Calibration and Rescaling Principles for Nonlinear Inverse Heat Conduction and Parameter Estimation Problems

This dissertation provides a systematic method for resolving nonlinear inverse heat conduction problems based on a calibration formulation and its accompanying principles. It is well-known that inverse heat conduction problems are ill-posed and hence subject to stability and uniqueness issues. Regularization methods are required to extract the best prediction based on a family of solutions. To date, most studies require sophisticated and combined numerical methods and regularization schemes for producing predictions. All thermophysical and geometrical properties must be provided in the simulations. The successful application of the numerical methods relies on the accuracy of the related system parameters as previously described. Due to the existence of uncertainties in the system parameters, these numerical methods possess bias of varying magnitudes. The calibration based approaches are proposed to minimize the systematic errors since system parameters are implicitly included in the mathematical formulation based on several calibration tests. To date, most calibration inverse studies have been based on the assumption of constant thermophysical properties. In contrast, this dissertation focuses on accounting for temperature-dependent thermophysical properties that produces a nonlinear heat equation. A novel rescaling principle is introduced for linearzing the system. This concept generates a mathematical framework similar to that of the linear formulation. Unlike the linear formulation, the present approach does require knowledge of thermophysical properties. However, all geometrical properties and sensor characterization are completely removed from the system. In this dissertation, a linear one-probe calibration method is first introduced as background. After that, the calibration method is generalized to the one-probe and two-probe, one-dimensional thermal system based on the assumption of temperature-dependent thermophysical properties. All previously proposed calibration equations are expressed in terms of a Volterra integral equation of the first kind for the unknown surface (net) heat flux and hence requires regularization owning to the ill-posed nature of first kind equations. A new strategy is proposed for determining the optimal regularization parameter that is independent of the applied regularization approach. As a final application, the described calibration principle is used for estimating unknown thermophysical properties above room temperature

Boundary element methods for solving inverse boundary conditions identification problems

This thesis explores various features of the boundary element method (BEM) used in solving heat transfer boundary conditions identification problems. In particular, we present boundary integral equation (BIE) formulations and procedures of the numerical computation for the approximation of the boundary temperatures, heat fluxes and space, time or temperature dependent heat transfer coefficients. There are many practical heat transfer situations where such problems occur, for example in high temperature regions or hostile environments, such as in combustion chambers, steel cooling processes, etc., in which the actual method of heat transfer on the surface is unknown. In such situations the boundary condition relating the heat flux to the difference between the boundary temperature and that of the surrounding fluid is represented by an unknown function which may depend on space, time, or temperature. In these inverse heat conduction problems (IHCP), the BEM is formulated as a minimization of some functional that measures the discrepancy between the measured data, say the average temperature on a portion of the boundary or at an instant over the whole domain. The minimization provides solutions that are consistent with the data. This indicates that the BEM algorithms for the IRCP are robust, stable and predict reliable results. When the input data is noisy, we have used the truncated singular value decomposition and the Tikhonov regularisation methods to stabilise the solution of the IRCI' boundary conditions identification. Numerical approximations have been obtained and, where possible, the results obtained are compared to the analytical solutions