scholarly journals An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Samaneh Soradi-Zeid ◽  
Mehdi Mesrizadeh ◽  
Thabet Abdeljawad
Keyword(s):  
Neumann Boundary ◽  
Nonlinear Problems ◽  
Delta Function ◽  
Collocation Methods ◽  
Laplacian Equation ◽  
Point Collocation ◽  
Kronecker Delta ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Element Free

Abstract This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

Solution of Material Mechanics by RPIM Meshless Method

2013 ◽  
Vol 740 ◽  
pp. 211-216
Author(s):  
Wen Sheng Dong ◽  
Zheng Lei ◽  
Xue Mei Liu
Keyword(s):  
Boundary Conditions ◽  
Compact Support ◽  
Shape Function ◽  
Interpolation Method ◽  
Delta Function ◽  
Additional Treatment ◽  
System Equation ◽  
Kronecker Delta ◽  
Radial Point Interpolation ◽  
Point Interpolation

With the compact support of radial point interpolation shape functions (RPIM), the stiffness matrix of system equation can be sparse, so its very suitable for meshless methods.This interpolation method have the Kronecker delta function property, so the essential boundary conditions can be enforced directly and accurately without any additional treatment. In this paper, the radial point interpolation shape function is used, calculate the shape function in different node distribution cases, analyse the compact support and Kronecker delta function,use penalty method to apply the essential boundary conditions. In this paper, the RPIM is applied to solving the deformation in one and two dimensional solids by different loads, the results demonstrated that the meshfree RPIM can effectively solve material mechanics problems.

Analysis of Transient Thermo-Elastic Problems Using a Cell-Based Smoothed Radial Point Interpolation Method

2016 ◽  
Vol 13 (05) ◽  
pp. 1650023 ◽  
Author(s):  
Gang Wu ◽  
Jian Zhang ◽  
Yuelin Li ◽  
Lairong Yin ◽  
Zhiqiang Liu
Keyword(s):  
Transfer Problem ◽  
Interpolation Method ◽  
Weak Form ◽  
Delta Function ◽  
Radial Point Interpolation Method ◽  
Kronecker Delta ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
A Cell ◽  
Point Interpolation

The transient thermo-elastic problems are solved by a cell-based smoothed radial point interpolation method (CS-RPIM). For this method, the problem domain is first discretized using triangular cells, and each cell is further divided into smoothing cells. The field functions are approximated using RPIM shape functions which have Kronecker delta function property. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form. At first, the temperature field is acquired by solving the transient heat transfer problem and it is then employed as an input for the mechanical problem to calculate the displacement and stress fields. Several numerical examples with different kinds of boundary conditions are investigated to verify the accuracy, convergence rate and stability of the present method.

The Local Point Interpolation–Boundary Element Method: Application to 3D Stationary Thermo-Piezoelectricity

2016 ◽  
Vol 13 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Richard Kouitat Njiwa
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Thermal Conduction ◽  
Linear Equations ◽  
Point Collocation ◽  
Simple Strategy ◽  
Local Point ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Element Method

This paper presents a simple strategy allowing to adapt well-established isotropic BEM approach for the solution of multi-physics problems with anisotropic material parameters. The method is based on the partition of the primary kinematical fields into complementary and particular parts. The isotropic linear equations for the complementary fields are solved by the conventional boundary element method. The particular fields are obtained by a point collocation of a strong form differential equation. Adopting local radial point interpolation, the effectiveness of the approach is proved by considering various examples of stationary thermal conduction, thermos-elasticity and thermos-piezoelectricity.

