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

Abstract

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.