Influence of selectable parameters in element-free Galerkin method: One-dimensional bar axially loaded problem

2008 ◽  
Vol 222 (9) ◽  
pp. 1621-1633 ◽  
Author(s):  
O F Valencia ◽  
F J Gómez-Escalonilla ◽  
J López Díez
Keyword(s):  
Galerkin Method ◽  
Computational Cost ◽  
Axial Loads ◽  
Element Free Galerkin Method ◽  
One Dimensional ◽  
Practical Applications ◽  
Element Free Galerkin ◽  
Support Size ◽  
Optimal Values ◽  
Element Free

Meshless methods (MMs) have become interesting and promising methods in solving partial differential equations, because of their flexibility in practical applications when compared with the standard finite-element method (e.g. crack propagation, large deformations, and so on). Implementation of these methods requires a good understanding of the influence of some specific selectable parameters. In this article, those parameters are analysed for one of the most popular MMs, the element-free Galerkin method, considering both accuracy and computational cost. Thus, the dependence of the solutions on grid irregularity, order of the polynomial basis, type of weight function, and the support size is investigated, and conclusions are drawn with respect to recommended or ‘optimal’ values for one-dimensional bar problems with applied axial loads.

Some Advantages of the Elliptic Weight Function for the Element Free Galerkin Method

2005 ◽  
Author(s):  
Reza Naghdabadi ◽  
Mohsen Asghari
Keyword(s):  
Weight Function ◽  
Galerkin Method ◽  
Computational Cost ◽  
Characteristic Parameter ◽  
Major Influence ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Influence Radius ◽  
Anisotropic Weight ◽  
Element Free

In this paper, an anisotropic weight function in the elliptic form is introduced for the Element Free Galerkin Method (EFGM). In the circular (isotropic) weight function, each node has one characteristic parameter that determines its domain of influence. In the elliptic weight function, each node has three characteristic parameters that are major influence radius, minor influence radius and the direction of the major influence. Using the elliptic weight function each point of the domain may be affected by a less number of nodes in certain conditions. Thus, the computational cost of the method is decreased. In addition, the dependency of the solution on the method that is used for the enforcement of the essential boundary conditions, decreases. As an application of the proposed elliptic weight function, some examples of elastostatic problems are solved and the results are compared with those available in the literature.

scholarly journals Element-Free Galerkin Method Based on Block-Pulse Wavelets Integration for Solving Fourth-Order Obstacle Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Azam ◽  
Khalid Parvez ◽  
Muhammad Omair
Keyword(s):  
Galerkin Method ◽  
Fourth Order ◽  
Computational Cost ◽  
Contact Problems ◽  
Weight Functions ◽  
Test Problems ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Element Free ◽  
Technique Comparison

We introduce improved element-free Galerkin method based on block pulse wavelet integration for numerical approximations to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. Moving least squares (MLS) approach is used to construct shape functions with optimized weight functions and basis. Numerical results for test problems are presented in this article to elaborate the pertinent features for the proposed technique. Comparison with existing techniques shows that our proposed method based on integration technique provides better approximation at reduced computational cost.

The influence of selectable parameters in the element-free Galerkin method: A one-dimensional beam-in-bending problem

2009 ◽  
Vol 223 (7) ◽  
pp. 1579-1590 ◽  
Author(s):  
O F Valencia ◽  
F J Gómez-Escalonilla ◽  
J López-Díez
Keyword(s):  
Galerkin Method ◽  
Least Squares ◽  
Least Squares Method ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Element Free Galerkin Method ◽  
One Dimensional ◽  
Element Free Galerkin ◽  
Axially Loaded ◽  
Element Free

Continuing with the analysis performed for the one-dimensional axially loaded bar problem, a beam in bending is analysed to understand the influence of the characteristic parameters that have any influence in the solution of this problem using the element-free Galerkin method (EFGM), one of the most popular meshless methods. Both accuracy and time cost are considered as the evaluation functions to perform such an analysis. Both functions provide a reasonable idea to consider EFGM as an adequate method to solve the problem considered in this article. As in a one-dimensional axially loaded bar problem, the parameters to be considered will be those that affect the solution: number of nodes in which the domain is modelled, the nodes scatter, the order of the polynomial base to generate shape functions, the order of the quadrature to solve integrals, and the support radius. Besides, as in a one-dimensional axially loaded problem, some cases with different loading and stiffness conditions are considered. However, in this analysis a generalized moving least squares method is used to create shape functions instead of the moving least squares.

Application of an Element-free Galerkin Method to Water Wave Propagation Problems

2014 ◽  
Vol 60 (1-4) ◽  
pp. 87-105 ◽  
Author(s):  
Ryszard Staroszczyk
Keyword(s):  
Wave Propagation ◽  
Galerkin Method ◽  
Present Model ◽  
Free Surface Elevation ◽  
Element Free Galerkin Method ◽  
Plane Flow ◽  
Element Free Galerkin ◽  
Proposed Model ◽  
Sloping Bed ◽  
Element Free

Abstract The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is solved numerically by applying an element-free Galerkin method. First, the proposed model is validated by comparing its predictions with experimental data for the plane flow in water of uniform depth. Then, as illustrations, results of numerical simulations performed for plane gravity waves propagating through a region with a sloping bed are presented. These results show the evolution of the free-surface elevation, displaying progressive steepening of the wave over the sloping bed, followed by its attenuation in a region of uniform depth. In addition, some of the results of the present model are compared with those obtained earlier by using the conventional finite element method.

Implementing of Displacement Boundary Conditions of Element-Free Galerkin Method and its Applications Used in Fracture Mechanics

2012 ◽  
Vol 629 ◽  
pp. 606-610
Author(s):  
Gang Cheng ◽  
Wei Dong Wang ◽  
Dun Fu Zhang
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Reproducing Kernel ◽  
Particle Method ◽  
Transformation Method ◽  
Computational Method ◽  
Reproducing Kernel Particle Method ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Element Free

The main draw back of the Moving Least Squares (MLS) approximate used in element free Galerkin method (EFGM) is its lack the property of the delta function. To alleviate difficulties in the treatment of essential boundary conditions in EFGM, the local transformation method and the boundary singular weight method, which are used in the reproducing kernel particle method, is combined with the element free Galerkin method. The computational method is given to analyze the stress intensity factors and the numerical simulation of crack propagation of two-dimentional problems of the elastic fracture analysis. The application examples reveal the effectiveness and feasibility of the present methods.

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