@inbook {2024, title = {An Online Stabilization Method for Parametrized Viscous Flows}, booktitle = {Reduction, Approximation, Machine Learning, Surrogates, Emulators and Simulators.}, year = {2024}, publisher = {Springer, Cham}, organization = {Springer, Cham}, abstract = {

The purpose of this work is to investigate the inf-sup stability of reduced basis (RB) method applied to parametric Stokes problem. While performing the Galerkin projection on the reduced space, the inf-sup approximation stability has always been a challenge for the RB community, even if the construction of reduced basis is done using a stable high-fidelity method. In this work we propose a new online stabilization strategy for RB approximation of parametrized Stokes problem. In this strategy, a stable high-fidelity method is used to construct the RB spaces, and then, online solution is improved by a post processing based on rectification method\ [8,\ 13,\ 16]. This approach involves the computation of less expensive (but less consistent) FE approximation during the online stage and hence the improvement of online solutions using a RB-based rectification method. The consistency of the RB solution is also improved. We compare this approach with existing\ offline-online stabilization\ approach presented in our earlier work [2]. All the numerical simulations are carried out using RBniCS\ [4,\ 14], an open-source reduced order modelling library, built on top of FEniCS\ [15].

}, isbn = {978-3-031-55060-7}, doi = {https://doi.org/10.1007/978-3-031-55060-7_1}, url = {https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1}, author = {Shafqat Ali and Francesco Ballarin and Gianluigi Rozza} } @article {2023, title = {An optimisation{\textendash}based domain{\textendash}decomposition reduced order model for the incompressible Navier-Stokes equations}, volume = {151}, year = {2023}, month = {2023/12/01/}, pages = {172 - 189}, abstract = {

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain{\textendash}decomposition (DD) methods and reduced{\textendash}order modelling (ROM). In particular, we consider an optimisation{\textendash}based domain{\textendash}decomposition algorithm for the parameter{\textendash}dependent stationary incompressible Navier{\textendash}Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal{\textendash}control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward{\textendash}facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain{\textendash}decomposition algorithm.

}, keywords = {Computational fluid dynamics, Domain decomposition, Optimal control, Proper orthogonal decomposition, Reduced order modelling}, isbn = {0898-1221}, url = {https://www.sciencedirect.com/science/article/pii/S0898122123004248}, author = {Ivan Prusak and Monica Nonino and Davide Torlo and Francesco Ballarin and Gianluigi Rozza} } @article {2023, title = {A reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations}, journal = {ADVANCES IN COMPUTATIONAL SCIENCE AND ENGINEERING}, volume = {1}, year = {2023}, pages = {180-221}, abstract = {

In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.

}, doi = {https://doi.org/10.3934/acse.2023008}, url = {https://publires.unicatt.it/en/publications/a-reduced-basis-stabilization-for-the-unsteady-stokes-and-navier-}, author = {Shafqat Ali and Francesco Ballarin and Gianluigi Rozza} } @article {2022, title = {Finite element based Model Order Reduction for parametrized one-way coupled steady state linear thermo-mechanical problems}, journal = {Finite Elements in Analysis and Design}, volume = {212}, year = {2022}, month = {12/2022}, abstract = {

This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled\ steady state\ linear thermo-mechanical problems in a\ finite element\ setting. We apply\ Proper Orthogonal Decomposition\ (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on\ Artificial Neural Network\ (ANN). We aim to compare POD-G and POD-ANN in terms of relevant features including errors and computational efficiency. In this context, both physical and geometrical parametrization are considered. We also carry out a validation of the Full Order Model (FOM) based on customized benchmarks in order to provide a complete computational pipeline. The framework proposed is applied to a relevant industrial problem related to the investigation of thermo-mechanical phenomena arising in\ blast furnace\ hearth walls.

