A reduced basis method by means of transport maps for a fluid–structure interaction problem with slowly decaying Kolmogorov $ n $-width
Monica Nonino, Francesco Ballarin, Gianluigi Rozza, Yvon Maday, mathLab, Mathematics Area, SISSA, Trieste, Italy, Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France
Abstract
The aim of this work is to present a Model Order Reduction (MOR) procedure that is carried out by means of a preprocessing of the snapshots in the offline phase, and to apply it to a Fluid–Structure Interaction (FSI) problem of interest, where the physical domain is two dimensional, the fluid is Newtonian and laminar, and the solid is one dimensional, linear and elastic. This problem exhibits a slow decay of the Kolmogorov $ n $-width: this is reflected, at the numerical level, by a slow decay in the magnitude of the eigenvalues returned by a Proper Orthogonal Decomposition on the solution manifold. By means of a preprocessing procedure, we show how we are able to control the decay of the Kolmogorov $ n $–width of the obtained solution manifold. The preprocessing employed in the manuscript is based on the composition of the snapshots with a map belonging to a family of smooth and invertible mappings from the physical domain into itself. In order to assess the capabilities and the performance of the proposed MOR strategy, we draw a comparison between the results of the novel offline stage and the standard one, as well as a comparison between the novel online phase and the standard one.