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 domain Ω ⊂ R and the manifoldM = {uμ : μ ∈ P} associated with the parameter domain P ⊂ R and the parametric field μ 7→ uμ ∈ L(Ω), our approach takes as input a set of snapshots {u}train k=1 ⊂ M and returns a parameter-dependent bijective mapping Φ : Ω × P → R: 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}dd 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 and for a potential flow past a rotating airfoil to demonstrate the effectiveness of our method.