We present a general—i.e., independent of the underlying equation—egistration procedure for parameterized model order reduction. Given the spatial domain \(\varOmega \subset \mathbb {R}^2\) and the manifold \(\mathcal {M}= \{ u_{\mu } : \mu \in \mathcal {P} \}\) associated with the parameter domain \(\mathcal {P} \subset \mathbb {R}^P\) and the parametric field \(\mu \mapsto u_{\mu } \in L^2(\varOmega )\), our approach takes as input a set of snapshots \(\{ u^k \}_{k=1}^{n_\mathrm{train}} \subset \mathcal {M}\) and returns a parameter-dependent bijective mapping \({\varPhi }: \varOmega \times \mathcal {P} \rightarrow \mathbb {R}^2\): the mapping is designed to make the mapped manifold \(\{ u_{\mu } \circ {\varPhi }_{\mu }: \, \mu \in \mathcal {P} \}\) 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 \(\{ \varOmega _{q} \}_{q=1} ^{N_\mathrm{dd}}\) of the domain \(\varOmega \) such that \(\varOmega _1,\ldots ,\varOmega _{N_\mathrm{dd}}\) 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.

我们提出了一个通用的——即独立于基础方程——参数化模型阶数减少的注册程序。给定空间域\(\varOmega \subset \mathbb {R}^2\)和流形\(\mathcal {M}= \{ u_{\mu } : \mu \in \mathcal {P} \}\ )与参数域\(\mathcal {P} \subset \mathbb {R}^P\)和参数域\(\mu \mapsto u_{\mu } \in L^2(\varOmega )\) 相关联，我们的方法将一组快照\(\{ u^k \}_{k=1}^{n_\mathrm{train}} \subset \mathcal {M}\) 作为输入，并返回一个依赖参数的双射映射\({\varPhi }: \varOmega \times \mathcal {P} \rightarrow \mathbb {R}^2\)：映射旨在使映射流形\(\{ u_{\mu } \circ {\varPhi }_{\mu }: \, \mu \in \mathcal {P} \}\)更适合线性压缩方法。在这项工作中，我们扩展并进一步分析了 [Taddei, SISC, 2020] 中提出的注册方法。当前工作的贡献是双重的。首先，我们通过引入适当的坐标系变换来扩展处理环形域的方法。其次，我们讨论一般二维几何的扩展：为此，我们引入了一个谱元素近似，它依赖于一个分区\(\{ \varOmega _{q} \}_{q=1} ^{N_ \mathrm{dd}}\)域\(\varOmega \)使得\(\varOmega _1,\ldots ,\varOmega _{N_\mathrm{dd}}\)与单位平方同构。我们进一步表明，我们的光谱元素近似可以处理参数化几何。我们提出了严格的数学分析来证明我们的建议；此外，我们提供了环形域中传热问题的数值结果，通过旋转对称翼型的势流和通过非对称翼型的无粘性跨音速可压缩流，以证明我们方法的有效性。