Finite block method for transient heat conduction analysis in functionally graded media

Based on the one-dimensional differential matrix derived from the Lagrange series interpolation, the finite block method is proposed first time to solve both stationary and transient heat conduction problems of anisotropic and functionally graded materials. The main idea is to establish the first order one-dimensional differential matrix constructed by using Lagrange series with uniformly distributed nodes. Then the higher order of derivative matrix for one-dimensional problem is obtained. By introducing the mapping technique, a block of quadratic type is transformed from Cartesian coordinate. (xyz) to normalised coordinate. (xi eta sigma) with 8 seeds or 20 seeds for two or three dimensions. Then the differential matrices in physical domain are determined from that in normalised transformed coordinate system. In addition, the time dependent partial differential equations are analysed in the Laplace transformed domain, and the Durbin inversion method is used to determine the values in time domain. Illustrative two-dimensional and three-dimensional numerical examples are given, and comparisons have been made with analytical solutions. Copyright (C) 2014 John Wiley & Sons, Ltd.