Preface

The Institute for Industrial Mathematics, which will become the first Fraunhofer Institute for applied mathematics in Germany, has focused its work on practical problems coming from industry. The following case study deals with a very old problem in the production of glass. It is part of a project of the ITWM with Schott Glas, the most important producer of specialized glasses in Germany. The famous physicist Josef von Fraunhofer faced the problem when trying to produce large lenses for astronomical binoculars – almost 200 years ago. During the cooling of the glass, thermal tensions tended to produce fractures, which might be avoided by a proper control of the cooling process. This can be achieved by regulation of the surrounding temperature. However, in order to control the process, it is necessary to understand and to predict the thermal tensions, i.e. the stresses set up by nonhomogeneous thermal contraction. This is complicated, because heat is transported through radiation. Glass is a semitransparent medium – each point in the glass is a new source of radiation. This property is modelled in the radiation equation – but this equation is not easy to solve. In contrast to other case studies in this volume, there is in general nothing flat or thin. The thermal tensions depend heavily on the three-dimensional object as a whole, and the object very often has no symmetries. Therefore, numerical methods are unavoidable, but still pose difficult problems: a straightforward solution of the three-dimensional radiative equation, say by ray tracing, discrete ordinates, or even P1-approximation does not work in real (i.e. real industrial) problems.