Reconstruction of boundary condition laws in heat conduction using the boundary element method

Physical problems involving heat exchange between the ends of a rod and the surrounding environment can be formulated as a set of equations representing the heat equation and boundary conditions relating the heat fluxes to the difference between the boundary temperatures and the temperature of the surrounding fluid through a function ff. When the heat transfer is purely convective, or solely radiative, then one assumes that ff is a linear functional (Newton’s law of cooling), or obeys a fourth-order power law (Stefan’s law), respectively. However, there are many practical heat transfer situations in which either the governing equation does not take a simple form or the actual method of heat transfer is unknown. In such cases the heat transfer coefficient depends on the boundary temperature and the dependence has a complicated or unknown structure. In this study, we investigate a one-dimensional inverse heat conduction problem with unknown nonlinear boundary conditions. We develop the boundary element method to construct and solve numerically the missing terms involving the boundary temperature, the heat flux and the function ff. To stabilise the solution we employ an engineering approach based on approximating the function ff by a polynomial function of temperature. Numerical results are presented and discussed.