Using the digamma function for basis functions in mesh-free computational methods

2021 ◽  
Vol 131 ◽  
pp. 218-227
Author(s):  
Bryce D. Wilkins ◽  
Theodore V. Hromadka
Keyword(s):  
Computational Methods ◽  
Basis Functions ◽  
Digamma Function ◽  
Mesh Free

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.

Mesh-Free Simulation of Transport Phenomena in Continuous Castings of Aluminium Alloys

2006 ◽  
Vol 508 ◽  
pp. 497-502
Author(s):  
Božidar Šarler ◽  
Robert Vertnik
Keyword(s):  
Heat Treatment ◽  
Aluminium Alloys ◽  
Numerical Scheme ◽  
Transport Phenomena ◽  
Basis Functions ◽  
Simplified Model ◽  
Mesh Free ◽  
Dc Casting ◽  
Spatial Discretisation ◽  
Volume Method

This paper introduces a general numerical scheme for solving convective-diffusive problems that appear in the solution of microscopic and macroscopic transport phenomena in continuous castings and the heat treatment of aluminium alloys. The numerical scheme is based on spatial discretisation that involves pointisation only. The solution is based on diffuse collocation with multi-quadric radial basis functions. The application of the method is demonstrated in a simplified model of a billet DC casting and verified by a comparison with the classical finite volume method.

scholarly journals Mesh Free Radial Point Interpolation Based Displacement Recovery Techniques for Elastic Finite Element Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1900
Author(s):  
Mohd. Ahmed ◽  
Devinder Singh ◽  
Saeed AlQadhi ◽  
Majed A. Alrefae
Keyword(s):  
Finite Element ◽  
Error Recovery ◽  
Basis Functions ◽  
Least Square ◽  
Element Solution ◽  
Displacement Error ◽  
Mesh Free ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Recovery Technique

The study develops the displacement error recovery method in a mesh free environment for the finite element solution employing the radial point interpolation (RPI) technique. The RPI technique uses the radial basis functions (RBF), along with polynomials basis functions to interpolate the displacement fields in a node patch and recovers the error in displacement field. The global and local errors are quantified in both energy and L2 norms from the post-processed displacement field. The RPI technique considers multi-quadrics/gaussian/thin plate splint RBF in combination with linear basis function for displacement error recovery analysis. The elastic plate examples are analyzed to demonstrate the error convergence and effectivity of the RPI displacement recovery procedures employing mesh free and mesh dependent patches. The performance of a RPI-based error estimators is also compared with the mesh dependent least square based error estimator. The triangular and quadrilateral elements are used for the discretization of plates domains. It is verified that RBF with their shape parameters, choice of elements, and errors norms influence considerably on the RPI-based displacement error recovery of finite element solution. The numerical results show that the mesh free RPI-based displacement recovery technique is more effective and achieve target accuracy in adaptive analysis with the smaller number of elements as compared to mesh dependent RPI and mesh dependent least square. It is also concluded that proposed mesh free recovery technique may prove to be most suitable for error recovery and adaptive analysis of problems dealing with large domain changes and domain discontinuities.

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

Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method

2019 ◽  
Vol 20 (6) ◽  
pp. 703-712
Author(s):  
Naraveni Rajashekar ◽  
Sudhakar Chaudhary ◽  
V.V.K. Srinivas Kumar
Keyword(s):  
Mesh Generation ◽  
Numerical Experiments ◽  
Basis Functions ◽  
Spline Method ◽  
Biharmonic Problem ◽  
Difficult Time ◽  
Mesh Free ◽  
Mesh Free Method ◽  
Low Dimensional ◽  
The Web

Abstract We describe and analyze the weighted extended b-spline (WEB-Spline) mesh-free finite element method for solving the p-biharmonic problem. The WEB-Spline method uses weighted extended b-splines as basis functions on regular grids and does not require any mesh generation which eliminates a difficult, time consuming preprocessing step. Accurate approximations are possible with relatively low-dimensional subspaces. We perform some numerical experiments to demonstrate the efficiency of the WEB-Spline method.

scholarly journals A numerical study of RBFs-DQ method for multi-asset option pricing problems

2018 ◽  
Vol 36 (1) ◽  
pp. 9
Author(s):  
Mojtaba Ranjbar ◽  
Leila Khodayari
Keyword(s):  
Numerical Study ◽  
Basis Functions ◽  
Weighted Sum ◽  
Physical Domain ◽  
Change Of Variables ◽  
Mesh Free ◽  
Black Scholes ◽  
Mixed Derivatives ◽  
Pricing Problems ◽  
Spatial Derivatives

In this paper, we propose a numerical scheme to solve multi-dimensional Black-Scholes equation using the global radial basis functions-based dierential quadrature (RBFs-DQ) method. Before applying the method, it is needed to remove mixed derivatives from the Black-Scholes equation by making an appropriate change of variables . Then, any spatial derivatives are approximated by a linear weighted sum of all the function values in the whole physical domain. In the RBFs-DQ method the weighting coecients are computed by RBFs. The method is very easy to implement and the non-singularity is ensured. The proposed method com bines the advantages of the conventional DQ method and the RBFs. It also remains mesh-free feature of RBFs.

scholarly journals A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 779 ◽  
Author(s):  
Fusun Yalcin ◽  
Yilmaz Simsek
Keyword(s):  
Bernoulli Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Moment Generating Function ◽  
Basis Functions ◽  
Integral Representations ◽  
Harmonic Numbers ◽  
Bernstein Type ◽  
Digamma Function ◽  
New Class

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

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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