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.

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.

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 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.

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.

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.

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.

scholarly journals A Quasi-Conforming Point Interpolation Method (QC-PIM) for Elasticity Problems

2016 ◽  
Vol 13 (05) ◽  
pp. 1650026 ◽  
Author(s):  
X. Xu ◽  
Z. Q. Zheng ◽  
Y. T. Gu ◽  
G. R. Liu
Keyword(s):  
Potential Energy ◽  
Displacement Field ◽  
Numerical Experiments ◽  
Interpolation Method ◽  
Problem Domain ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation ◽  
Discontinuous Displacement ◽  
Quadratic Models

It is well known that a high-order point interpolation method (PIM) based on the standard Galerkin formations is not conforming, and thus the solution may not always be convergent. This paper proposes a new interesting technique called quasi-conforming point interpolation method (QC-PIM) for solving elasticity problems, by devising a novel scheme that smears the discontinuity. In the QC-PIM, the problem domain is first discretized by a set of background cells (typically triangles that can be automatically generated), and the average displacements on the interfaces of the two neighboring cells are assumed to be equal. We prove that when the size of background cells approaches to zero, all the additional potential energy coming from the discontinuous displacement field becomes zero, which ensures the pass of the standard patch test and hence the convergence. Numerical experiments verify that QC-PIM can produce the convergent solutions with higher accuracy and convergent rate that is in between fully conforming linear and quadratic models.

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.

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

2009 ◽  
Vol 01 (01) ◽  
pp. 233-258 ◽  
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.

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