as for the free decaying swirling flow presented in this study. The three-dimensional motion involved induces a spatial dependent velocity-field. Thus, the interpolating method has to be three-dimensional and to take into account possible flow nonlinearity, making common methods unsuitable. A different interpolation method is thus proposed, based on a neural network algorithm with Radial Basis Functions.

scholarly journals Radial Basis Point Interpolation Method with Reordering Gauss Domains for 2D Plane Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Shi-Chao Yi ◽  
Fu-jun Chen ◽  
Lin-Quan Yao
Keyword(s):  
Interpolation Method ◽  
Computational Time ◽  
Integration Scheme ◽  
Cpu Time ◽  
Integration Schemes ◽  
Basis Point ◽  
Radial Basis ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Gauss Points

We present novel Gauss integration schemes with radial basis point interpolation method (RPIM). These techniques define new Gauss integration scheme, researching Gauss points (RGD), and reconstructing Gauss domain (RGD), respectively. The developments lead to a curtailment of the elapsed CPU time without loss of the accuracy. Numerical results show that the schemes reduce the computational time to 25% or less in general.

Sign in / Sign up

Export Citation Format

Share Document