Nonlinear heat equation for nonhomogeneous anisotropic materials: A dual‐reciprocity boundary element solution

Abstract

A dual-reciprocity boundary element method is presented for the numerical solution of initial-boundary value problems governed by a nonlinear partial differential equation for heat conduction in nonhomogeneous anisotropic materials. To assess the validity and accuracy of the method, some specific problems are solved.

A dual-reciprocity boundary element approach for axisymmetric nonlinear time-dependent heat conduction in a nonhomogeneous solid

“…The works in [2,5] are also applicable to linear heat conduction in nonhomogeneous media such as functionally graded materials. The analysis of functionally graded materials is a topic of special interest in boundary element methods.…”

“…Earlier works on boundary element methods, such as Kikuta et al [12] and Goto and Suzuki [9], assume that the solids are thermally isotropic and have density, specific heat capacity and thermal conductivity which are functions of temperature alone. Clements and Budhi [8], Azis and Clements [5] and, more recently, Ang and Clements [2] have proposed boundary element procedures for thermally anisotropic solids with material properties that vary with temperature and spatial coordinates.…”

“…Such a problem is of practical interest as axisymmetric structures can be found in many engineering applications (such as pressure vessels and piping components). The analyses in Ang and Clements [2], Azis and Clements [5] and Brebbia et al [6] are used as a guide to convert the nonlinear partial differential equation governing the axisymmetric heat conduction into a suitable integro-differential equation. In addition to a boundary integral over a curve on an appropriate coordinate plane, the integrodifferential equation also contains a domain integral.…”

A Multipoint Flux Approximation of the Steady-State Heat Conduction Equation in Anisotropic Media

“…For example, Sladek et al [11] introduced a local boundary integral equation method (a class of boundary element method) with the moving least square approximation for spatial variations of physical fields together with using the Laplace transform technique for time variable for the transient heat conduction problem in 3D axisymmetric solid body with continuously nonhomogeneous and anisotropic material properties. Ang and Clements [12] considered the problem of the nonlinear heat equation in nonhomogeneous anisotropic materials using boundary element method. They use Kirchhoff's transformation with substitution of variables to formulate the problem in terms of integrodifferential equation for the development of a dual-reciprocity boundary element method.…”

Fundamental-Solution-Based Hybrid Element Model for Nonlinear Heat Conduction Problems with Temperature-Dependent Material Properties

“…Different to analytical solutions, numerical results can be easily obtained by numerical methods for such nonlinear heat transfer problems, for example, the finite element method (FEM) [1,2], the boundary element method (BEM) [3][4][5], or the dual-reciprocity BEM (DRBEM) [6,7], the method of fundamental solution (MFS) [8,9], and the meshless element free Galerkin method [10,11]. As an alternative to numerical approaches mentioned above, the fundamentalsolution-based hybrid finite element method, named as HFS-FEM, was initially developed for linear heat transfer problem [12] and then was extended to complicated thermal analysis in composite structures [13] and biological tissues [14].…”