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.

scholarly journals The Improved Element-Free Galerkin Method for 3D Helmholtz Equations

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 14
Author(s):  
Heng Cheng ◽  
Miaojuan Peng
Keyword(s):  
Weak Form ◽  
Weight Functions ◽  
Computational Accuracy ◽  
Element Free Galerkin Method ◽  
Helmholtz Equations ◽  
Scale Parameters ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Computational Speed ◽  
Element Free

The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.

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.

The Improved Element-Free Galerkin Method for Diffusional Drug Release Problems

2020 ◽  
Vol 12 (08) ◽  
pp. 2050096
Author(s):  
Guodong Zheng ◽  
Yumin Cheng
Keyword(s):  
Drug Release ◽  
Weak Form ◽  
Equation System ◽  
Least Square ◽  
Moving Least Square ◽  
Scale Parameters ◽  
Element Free Galerkin ◽  
Penalty Factor ◽  
The Difference ◽  
Element Free

By using the improved moving least-square (IMLS) approximation to present the shape function, the improved element-free Galerkin (IEFG) method is investigated to solve diffusional drug release problems in this paper. In order to get the discretized equation system, Galerkin weak form of a diffusional drug release problem is used with applying essential boundary conditions using the penalty method. The difference method is applied for discretization of time domain. Then the formulae of IEFG method for solving diffusional drug release problems are presented. Three numerical example problems are given to study the convergence of solutions of IEFG method in this paper. The influences of scale parameters of influence domain, penalty factor and node distribution on the accuracy of the solutions of IEFG method are discussed. Compared with finite element method, the correctness of IEFG method in this paper is shown.

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.

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