An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

The so-called Dual Reciprocity Boundary Element Method (DRBEM) has been a popular alternative scheme designed to alleviate problems encountered when using the traditional BEM for numerically solving engineering problems that are described by PDEs. The method starts with writing the right-hand-side of Poisson equation as a summation of a pre-chosen multivariate function known as ‘Radial Basis Function (RBF)’. Nevertheless, a common undesirable feature of using RBFs is the appearance of the so-called ‘shape parameter’ whose value greatly affects the solution accuracy. In this work, a new form of RBF containing no shape (so that it can be called ‘shapefree/shapeless’) is invented, proposed and applied in conjunction with DRBEM is validated numerically. The solutions obtained are compared against both exact ones and those presented in literature where appropriate, for validation. It is found that reasonably and comparatively good approximated solutions of PDEs can still be obtained without the difficulty of choosing a good shape for RBF used.

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From Global to Local Radial Basis Function Collocation Method for Transport Phenomena

Advances in Meshfree Techniques ◽

10.1007/978-1-4020-6095-3_14 ◽

2007 ◽

pp. 257-282 ◽

Cited By ~ 17

Author(s):

Božidar Šarler

Keyword(s):

Radial Basis Function ◽

Collocation Method ◽

Basis Function ◽

Transport Phenomena ◽

Radial Basis

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Nonlinear Bending Analysis of Hyperelastic Plates Using FSDT and Meshless Collocation Method Based on Radial Basis Function

International Journal of Applied Mechanics ◽

10.1142/s1758825121500071 ◽

2021 ◽

Vol 13 (01) ◽

pp. 2150007

Author(s):

Shahram Hosseini ◽

Gholamhossein Rahimi

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Boundary Conditions ◽

Radial Basis Function ◽

Collocation Method ◽

Basis Function ◽

The Finite Element Method ◽

Radial Basis ◽

Meshless Collocation ◽

Element Method

This paper investigates the nonlinear bending analysis of a hyperelastic plate via neo-Hookean strain energy function. The first-order shear deformation plate theory (FSDPT) is used for the formulation of the field variables. Also, the nonlinear Lagrangian strains are considered via the right Cauchy–Green tensor. The governing equations and nonlinear boundary conditions are derived using Euler–Lagrange relations. The meshless collocation method based on radial basis function is used to discretize the governing equations of the hyperelastic plate. Square and circular plates are studied to evaluate the accuracy of the meshless collocation method based on thin-plate spline (TPS) and multiquadric (MQ) and logarithmic thin-plate spline (LTPS) radial basis function. Also, the results of the meshless method are compared to those of the finite element method. In some cases, the meshless method is more efficient than the finite element method due to no meshing. The linear and nonlinear natural boundary conditions are directly imposed on the stiffness matrix and are compared to each other. The maximum differences between linear and nonlinear natural boundary conditions are 1.43%. The von-Mises stress using meshless collocation method based on TPS basis function is compared to those of the finite element method.

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