}, keywords = {Artificial Neural Network, Blast furnace, Finite Element Method, Galerkin projection, Geometric and physical parametrization, Proper orthogonal decomposition, Thermo-mechanical problems}, issn = {0168-874X}, doi = {https://doi.org/10.1016/j.finel.2022.103837}, url = {https://www.sciencedirect.com/science/article/abs/pii/S0168874X2200110X}, author = {Nirav Shah and Michele Girfoglio and Peregrina Quintela and Gianluigi Rozza and Alejandro Lengomin and Francesco Ballarin and Patricia Barral} } @booklet {2022, title = {Projection based semi{\textendash}implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid{\textendash}Structure Interaction problems}, year = {2022}, abstract = {

The goal of this manuscript is to present a partitioned Model Order Reduction method that is based on a semi-implicit projection scheme to solve multiphysics problems. We implement a Reduced Order Method based on a Proper Orthogonal Decomposition, with the aim of addressing both time-dependent and time-dependent, parametrized Fluid-Structure Interaction problems, where the fluid is incompressible and the structure is thick and two dimensional.

}, author = {Monica Nonino and Francesco Ballarin and Gianluigi Rozza and Yvon Maday} } @article {2022, title = {Space-time POD-Galerkin approach for parametric flow control}, journal = {Handbook of Numerical Analysis}, volume = {23}, year = {2022}, chapter = {307}, author = {Francesco Ballarin and Gianluigi Rozza and Maria Strazzullo} } @unpublished {2021, title = {An artificial neural network approach to bifurcating phenomena in computational fluid dynamics}, year = {2021}, author = {Federico Pichi and Francesco Ballarin and Gianluigi Rozza and Jan S Hesthaven} } @unpublished {2021, title = {A CERTIFIED REDUCED BASIS Method FOR LINEAR PARAMETRIZED PARABOLIC OPTIMAL CONTROL PROBLEMS IN SPACE-TIME FORMULATION}, year = {2021}, author = {Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza} } @unpublished {2021, title = {Consistency of the full and reduced order models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows}, year = {2021}, author = {Maria Strazzullo and Michele Girfoglio and Francesco Ballarin and T. Iliescu and Gianluigi Rozza} } @article {2021, title = {Non-intrusive data-driven ROM framework for hemodynamics problems}, journal = {Acta Mechanica Sinica}, volume = {37}, year = {2021}, pages = {1183{\textendash}1191}, author = {Michele Girfoglio and Leonardo Scandurra and Francesco Ballarin and Giuseppe Infantino and Francesca Nicol{\`o} and Andrea Montalto and Gianluigi Rozza and Roberto Scrofani and Marina Comisso and Francesco Musumeci} } @article {2021, title = {A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences}, journal = {Computers and Mathematics with Applications}, volume = {102}, year = {2021}, pages = {261-276}, doi = {10.1016/j.camwa.2021.10.020}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85117948561\&doi=10.1016\%2fj.camwa.2021.10.020\&partnerID=40\&md5=cb57d59a6975a35315b2cf5d0e3a6001}, author = {G. Carere and Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza and R. Stevenson} } @article {2020, title = {POD{\textendash}Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation}, journal = {Journal of Scientific Computing}, volume = {83}, year = {2020}, author = {Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza} } @article {2020, title = {Stabilized reduced basis methods for parametrized steady Stokes and Navier{\textendash}Stokes equations}, journal = {Computers \& Mathematics with Applications}, volume = {80}, year = {2020}, month = {1 December 2020}, pages = {2399-2416}, abstract = {

It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf{\textendash}sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in\ computational fluid dynamics, the lack of inf{\textendash}sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf{\textendash}sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the\ Finite Element\ literature, such as Brezzi{\textendash}Pitkaranta, Franca{\textendash}Hughes, streamline upwind Petrov{\textendash}Galerkin, Galerkin\ Least Square. In the spirit of\ offline{\textendash}online\ reduced basis computational splitting, two such options are proposed, namely\ offline-only stabilizationand\ offline{\textendash}online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced\ basis spaces\ (i.e., neglecting supremizer enrichment) for which a modified inf{\textendash}sup stability is still preserved at the reduced order level.

}, issn = {0898-1221}, doi = {https://doi.org/10.1016/j.camwa.2020.03.019.}, url = {https://www.sciencedirect.com/science/article/pii/S0898122120301231}, author = {Shafqat Ali and Francesco Ballarin and Gianluigi Rozza} }