| Funding agency | Austrian Science Fund (FWF) |
|---|---|
| Project number | P 33956 |
| Proposer and principal inverstigator (until Dec 2022) | Clemens Hofreither |
| Principal investigator (since Jan 2023) | Stefan Takacs |
| Project employees | Stefan Tyoler Ludwig Miter |
| Duration | 2021-01-01 – 2025-12-31 |
Computer simulations of various physical processes, like elastic deformations of solid bodies or fluid flow, play an important role in many applications in science and technology. Several mathematical methods are used towards this goal; in the project, we have focused on “Isogeometric Analysis” (IgA). This is a relatively young approach which has attracted attention since it can directly deal with the representation of bodies from Computer Aided Design software – imagine a car chassis or an airplane wing. Traditional approaches often required the approximation of these objects in other forms, like straight-lined elements, which loses the exact curvature and is often a tedious procedure.
An important feature of modern simulation techniques is adaptivity. This is 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. One classical approach is based on local mesh refinement. As an example, consider the flow of air around an airplane wing 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 have investigated how to make adaptive simulations in the area of IgA as efficient as possible. The main advantage of mesh refinement over alternatives is its conceptual simplicity. Additionally, it usually leads to the algorithms with the smallest computational effort. In the project, we have developed a novel method for an adaptive mesh refinement method that is efficient and preserves the desirable properties of IgA.
One alternative approach to adaptivity are reduced basis methods. Algorithms are developed and used that analyze the solutions from high-fidelity simulations for various configurations in order to determine common characteristics of all solutions. This information is then used to significantly speed up following simulations. The beginnings of these methods date back more than four decades. Since the first stage of high-fidelity simulations can be interpreted as a “learning phase”, such approaches have also many similarities to modern machine learning techniques. We have proposed such a reduced basis method for so-called fractional order diffusion problems that can be used, for example, for the simulation of flows through porous media, where approaches based on adaptive mesh refinement are not as helpful because of aspects of the problem structure.
For both approaches, we have been able to give a mathematical convergence analysis that guarantees that the approximation errors do not exceed certain limits. Moreover, we have provided our methods to open source software libraries for better accessibility for researchers and other interested parties.