Non‐linear heat conduction in composite bodies: A boundary element formulation

Abstract

SUMMARYThe present paper discusses the numerical solution of steady-state non-linear heat conduction problems in composite bodies by using the boundary element method. Two kinds of non-linearities are considered: the temperature dependence of the thermal conductivity and boundary conditions of the radiative type. By introducing the integral of conductivity as a new variable the governing equation of the problem becomes linear in the transform space. Transformed boundary conditions of the Dirichlet and Neumann types are also linear but convective boundary conditions become non-linear. Also, discontinuities arise in the value of the integral of conductivity across the interface between materials with different properties since continuity of temperature is imposed. The problem is numerically solved by discretizing the external and interface boundaries of the region under consideration with constant boundary elements and applying an iterative scheme of the Newton-Raphson type.

Identification of coating defects in cathodically protected underground pipelines

“…For non-linear boundary conditions, a NewtonRaphson method can be used to solve the resulting non-linear system of equations [8,9].…”

A scalable and implicit meshless RBF method for the 3D unsteady nonlinear Richards equation with single and multi‐zone domains

“…However, it is important to note that at each zone interface centre two distinct matching conditions are simultaneously applied, and as such each zone interface centre will result in two entries within the local interpolation system (55). A glossary of terms used for the classification of the various data-centre types is given in Table I.…”

“…The optimal way of including this integral matching equation into the LHI formulation is currently under investigation, and the results will be reported in the literature when available. The existing work on modelling heterogeneous materials with the Kirchhoff transformation, using the boundary element method applied to nonlinear heat conduction problems, can be found in [55] and [56]. Although the difference in K r can be important in many circumstances (see for example [57]), the discontinuity in K s often plays the larger role in the solution behaviour.…”

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“…Also the corresponding values of the temperature, T( ỹ j , t i ), for j ϭ 1, N 0 , are required in order Piecewise linear or quadratic space functions may also be to calculate the new constant value of the thermal diffusiv-introduced (see [23]), but this approach will be investigated ity a iϩ1 , given by expression (8) at the time t iϩ1 . At this in another study.…”