Interfaces and free boundaries

A broad spectrum of mathematical problems in the applied sciences and in medicine concerns the determination of interfaces or free boundaries. For instance, in fracture mechanics one is interested in the contact behaviour of opposite crack faces in order to detect structural damage; in mathematical ?nance one would like to determine the exercise boundary of an option; and in medicine the fast segmentation of structural information in magnetic resonance (MR) images (like the location of a tumour in an intra-operative reassessment) and the tomographic reconstruction of tissue structure by means of non-invasive measurements are of importance. Moreover, there exists a myriad of applications, which require to control a system in such a way that its state optimally ful?lls a prescribed criterion. For instance, in the formation of polymer membranes one is interested in achieving a porosity structure which favours certain ?ltration or permeability properties.The START project concentrated on the development of the mathematical foundations for this broad class of problems. On the one hand, for optimization problems with equilibrium constraints (like the problems of fracture mechanics or of the polymer membrane mentioned above) characterizations of solutions were achieved by means of novel mathematical tools and techniques in set-valued analysis.On the other hand, novel methods for the optimization of forms and topological structures (like in image segmentation and tomography) were introduced. Our new analytical insights were put into practice by means of novel numerical algorithms. In this context, special attention was paid to the ef?cient reduction of the complex mathematical models suitable for computer-oriented realization.This goal was achieved by the rigorous derivation of error estimators and local model adaptation. In mathematical image processing, the project work led to a unifying framework for impedance, ?uorescent optical, and magnetic induction tomography. For this purpose, topological derivatives were used for the detection of the location and number of hidden objects. Subsequently, shape sensitivity was applied in order to locally adjust the shape of the detected inclusions. This methodology can be used in medical imaging as well as in material sciences. Another application area is the detection of buried land mines from re?ections of elastic waves. Further, an ef?cient method for multi-phase segmentation of image information was developed. It allows to process a formidable data volume, as produced by MR imaging, in a short time. The robustness of our new method and its ef?ciency reduce waiting time for patients, which is of interest in medical applications.Shape and topology optimization was also used for the computation of optimal air ducts in diesel engines. In this way, ef?ciency gains in total pressure loss and uniform out?ow of more than 10% could be obtained. The resulting software is used for prototyping in the automotive industry.