Refinable Freeform Splines via Polynomial Reproduction

This project aims to improve methods for solving complex mathematical problems known as partial differential equations (PDEs) on different shapes. Our main focus is on "isogeometric analysis", a technique that employs spline functions for both the design and analysis of complicated shapes. Partial differential equations are crucial in fields such as engineering and physics, but their isogeometric discretisation becomes challenging when dealing with certain points called extraordinary vertices. At these points, it is important to ensure smoothness and accuracy for effective simulations. We present a new type of spline called RFF-Splines (Refinable Freeform Splines) which will support numerical methods without restrictions on shape design. The project encompasses everything from the theoretical concepts to their practical implementation in code (specifically C++) and testing in demanding scenarios that involve high-order PDEs. This comprehensive approach ensures that our solutions are not only robust but also effective. What sets our work apart is the innovative way we develop the foundation for these spline functions. This includes easy-to-use evaluation and integration methods, the ability to adjust based on local needs (using a technique called local refinement), and strong theoretical backing that guarantees the effectiveness of our splines. The main inspiration for RFF-Splines comes from the earlier research of Hartmut Prautzsch, blending polynomial mappings with spline designs. An international team is driving this project, with contributions from JKU Linz (Austria) and INRIA (France). Together, we aim to advance the field of mathematical modelling and simulation through this innovative approach.