Publications

2018

  1. N. Demo, M. Tezzele, A. Mola, and G. Rozza, “An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment”, in The 28th International Ocean and Polar Engineering Conference, 2018.
    [BibTeX] [Abstract] [Download preprint]
    In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is a common benchmark in ship hydrodynamics simulations.

    @inproceedings{DemoTezzeleMolaRozza2018,
    author = {Demo, Nicola and Tezzele, Marco and Mola, Andrea and Rozza, Gianluigi},
    title = {An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment},
    booktitle = {The 28th International Ocean and Polar Engineering Conference},
    year = {2018},
    preprint = {https://arxiv.org/abs/1801.06369},
    abstract = {In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is a common benchmark in ship hydrodynamics simulations.}
    }

  2. N. Demo, M. Tezzele, G. Gustin, G. Lavini, and G. Rozza, “Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition”, in Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018, p. pp. 212–219.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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

    @inproceedings{DemoTezzeleGustinLaviniRozza2018,
    author = {Demo, Nicola and Tezzele, Marco and Gustin, Gianluca and Lavini, Gianpiero and Rozza, Gianluigi},
    booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research},
    title = {Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition},
    year = {2018},
    preprint = {https://arxiv.org/abs/1803.07368},
    doi = {10.3233/978-1-61499-870-9-212},
    pages = {212--219},
    publisher = {IOS Press},
    abstract = {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}
    }

  3. M. W. Hess and G. Rozza, “A Spectral Element Reduced Basis Method in Parametric CFD”, in Numerical Mathematics and Advanced Applications – ENUMATH 2017), Springer, vol. 126, 2018.
    [BibTeX] [Abstract] [Download preprint]
    We 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 solu- tions 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 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.

    @inbook{HessRozza2017,
    author = {Hess, Martin W. and Rozza, Gianluigi},
    year = {2018},
    chapter = {A Spectral Element Reduced Basis Method in Parametric CFD},
    booktitle = {Numerical Mathematics and Advanced Applications - ENUMATH 2017},
    volume = {126},
    publisher = {Springer},
    preprint = {https://arxiv.org/abs/1712.06432},
    abstract = {We 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 solu- tions 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 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.}
    }

  4. M. Hess, A. Alla, A. Quaini, G. Rozza, and M. Gunzburger, “A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

    @unpublished{HessAllaQuainiRozzaGunzburger2018,
    author = {Hess, Martin and Alla, Alessandro and Quaini, Annalisa and Rozza, Gianluigi and Gunzburger, Max},
    title = {A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions},
    year = {2018},
    preprint = {https://arxiv.org/abs/1807.08851},
    abstract = {Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.}
    }

  5. S. Hijazi, S. Ali, G. Stabile, F. Ballarin, and G. Rozza, “The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    We present two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.

    @unpublished{HijaziAliStabileBallarinRozza2018,
    author = {Hijazi, Saddam and Ali, Shafqat and Stabile, Giovanni and Ballarin, Francesco and Rozza, Gianluigi},
    title = {The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows},
    year = {2018},
    preprint = {https://arxiv.org/abs/1807.11370},
    abstract = {We present two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.}
    }

  6. D. B. P. Huynh, F. Pichi, and G. Rozza, “Reduced basis approximation and a posteriori error estimation: applications to elasticity problems in several parametric settings”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations - rapid convergence, an a posteriori error estimation procedures - rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor, and Offline-Online computational decomposition strategies - minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations - with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

    @unpublished{HuynhPichiRozza2018,
    author = {Huynh, Dinh Bao Phuong and Pichi, Federico and Rozza, Gianluigi},
    title = {Reduced basis approximation and a posteriori error estimation: applications to elasticity problems in several parametric settings},
    year = {2018},
    preprint = {https://arxiv.org/abs/1801.06553},
    abstract = {In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations - rapid convergence, an a posteriori error estimation procedures - rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor, and Offline-Online computational decomposition strategies - minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities - to model the materials and loads - and geometrical parameters - to model different geometrical configurations - with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.}
    }

  7. E. N. Karatzas, G. Stabile, N. Atallah, G. Scovazzi, and G. Rozza, “A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.

    @unpublished{KaratzasStabileNouveauScovazziRozza2018,
    author = {Karatzas, Efthymios N. and Stabile, Giovanni and Atallah, Nabil and Scovazzi, Guglielmo and Rozza, Gianluigi},
    title = {A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries},
    year = {2018},
    preprint = {https://arxiv.org/abs/1807.07753},
    abstract = {A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.}
    }

  8. E. N. Karatzas, G. Stabile, L. Nouveau, G. Scovazzi, and G. Rozza, “A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

    @unpublished{KaratzasStabileNouveauScovazziRozza2018,
    author = {Karatzas, Efthymios N. and Stabile, Giovanni and Nouveau, Leo and Scovazzi, Guglielmo and Rozza, Gianluigi},
    title = {A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics},
    year = {2018},
    preprint = {https://arxiv.org/abs/1807.07790},
    abstract = {We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.}
    }

  9. F. Pichi and G. Rozza, “Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    This work focuses on the detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the computational reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution.

