scholarly journals Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature

2019 ◽  
Vol 34 (2) ◽  
pp. 119-126
Author(s):  
Martin W. Hess ◽  
Annalisa Quaini ◽  
Gianluigi Rozza
Keyword(s):  
Model Order Reduction ◽  
Stokes Equations ◽  
Order Reduction ◽  
Navier Stokes ◽  
Reduced Basis ◽  
Model Order ◽  
Navier Stokes Equations

scholarly journals Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations

2019 ◽  
Vol 24 (2) ◽  
pp. 45 ◽  
Author(s):  
Nissrine Akkari ◽  
Fabien Casenave ◽  
Vincent Moureau
Keyword(s):  
Stokes Equations ◽  
A Priori ◽  
Order Reduction ◽  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Navier Stokes ◽  
Reduced Basis ◽  
Order Model ◽  
Navier Stokes Equations ◽  
Reduced Order

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of 2 . 2 ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC.

scholarly journals Model order reduction for the simulation of parametric interest rate models in financial risk analysis

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andreas Binder ◽  
Onkar Jadhav ◽  
Volker Mehrmann
Keyword(s):  
Model Order Reduction ◽  
Financial Risk ◽  
Large Scale ◽  
Order Reduction ◽  
Parametric Models ◽  
Diffusion Reaction ◽  
High Dimensional ◽  
Reduced Basis ◽  
Model Order ◽  
Reduction Approach

AbstractThis paper presents a model order reduction approach for large scale high dimensional parametric models arising in the analysis of financial risk. To understand the risks associated with a financial product, one has to perform several thousand computationally demanding simulations of the model which require efficient algorithms. We establish a model reduction approach based on a variant of the proper orthogonal decomposition method to generate small model approximations for the high dimensional parametric convection-diffusion-reaction partial differential equations. This approach requires to solve the full model at some selected parameter values to generate a reduced basis. We propose an adaptive greedy sampling technique based on surrogate modeling for the selection of the sample parameter set. The new technique is analyzed, implemented, and tested on industrial data of a floater with cap and floor under the Hull–White model. The results illustrate that the reduced model approach works well for short-rate models.

scholarly journals Model Order Reduction of Nonlinear Magnetodynamics with Manifold Interpolation

2015 ◽  
Author(s):  
Y. Paquay ◽  
O. Brüls ◽  
C. Geuzaine
Keyword(s):  
Model Order Reduction ◽  
Interpolation Method ◽  
Order Reduction ◽  
Nonlinear Problems ◽  
Single Shot ◽  
Reduced Basis ◽  
Model Order ◽  
Core System ◽  
Input Parameters ◽  
Speed Up

In the model order reduction community, linear systems have been widely studied and reduced thanks to various techniques. Proper Orthogonal Decomposition (POD) in particular has been very successful, and has recently been gaining popularity in computational electromagnetics. However, the efficiency of POD degrades considerably for nonlinear problems, in particular for nonlinear magnetodynamic models---necessary for designing most of today's electrical machines and drives. We propose to investigate an algorithm which first applies the POD to construct reduced order models of nonlinear magnetodynamic problems for discrete sets of values of the input parameters. Then, for a new set of values of the input parameters, a nonlinear interpolation on manifolds is performed to determine the reduced basis. This interpolation method is based on the theory previously studied for aerodynamic problems. The goal here is not to speed up single shot calculations, but to be able to determine efficiently reduced models for nonlinear problems based on previous offline computations. As a simple application, we apply the procedure to a nonlinear inductor-core system, solved using a classical finite element method. In order to gauge the interest of manifold interpolation we compare the proposed approach to the direct use of a precomputed reduced basis, as well as with the use of standard Lagrange interpolation.

scholarly journals Reduced order methods for nonlinear parametric problems with branching solutions

2018 ◽  
Author(s):  
Federico Pichi ◽  
Martin Wilfried Hess ◽  
Annalisa Quaini ◽  
Gianluigi Rozza
Keyword(s):  
Stokes Equations ◽  
Spectral Element ◽  
Navier Stokes ◽  
Reduced Basis ◽  
State Equations ◽  
Order Error ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Plate Equations ◽  
Basis Method

The aim of this work is to show the applicability of the reduced basis model reduction in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e., following the different bifurcating branches, as well as determining the bifurcation point itself, is a complex computational task. Reduced Order Models (ROM) can potentially reduce the computational burden by several orders of magnitude, in particular in conjunction with sampling techniques. In the first task we focus on nonlinear structural mechanics, and we deal with an application of ROM to Von Kármán plate equations, where the buckling effect arises, adopting reduced basis method. Moreover, in the search of the bifurcation points, it is crucial to supplement the full problem with a reduced generalized parametric eigenvalue problem, properly paired with state equations and also a reduced order error analysis. These studies are carried out in view of vibroacoustic applications (in collaboration with A.T. Patera at MIT). As second task we consider the incompressible Navier-Stokes equations, discretized with the spectral element method, in a channel and a cavity. Both system undergo bifurcations with increasing Reynolds - and Grashof - number, respectively. Applications of this model are contraction-expansion channels, found in many biological systems, such as the human heart, for instance, or crystal growth in cavities, used in semiconductor production processes. This last task is in collaboration with A. Alla and M. Gunzburger (Florida State University).

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