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 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.

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.

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 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.

Coupling BEM and the Local Point Interpolation for the Solution of Anisotropic Elastic Nonlinear, Multi-Physics and Multi-Fields Problems

2019 ◽  
Vol 17 (09) ◽  
pp. 1950067
Author(s):  
Richard Kouitat Njiwa ◽  
Gael Pierson ◽  
Arnaud Voignier
Keyword(s):  
Solid Mechanics ◽  
Interpolation Method ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Local Point ◽  
Problem Dimension ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation ◽  
Anisotropic Elastic

The pure boundary element method (BEM) is effective for the solution of a large class of problems. The main appeal of this BEM (reduction of the problem dimension by one) is tarnished to some extent when a fundamental solution to the governing equations does not exist as in the case of nonlinear problems. The easy to implement local point interpolation method applied to the strong form of differential equations is an attractive numerical approach. Its accuracy deteriorates in the presence of Neumann-type boundary conditions which are practically inevitable in solid mechanics. The main appeal of the BEM can be maintained by a judicious coupling of the pure BEM with the local point interpolation method. The resulting approach, named the LPI-BEM, seems versatile and effective. This is demonstrated by considering some linear and nonlinear elasticity problems including multi-physics and multi-field problems.

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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