Registration-based model reduction in complex two-dimensional geometries
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We present a general – i.e., independent of the underlying equation – registration procedure for parameterized model order reduction. Given the spatial domain Ω⊂ℝ^2 and the manifold ℳ= { u_μ : μ∈𝒫} associated with the parameter domain 𝒫⊂ℝ^P and the parametric field μ↦ u_μ∈ L^2(Ω), our approach takes as input a set of snapshots { u^k }_k=1^n_ train⊂ℳ and returns a parameter-dependent bijective mapping Φ: Ω×𝒫→ℝ^2: the mapping is designed to make the mapped manifold { u_μ∘Φ_μ: 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 {Ω_q}_q=1 ^N_ dd of the domain Ω such that Ω_1,…,Ω_N_ dd 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.
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