Meshless Element Free Galerkin Research Articles

Meshless Method for Numerical Modeling of Pulsed Eddy Currents

Meshless methods have attracted great attention due to their advantage in geometric representation. In this paper, a meshless element-free Galerkin method is applied for the first time to solve pulsed eddy-current problems. Detailed mathematical derivations and the numerical implementation are discussed. The model is validated against analytic solutions for two canonical cases.

MESHLESS EFG METHOD IN THREE-DIMENSIONAL HEAT TRANSFER PROBLEMS: A NUMERICAL COMPARISON, COST AND ERROR ANALYSIS

The present analysis deals with the numerical solution of three-dimensional heat transfer problems using a meshless element-free Galerkin (EFG) method. The EFG method utilizes the moving least-square (MLS) approximation to approximate the unknown function of temperature T(x) with T h (x). The approximants are constructed by using a weight function, a monomial basis, and a set of coefficients that depends on position. A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced using the Lagrange multiplier method. MATLAB codes have been developed to obtain the numerical results for a model problem of three-dimensional heat transfer in orthotropic materials using different EFG weight functions. Three new weight functions, exponential, elliptical, and cosine, are proposed. A numerical comparison is made among the results obtained using proposed (exponential, elliptical, and cosine) and existing (quadratic) weight functions for a model problem. L2 error is calculated for the proposed and existing EFG weight functions using 125 nodes. FORTRAN software has also been developed and executed on a PARAM 10000 supercomputing machine to obtain the computational cost of the EFG method. The computational cost of the EFG method is obtained for different orders of Gaussian quadrature and for different values of scaling parameter. The effect of scaling parameter on EFG results (temperature values) is also discussed in detail. The effectiveness of EFG method is shown by comparing the EFG results with those obtained by the finite-element method.

Application of meshless EFG method in fluid flow problems

This paper deals with the solution of two-dimensional fluid flow problems using the meshless element-free Galerkin method. The unknown function of velocity u(x) is approximated by moving least square approximants uh(x). These approximants are constructed by using a weight function, a monomial basis function and a set of non-constant coefficients. The variational method is used for the development of discrete equations. The Lagrange multiplier technique has been used to enforce the essential boundary conditions. A new exponential weight function has been proposed. The results are obtained for a two-dimensional model problem using different EFG weight functions and compared with the results of finite element and exact methods. The results obtained using proposed weight functions (exponential) are more promising as compared to those obtained using existing weight functions (quartic spline and Gaussian)

A numerical solution of composite heat transfer problems using meshless method

This paper deals with the three-dimensional numerical solution of composite heat transfer problems using meshless element free Galerkin method (EFG). The EFG method utilizes moving least square (MLS) approximants to approximate the unknown function of temperature T( x ) with T h( x ) . These MLS approximants are constructed by using a weight function, a basis function and a set of non-constants coefficients. Variational method is utilized for the discretization of the governing equations. The essential boundary conditions are enforced using Lagrange multiplier technique. The MATLAB codes have been developed to obtain the numerical solution. The EFG results are obtained for a model problem using different weight functions. Three new weight functions namely exponential, rational and cosine have been proposed. A comparison is made among the results obtained using proposed (exponential, rational and cosine) and existing (R&R, cubic spline, quartic spline, Gaussian, quadratic and hyperbolic) EFG weight functions with finite element method (FEM) for a three-dimensional composite heat transfer model problem. The validation of the EFG code has been achieved by comparing the EFG results with those obtained by FEM. The effect of scaling parameter on EFG results has also been discussed.

HEAT TRANSFER ANALYSIS OF TWO-DIMENSIONAL FINS USING MESHLESS ELEMENT FREE GALERKIN METHOD

This article deals with the transient and steady-state solution of two-dimensional heat transfer through the fins using a meshless element free Galerkin method. Moving least-square approximants are used to approximate the unknown function of temperature T h (x) with Th (x) . These approximants are constructed by using a linear basis, a weight function, and a set of nonconstant coefficients. The variational method has been used for the development of discrete equations. Essential boundary conditions are enforced by using Lagrange multipliers. A hyperbolic weight function has been proposed. The results are obtained for a two-dimensional model and compared with the results of the finite-element method.

THE ELEMENT FREE GALERKIN METHOD IN THREE DIMENSIONAL STEADY STATE HEAT CONDUCTION

This paper deals with the solution of three dimensional steady state heat conduction problems using a meshless Element Free Galerkin Method. The unknown function T(x, y, z) is approximated by using moving least square approximants. These approximants are constructed using a weight function, a monomial basis and a set of non-constant coefficients. The governing equations are developed using a variational technique and essential boundary conditions are imposed by using Lagrange multiplier method. A hyperbolic weight function has been proposed. The results are obtained for a three dimensional model by taking different types of weight functions. the Element Free Galerkin Method has been validated by comparing the results with those obtained by finite element and analytical methods.

Topological aspects of meshless methods and nodal ordering for meshless discretizations

AbstractThe meshless element‐free Galerkin method (EFGM) is considered and compared to the finite‐element method (FEM). In particular, topological aspects of meshless methods as the nodal connectivity and invertibility of matrices are studied and compared to those of the FE method. We define four associated graphs for meshless discretizations of EFGM and investigate their connectivity. The ways that the associated graphs for coupled FE‐EFG models might be defined are recommended. The associated graphs are used for nodal ordering of meshless models in order to reduce the bandwidth, profile, maximum frontwidth, and root‐mean‐square wavefront of the corresponding matrices. Finally, the associated graphs are numerically compared. Copyright © 2001 John Wiley & Sons, Ltd.