Uncertainty Propagation of Initial Conditions in Thermal Models

The operation of machine tools often demands a highly accurate knowledge of the tool center point's (TCP) position. The displacement of the TCP over time can be inferred from thermal models, which comprise a set of geometrically coupled heat equations. Each of these equations represents the temperature in part of the machine, and they are often formulated on complicated geometries. The accuracy of the TCP prediction depends highly on the accuracy of the model parameters, such as heat exchange parameters, and the initial temperature. Thus it is of utmost interest to determine the influence of these parameters on the TCP displacement prediction. In turn, the accuracy of the parameter estimate is essentially determined by the measurement accuracy and the sensor placement. Determining the accuracy of a given sensor configuration is a key prerequisite of optimal sensor placement. We develop here a thermal model for a particular machine tool. On top of this model we propose two numerical algorithms to evaluate any given thermal sensor configuration with respect to its accuracy. We compute the posterior variances from the posterior covariance matrix with respect to an uncertain initial temperature field. The full matrix is dense and potentially very large, depending on the model size. Thus, we apply a low-rank method to approximate relevant entries, i.e. the variances on its diagonal. We first present a straightforward way to compute this approximation which requires computation of the model sensitivities with with respect to the initial values. Additionally, we present a low-rank tensor method which exploits the underlying system structure. We compare the efficiency of both algorithms with respect to runtime and memory requirements and discuss their respective advantages with regard to optimal sensor placement problems.