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

SUMMARY 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 (ξης) 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 © 2014 John Wiley & Sons, Ltd.

[1]  Y. G. Xu,et al.  Inverse heat conduction problems by using particular solutions , 2011 .

[2]  Li,et al.  Moving least-square reproducing kernel methods (I) Methodology and convergence , 1997 .

[3]  S. Li,et al.  Synchronized reproducing kernel interpolant via multiple wavelet expansion , 1998 .

[4]  Junji Tani,et al.  Surface Waves in Functionally Gradient Piezoelectric Plates , 1994 .

[5]  Satya N. Atluri,et al.  Meshless Local Petrov-Galerkin Method for Heat Conduction Problem in an Anisotropic Medium , 2004 .

[6]  K. Y. Lam,et al.  TRANSIENT WAVES IN A FUNCTIONALLY GRADED CYLINDER , 2001 .

[7]  K. Y. Lam,et al.  Characteristics of waves in a functionally graded cylinder , 2002 .

[8]  K. Y. Lam,et al.  Transient waves in plates of functionally graded materials , 2001 .

[9]  G. Paulino,et al.  ISOPARAMETRIC GRADED FINITE ELEMENTS FOR NONHOMOGENEOUS ISOTROPIC AND ORTHOTROPIC MATERIALS , 2002 .

[10]  直 大好,et al.  傾斜機能材料平板におけるラム波伝播とその衝撃応答 : 第一報,解析手法 , 1991 .

[11]  M. H. Aliabadi,et al.  An improved meshless collocation method for elastostatic and elastodynamic problems , 2007 .

[12]  Ted Belytschko,et al.  Arbitrary Lagrangian-Eulerian formulation for element-free Galerkin method , 1998 .

[13]  Xu Han,et al.  A QUADRATIC LAYER ELEMENT FOR ANALYZING STRESS WAVES IN FGMS AND ITS APPLICATION IN MATERIAL CHARACTERIZATION , 2000 .

[14]  Ted Belytschko,et al.  Smoothing, enrichment and contact in the element-free Galerkin method , 1999 .

[15]  T. Belytschko,et al.  Thermal softening induced plastic instability in rate-dependent materials , 2009 .

[16]  K. Y. Lam,et al.  Stress waves in functionally gradient materials and its use for material characterization , 1999 .

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

[18]  Ch. Zhang,et al.  Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients , 2005 .

[19]  Vladimir Sladek,et al.  Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method , 2003 .

[20]  Theodosios Korakianitis,et al.  Finite integration method for partial differential equations , 2013 .

[21]  Yoshinari Miyamoto,et al.  Functionally Graded Materials. , 1995 .

[22]  Shaofan Li,et al.  Reproducing kernel hierarchical partition of unity, Part I—formulation and theory , 1999 .

[23]  F. Durbin,et al.  Numerical Inversion of Laplace Transforms: An Efficient Improvement to Dubner and Abate's Method , 1974, Comput. J..

[24]  Ted Belytschko,et al.  EFG approximation with discontinuous derivatives , 1998 .

[25]  Y. Hon,et al.  Finite integration method for nonlocal elastic bar under static and dynamic loads , 2013 .

[26]  H. A. Watts,et al.  Computational Solution of Linear Two-Point Boundary Value Problems via Orthonormalization , 1977 .

[27]  Wing Kam Liu,et al.  Reproducing kernel particle methods for structural dynamics , 1995 .

[28]  Ch. Zhang,et al.  Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids , 2003 .