Registration-based model reduction in complex two-dimensional geometries

We present a general—i.e., independent of the underlying equation— registration procedure for parameterized model order reduction. Given the spatial domainΩ ⊂ R2 and the manifold M= {uμ : μ ∈ P} associated with the parameter domain P ⊂ RP and the parametric field μ  → uμ ∈ L2(Ω), our approach takes as input a set of snapshots {uk }ntrain k=1⊂Mand returns a parameter-dependent bijectivemappingΦ : Ω×P → R2: the mapping is designed to make the mapped manifold {uμ ◦Φμ : μ ∈ P} more amenable for linear compression methods. In this work,we extend and further analyze the registration approach proposed in [Taddei, SISC,2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition {Ωq }Ndd q=1 of the domain Ω such that Ω1, . . . , ΩNdd are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.