Computer Methods in Applied Mechanics and Engineering
Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions
Introduction
In many heat transfer problems the assumption of constant thermal conductivity, i.e. that the heat conductors are homogeneous within the whole temperature variation interval, may lead to unacceptable errors in high-temperature environments or if large temperature differences are present, see [31]. In the steady-state situation, the nonlinearity associated with the temperature dependence of the thermal conductivity can be removed by employing the Kirchhoff transformation, which replaces the original nonlinear partial differential equation in divergence form by the Laplace equation in the transformed space, see [11]. Boundary conditions of the Dirichlet (first kind) or Neumann (second kind) types pose no problem for the transformation, but the Robin convective (third kind) boundary conditions become non-linear. Although this non-linearity is not strong, convergence problems may arise if radiative heat transfer (fourth kind) boundary conditions are also present, see [12]. Since all the non-linearities are transferred to the boundary conditions, the Kirchhoff transformation approach is very well-suited for applying the boundary element method (BEM) [7], [20], or, more recently, the method of fundamental solutions (MFS) [22], [23]. In the same manner these techniques can be extended to composite bodies through the subregion technique. In it, each region is dealt with separately and then the whole body is linked together by applying compatibility and equilibrium conditions along the interfaces between subregions.
Two-dimensional boundary value problems of heat conduction in nonlinear composite materials have been the subject of several studies using the BEM [2], [6], [8]. However, the implementation of the BEM becomes quite tedious, especially in three-dimensional irregular domains. Moreover, the evaluation of the gradient of the temperature solution on the boundary requires the use of finite differences or the evaluation of hypersingular integrals. In order to alleviate some of these difficulties, we propose the use of the MFS. The merits and drawbacks of the MFS over the BEM for solving elliptic boundary value problems are thoroughly discussed in Refs. [10], [13], [14], [19], [25], [32]. Recently, the MFS has been made applicable to inhomogeneous elliptic equations [1] and inverse problems [33].
Prior to this study, the MFS was used for the solution of problems of heat conduction in linear layered materials with linear boundary conditions [3]. It is the purpose of this paper to extend this analysis to nonlinear materials with nonlinear boundary conditions.
The mathematical formulation of the problem is given in Section 2. The MFS and its implementation are described in Section 3. Numerical results are presented in Section 4 and in Section 5 some conclusions and ideas for future applications are given.
Section snippets
Mathematical formulation
Consider a bounded domain , , with piecewise smooth boundary , formed from two (or more) subregions and separated by the interfacial surface . The material of subregion has a temperature dependent thermal conductivity and the material of subregion has a different thermal conductivity . The governing steady-state heat conduction equations arewhere is the temperature solution in domain , , and, for the sake of
The method of fundamental solutions (MFS)
As the sources of nonlinearity are associated with the boundary conditions (2.14), (2.15), (2.17), (2.18) only, the boundary value problem (2.11), (2.12), (2.13), (2.14), (2.15), (2.16), (2.17), (2.18) for each subregion can, following [23], be converted into a minimization problem, or equivalently an algebraic system of nonlinear equations, using the MFS.
From [9], [26], the MFS approximations for the solutions and of the Laplace Eq. (2.11) have the form
Numerical results and discussion
In this section, we present numerical results obtained from the application of the MFS described in the previous section.
Conclusions
In this paper, the application of the MFS to steady-state non-linear heat conduction problems in composite heat conductors has been investigated. The MFS is used in conjunction with a domain decomposition technique and the method recasts the problem as a non-linear minimization problem. The numerical results obtained are in good agreement with the available analytical solutions showing high accuracy and stable convergence, and probably with the BEM results of [8] if these were to be corrected.
Acknowledgement
The authors would like to thank the UK Royal Society for supporting this research.
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