264 research outputs found
Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains
In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model
Space-time reduced basis methods for parametrized unsteady Stokes equations
In this work, we analyse space-time reduced basis methods for the efficient
numerical simulation of hemodynamics in arteries. The classical formulation of
the reduced basis (RB) method features dimensionality reduction in space, while
finite differences schemes are employed for the time integration of the
resulting ordinary differential equation (ODE). Space-time reduced basis
(ST-RB) methods extend the dimensionality reduction paradigm to the temporal
dimension, projecting the full-order problem onto a low-dimensional
spatio-temporal subspace. Our goal is to investigate the application of ST-RB
methods to the unsteady incompressible Stokes equations, with a particular
focus on stability. High-fidelity simulations are performed using the Finite
Element (FE) method and BDF2 as time marching scheme. We consider two different
ST-RB methods. In the first one - called ST-GRB - space-time model order
reduction is achieved by means of a Galerkin projection; a spatio-temporal
velocity basis enrichment procedure is introduced to guarantee stability. The
second method - called ST-PGRB - is characterized by a Petrov--Galerkin
projection, stemming from a suitable minimization of the FOM residual, that
allows to automatically attain stability. The classical RB method - denoted as
SRB-TFO - serves as a baseline for the theoretical development. Numerical tests
have been conducted on an idealized symmetric bifurcation geometry and on the
patient-specific one of a femoropopliteal bypass. The results show that both
ST-RB methods provide accurate approximations of the high-fidelity solutions,
while considerably reducing the computational cost. In particular, the ST-PGRB
method exhibits the best performance, as it features a better computational
efficiency while retaining accuracies in accordance with theoretical
expectations.Comment: 30 pages (25 + 5 in appendix), 4 figures, 4 tables. To appear on SIAM
Journal on Scientific Computing (SISC
On the application of the Reduced Basis Method to Fluid-Structure Interaction problems
With this thesis the author aims at giving an extensive overview on the application of the Reduced Basis Method to Fluid\u2013Structure Interaction (FSI) problems. The work exposed is divided into three main research directions: the First two methods presented are based on a standard Finite Element discretization of the problem of interest, whereas the third method presented differs from the other two because it is based on an embedded Finite Element discretization. In this way the author wants to show the advantages of pursuing a model order reduction with either a standard Finite Element method or with a Cut Finite Element method, depending on the particular problem of interest: throughout the Chapters it will be shown that a reduction method based on a classical Finite Element discretization is well suited for multiphysics problems where the geometry of the domain does not change significantly; on the contrary, a Cut Finite Element approach shows its full potentiality in situations where, for example, the structure undergoes a large deformation. The algorithms presented in this thesis are: a partitioned (or segregated) Reduced Basis Method that is based on a Chorin\u2013Temam projection scheme with semi\u2013implicit coupling of the solid and the fluid problem, a Reduced Basis Method enriched with a preprocessing of the snapshots during the offline phase, and lastly a Reduced Order Method in a Cut Finite Element framework. According to the approach adopted to adress the particular problem of interest, the thesis proposes a modification and an improvement of the Reduced Basis Method in order to obtain a complete model order reduction procedure. Several test cases are considered throughout the work: a toy problem that describes the deformation of two leaflets under the influence of the jet of a fluid; a Fluid\u2013 Structure Interaction problem whose solution exhibits a transport dominated behaviour, and, in addition, some Computational Fluid Dynamics toy problems, also in the case of parameter dependence. For each one of the test cases considered, first there is an introduction to the problem formulation, and then the proposed model order reduction procedure follows
A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier-Stokes problems
We focus on steady and unsteady Navier-Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1 -3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past
Model order reduction with novel discrete empirical interpolation methods in space-time
This work proposes novel techniques for the efficient numerical simulation of
parameterized, unsteady partial differential equations. Projection-based
reduced order models (ROMs) such as the reduced basis method employ a
(Petrov-)Galerkin projection onto a linear low-dimensional subspace. In
unsteady applications, space-time reduced basis (ST-RB) methods have been
developed to achieve a dimension reduction both in space and time, eliminating
the computational burden of time marching schemes. However, nonaffine
parameterizations dilute any computational speedup achievable by traditional
ROMs. Computational efficiency can be recovered by linearizing the nonaffine
operators via hyper-reduction, such as the empirical interpolation method in
matrix form. In this work, we implement new hyper-reduction techniques
