Meshfree Method Analysis of Biot's Consolidation Using Cell-Based Smoothed Point Interpolation Method

2016 ◽  
Vol 846 ◽  
pp. 409-414
Author(s):  
Arash Tootoonchi ◽  
Arman Khoshghalb ◽  
Nasser Khalili
Keyword(s):  
Interpolation Method ◽  
Meshfree Method ◽  
Delta Function ◽  
Shape Functions ◽  
Point Interpolation Method ◽  
Background Mesh ◽  
Point Interpolation ◽  
Support Domain ◽  
Biot’S Equations ◽  
Method Analysis

A set of cell-based smoothed point interpolation methods are proposed for the numerical analysis of Biot’s formulation. In the proposed methods, the problem domain is discretized using a triangular background mesh. Shape functions are constructed using either polynomial or radial point interpolation method (PIM), leading to the delta function property of shape functions and consequently, easy implementation of essential boundary conditions. The Biot’s equations are discretised in space and time. A variety of support domain selection schemes (T-schemes) are investigated. The accuracy and convergence rate of the proposed methods are examined by comparing the numerical results with the analytical solution for the benchmark problem of one dimensional consolidation.

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.

A Coupled FE-Meshfree Triangular Element for Acoustic Radiation Problems

2020 ◽  
pp. 2041002
Author(s):  
Wei Li ◽  
Qifan Zhang ◽  
Qiang Gui ◽  
Yingbin Chai
Keyword(s):  
Acoustic Radiation ◽  
Interpolation Method ◽  
Partition Of Unity ◽  
Local Approximation ◽  
Triangular Element ◽  
Two Dimensional ◽  
Shape Functions ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation

To improve the accuracy of the standard finite element (FE) solutions for acoustic radiation computation, this work presents the coupling of a radial point interpolation method (RPIM) with the standard FEM based on triangular (T3) mesh to give a coupled “FE-Meshfree” Trig3-RPIM element for two-dimensional acoustic radiation problems. In this coupled Trig3-RPIM element, the local approximation (LA) is represented by the polynomial-radial basis functions and the partition of unity (PU) concept is satisfied using the standard FEM shape functions. Incorporating the present coupled Trig3-RPIM element with the appropriate non-reflecting boundary condition, the two-dimensional acoustic radiation problems in exterior unbounded domain can be successfully solved. The numerical results demonstrate that the present coupled Trig3-RPIM have significant superiorities over the standard FEM and can be regarded as a competitive numerical techniques for exterior acoustic computation.

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

2014 ◽  
Vol 969 ◽  
pp. 55-62 ◽  
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.

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.

Shear Locking Analysis of Plate Bending by Using Meshless Local Radial Point Interpolation Method

2012 ◽  
Vol 166-169 ◽  
pp. 2867-2870 ◽  
Author(s):  
Ping Xia ◽  
Ke Xiang Wei
Keyword(s):  
Boundary Conditions ◽  
Interpolation Method ◽  
Delta Function ◽  
Plate Bending ◽  
Shear Locking ◽  
Radial Point Interpolation Method ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Two Sides

The shape function of the meshless local radial point interpolation method is constructed by using the radial basis functions and possesses Kronecker delta function properties. Therefore, the essential boundary conditions can be easily imposed. Causation of shear locking occur in plate bending is analyzed. Bending problems for plate with two sides simply supported, the other two sides clamped boundary conditions, are analyzed by the meshless local radial point interpolation method. The shear locking is easier avoided in the meshless method than in the finite element method, and the measure of avoiding the shear locking are presented.

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.

Dispersion Error Reduction for Acoustic Problems Using the Finite Element-Least Square Point Interpolation Method

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
L. Y. Yao ◽  
J. W. Zhou ◽  
Z. Zhou ◽  
L. Li
Keyword(s):  
Finite Element ◽  
Interpolation Method ◽  
Local Approximation ◽  
Least Square ◽  
Numerical Dispersion ◽  
Benchmark Problems ◽  
Shape Functions ◽  
Dispersion Error ◽  
Point Interpolation Method ◽  
Point Interpolation

The shape function of the finite element-least square point interpolation method (FE-LSPIM) combines the quadrilateral element for partition of unity and the least square point interpolation method (LSPIM) for local approximation, and inherits the completeness properties of meshfree shape functions and the compatibility properties of FE shape functions, and greatly reduces the numerical dispersion error. This paper derives the formulas and performs the dispersion analysis for the FE-LSPIM. Numerical results for benchmark problems show that, the FE-LSPIM yields considerably better results than the finite element method (FEM) and element-free Galerkin method (EFGM).

On the Usage of Tetrahedral Background Cells in Nodal Integration of RPIM for 3D Elasto-Static Problems

2015 ◽  
Vol 12 (06) ◽  
pp. 1550036
Author(s):  
M. M. Yavuz ◽  
B. Kanber
Keyword(s):  
Taylor Series ◽  
Interpolation Method ◽  
Shape Parameters ◽  
Radial Point Interpolation Method ◽  
Nodal Integration ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Support Domain ◽  
Fifth Order

In this paper, tetrahedral background cells are used in nodal integration of radial point interpolation method (RPIM). The nodal integration is based on Taylor series terms and it is originally applied for the solutions of 2D problems in literature. Therefore, in this study, it is attempted that the tetrahedral integration cells are used in the solution of 3D elasto-static problems. The accuracy is seriously affected by order of Taylor series terms and it is investigated up to fifth order. A methodology is developed for prevention of negative volumes and calculation problems in subdivision of integration cells for each node. Three different case studies are solved with different support domain sizes and shape parameters. The best accuracy is achieved with fourth-order Taylor terms in nodal integration radial point interpolation method (NI-RPIM). [Formula: see text]-value of 3.00 and [Formula: see text] value of 1.03 in radial basis functions give good results in all cases.

scholarly journals An improved radial point interpolation method applied for elastic problem with functionally graded material

2015 ◽  
Vol 18 (2) ◽  
pp. 131-138
Author(s):  
Viet Quoc Phung ◽  
Nha Thanh Nguyen ◽  
Thien Tich Truong
Keyword(s):  
Interpolation Method ◽  
Functionally Graded ◽  
Thin Plates ◽  
Graded Materials ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Graded Material ◽  
Point Interpolation ◽  
Support Domain ◽  
The One

A meshless method based on radial point interpolation was developed as an effective tool for solving partial differential equations, and has been widely applied to a number of different problems. Besides its advantages, in this paper we introduce a new way to improve the speed and time calculations, by construction and evaluation the support domain. From the analysis of two-dimensional thin plates with different profiles, structured conventional isotropic materials and functional graded materials (FGM), the results are compared with the results had done before that indicates: on the one hand shows the accuracy when using the new way, on the other hand shows the time count as more economical

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.

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