We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization. It is shown that an expansion in element-wise local degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

%B Lecture Notes in Computational Science and Engineering %V 134 %P 561-571 %G eng %R 10.1007/978-3-030-39647-3_45 %0 Journal Article %J Computers and Mathematics with Applications %D 2020 %T Stabilized reduced basis methods for parametrized steady Stokes and Navier–Stokes equations %A Shafqat Hyder Ali %A Francesco Ballarin %A Gianluigi Rozza %XIt 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–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–sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf–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–Pitkaranta, Franca–Hughes, streamline upwind Petrov–Galerkin, Galerkin Least Square. In the spirit of offline–online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline–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–sup stability is still preserved at the reduced order level.

%B Computers and Mathematics with Applications %V 80 %P 2399-2416 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85083340115&doi=10.1016%2fj.camwa.2020.03.019&partnerID=40&md5=7ace96eee080701acb04d8155008dd7d %R 10.1016/j.camwa.2020.03.019 %0 Unpublished Work %D 2020 %T A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems %A Nicola Demo %A Marco Tezzele %A Gianluigi Rozza %XIn this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space. In many cases, GA requires in fact more function evaluations than others optimization method to converge to the optimum. Thus, complex and high-dimensional functions may result intractable with the standard algorithm. To address this issue, we propose to linearly map the input parameter space of the original function onto its AS before the evolution, performing the mutation and mate processes in a lower dimensional space. In this contribution, we describe the novel method called ASGA, presenting differences and similarities with the standard GA method. We test the proposed method over n-dimensional benchmark functions – Rosenbrock, Ackley, Bohachevsky, Rastrigin, Schaffer N. 7, and Zakharov – and finally we apply it to an aeronautical shape optimization problem.

%G eng %U https://arxiv.org/abs/2006.07282 %0 Conference Paper %B 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 %D 2019 %T Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces %A Marco Tezzele %A Nicola Demo %A Gianluigi Rozza %XWe propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape.

%B 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075390244&partnerID=40&md5=3e1f2e9a2539d34594caff13766c94b8 %0 Journal Article %J Lecture Notes in Computational Science and Engineering %D 2019 %T A spectral element reduced basis method in parametric CFD %A Hess, Martin %A Gianluigi Rozza %XWe consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

%B Lecture Notes in Computational Science and Engineering %V 126 %P 693-701 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85060005503&doi=10.1007%2f978-3-319-96415-7_64&partnerID=40&md5=d1a900db8ddb92cd818d797ec212a4c6 %R 10.1007/978-3-319-96415-7_64 %0 Conference Paper %B Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research %D 2018 %T Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition %A Nicola Demo %A Marco Tezzele %A Gianluca Gustin %A Gianpiero Lavini %A Gianluigi Rozza %X Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship. %B Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research %I IOS Press %C Trieste, Italy %G eng %U http://ebooks.iospress.nl/publication/49229 %& 212 %R 10.3233/978-1-61499-870-9-212 %0 Conference Paper %B Technology and Science for the Ships of the Future - Proceedings of NAV 2018: 19th International Conference on Ship and Maritime Research %D 2018 %T SRTP 2.0 - The evolution of the safe return to port concept %A D. Cangelosi %A A. Bonvicini %A M. Nardo %A Andrea Mola %A A. Marchese %A Marco Tezzele %A Gianluigi Rozza %XIn 2010 IMO (International Maritime Organisation) introduced new rules in SOLAS with the aim of intrinsically increase the safety of passenger ships. This requirement is achieved by providing safe areas for passengers and essential services for allowing ship to Safely Return to Port (SRtP). The entry into force of these rules has changed the way to design passenger ships. In this respect big effort in the research has been done by industry to address design issues related to the impact on failure analysis of the complex interactions among systems. Today the research activity is working to bring operational matters in the design stage. This change of research focus was necessary because human factor and the way to operate the ship itself after a casualty on board may have a big impact in the design of the ship/systems. Also the management of the passengers after a casualty is becoming a major topic for safety. This paper presents the state of the art of Italian knowledge in the field of system engineering applied to passenger ship address to safety improvement and design reliability. An overview of present tools and methodologies will be offered together with future focuses in the research activity.

%B Technology and Science for the Ships of the Future - Proceedings of NAV 2018: 19th International Conference on Ship and Maritime Research %G eng %R 10.3233/978-1-61499-870-9-665 %0 Journal Article %J SIAM-ASA Journal on Uncertainty Quantification %D 2018 %T Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs %A D. Torlo %A Francesco Ballarin %A Gianluigi Rozza %XIn this work, we propose viable and eficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov-Galerkin (SUPG) in our case) in the ofine-online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high-fdelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

%B SIAM-ASA Journal on Uncertainty Quantification %V 6 %P 1475-1502 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85058246502&doi=10.1137%2f17M1163517&partnerID=40&md5=6c54e2f0eb727cb85060e988486b8ac8 %R 10.1137/17M1163517 %0 Journal Article %D 2015 %T Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations %A Francesco Ballarin %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %X In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number. %I Wiley %G en %U http://urania.sissa.it/xmlui/handle/1963/34701 %1 34915 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-10-22T11:39:58Z No. of bitstreams: 1 IJNME_2014.pdf: 6761966 bytes, checksum: ad65f2c4d2dbd30a4a1590ff42ee49a0 (MD5) %R 10.1002/nme.4772 %0 Journal Article %D 2014 %T Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows %A Francesco Ballarin %A Andrea Manzoni %A Gianluigi Rozza %A Sandro Salsa %X Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature. %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34698 %1 34914 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-10-22T11:24:27Z (GMT) No. of bitstreams: 0 %R 10.1007/s10915-013-9807-8 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2014 %T Stabilized reduced basis method for parametrized advection-diffusion PDEs %A Pacciarini, P. %A Gianluigi Rozza %XIn this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the ``double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

%B Computer Methods in Applied Mechanics and Engineering %V 274 %P 1–18 %G eng %R 10.1016/j.cma.2014.02.005 %0 Conference Paper %B 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 %D 2014 %T Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts %A Pacciarini, P. %A Gianluigi Rozza %XAdvection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

%B 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 %P 5614–5624 %G eng %U https://infoscience.epfl.ch/record/203327/files/ECCOMAS_PP_GR.pdf %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T Stochastic optimal robin boundary control problems of advection-dominated elliptic equations %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. %B SIAM Journal on Numerical Analysis %V 51 %P 2700–2722 %G eng %R 10.1137/120884158 %0 Journal Article %J International Journal Numerical Methods Biomedical Engineering %D 2012 %T Simulation-based uncertainty quantification of human arterial network hemodynamics %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %K uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction %X This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time. %B International Journal Numerical Methods Biomedical Engineering %I Wiley %G en %1 6467 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-03-07T12:33:02Z\nNo. of bitstreams: 1\nreport.pdf: 1605405 bytes, checksum: bb12cf074ce32a80567a0cde0c0861db (MD5)