Comments on ‘Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM)’ by G. R. Liu and G. Y. Zhang,International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme.2204 [1]

2009 ◽  
Vol 77 (7) ◽  
pp. 1046-1050
Author(s):  
G. R. Liu ◽  
Guiyong Zhang
Keyword(s):  
Numerical Methods ◽  
Upper Bound ◽  
Interpolation Method ◽  
Unique Property ◽  
Bound Solution ◽  
Upper Bound Solution ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation

scholarly journals THE UPPER BOUND PROPERTY FOR SOLID MECHANICS OF THE LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD (LC-RPIM)

2007 ◽  
Vol 04 (03) ◽  
pp. 521-541 ◽  
Author(s):  
G. Y. ZHANG ◽  
G. R. LIU ◽  
T. T. NGUYEN ◽  
C. X. SONG ◽  
X. HAN ◽  
...  
Keyword(s):  
Exact Solution ◽  
Lower Bounds ◽  
Upper Bound ◽  
Interpolation Method ◽  
Energy Norm ◽  
Upper And Lower Bounds ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation

It has been proven by the authors that both the upper and lower bounds in energy norm of the exact solution to elasticity problems can now be obtained by using the fully compatible finite element method (FEM) and linearly conforming point interpolation method (LC-PIM). This paper examines the upper bound property of the linearly conforming radial point interpolation method (LC-RPIM), where the Radial Basis Functions (RBFs) are used to construct shape functions and node-based smoothed strains are used to formulate the discrete system equations. It is found that the LC-RPIM also provides the upper bound of the exact solution in energy norm to elasticity problems, and it is much sharper than that of LC-PIM due to the decrease of stiffening effect. An effective procedure is also proposed to determine both upper and lower bounds for the exact solution without knowing it in advance: using the LC-RPIM to compute the upper bound, using the standard fully compatible FEM to compute the lower bound based on the same mesh for the problem domain. Numerical examples of 1D, 2D and 3D problems are presented to demonstrate these important properties of LC-RPIM.

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 Combination of Singular Cell-Based Smoothed Radial Point Interpolation Method and FEM in Solving Fracture Problem

2018 ◽  
Vol 15 (08) ◽  
pp. 1850079 ◽  
Author(s):  
Guiyong Zhang ◽  
Yaomei Wang ◽  
Yong Jiang ◽  
Yichen Jiang ◽  
Zhi Zong
Keyword(s):  
Strain Energy ◽  
Interpolation Method ◽  
The Finite Element Method ◽  
Radial Point Interpolation Method ◽  
Bound Solution ◽  
Numerical Studies ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Lower Bound Solution

The singular cell-based smoothed radial point interpolation method (CS-RPIM) has been previously proposed and shown good performance in solving fracture problems. Motivated from the fact that CS-RPIM performs over softly by providing an upper bound solution and the finite element method (FEM) is overly stiff by providing a lower bound solution, this work proposes a combination of singular CS-RPIM and FEM with a correlation coefficient [Formula: see text], and [Formula: see text] has been recommended through intensive numerical studies. Several numerical examples have been studied and the proposed method has been found perform quite well from both stress intensity factors and strain energy.

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.

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