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).

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.

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.

The natural neighbor radial point interpolation method in the Elasto-Static analysis of Honeycomb-Shaped foams

2021 ◽  
pp. 2150014
Author(s):  
N. A. Nascimento ◽  
J. Belinha ◽  
R. M. Natal Jorge ◽  
D. E. S. Rodrigues
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Interpolation Method ◽  
The Finite Element Method ◽  
Solid Materials ◽  
Cellular Solid ◽  
Natural Neighbor ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation

Cellular solid materials are progressively becoming more predominant in lightweight structural applications as more technologies realize these materials can be improved in terms of performance, quality control, repeatability and production costs, when allied with fast developing manufacturing technologies such as Additive Manufacturing. In parallel, the rapid advances in computational power and the use of new numerical methods, such as Meshless Methods, in addition to the Finite Element Method (FEM) are highly beneficial and allow for more accurate studies of a wide range of topologies associated with the architecture of cellular solid materials. Since these materials are commonly used as the cores of sandwich panels, in this work, two different topologies were designed — conventional honeycombs and re-entrant honeycombs — for 7 different values of relative density, and tested on the linear-elastic domain, in both in-plane directions, using the Natural Neighbor Radial Point Interpolation Method (NNRPIM), a newly developed meshless method, and the Finite Element Method (FEM) for comparison purposes.

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.

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