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.