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.

scholarly journals Solving the forward-backward heat equation in two-dimension by Radial basis functions mesh free method

2020 ◽  
Vol 22 (2) ◽  
pp. 305-318
Author(s):  
Siamak Banei ◽  
◽  
Kamal Shanazari ◽  
Yaqub Azari ◽  
◽  
...  
Keyword(s):  
Heat Equation ◽  
Radial Basis Functions ◽  
Basis Functions ◽  
Two Dimension ◽  
Mesh Free ◽  
Radial Basis ◽  
Mesh Free Method

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.

Solving PDEs with radial basis functions

Acta Numerica ◽  
2015 ◽  
Vol 24 ◽  
pp. 215-258 ◽  
Author(s):  
Bengt Fornberg ◽  
Natasha Flyer
Keyword(s):  
Radial Basis Functions ◽  
Finite Differences ◽  
Large Scale ◽  
Numerical Approach ◽  
Basis Functions ◽  
Integration Time ◽  
Small Scale ◽  
Major Step ◽  
Mesh Free ◽  
Radial Basis

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

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.

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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