Isogeometric Analysis on Multiple Patches for Aerospace Applications

Abstract

Computer-driven simulations in the context of aerospace research are of paramount importance. These simulations are done by solving partial differential equations approximately with numerical methods. One famous and recent approach in this context is the isogeometric analysis (IGA) which is able to simulate directly on smooth geometries. Since in most real-world applications the geometries are given as CAD files, the isogeometric approach has to be extended to more complicated geometries, i.e. NURBS-based multiple patches. The contributions of this thesis are that we derive an isogeometric method on multiple patches. Since this a restricted variational problem for elliptic equations, we solve it with the generalized Lagrange multipliers method. Furthermore, we give an estimation for the integration complexity of IGA and compare it to the integration complexity of the classical finite element method (FEM). We demonstrate that the a-priori error of IGA on single patches is the same as encountered in the a-priori error analysis of FEM. In addition, we show that the condition number of IGA increases exponentially with increasing polynomial degree and that it is asymptotically of the same order as the condition number of FEM under h-refinement. Finally, we simulate a potential flow around a NACA airfoil in two dimensions using the isogeometric method on multiple patches. This provides information about the problems and challenges in the application of IGA in modern aerospace science.