Resolution of an inverse parabolic problem using sinc-galerkin method

In this paper, a numerical method is proposed to solve an Inverse Heat Conduction Problem (IHCP) using noisy data based on Sinc-Galerkin method. A stable numerical solution is determined for the problem. To do this, we use a sensor located at a point inside the body and measure u(x, t) at a point x = a, where 0 < a < 1. We also show that the rate of convergence of the method is as exponential. The numerical results show the efficiency of our approach to estimate the unknown functions of the inverse problem. The function can be computed within a couple of minutes CPU time at pentium IV-2.7 GHz PC.Publisher's Versio

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

Efficient computational strategies for the control process of continuous casting machines

In continuous casting machineries, monitoring the mold is essential for the safety and quality of the process. Then, the objective of this thesis is to develop mathematical tools for the real-time estimation of the mold-steel heat flux which is the quantity of interest when controlling the mold behaviour. We approach this problem by first considering the mold modelling problem (direct problem). Then, we plant the heat flux estimation problem as the inverse problem of estimating a Neumann boundary condition having as data pointwise temperature measurements in the interior of the mold domain given by the thermocouples that are buried inside the mold plates. In formulating the inverse problem, we consider both the steady and unsteady-state case. For the numerical solution of these problems, we develop several methodologies. We consider traditional methods such as Alifanov's regularization as well as novel methodologies that exploit the parametrization of the sought heat flux. We develop the latter methods to have an offline-online decomposition with a computationally efficient online part. Moreover, in the unsteady-state case, we propose a novel, incremental, data-driven model order reduction technique to achieve the real-time performance of the online phase. Finally, we test all discussed methods on academic and industrial benchmark cases. The results show that the proposed novel numerical tools outclass traditional methods both in performance and computational cost. Moreover, they prove to be robust with respect to the measurements noise and confirm that the computational cost is suitable for real-time estimation of the heat flux