MESHFREE CELL-BASED SMOOTHED POINT INTERPOLATION METHOD USING ISOPARAMETRIC PIM SHAPE FUNCTIONS AND CONDENSED RPIM SHAPE FUNCTIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 705-730 ◽  
Author(s):  
G. Y. ZHANG ◽  
G. R. LIU
Keyword(s):  
Computational Efficiency ◽  
Interpolation Method ◽  
Shape Functions ◽  
Generalized Gradient ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
System Equations ◽  
Gradient Smoothing

This paper presents two novel and effective cell-based smoothed point interpolation methods (CS-PIM) using isoparametric PIM (PIM-Iso) shape functions and condensed radial PIM (RPIM-Cd) shape functions respectively. These two types of PIM shape functions can successfully overcome the singularity problem occurred in the process of creating PIM shape functions and make the constructed CS-PIM models work well with the three-node triangular meshes. Smoothed strains are obtained by performing the generalized gradient smoothing operation over each triangular background cells, because the nodal PIM shape functions can be discontinuous. The generalized smoothed Galerkin (GS-Galerkin) weakform is used to create the discretized system equations. Some numerical examples are studied to examine various properties of the present methods in terms of accuracy, convergence, and computational efficiency.

A cell-based smoothed point interpolation method (CS-PIM) for 2D thermoelastic problems

2017 ◽  
Vol 27 (6) ◽  
pp. 1249-1265 ◽  
Author(s):  
Yijun Liu ◽  
Guiyong Zhang ◽  
Huan Lu ◽  
Zhi Zong
Keyword(s):  
Thermal Stresses ◽  
Interpolation Method ◽  
Content Type ◽  
Novel Approach ◽  
Convergence Results ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Thermoelastic Problems ◽  
Gradient Smoothing ◽  
Practical Implications

Purpose Due to the strong reliance on element quality, there exist some inherent shortcomings of the traditional finite element method (FEM). The model of FEM behaves overly stiff, and the solutions of automated generated linear elements are generally of poor accuracy about especially gradient results. The proposed cell-based smoothed point interpolation method (CS-PIM) aims to improve the results accuracy of the thermoelastic problems via properly softening the overly-stiff stiffness. Design/methodology/approach This novel approach is based on the newly developed G space and weakened weak (w2) formulation, and of which shape functions are created using the point interpolation method and the cell-based gradient smoothing operation is conducted based on the linear triangular background cells. Findings Owing to the property of softened stiffness, the present method can generally achieve better accuracy and higher convergence results (especially for the temperature gradient and thermal stress solutions) than the FEM does by using the simplest linear triangular background cells, which has been examined by extensive numerical studies. Practical implications The CS-PIM is capable of producing more accurate results of temperature gradients as well as thermal stresses with the automated generated and unstructured background cells, which make it a better candidate for solving practical thermoelastic problems. Originality/value It is the first time that the novel CS-PIM was further developed for solving thermoelastic problems, which shows its tremendous potential for practical implications.

A CONFORMING POINT INTERPOLATION METHOD (CPIM) BY SHAPE FUNCTION RECONSTRUCTION FOR ELASTICITY PROBLEMS

2010 ◽  
Vol 07 (03) ◽  
pp. 369-395 ◽  
Author(s):  
X. XU ◽  
G. R. LIU ◽  
Y. T. GU ◽  
G. Y. ZHANG
Keyword(s):  
Convergence Rate ◽  
Computational Efficiency ◽  
Shape Function ◽  
Interpolation Method ◽  
Numerical Studies ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation ◽  
Continuous Displacement ◽  
High Computational Efficiency

A conforming point interpolation method (CPIM) is proposed based on the Galerkin formulation for 2D mechanics problems using triangular background cells. A technique for reconstructing the PIM shape functions is proposed to create a continuous displacement field over the whole problem domain, which guarantees the CPIM passing the standard patch test. We prove theoretically the existence and uniqueness of the CPIM solution, and conduct detailed analyses on the convergence rate; computational efficiency and band width of the stiffness matrix of CPIM. The CPIM does not introduce any additional degrees of freedoms compared to the linear FEM and original PIM; while convergence rate of quadratic CPIM is in between that of linear FEM and quadratic FEM which results in the high computational efficiency. Intensive numerical studies verify the properties of the CPIM.