explicitly tailored to deal with unsteady problems and embed them in a ST-RB
framework. For each of the proposed methods, we develop a posteriori error
bounds. We run numerical tests to compare the performance of the proposed ROMs
against high-fidelity simulations, in which we combine the finite element
method for space discretization on 3D geometries and the Backward Euler time
integrator. In particular, we consider a heat equation and an unsteady Stokes
equation. The numerical experiments demonstrate the accuracy and computational
efficiency our methods retain with respect to the high-fidelity simulations
Stabilized reduced basis methods for the approximation of parametrized viscous flows
In Reduced Basis (RB) method, the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if the stable Taylor-Hood Finite Element pair is chosen. Therefore in this PhD thesis we aim to build a stabilized RB method suitable for the approximation of parametrized viscous flows. Starting from the state of the art we study the residual based stabilization techniques for parametrized viscous flows in a RB setting. We are interested in the approximation of the velocity and pressure. extit{Offline-online} computational splitting is implemented and extit{offline-only stabilization}, and extit{offline-online stabilization} are compared (as well as without a stabilization approach). Different test cases are illustrated and several classical stabilization approaches like Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square are recast into a parametric reduced order setting. The RB method is introduced as a Galerkin projection into reduced spaces, generated by basis functions chosen through a greedy (steady cases) and POD-greedy (unsteady cases) algorithms. This approach is then compared with the supremizer options to guarantee the approximation stability by increasing the corresponding parametric inf-sup condition. We also implement a rectification method to correct the consistency of extit{offline-only stabilization} approach. Several numerical results for both steady and unsteady problems are presented and compared. The goal is two-fold: to guarantee the RB inf-sup stability and to guarantee online computational savings by reducing the dimension of the online reduced basis system
Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters
Geometrically parametrized Partial Differential Equations are nowadays widely used in many different fields as, for example, shape optimization processes or patient specific surgery studies. The focus of this work is on some advances for this topic, capable of increasing the accuracy with respect to previous approaches while relying on a high cost-benefit ratio performance. The main scope of this paper is the introduction of a new technique mixing up a classical Galerkin-projection approach together with a data-driven method to obtain a versatile and accurate algorithm for the resolution of geometrically parametrized incompressible turbulent Navier-Stokes problems. The effectiveness of this procedure is demonstrated on two different test cases: a classical academic back step problem and a shape deformation Ahmed body application. The results show into details the properties of the architecture we developed while exposing possible future perspectives for this work
Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models
Deep learning-based reduced order models (DL-ROMs) have been recently
proposed to overcome common limitations shared by conventional ROMs - built,
e.g., exclusively through proper orthogonal decomposition (POD) - when applied
to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can
achieve extreme efficiency in the training stage and faster than real-time
performances at testing, thanks to a prior dimensionality reduction through POD
and a DL-based prediction framework. Nonetheless, they share with conventional
ROMs poor performances regarding time extrapolation tasks. This work aims at
taking a further step towards the use of DL algorithms for the efficient
numerical approximation of parametrized PDEs by introducing the -POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM
framework by adding a two-fold architecture taking advantage of long short-term
memory (LSTM) cells, ultimately allowing long-term prediction of complex
systems' evolution, with respect to the training window, for unseen input
parameter values. Numerical results show that this recurrent architecture
enables the extrapolation for time windows up to 15 times larger than the
training time domain, and achieves better testing time performances with
respect to the already lightning-fast POD-DL-ROMs.Comment: 28 page
Projection based semi--implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid--Structure Interaction problems
We present a partitioned Model Order Reduction method for multiphysics
problems, that is based on a semi-implicit treatment of the coupling
conditions, and on a projection scheme. The proposed Reduced Order Method is
based on the Proper Orthogonal Decomposition and on a Galerkin projection onto
the reduced basis spaces; we aim of addressing both time-dependent and
time-dependent, parametrized Fluid-Structure Interaction problems, where the
fluid is incompressible and the structure is linear, elastic and two
dimensional
Projection-based reduced order models for a cut finite element method in parametrized domains
This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail. \ua9 2019 Elsevier Lt
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