2021/01/25 by Taddei, Tommaso, Zhang, Lei · 1 citation
#35J20 #41A45 #65N30 #90C26 #FOS: Mathematics #G.1.8 #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2101.10259
We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain Ω⊂ 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 \k=1^n\rm train ⊂ M and returns a parameter-dependent bijective mapping Φ: 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 \q=1 ^N\rm dd of the domain Ω such that Ω1,…,Ω_N\rm 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, 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.