Weak and approximate C1-smoothness in isogeometric analysis

Since its introduction computer aided design (CAD) software has been developed to help design and construct technical objects composed of simple geometric shapes. In technical applications, the designed objects should have certain desired physical properties. In that case, one has to transfer the information from the CAD system to different software to perform the simulation of the underlying physical processes. This transfer of information is usually not possible without loss of information and is often time consuming and inefficient. In the last decade it has been the aim of many research groups to incorporate simulation capabilities directly into the CAD systems. This requires a completely different approach, called isogeometric analysis (IGA), which expands the CAD software from a simple design tool to a fully incorporated simulation program. This project entitled weak and approximate C1-smoothness in isogeometric analysis is exploring this new idea. It is based upon ongoing research at the Institute of Applied Geometry at the JKU Linz. In the project we extend the applicability of the IGA framework substantially to a wide range of applications. We design methods that are more robust and geometrically more flexible than standard methods that are currently in use. In the future these methods may be incorporated into CAD systems, which will then simplify and accelerate design, simulation and manufacturing processes. The obtained results have implications on a wide range of research areas, from biomedical applications, to the design of architectural structures or the development of industrial components. This may help in designing more durable structures or give an indication on whether or not surgery is needed. The project will be conducted within an outstanding research group at JKU Linz in close cooperation with researchers from the Johann Radon Institute of Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences and with the University of Pavia in Italy. In addition we develop and extend existing open-source software packages. The developed methods and obtained results will thus be made available to other researchers and companies.

Computer simulations are of great importance in applied research. In this project we dealt with simulation methods for a special class of problems, so-called partial differential equations. Such problems arise in mathematical modeling in many application areas, such as physics, chemistry, biology, or engineering applications. Specifically, we designed methods that are based on CAD (Computer Aided Design) models. In many fields of application CAD software is used to create computer models of real, physical objects. In such a CAD model, a complex object is composed of simple geometric shapes. In many applications the question arises, which physical properties the examined objects have. In order to perform simulations of the underlying physical processes, the geometric information about the object must be transferred from the CAD system to another software system. This information transfer usually leads to data loss and is often time consuming and inefficient. In this project, entitled "Weak and approximate smoothness in isogeometric analysis", we set out to develop new methods for generating CAD models as well as suitable simulation methods. Thus, methods were developed that are robust and have a high geometric flexibility. The basic idea of the project was to reduce the requirements for the smoothness of the geometric representation. Instead of generating geometric models whose surface is perfectly smooth, the methods developed in this project allow for certain tolerances that are negligible in many applications or disappear as simulation accuracy improves. Thus, IGA simulation methods could be extended to a wide range of different applications. In the future, the newly developed methods can be integrated into CAD systems, to simplify and accelerate the design and development process. The research work was carried out at the Institute of Applied Geometry at JKU Linz, in cooperation with researchers from the Johann Radon Institute of Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences, the University of Pavia, the Carinthia University of Applied Sciences, the TU Delft and the University of Ljubljana. In addition, open-source software was developed as part of the project. The knowledge gained has implications for a wide range of research areas, from biomedical applications to architectural design and the development of industrial components. The methods can, for example, help to design more stable or lighter components.