Geometric modeling, isogeometric analysis and the finite cell method

The advent of isogeometric analysis (IGA) using the same basis functions for design and analysis constitutes a milestone in the unification of geometric modeling and numerical simulation. However, an important class of geometric models based on the CSG (Constructive Solid Geometry) concept such as trimmed NURBS surfaces do not fully support the isogeometric paradigm, since basis functions do not explicitly represent the boundary. The finite cell method (FCM) is a high-order fictitious domain method, which offers simple meshing of potentially complex domains into a structured grid of cuboid cells, while still achieving exponential rates of convergence for smooth problems. In the present paper, we first discuss the possibility to directly couple the finite cell method to CSG, without any necessity for meshing the three-dimensional domain, and then explore a combination of the best of the two approaches IGA and FCM, closely following ideas of the recently introduced shell FCM. The resulting finite cell extension to isogeometric analysis achieves a truly straightforward transfer of a trimmed NURBS surface into an analysis suitable NURBS basis, while benefiting from the favorable properties of the high-order and high-continuity basis functions. Accuracy and efficiency of the new approach are demonstrated by a numerical benchmark, and its versatility is outlined by the analysis of different trimmed design variants of a brake disk.

► We emphasize that constructive solid modeling (CSG) needs to be better integrated with numerical analysis.
► We introduce the Finite Cell Method (FCM) as a high order fictitious domain approach.
► We demonstrate that the FCM fits very well to CSG.
► We show that geometrical and even topological model changes require no re-meshing.
► We demonstrate that even very complex geometric models can directly be computed by FCM without meshing.