Fast Methods for Adaptive Isogeometric Analysis

Computer simulations of various physical processes, such as elasticity of solid bodies or fluid flow, play an important role in many modern problems in science and technology. A number of mathematical methods are commonly used towards this goal; in the present project, we are interested in Isogeometric Analysis. This is a relatively young approach which has attracted attention due to the fact that it can directly deal with curved surfaces and bodies - imagine, for instance, a car chassis or the profile of an airplane wing. In traditional approaches, it was often necessary to approximate these curved objects using straight-lined elements, which loses the exact curvature and is often a tedious procedure. An important feature of modern simulation techniques is known as adaptivity. By this we mean the ability of distributing computational effort in such a way that areas of the simulation which exhibit higher degrees of detail are treated at a higher resolution. As an example, consider the flow of a fluid which is calm at first but develops turbulence at some point; it makes sense to invest more computational effort into simulating these more complex turbulent effects at a higher resolution. This intelligent distribution of computer power allows us to save on overall computing time without degrading the quality of the result. We are interested in how to make adaptive simulations in the area of Isogeometric Analysis as efficient as possible. To that end, we are designing new algorithms which are based on the idea of so-called multigrid solvers. Roughly speaking, the idea is to consider the problem simultaneously at multiple different levels of resolution, not dissimilar to how you could view a video at different resolutions. By transferring information between coarse and fine resolutions in a clever way, it is actually possible to obtain solvers which are significantly faster than those which look at only one fixed level of detail. How these levels of detail of the multigrid solver interact with the locally varying levels of resolution which occur due to adaptivity is one of our research questions. By understanding these questions better, we hope to develop new adaptive simulation methods which outperform the state of the art.