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Register a new userA discretize-then-map approach for the treatment of parameterized geometries in model order reduction
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Tommaso Taddei
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains ${ Omega_{mu}: mu in mathcal{P} } subset mathbb{R}^D$ where $mu in mathcal{P} subset mathbb{R}^P$ denotes a vector of parameters, (ii) a parameterized mapping ${Phi}_{mu}$ between a reference domain $Omega$ and the parameter-dependent domain $Omega_{mu}$, and (iii) a finite element triangulation of $Omega$, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to move the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard map-then-discretize approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.
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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 $Omega subset mathbb{R}^2$ 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 $mu mapsto u_{mu} in L^2(Omega)$, our approach takes as input a set of snapshots ${ u^k }_{k=1}^{n_{rm train}} subset mathcal{M}$ and returns a parameter-dependent bijective mapping ${Phi}: Omega times mathcal{P} to mathbb{R}^2$: the mapping is designed to make the mapped manifold ${ u_{mu} circ {Phi}_{mu}: , mu in mathcal{P} }$ 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 ${ Omega_{q} }_{q=1} ^{N_{rm dd}}$ of the domain $Omega$ such that $Omega_1,ldots,Omega_{N_{rm dd}}$ 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.
We propose a general --- i.e., independent of the underlying equation --- registration method for parameterized Model Order Reduction. Given the spatial domain $Omega subset mathbb{R}^d$ and a set of snapshots ${ u^k }_{k=1}^{n_{rm train}}$ over $Omega$ associated with $n_{rm train}$ values of the model parameters $mu^1,ldots, mu^{n_{rm train}} in mathcal{P}$, the algorithm returns a parameter-dependent bijective mapping $boldsymbol{Phi}: Omega times mathcal{P} to mathbb{R}^d$: the mapping is designed to make the mapped manifold ${ u_{mu} circ boldsymbol{Phi}_{mu}: , mu in mathcal{P} }$ more suited for linear compression methods. We apply the registration procedure, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowly-decaying Kolmogorov $N$-widths; we also consider the application to problems in parameterized geometries. We present a theoretical result to show the mathematical rigor of the registration procedure. We further present numerical results for several two-dimensional problems, to empirically demonstrate the effectivity of our proposal.
We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping $Phi$ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.
We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.
The Kolmogorov $n$-width of the solution manifolds of transport-dominated problems can decay slowly. As a result, it can be challenging to design efficient and accurate reduced order models (ROMs) for such problems. To address this issue, we propose a new learning-based projection method to construct nonlinear adaptive ROMs for transport problems. The construction follows the offline-online decomposition. In the offline stage, we train a neural network to construct adaptive reduced basis dependent on time and model parameters. In the online stage, we project the solution to the learned reduced manifold. Inheriting the merits from both deep learning and the projection method, the proposed method is more efficient than the conventional linear projection-based methods, and may reduce the generalization error of a solely learning-based ROM. Unlike some learning-based projection methods, the proposed method does not need to take derivatives of the neural network in the online stage.
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