Reduced‐order modeling of parameterized PDEs using time–space‐parameter principal component analysis

Audouze, Christophe; Vuyst, Florian De; Nair, Prasanth B.

doi:10.1002/nme.2540

Cited by 118 publications

(77 citation statements)

References 27 publications

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“…Another set of nonintrusive approaches represents the parametric solution in a reduced subspace (usually using POD) and then interpolates those solutions without recourse to the underlying full system. Interpolation can be achieved using polynomial or spline interpolation [60,163,69], least squares fitting [59,178], or radial basis function models [20].…”

Section: Equation-free Model Reductionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Equation-free Model Reductionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…viscosity, material property) vary in space and time. Audouze et al presented a proper orthogonal decomposition (POD) non-intrusive reduced order model for part of nonlinear parameterized PDEs [13,3]. The key idea underpinning the proposed method in [13] is to split the reduced-order approximation into two terms.…”

Section: Introductionmentioning

confidence: 99%

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

Xiao¹,

Fang²,

Pain³

et al. 2017

Computer Methods in Applied Mechanics and Engineering

102

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A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ω p ∈ R P ). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ω p can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique [50,53] where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier-Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

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“…A number of non-intrusive reduced order methods have been proposed, such as a black-box stencil interpolation method [32], a POD-RBF method for unsteady fluid flows [33], a Taylor series and Smolyak sparse grid method for the NavierStokes equations [34], a two-level NIROM based on POD-RBF method for nonlinear parametrized PDEs [35,36,33], a POD-RBF for the Navier-Stokes equations [37]. NIROMs have also been applied to realistic problems such as multi-phase flow in porous media problems [38] and incompressible fluids and solids without fracturing problems [39].…”

Section: Introductionmentioning

confidence: 99%

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

Xiao¹,

Yang²,

Fang

et al. 2017

Journal of Computational Physics

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This work presents the first application of a non-intrusive reduced order method to model solid interacting with compressible fluid flows to simulate crack initiation and propagation. In the high fidelity model, the coupling process is achieved by introducing a source term into the momentum equation, which represents the effects of forces of the solid on the fluid. A combined single and smeared crack model with the MohrCoulomb failure criterion is used to simulate crack initiation and propagation. The non-intrusive reduced order method is then applied to compressible fluid and fractured solid coupled modelling where the computational cost involved in the full high fidelity simulation is high. The non-intrusive reduced order model (NIROM) developed here is constructed through proper orthogonal decomposition (POD) and a radial basis function (RBF) multi-dimensional interpolation method.The performance of the NIROM for solid interacting with compressible fluid flows, in the presence of fracture models, is illustrated by two complex test cases: an immersed wall in a fluid and a blasting test case. The numerical simulation results show that the NIROM is capable of capturing the details of compressible fluids and fractured solids while the CPU time is reduced by several orders of magnitude. In addition, the issue of whether or not to subtract the mean from the snapshots before applying POD is discussed in this paper. It is shown that solutions of the NIROM, without mean subtracted before constructing the POD basis, captured more details than the NIROM with mean subtracted from snapshots.

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Reduced‐order modeling of parameterized PDEs using time–space‐parameter principal component analysis

Cited by 118 publications

References 27 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

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