EDGE-BASED SMOOTHED POINT INTERPOLATION METHODS

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.

Download Full-text

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

International Journal of Computational Methods ◽

10.1142/s0219876212400336 ◽

2012 ◽

Vol 09 (02) ◽

pp. 1240033 ◽

Cited By ~ 4

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.

Download Full-text

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

International Journal of Computational Methods ◽

10.1142/s0219876209001796 ◽

2009 ◽

Vol 06 (01) ◽

pp. 147-179 ◽

Cited By ~ 96

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.

Download Full-text

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

International Journal of Computational Methods ◽

10.1142/s0219876211002836 ◽

2011 ◽

Vol 08 (04) ◽

pp. 801-812 ◽

Cited By ~ 2

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.

Download Full-text

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

International Journal of Computational Methods ◽

10.1142/s0219876205000661 ◽

2005 ◽

Vol 02 (04) ◽

pp. 645-665 ◽

Cited By ~ 176

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.

Download Full-text

A LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD FOR SOLID MECHANICS PROBLEMS

International Journal of Computational Methods ◽

10.1142/s0219876206001132 ◽

2006 ◽

Vol 03 (04) ◽

pp. 401-428 ◽

Cited By ~ 106

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.

Download Full-text

A NOVEL SCHEME OF STRAIN-CONSTRUCTED POINT INTERPOLATION METHOD FOR STATIC AND DYNAMIC MECHANICS PROBLEMS

International Journal of Applied Mechanics ◽

10.1142/s1758825109000083 ◽

2009 ◽

Vol 01 (01) ◽

pp. 233-258 ◽

Cited By ~ 30

Author(s):

G. R. LIU ◽

G. Y. ZHANG

Keyword(s):

Forced Vibration ◽

Present Method ◽

Degrees Of Freedom ◽

Interpolation Method ◽

Linear Strain ◽

Generalized Gradient ◽

Strain Fields ◽

Point Interpolation Method ◽

Point Interpolation ◽

Edge Based

This paper presents a new scheme of strain-constructed point interpolation method (SC-PIM) for static, free and forced vibration analysis of solids and structures using triangular cells. In the present scheme, displacement fields are assumed using shape functions created via the point interpolation method (PIM), which possess the Kronecker delta property facilitating the straightforward enforcement of displacement boundary conditions. Using the generalized gradient smoothing technique, the "smoothed" strains at the middle points of the cells edges are first obtained using the corresponding edge-based smoothing domains and the assumed displacement field. In each triangular background cell, the strains at the vertices are assigned using these smoothed strains in a proper manner, and then piecewisely linear strain fields are constructed by the linear interpolation for each sub-triangular cell using the edge-based "smoothed" strains. With the assumed displacements and constructed linear strain fields, the discretized system equations are created using the Strain Constructed Galerkin (SC-Galerkin) weak form. A number of benchmark numerical examples, including the standard patch test, static, free and forced vibration problems, have been studied and intensive numerical results have demonstrated that the present method possesses the following properties: (1) it works well with the simplest triangular mesh, no additional degrees of freedom and parameters are introduced and very easy to implement; (2) it is at least linearly conforming; (3) it possesses a close-to-exact stiffness: it is much stiffer than the "overly-soft" node-based smoothed point interpolation method (NS-PIM) and much softer than the "overly-stiff" FEM model; (4) the results of the present method are of superconvergence and ultra-accuracy: about one order of magnitude more accurate than those of the linear FEM; (5) there are no spurious non-zeros energy modes found and it is also temporally stable, hence the present method works well for dynamic problems.

Download Full-text

An edge-based smoothed point interpolation method for elasto-plastic coupled hydro-mechanical analysis of saturated porous media

Computers and Geotechnics ◽

10.1016/j.compgeo.2016.10.002 ◽

2017 ◽

Vol 82 ◽

pp. 99-109 ◽

Cited By ~ 8

Author(s):

O. Ghaffaripour ◽

A. Khoshghalb ◽

N. Khalili

Keyword(s):

Porous Media ◽

Interpolation Method ◽

Mechanical Analysis ◽

Saturated Porous Media ◽

Point Interpolation Method ◽

Point Interpolation ◽

Edge Based

Download Full-text

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.969.55 ◽

2014 ◽

Vol 969 ◽

pp. 55-62 ◽

Cited By ~ 1

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.

Download Full-text