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.

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.

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.

Pseudospectral Meshless Radial Point Hermit Interpolation Versus Pseudospectral Meshless Radial Point Interpolation

2019 ◽  
Vol 17 (07) ◽  
pp. 1950023 ◽  
Author(s):  
Elyas Shivanian
Keyword(s):  
Boundary Conditions ◽  
Interpolation Method ◽  
Meshless Methods ◽  
Basis Functions ◽  
Elliptic Pdes ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Impedance Boundary ◽  
Radial Point Interpolation ◽  
Point Interpolation

This paper develops pseudospectral meshless radial point Hermit interpolation (PSMRPHI) and pseudospectral meshless radial point interpolation (PSMRPI) in order to apply to the elliptic partial differential equations (PDEs) held on irregular domains subject to impedance (convective) boundary conditions. Elliptic PDEs in simplest form, i.e., Laplace equation or Poisson equation, play key role in almost all kinds of PDEs. On the other hand, impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems, are nearly more complicated forms of the boundary conditions in boundary value problems (BVPs). Based on this problem, we aim also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods. PSMRPI method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of PSMRPI and PSMRPHI methods. While the latter one has been rarely used in applications, we observe that is more accurate and reliable than PSMRPI method.

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.

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.

Using Meshfree Weak-Strong Form Method for a 2-D Heat Transfer Problem

2009 ◽  
Author(s):  
S. Zahiri ◽  
F. Daneshmand ◽  
M. H. Akbari
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Transfer Problem ◽  
Interpolation Method ◽  
Meshfree Method ◽  
Weak Form ◽  
Heat Transfer Problem ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Local Weak Form

In this work, the numerical simulation of 2-D heat transfer problem is studied by using a meshfree method. The method is based on the local weak form collocation and the meshfree weak-strong (MWS) form. The goal of the paper is to find the temperature distribution in a rectangular plate. The results obtained are compared by those obtained by use of other numerical methods. Two types of boundary conditions are considered in this paper: Dirichlet and Neumann types. The Local Radial Point Interpolation Method (LRPIM) is used as the meshfree method. It is shown that the essential boundary conditions can be easily enforced as in the Finite Element Method (FEM), since the radial point interpolation shape functions posses the Kronecker delta property. It is also shown that the natural (derivative) boundary conditions can be satisfied by using the MWS method and no additional equation or treatment are needed. The MWS method as presented in this paper works well with local quadrature cells for nodes on the natural boundary and can be generated without any difficulty.

A LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD FOR SOLID MECHANICS PROBLEMS

2006 ◽  
Vol 03 (04) ◽  
pp. 401-428 ◽  
Author(s):  
G. R. LIU ◽  
Y. LI ◽  
K. Y. DAI ◽  
M. T. LUAN ◽  
W. XUE
Keyword(s):  
Solid Mechanics ◽  
Interpolation Method ◽  
Weak Form ◽  
Delta Function ◽  
Field Function ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Integration Techniques ◽  
Point Interpolation Method ◽  
Point Interpolation

A linearly conforming radial point interpolation method (LC-RPIM) is presented for stress analysis of two-dimensional solids. In the LC-RPIM method, each field node is enclosed by a Voronoi polygon, and the displacement field function is approximated using RPIM shape functions of Kronecker delta function property created by simple interpolation using local nodes and radial basis functions augmented with linear polynomials to guarantee linear consistency. The system equations are then derived using the Galerkin weak form and nodal integration techniques, and the essential boundary conditions are imposed directly as in the finite element method. The LC-RPIM method is verified via various numerical examples and an extensive comparison study is conducted with the conventional RPIM, analytical approach and FEM. It is found that the presented LC-RPIM is more stable, more accurate in stress and more efficient than the conventional RPIM.

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