A Coupled FE-Meshfree Triangular Element for Acoustic Radiation Problems

2020 ◽  
pp. 2041002
Author(s):  
Wei Li ◽  
Qifan Zhang ◽  
Qiang Gui ◽  
Yingbin Chai
Keyword(s):  
Acoustic Radiation ◽  
Interpolation Method ◽  
Partition Of Unity ◽  
Local Approximation ◽  
Triangular Element ◽  
Two Dimensional ◽  
Shape Functions ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation

To improve the accuracy of the standard finite element (FE) solutions for acoustic radiation computation, this work presents the coupling of a radial point interpolation method (RPIM) with the standard FEM based on triangular (T3) mesh to give a coupled “FE-Meshfree” Trig3-RPIM element for two-dimensional acoustic radiation problems. In this coupled Trig3-RPIM element, the local approximation (LA) is represented by the polynomial-radial basis functions and the partition of unity (PU) concept is satisfied using the standard FEM shape functions. Incorporating the present coupled Trig3-RPIM element with the appropriate non-reflecting boundary condition, the two-dimensional acoustic radiation problems in exterior unbounded domain can be successfully solved. The numerical results demonstrate that the present coupled Trig3-RPIM have significant superiorities over the standard FEM and can be regarded as a competitive numerical techniques for exterior acoustic computation.

Application of the Meshless Local Petrov-Galerkin Method for Subsoil Settlement Analysis

2014 ◽  
Vol 969 ◽  
pp. 55-62 ◽  
Author(s):  
Juraj Mužík ◽  
Dana Sitányiová
Keyword(s):  
Galerkin Method ◽  
Interpolation Method ◽  
Meshless Methods ◽  
Deformation Analysis ◽  
Shape Functions ◽  
Equilibrium Equations ◽  
Weak Formulation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Set Of Points

The paper deals with use of the meshless method for soil stress-deformation analysis. There are many formulations of the meshless methods. The article presents the Meshless Local Petrov-Galerkin method (MLPG) local weak formulation of the equilibrium equations. The main difference between meshless methods and the conventional finite element method (FEM) is that meshless shape functions are constructed using randomly scattered set of points without any relation between points. The shape function construction is the crucial part of the meshless numerical analysis in the construction of shape functions. The article presents the radial point interpolation method (RPIM) for the shape functions construction.

scholarly journals A NODE-BASED SMOOTHED CONFORMING POINT INTERPOLATION METHOD (NS-CPIM) FOR ELASTICITY PROBLEMS

2011 ◽  
Vol 08 (04) ◽  
pp. 801-812 ◽  
Author(s):  
X. XU ◽  
Y. T. GU ◽  
X. YANG
Keyword(s):  
Strain Energy ◽  
Solid Mechanics ◽  
Interpolation Method ◽  
Weak Form ◽  
Patch Tests ◽  
Numerical Studies ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation ◽  
System Equations

This paper formulates a node-based smoothed conforming point interpolation method (NS-CPIM) for solid mechanics. In the proposed NS-CPIM, the higher-order CPIM shape functions have been constructed to produce a continuous and piecewise quadratic displacement field over the whole problem domain, whereby the smoothed strain field was obtained through smoothing operation over each smoothing domain associated with domain nodes. The smoothed Galerkin weak form was then developed to create the discretized system equations. Numerical studies have demonstrated the following good properties: NS-CPIM (1) can pass both standard and quadratic patch tests; (2) provides an upper bound of strain energy; (3) avoids the volumetric locking; and (4) provides the higher accuracy than those in the node-based smoothed schemes of the original PIMs.

A NOVEL QUADRATIC EDGE-BASED SMOOTHED CONFORMING POINT INTERPOLATION METHOD (ES-CPIM) FOR ELASTICITY PROBLEMS

2012 ◽  
Vol 09 (02) ◽  
pp. 1240033 ◽  
Author(s):  
X. XU ◽  
G. R. LIU ◽  
Y. T. GU
Keyword(s):  
Displacement Field ◽  
Degrees Of Freedom ◽  
Solid Mechanics ◽  
Interpolation Method ◽  
Weak Form ◽  
Shape Functions ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation ◽  
Edge Based

This paper formulates an edge-based smoothed conforming point interpolation method (ES-CPIM) for solid mechanics using the triangular background cells. In the ES-CPIM, a technique for obtaining conforming PIM shape functions (CPIM) is used to create a continuous and piecewise quadratic displacement field over the whole problem domain. The smoothed strain field is then obtained through smoothing operation over each smoothing domain associated with edges of the triangular background cells. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. Numerical studies have demonstrated that the ES-CPIM possesses the following good properties: (1) ES-CPIM creates conforming quadratic PIM shape functions, and can always pass the standard patch test; (2) ES-CPIM produces a quadratic displacement field without introducing any additional degrees of freedom; (3) The results of ES-CPIM are generally of very high accuracy.

