Determination of thermal properties: Applications of regularized output least squares method

The process of numerical analysis in structural fire design comprises three main components: determination of the fire exposure, the thermal analysis and the structural analysis. The thermal analysis requires well-defined input information on thermal material properties for determining the transient temperature state of the fire-exposed structure. This work presents a systematic methodology to treat identification of temperature dependent thermal properties from test results. This method is known as the Regularized Output Least Squares Method (ROLS). Applications of the method to identification of thermal properties in different cases are presented. For each problem, the direct problem is formulated as a system of one or several ordinary differential equations which are semi-discretized via the variational form of the general heat conduction problem. The solution of the direct problem is obtained by time-integrating the semi-discrete equations by means of numerical quadrature. The problem of identification of the parameters appearing in the formulation of the direct problem is know as an inverse problem. A common feature of inverse problems is instability, that is, small changes in the data which may give rise to large changes in the solution. Small finite dimensional problems are typically stable, however, as the discretization is refined, the number of variables increases and the instability of the original problem increases. Therefore regularization is needed. Both mesh coarsing and Tikhonov-regularization have been adapted in order to achieve a stabilized solution. The available a priori known physical constraints on the parameters are taken into account in the minimization. The distributed parameters are discretized. The thermal properties are approximated as piece-wise linear functions of temperature. The unknowns are found by minimizing a constrained and regularized functional which is the sum of the residual norm of the errors (data - model) plus the norm of the second derivatives of the properties with respect to the temperature. An appropriate balance between the need to describe the measurements well and the need to achieve a stable solution is reached by finding an optimal regularization parameter. Both Newton and conjugate gradient methods have been used in the minimization. The Morozov discrepancy principle is used to find a reasonable value for the regularization parameter.