A Numerical Study of Weight Functions, Scaling, and Penalty Parameters for Heat Transfer Applications

ABSTRACT This article describes a numerical study of weight functions, scaling, and penalty parameters for heat transfer problems. The numerical analysis is carried out using a meshless element-free Galerkin (EFG) method, which utilizes moving least-square (MLS) approximants to approximate the unknown function of temperature. These MLS approximants are constructed by using a weight function, a basis function, and a set of coefficients that depend on position. Lagrange multiplier and penalty methods are used to enforce the essential boundary conditions. MATLAB software is developed to obtain the EFG results. A new rational weight function is proposed. Comparisons are made among the results obtained using cubic spline, quartic spline, Gaussian, quadratic, hyperbolic, rational, exponential and, cosine weight functions in one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) heat transfer problems. The L2 error norm and rate of convergence are evaluated for different EFG weight functions and the finite-element method (FEM). The effect of scaling and penalty parameters on EFG results is discussed in detail. The results obtained by the EFG method are compared with those obtained by finite-element and analytical methods.

[1]  Lalita Udpa,et al.  Meshless element-free Galerkin method in NDT applications , 2002 .

[2]  Ted Belytschko,et al.  THE ELEMENT FREE GALERKIN METHOD FOR DYNAMIC PROPAGATION OF ARBITRARY 3-D CRACKS , 1999 .

[3]  D. O. Thompson,et al.  Review of Progress in Quantitative Nondestructive Evaluation , 1985 .

[4]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[5]  I. Singh,et al.  HEAT TRANSFER ANALYSIS OF TWO-DIMENSIONAL FINS USING MESHLESS ELEMENT FREE GALERKIN METHOD , 2003 .

[6]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[7]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[8]  Ted Belytschko,et al.  Element-free Galerkin method for wave propagation and dynamic fracture , 1995 .

[9]  Ted Belytschko,et al.  Element-free Galerkin methods for dynamic fracture in concrete , 2000 .

[10]  YuanTong Gu,et al.  A coupled Element Free Galerkin / Boundary Element method for stress analysis of two-dimensional solids , 2001 .

[11]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[12]  T. Belytschko,et al.  Analysis of thin shells by the Element-Free Galerkin method , 1996 .

[13]  Ted Belytschko,et al.  Enforcement of essential boundary conditions in meshless approximations using finite elements , 1996 .

[14]  Indra Vir Singh,et al.  MESHLESS EFG METHOD IN THREE-DIMENSIONAL HEAT TRANSFER PROBLEMS: A NUMERICAL COMPARISON, COST AND ERROR ANALYSIS , 2004 .

[15]  YuanTong Gu,et al.  Boundary meshfree methods based on the boundary point interpolation methods , 2002 .

[16]  T. Belytschko,et al.  Fracture and crack growth by element free Galerkin methods , 1994 .

[17]  Ted Belytschko,et al.  An element-free Galerkin method for three-dimensional fracture mechanics , 1997 .

[18]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[19]  Sharif Rahman,et al.  An element-free Galerkin method for probabilistic mechanics and reliability , 2001 .

[20]  T. Belytschko,et al.  DYNAMIC FRACTURE USING ELEMENT-FREE GALERKIN METHODS , 1996 .

[21]  K. Bathe,et al.  The method of finite spheres , 2000 .

[22]  Ted Belytschko,et al.  A Petrov-Galerkin Diffuse Element Method (PG DEM) and its comparison to EFG , 1997 .

[23]  S. Rahman,et al.  Probabilistic fracture mechanics by Galerkin meshless methods – part I: rates of stress intensity factors , 2002 .

[24]  Guangyao Li,et al.  Element‐free Galerkin method for contact problems in metal forming analysis , 2001 .

[25]  Ted Belytschko,et al.  Analysis of thin plates by the element-free Galerkin method , 1995 .

[26]  T. Belytschko,et al.  THE NATURAL ELEMENT METHOD IN SOLID MECHANICS , 1998 .

[27]  Ted Belytschko,et al.  ESFLIB: A library to compute the element free Galerkin shape functions , 2001 .

[28]  Antonio Huerta,et al.  Imposing essential boundary conditions in mesh-free methods , 2004 .

[29]  Sharif Rahman,et al.  Probabilistic fracture mechanics by Galerkin meshless methods – part II: reliability analysis , 2002 .

[30]  Ted Belytschko,et al.  A coupled finite element-element-free Galerkin method , 1995 .

[31]  T. Belytschko,et al.  A new implementation of the element free Galerkin method , 1994 .

[32]  T. Belytschko,et al.  Crack propagation by element-free Galerkin methods , 1995 .

[33]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[34]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[35]  Joseph J Monaghan,et al.  An introduction to SPH , 1987 .

[36]  Sunil Saigal,et al.  AN IMPROVED ELEMENT FREE GALERKIN FORMULATION , 1997 .

[37]  S. Atluri,et al.  A meshless local boundary integral equation (LBIE) method for solving nonlinear problems , 1998 .

[38]  Vlatko Cingoski,et al.  Element-free Galerkin method for electromagnetic field computations , 1998 .

[39]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[40]  YuanTong Gu,et al.  Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation , 2000 .

[41]  T. Belytschko,et al.  Element-free galerkin methods for static and dynamic fracture , 1995 .

[42]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[43]  Ravi Prakash,et al.  The Element Free Galerkin Method in Three Dimensional Steady State Heat Conduction , 2002, Int. J. Comput. Eng. Sci..

[44]  S. Rahman,et al.  An efficient meshless method for fracture analysis of cracks , 2000 .

[45]  G. Yagawa,et al.  Free mesh method: A new meshless finite element method , 1996 .