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.

EDGE-BASED SMOOTHED POINT INTERPOLATION METHODS

2008 ◽  
Vol 05 (04) ◽  
pp. 621-646 ◽  
Author(s):  
G. R. LIU ◽  
G. Y. ZHANG
Keyword(s):  
Degrees Of Freedom ◽  
Solid Mechanics ◽  
Interpolation Method ◽  
Weak Form ◽  
Dirichlet Boundary Conditions ◽  
Triangular Meshes ◽  
Weak Formulation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Edge Based

This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for solid mechanics using three-node triangular meshes. In the ES-PIM, displacement fields are construed using the point interpolation method (polynomial PIM or radial PIM), and hence the shape functions possess the Kronecker delta property, facilitates the enforcement of Dirichlet boundary conditions. Strains are 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 and the formation is weakened weak formulation. Four schemes of selecting nodes for interpolation using the PIM have been introduced in detail and ES-PIM models using these four schemes have been developed. Numerical studies have demonstrated that the ES-PIM possesses the following good properties: (1) the ES-PIM models have a close-to-exact stiffness, which is much softer than for the overly-stiff FEM model and much stiffer than for the overly-soft node-based smoothed point interpolation method (NS-PIM) model; (2) results of ES-PIMs are generally of superconvergence and "ultra-accurate"; (3) no additional degrees of freedom are introduced, the implementation of the method is straightforward, and the method can achieve much better efficiency than the FEM using the same set of triangular meshes.

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.

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 LINEARLY CONFORMING POINT INTERPOLATION METHOD (LC-PIM) FOR 2D SOLID MECHANICS PROBLEMS

2005 ◽  
Vol 02 (04) ◽  
pp. 645-665 ◽  
Author(s):  
G. R. LIU ◽  
G. Y. ZHANG ◽  
K. Y. DAI ◽  
Y. Y. WANG ◽  
Z. H. ZHONG ◽  
...  
Keyword(s):  
Solid Mechanics ◽  
Interpolation Method ◽  
Weak Form ◽  
Integration Scheme ◽  
Strain Smoothing ◽  
Monotonic Convergence ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
System Equations ◽  
The Moment

A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

MESHFREE CELL-BASED SMOOTHED ALPHA RADIAL POINT INTERPOLATION METHOD (CS-α RPIM) FOR SOLID MECHANICS PROBLEMS

2013 ◽  
Vol 10 (04) ◽  
pp. 1350020 ◽  
Author(s):  
G. R. LIU ◽  
G. Y. ZHANG ◽  
Z. ZONG ◽  
M. LI
Keyword(s):  
Solid Mechanics ◽  
Interpolation Method ◽  
Shape Functions ◽  
Radial Point Interpolation Method ◽  
Practical Procedure ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Factor Α ◽  
2D And 3D

This paper presents a novel and effective cell-based smoothed alpha radial point interpolation method (CS-αRPIM) using αPIM shape functions for approximating displacement and cell-based smoothed strains for displacement gradient construction. Using a scaling factor α ∈ [0, 1], the αPIM shape functions are combinations of the condensed RPIM (RPIM-Cd) shape functions and the linear PIM shape functions, where the former often leads to a "softer" CS-RPIM model, and the latter a "stiffer" linear CS-RPIM model (which is as same as linear FEM), compared to the exact one. Through adjusting the value of α in our new CS-αRPIM, the stiffness of the model can be "designed" for desired purposes, such as for seeking nearly exact solutions in strain energy norm (or possibly other norms). A simple and practical procedure to search for such an α has also been presented. Some 2D and 3D numerical examples are studied to examine various properties of the present method in terms of accuracy, convergence and computational efficiency.

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.

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.

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.

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