The simulation of heat transport for a single device is easily tackled by current computational resources, even for a complex, finely structured geometry; however, the calculation of a multi-scale system consisting of a large number of those devices, e.g., assembled printed circuit boards, is still a challenge. A further problem is the large change in heat conductivity of many semiconductor materials with temperature. We model the heat transfer along a 1D beam that has a nonlinear heat capacity which is represented by a polynomial of arbitrary degree as a function of the temperature state. For accurate modeling of the temperature distribution, the resulting model requires many state variables to be described adequately. The resulting complexity, i.e., number of first order differential equations and nonlinear parts, is such that a simplification or model reduction is needed in order to perform a simulation in an acceptable amount of time for the applications at hand.
In this paper, we describe the modeling considerations leading to a large nonlinear system of equations. Sample results from this model and examples of successful model order reduction can be found in [YLLK04] and the corresponding benchmark document, available online on the Oberwolfach Model Reduction Benchmark Collection website [OBC] (“Nonlinear heat transfer modeling”).