A NORMED G SPACE AND WEAKENED WEAK (W2) FORMULATION OF A CELL-BASED SMOOTHED POINT INTERPOLATION METHOD

2009 ◽  
Vol 06 (01) ◽  
pp. 147-179 ◽  
Author(s):  
G. R. LIU ◽  
G. Y. ZHANG
Keyword(s):  
Solid Mechanics ◽  
Interpolation Method ◽  
Temporal Stability ◽  
Weak Form ◽  
Dirichlet Boundary Conditions ◽  
Shape Functions ◽  
Displacement Fields ◽  
Point Interpolation Method ◽  
A Cell ◽  
Point Interpolation

This paper presents a normed G1 space and a weakened weak (W2) formulation of a cell-based smoothed point interpolation method (CS-PIM) for 2D solid mechanics problems using three-node triangular cells. Displacement fields in the CS-PIM are constructed using the point interpolation method (polynomial PIM or radial PIM) and hence the shape functions possess the Kronecker delta property facilitating the easy enforcement of Dirichlet boundary conditions. The edge-based T-schemes are introduced for selecting supporting nodes for creating the PIM shape functions and an adaptive coordinate transformation (CT) technique is proposed to solve the singularity problem for the moment matrix. Smoothed strains are obtained by performing the generalized smoothing operation over each triangular background cell. Because the nodal PIM shape functions can be discontinuous, a W2 formulation of generalized smoothed Galerkin (GS-Galerkin) weak form is then used to create the discretized system equations. Numerical examples including static, free and forced vibration problems have been studied to examine the present method in terms of accuracy, convergence, efficiency and temporal stability.

Dispersion Error Reduction for Acoustic Problems Using the Finite Element-Least Square Point Interpolation Method

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
L. Y. Yao ◽  
J. W. Zhou ◽  
Z. Zhou ◽  
L. Li
Keyword(s):  
Finite Element ◽  
Interpolation Method ◽  
Local Approximation ◽  
Least Square ◽  
Numerical Dispersion ◽  
Benchmark Problems ◽  
Shape Functions ◽  
Dispersion Error ◽  
Point Interpolation Method ◽  
Point Interpolation

The shape function of the finite element-least square point interpolation method (FE-LSPIM) combines the quadrilateral element for partition of unity and the least square point interpolation method (LSPIM) for local approximation, and inherits the completeness properties of meshfree shape functions and the compatibility properties of FE shape functions, and greatly reduces the numerical dispersion error. This paper derives the formulas and performs the dispersion analysis for the FE-LSPIM. Numerical results for benchmark problems show that, the FE-LSPIM yields considerably better results than the finite element method (FEM) and element-free Galerkin method (EFGM).

Elastic-plastic analysis of multi-material structures using edge-based smoothed point interpolation method

2018 ◽  
Vol 233 (4) ◽  
pp. 1289-1298
Author(s):  
SZ Feng ◽  
YH Cheng ◽  
AM Li
Keyword(s):  
Interpolation Method ◽  
Shape Functions ◽  
Integral Domains ◽  
Elastic Plastic ◽  
Plastic Analysis ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Triangular Elements ◽  
Standard Finite Element Method ◽  
Edge Based

An edge-based smoothed point interpolation method is formulated to deal with elastic-plastic analysis of multi-material structures. The problem domain is discretized using triangular elements and field functions are approximated using point interpolation method shape functions. Edge-based smoothing domains are constructed based on the edge of triangular cells and smoothing operations are then performed in these integral domains. Numerical examples with different kinds of material models are investigated to fully verify the validity of the present method. It is observed that all edge-based smoothed point interpolation method models can achieve much better accuracy and higher convergence rate than the standard finite element method, when dealing with elastic-plastic analysis of multi-material structures.

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