Towards Automatic and Reliable Localized Model Order Reduction

Abstract: Finite element based simulation of phenomena governed by partial differential equations is a standard tool in many engineering workflows today. However, the simulation of complex geometries is computationally expensive. Many engineering workflows require multiple simulations with small, non parametric changes in between. The use of localized model order reduction for subsequent simulations of geometries with localized changes is very promising. It produces lots of computational tasks with little dependencies and thus parallelizes well. Furthermore, the possibility to reuse intermediary results in the subsequent simulations can lead to large computational savings. In this thesis, we investigate different aspects of localized model order reduction and propose various improvements. A simulation methodology named ArbiLoMod, comprising a localized training, a localized a posteriori error estimator and an enrichment procedure is proposed. A new localized a posteriori error estimator with computable constants is presented and analyzed. A new training algorithm which is based on a transfer operator is derived. It can be shown to converge nearly as fast as the singular value decay of this operator. The transfer operator's spectrum is observed to decay fast in electromagnetic simulations in printed circuit boards. New online enrichment algorithms are proposed. All results are supported by numerical experiments, for which the source code for reproduction is provided.