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Session Overview
Session
MS 28a: Adaptive and high-order approximation based on Reduced Order Methods
Time:
Wednesday, 14/July/2021:
4:10pm - 6:10pm

Session Chair: Francesca Bonizzoni
Session Chair: Gianluigi Rozza
Virtual location: Zoom 3


Session Abstract

Reduced Order Models (ROMs) aim at providing fast, accurate and robust numerical

solution of parametrized partial differential equations (PDEs).

These techniques are important in the multi-query scenario, where the PDE has to be solved for

many different values of the parameters, as well as in the real-time scenario, where the PDE has be

to solved in few seconds.

The purpose of this minisymposium is to collect the most recent results, as well as to provide a plat-

form for exchange of new concepts and ideas on high-order and adaptive ROMs for parametrized

PDEs.


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Presentations
4:10pm - 4:40pm

Large eddy simulation reduced order models (LES-ROMs)

Traian Iliescu

Virginia Tech, United States of America

Large eddy simulation (LES) is one of the most popular methods for the numerical simulation of turbulent flows. In this talk, we survey our group's efforts over the last decade to develop a large eddy simulation reduced order modeling (LES-ROM) framework for the numerical simulation of turbulent flows.

First, we define ROM spatial filters (e.g., the ROM projection and the ROM differential filter) that allow the definition of the large spatial scales that can be approximated by ROMs in under-resolved numerical simulations (i.e., when the number of ROM modes is not large enough to represent the complex flow dynamics). Then, we describe the ROM closure problem (i.e., modeling the interaction between the large, resolved scales and the small, unresolved scales), which represents one of the main obstacles in the development of ROMs for realistic, turbulent flows. To solve the ROM closure problem, we construct three types of ROM closures: (i) functional ROM closures, which are developed by using physical arguments; (ii) structural ROM closures, which are based on mathematical arguments; and (iii) data-driven ROM closures, which are constructed by using data-driven modeling. We present results for these LES-ROMs in the numerical simulation of under-resolved engineering flows (e.g., flow past a cylinder and turbulent channel flow) and the quasi-geostrophic equations (which model the large scale ocean circulation). Finally, we present numerical analysis results for these LES-ROMs, e.g., stability, convergence, and verifiability.



4:40pm - 5:10pm

NektarROM - A model order reduction framework based on the spectral/hp element solver Nektar++

Martin W. Hess1, Andrea Lario1, Gianmarco Mengaldo2, Gianluigi Rozza1

1SISSA, Scuola Internazionale Superiore di Studi Avanzati; 2National University of Singapore

We present the reduced order modeling (ROM) framework NektarROM which implements common model reduction techniques based on the proper orthogonal decomposition (POD) and is linked against the spectral/hp element library Nektar++ [1]. Parametric model reduction algorithms are implemented for both the incompressible and compressible Navier-Stokes equations.

The ROM module for incompressible computational fluid dynamics (CFD) allows the parametrization of physical quantities such as kinematic viscosity and Grashof number as well as affinely parametrized geometries for the steady state equations. In the unsteady case, data-driven ROM algorithms based on artificial neural networks (ANN) are used. A reduced order trajectory is given as training data to a Tensorflow ANN.

The ROM module for compressible CFD based on the Discontinuous Galerkin method allows the parametrization of physical quantities for the unsteady compressible Navier-Stokes equations. A Stability-Preserving Symmetry Inner Product is adopted instead of the L2 norm in order to improve the stability of the final reduced order system.

The NektarROM library uses the current master branch of Nektar++ and

is derived from the ITHACA-SEM (https://mathlab.sissa.it/ITHACA-SEM) and ITHACA-DG libraries.

The intended audience are developers who are looking for a solid base to develop their own algorithms for parametric ROM in CFD as well as users who want to readily

use state-of-the-art ROM algorithms in conjunction with spectral

element methods. The ROM framework comes with an extensive

example suite [2] that can serve users and developers

alike as a starting point to address their specific problems at hand.

A corresponding publication is in preparation [3].

[1]

Karniadakis, G., Sherwin, S. 2005.

Spectral/hp Element Methods for Computational Fluid Dynamics.

Oxford University Press, 2nd ed. (2005).

[2]

Hess, M.W., Quaini, A., Rozza, G. 2020.

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

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, Springer International Publishing, pp. 561--571, (2020).

[3]

Hess, M.W., Lario, A., Mengaldo, G., Rozza, G. 2021.

NektarROM: A Spectral Element Based Reduced Order Modeling Software Framework.

In preparation, (2021).



5:10pm - 5:40pm

Solving the Space-Time PGD of the incompressible Stokes equations using hybridizable DG

Hashim Elzaabalawy, Michel Visonneau

CNRS, LHEEA, France

A theoretical space-time decomposition is proposed based on the proper generalized decomposition (PGD) method for the Stokes equations. The decomposed formulation leads to a dual saddle point problem that can be solved simultaneously to obtain separate velocity and pressure temporal and spatial modes. A specific focus is given on the treatment of the pressure-velocity coupling or the incompressibility constraint in the PGD framework, such that the decomposed formulation respects the physical properties of incompressible flows. In this work, high-order hybridizable discontinuous Galerkin discretization is applied to compute the spatial modes. Owing to the high-order polynomial approximation in space, the gradients of the spatial modes can be calculated locally to be used to evaluate the temporal PGD coefficients. Similarly, hybridizable discontinuous Galerkin is applied to discretize the temporal problem. With this formulation, the traditional incremental approach for time can be discarded, and, consequently, the computational time needed for the simulation of complex unsteady flows may be reduced drastically.



 
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