A Spectral Element Reduced Basis Method for Navier–Stokes Equations with Geometric Variations

Hess, Martin W.; Quaini, Annalisa; Rozza, Gianluigi

doi:10.1007/978-3-030-39647-3_45

Cited by 11 publications

(10 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To the best of our knowledge, no work other than [7] addresses the use of local ROM basis for bifurcation problems. This is the continuation of previous work on model reduction with spectral element methods [10] and including parametric variations of the geometry [9], [8]. This paper aims at comparing one global ROM approach, our local ROM approach with the "best" criterion to select the local basis, and a recently proposed RB method that uses neural networks to accurately approximate the coefficients of the reduced model [11].…”

Section: Introductionmentioning

confidence: 86%

A localized reduced-order modeling approach for PDEs with bifurcating solutions

Hess

Alla

Quaini

et al. 2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE. 1 For finite-dimensional spaces, we use the nomenclature "points" and "vectors" interchangeably.2 In §3, we use the Navier-Stokes equations as a concrete setting to illustrate our methodology. 1 arXiv:1807.08851v1 [math.NA] 23 Jul 20183 In general, the forms N and F themselves are also discretized, e.g., because quadrature rules are used to approximate integrals appearing in their definition. However, here, we ignore such approximations, again to keep the exposition simple.4 Equation (1.2) represents a Galerkin type setting in which the trial function u N and test function v belong to the same space V N . We could easily generalize our discussion to the Petrov-Galerkin case for which these function would belong to different approximating spaces.

show abstract

Section: Introductionmentioning

confidence: 86%

A localized reduced-order modeling approach for PDEs with bifurcating solutions

Hess

Alla

Quaini

et al. 2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…This is analogous to reduced order assembly of the nonlinear term in the Navier-Stokes equations. For more details and applications, see [8].…”

Section: Reduced Order Modeling For Continuous Galerkin Methodsmentioning

confidence: 99%

“…Numerical examples where a parametrized geometry is considered can be found in [7] and [8]. In case of an affine parameter dependency, this parameter dependency can be made explicit in the system matrix as…”

Section: Parametric Variation In Geometrymentioning

confidence: 99%

Reduced order modeling for spectral element methods: current developments in Nektar++ and further perspectives

Hess¹,

Lario²,

Mengaldo³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we present recent efforts to develop reduced order modeling (ROM) capabilities for spectral element methods (SEM). Namely, we detail the implementation of ROM for both continuous Galerkin and discontinuous Galerkin methods in the spectral/hp element library Nektar++. The ROM approaches adopted are intrusive methods based on the proper orthogonal decomposition (POD). They admit an offline-online decomposition, such that fast evaluations for parameter studies and many-queries are possible. An affine parameter dependency is exploited such that the reduced order model can be evaluated independent of the large-scale discretization size. The implementation in the context of SEM can be found in the open-source model reduction software ITHACA-SEM.

show abstract

“…1. Some more field solutions are shown in [13] and closely related models have been computed also in [11], [14] and [18]. The geometry is decomposed into 36 triangular spectral elements and the velocity is resolved with modal Legendre polynomials of order 11.…”

Section: Channel With a Narrowing Of Varying Widthmentioning

confidence: 99%

“…Two numerical models serve to access the accuracy of the low-order approximation computed from the sparse polynomial interpolation method. More specifically, both models have previous results available obtained with the RB method [13], [12], which allows not only to compare the accuracy between both methods, but also the run time, implementation effort and other desirable features, such as offline-online splitting.…”

Section: Introductionmentioning

confidence: 99%

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations

Hess¹,

Rozza²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

This work investigates the use of sparse polynomial interpolation as a model order reduction method for the incompressible Navier-Stokes equations. Numerical results are presented underscoring the validity of sparse polynomial approximations and comparing with established reduced basis techniques. Two numerical models serve to access the accuracy of the reduced order models (ROMs), in particular parametric nonlinearities arising from curved geometries are investigated in detail. Besides the accuracy of the ROMs, other important features of the method are covered, such as offline-online splitting, run time and ease of implementation. The findings establish sparse polynomial interpolation as another instrument in the toolbox of methods for breaking the curse of dimensionality.

show abstract

A Spectral Element Reduced Basis Method for Navier–Stokes Equations with Geometric Variations

Cited by 11 publications

References 20 publications

A localized reduced-order modeling approach for PDEs with bifurcating solutions

A localized reduced-order modeling approach for PDEs with bifurcating solutions

Reduced order modeling for spectral element methods: current developments in Nektar++ and further perspectives

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations

Contact Info

Product

Resources

About