Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems

TL;DR

This paper introduces a preprocessing technique using transport maps to improve reduced order models for fluid dynamics problems with slow Kolmogorov n-width decay, enhancing basis efficiency.

Contribution

It proposes a novel offline preprocessing method with transport maps to produce smaller reduced basis spaces for complex fluid problems.

Findings

Transport maps reduce basis size needed for accurate models.

Improved reduced order models for fluid-structure interaction.

Comparison shows enhanced efficiency over standard methods.

Abstract

In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov $n$-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.