ASSESSMENT OF RPIM SHAPE PARAMETERS FOR SOLUTION ACCURACY OF 2D GEOMETRICALLY NONLINEAR PROBLEMS

2013 ◽  
Vol 10 (03) ◽  
pp. 1350003 ◽  
Author(s):  
O. YAVUZ BOZKURT ◽  
BAHATTİN KANBER ◽  
M. ZÜLFÜ AŞIK
Keyword(s):  
Basis Function ◽  
Interpolation Method ◽  
Nonlinear Problems ◽  
Geometrically Nonlinear ◽  
Shape Parameters ◽  
Monomial Basis ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Solution Accuracy

This study discussed the effects of shape parameters on the radial point interpolation method (RPIM) accuracy in 2D geometrically nonlinear problems. Four finite deformation problems with compressible Neo-Hookean material are numerically solved with the RPIM algorithm using the multi-quadric (MQ) radial basis function. Both regular and irregular node distributions are used. Their displacements and Cauchy stresses are compared for different values of shape parameters and monomial basis. It is found that the shape parameters proposed for linearly elastic problems (q = 1.03, αc = 4) can still be applicable to 2D geometrically nonlinear problems but careful selections should be made for the calculation of stress. For example, when q is used as 1.75 with irregular node distributions, stresses can be calculated more precisely.

Shear Locking Analysis of Plate Bending by Using Meshless Local Radial Point Interpolation Method

2012 ◽  
Vol 166-169 ◽  
pp. 2867-2870 ◽  
Author(s):  
Ping Xia ◽  
Ke Xiang Wei
Keyword(s):  
Boundary Conditions ◽  
Interpolation Method ◽  
Delta Function ◽  
Plate Bending ◽  
Shear Locking ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Two Sides

The shape function of the meshless local radial point interpolation method is constructed by using the radial basis functions and possesses Kronecker delta function properties. Therefore, the essential boundary conditions can be easily imposed. Causation of shear locking occur in plate bending is analyzed. Bending problems for plate with two sides simply supported, the other two sides clamped boundary conditions, are analyzed by the meshless local radial point interpolation method. The shear locking is easier avoided in the meshless method than in the finite element method, and the measure of avoiding the shear locking are presented.

On the Treatment of Neumann Boundary Conditions in Collocation-Based Meshless Methods

2013 ◽  
Vol 423-426 ◽  
pp. 1757-1762
Author(s):  
Xiang Dong Zhang ◽  
Lei Wang ◽  
Da Wei Teng
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Shape Functions ◽  
Numerical Accuracy ◽  
The Poor ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Methods Of Treatment

The existence of Neumann boundary is a major cause of the poor accuracy and instability of collocation-based methods. Taking a Poisson equation with Neumann boundary condition as the model, the present paper studies the effects of two different radial point interpolation shape functions and their parameters on the accuracy of numerical solutions of the equation. We also study the effects of methods including fictious point method, nodes densification method and Hermite collocation method on the improvement of numerical accuracy. By comparison of analytic and numerical solutions computed using a program developed during research, we obtain parameters of shape functions and methods of treatment of Neumann boundary conditions that can be adopted to give better numerical accuracy.

ANALYSIS OF MINDLIN–REISSNER PLATES USING CELL-BASED SMOOTHED RADIAL POINT INTERPOLATION METHOD

2010 ◽  
Vol 02 (03) ◽  
pp. 653-680 ◽  
Author(s):  
X. Y. CUI ◽  
G. R. LIU ◽  
G. Y. LI
Keyword(s):  
Plate Theory ◽  
Interpolation Method ◽  
Free Vibration Analysis ◽  
Delta Function ◽  
Shear Locking ◽  
Radial Point Interpolation Method ◽  
First Order ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation

In this paper, a formulation for the static and free vibration analysis of Mindlin–Reissner plates is proposed using the cell-based smoothed radial point interpolation method (CS-RPIM) with sub-domain smoothing operations. The radial basis functions augmented with polynomial basis are employed to construct the shape functions that have the Delta function property. The generalized smoothed Galerkin (GS-Galerkin) weakform is adopted to discretize the governing differential equations, and the curvature smoothing is performed to relax the continuity requirement and to improve the accuracy and the rate of convergence of the solution. The present CS-RPIM formulation is based on the first-order shear deformation plate theory, with effective treatment for shear-locking and hence is applicable to both thin and relatively thick plates. To verify the accuracy and stability of the present formulation, intensive comparison studies are conducted with existing results available in the literature and good agreements are obtained. The numerical examples confirm that the present method is shear-locking free and very stable and accurate even using extremely distributed nodes.

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