scholarly journals The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
J. F. Wang ◽  
S. Y. Hao ◽  
Y. M. Cheng
Keyword(s):  
Boundary Value Problems ◽  
Convergence Rates ◽  
Boundary Value ◽  
Weak Form ◽  
Point Boundary ◽  
Element Free Galerkin Method ◽  
Original Solution ◽  
Element Free Galerkin ◽  
Element Free ◽  
Derivatives Of

The interpolating moving least-squares (IMLS) method is discussed in detail, and a simpler formula of the shape function of the IMLS method is obtained. Then, based on the IMLS method and the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method for two-point boundary value problems is presented. The IEFG method has high computing speed and precision. Then error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theories show that, if the original solution is sufficiently smooth and the order of the basis functions is big enough, the solution of the IEFG method and its derivatives are convergent to the exact solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, two selected numerical examples are given to confirm the theories.

scholarly journals Analyzing 3D Advection-Diffusion Problems by Using the Improved Element-Free Galerkin Method

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Heng Cheng ◽  
Guodong Zheng
Keyword(s):  
Penalty Method ◽  
Weak Form ◽  
Singular Matrix ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Advection Diffusion ◽  
Computational Speed ◽  
The Difference ◽  
Element Free ◽  
Diffusion Problems

In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.

The Element Free Galerkin Method for Fully Developed, Open-Channel Magnetohydrodynamic Flow

2012 ◽  
Vol 232 ◽  
pp. 111-114
Author(s):  
Xing Hui Cai ◽  
Guo Xun Ji ◽  
Peng Xu ◽  
Man Lin Zhu ◽  
Jiang Ren Lu
Keyword(s):  
Liquid Metal ◽  
Galerkin Method ◽  
Open Channel ◽  
Metal Flow ◽  
Weak Form ◽  
Governing Equations ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Liquid Metal Flow ◽  
Element Free

In this paper, an element-free Galerkin method is presented to simulate the liquid metal flow in an open channel under external magnetic field. The global weak form of governing equations is obtained for the case of same size of the height of the liquid film and width of the open channel. Numerical simulations are carried out for some cases of liquid metal flow in an open channel. Results show that the element-free Galerkin method may steadily compute this kind of problem in some cases.

The Improved Interpolating Complex Variable Element Free Galerkin Method for Two-Dimensional Potential Problems

2019 ◽  
Vol 11 (10) ◽  
pp. 1950104 ◽  
Author(s):  
Yajie Deng ◽  
Xiaoqiao He ◽  
Ying Dai
Keyword(s):  
Complex Variable ◽  
Basis Function ◽  
Potential Problem ◽  
Shape Function ◽  
Weak Form ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Potential Problems ◽  
Element Free

In this paper, the improved interpolating complex variable moving least squares (IICVMLS) method is applied, in which the complete basis function is introduced and combined with the singular weight function to achieve the orthometric basis function. Then, the interpolating shape function is achieved to construct the interpolating trial function. Incorporating the IICVMLS method and the Galerkin integral weak form, an improved interpolating complex variable element free Galerkin (IICVEFG) method is proposed to solve the 2D potential problem. Because the essential boundary conditions can be straightaway imposed in the above method, the expressions of final dispersed matrices are more concise in contrast to the non-interpolating complex variable meshless methods. Through analyzing four specific potential problems, the IICVEFG method is validated with greater computing precision and efficiency.

scholarly journals A Meshfree Method for Simulating Myocardial Electrical Activity

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Heye Zhang ◽  
Huajun Ye ◽  
Wenhua Huang
Keyword(s):  
Shape Function ◽  
Meshfree Method ◽  
Weak Form ◽  
Solution Technique ◽  
Heart Model ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Monodomain Model ◽  
Element Free ◽  
Myocardial Electrical Activity

An element-free Galerkin method (EFGM) is proposed to simulate the propagation of myocardial electrical activation without explicit mesh constraints using a monodomain model. In our framework the geometry of myocardium is first defined by a meshfree particle representation that is, a sufficient number of sample nodes without explicit connectivities are placed in and inside the surface of myocardium. Fiber orientations and other material properties of myocardium are then attached to sample nodes according to their geometrical locations, and over the meshfree particle representation spatial variation of these properties is approximated using the shape function of EFGM. After the monodomain equations are converted to their Galerkin weak form and solved using EFGM, the propagation of myocardial activation can be simulated over the meshfree particle representation. The derivation of this solution technique is presented along a series of numerical experiments and a solution of monodomain model using a FitzHugh-Nagumo (FHN) membrane model in a canine ventricular model and a human-heart model which is constructed from digitized virtual Chinese dataset.

A NEW ELEMENT-FREE GALERKIN METHOD BASED ON IMPROVED COMPLEX VARIABLE MOVING LEAST-SQUARES APPROXIMATION FOR ELASTICITY

2012 ◽  
Vol 01 (01) ◽  
pp. 1250011 ◽  
Author(s):  
HONGPING REN ◽  
YUMIN CHENG
Keyword(s):  
Least Squares ◽  
Complex Variable ◽  
Weak Form ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Elasticity Problems ◽  
Element Free

In this paper, by constructing a new functional, an improved complex variable moving least-squares (ICVMLS) approximation is presented. Based on element-free Galerkin (EFG) method and the ICVMLS approximation, a new complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented. Galerkin weak form is used to obtain the discretized equations and the essential boundary conditions are applied with Lagrange multiplier. Then the formulae of the new CVEFG method for two-dimensional elasticity problems are obtained. Compared with the conventional EFG method, the new CVEFG method has greater computational precision and efficiency. For the purposes of demonstration, some selected numerical examples are solved using the ICVEFG method.

scholarly journals An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
F. X. Sun ◽  
C. Liu ◽  
Y. M. Cheng
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Mls Approximation ◽  
Element Free

Based on the moving least-squares (MLS) approximation, an improved interpolating moving least-squares (IIMLS) method based on nonsingular weight functions is presented in this paper. Then combining the IIMLS method and the Galerkin weak form, an improved interpolating element-free Galerkin (IIEFG) method is presented for two-dimensional potential problems. In the IIMLS method, the shape function of the IIMLS method satisfies the property of Kroneckerδfunction, and there is no difficulty caused by singularity of the weight function. Then in the IIEFG method presented in this paper, the essential boundary conditions are applied naturally and directly. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the MLS approximation; then under the same node distribution, the IIEFG method has higher computational precision than element-free Galerkin (EFG) method and interpolating element-free Galerkin (IEFG) method. Four selected numerical examples are presented to show the advantages of the IIMLS and IIEFG methods.

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