Geometry + Simulation

The proposed National Research Network (NFN) is devoted to the interaction of the two, mostly disjointly operated disciplines, applied geometry (in particular its descendant Computer Aided Design), which deals with questions of size and shape, and numerical simulation, which is concerned with discretization and approximation methods and their efficient implementation. In general, intrinsic geometric structures - such as the curved surface of an object - are not preserved but only approximated in today`s simulation technology. In recent years, and particularly since the advent of isogeometric analysis (IGA), the interaction between geometry and simulation has become the subject of a substantial number of research activities, taking place all around the world. It has already been demonstrated by these activities that the interaction of the previously disjoint scientific communities possesses significant potential. On the other hand, several challenging issues are still open in this field. The proposed research network will explore the new potential and address some of the challenging problems that are created by the cooperation of the scientific communities dealing with geometry and simulation. Practically relevant problems cannot be addressed satisfactorily by individual groups representing only one of the two communities. Moreover, the two fields will challenge each other to extend their research in new scientific directions. The proposed NFN will provide a coordinated effort to combine expertise from both fields in order to advance the state of the art, concerning both the theory and applications. The formation of the team for the planned NFN has been carefully designed to cover the main research areas involved, and so that it reflects existing strengths and cooperations in the Austrian research community. In view of the wide range of different topics and involved fields, ranging from applied geometry to image processing and numerical analysis, the joint knowledge of all groups will be needed for the successful implementation of this NFN. The six projects in the proposed NFN will be devoted to multigrid methods for isogeometric analysis, discontinuous Galerkin domain decomposition methods in isogeometric analysis, variational methods for imaging on manifolds, geodesic paths in shape space, statistical pattern analysis for advanced geometric descriptors, and geometric modelling for numerical simulation. One of the main goals of the proposed NFN is the initialization of a close cooperation between different research groups working in various areas of geometry and simulation. It is necessary to look at central problems of geometry and simulation from different viewpoints, to gain new tools and insight and to perform the work together within the same big project. An attempt to realize this scenario is the present NFN proposal.