    @unpublished{PichiRozza2018,
    author = {Pichi, Federico and Rozza, Gianluigi},
    title = {Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations},
    year = {2018},
    preprint = {https://arxiv.org/abs/1804.02014},
    abstract = {This work focuses on the detection of the buckling phenomena and bifurcation analysis of the parametric Von K\'arm\'an plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the computational reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution.}
    }

  10. G. Stabile, F. Ballarin, G. Zuccarino, and G. Rozza, “A reduced order variational multiscale approach for turbulent flows”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

    @unpublished{StabileBallarinZuccarinoRozza2018,
    author = {Stabile, Giovanni and Ballarin, Francesco and Zuccarino, Giacomo and Rozza, Gianluigi},
    title = {A reduced order variational multiscale approach for turbulent flows},
    year = {2018},
    preprint = {https://arxiv.org/abs/1809.11101},
    abstract = {The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.}
    }

  11. G. Stabile and G. Rozza, “Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations”, Computers & Fluids, 2018.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

    @article{StabileRozza2018,
    title = {Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier--Stokes equations},
    journal = {Computers & Fluids},
    year = {2018},
    abstract = {In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.},
    preprint = {https://arxiv.org/abs/1710.11580},
    author = {Stabile, Giovanni and Gianluigi Rozza},
    doi = {10.1016/j.compfluid.2018.01.035}
    }

  12. M. Strazzullo, F. Ballarin, R. Mosetti, and G. Rozza, “Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering”, SIAM Journal on Scientific Computing, 40(4), p. pp. B1055-B1079, 2018.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.

    @article{StrazzulloBallarinMosettiRozza2017,
    author = {Strazzullo, Maria and Ballarin, Francesco and Mosetti, Renzo and Rozza, Gianluigi},
    title = {Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering},
    journal = {SIAM Journal on Scientific Computing},
    volume = {40},
    number = {4},
    pages = {B1055-B1079},
    year = {2018},
    preprint = {https://arxiv.org/abs/1710.01640},
    doi = {10.1137/17M1150591},
    abstract = {We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.}
    }

  13. M. Tezzele, N. Demo, M. Gadalla, A. Mola, and G. Rozza, “Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics”, in Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018, p. pp. 569–576.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the high fidelity solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.

    @inproceedings{TezzeleDemoGadallaMolaRozza2018,
    author = {Tezzele, Marco and Demo, Nicola and Gadalla, Mahmoud and Mola, Andrea and Rozza, Gianluigi},
    booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research},
    doi = {10.3233/978-1-61499-870-9-569},
    pages = {569--576},
    publisher = {IOS Press},
    title = {Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics},
    year = {2018},
    preprint = {https://arxiv.org/abs/1803.07377},
    abstract = {We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the high fidelity solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.}
    }

  14. M. Tezzele, F. Salmoiraghi, A. Mola, and G. Rozza, “Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems”, Advanced Modeling and Simulation in Engineering Sciences, 5(1), p. pp. 25, 2018.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.

    @article{TezzeleSalmoiraghiMolaRozza2018,
    author = {Tezzele, Marco and Salmoiraghi, Filippo and Mola, Andrea and Rozza, Gianluigi},
    title = {Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems},
    journal = {Advanced Modeling and Simulation in Engineering Sciences},
    number = {1},
    pages = {25},
    doi = {10.1186/s40323-018-0118-3},
    preprint = {http://arxiv.org/abs/1709.03298},
    volume = {5},
    day = {10},
    month = {Sep},
    year = {2018},
    abstract = {We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.},
    }

  15. L. Venturi, F. Ballarin, and G. Rozza, “A weighted POD method for elliptic PDEs with random inputs”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.

    @unpublished{VenturiBallarinRozza2018,
    author = {Venturi, Luca and Ballarin, Francesco and Rozza, Gianluigi},
    title = {A weighted POD method for elliptic PDEs with random inputs},
    year = {2018},
    preprint = {https://arxiv.org/abs/1802.08724},
    abstract = {In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions.
    We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.}
    }

  16. L. Venturi, D. Torlo, F. Ballarin, and G. Rozza, “Weighted reduced order methods for parametrized partial differential equations with random inputs”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

    @unpublished{VenturiTorloBallarinRozza2018,
    author = {Venturi, Luca and Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Weighted reduced order methods for parametrized partial differential equations with random inputs},
    year = {2018},
    preprint = {https://arxiv.org/abs/1805.00828},
    abstract = {In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.}
    }

2017

  1. F. Ballarin, G. Rozza, and Y. Maday, “Reduced-order semi-implicit schemes for fluid-structure interaction problems”, in Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, G. Rozza, and K. Urban (eds.), Springer International Publishing, p. pp. 149–167, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

    @INBOOK{BallarinRozzaMaday2017,
    chapter = {Reduced-order semi-implicit schemes for fluid-structure interaction problems},
    year = {2017},
    author = {Ballarin, Francesco and Rozza, Gianluigi and Maday, Yvon},
    editor = {Benner, Peter and Ohlberger, Mario and Patera, Anthony and Rozza, Gianluigi and Urban, Karsten},
    booktitle = {Model Reduction of Parametrized Systems},
    publisher = {Springer International Publishing},
    pages = {149--167},
    abstract = {POD--Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.},
    doi = {10.1007/978-3-319-58786-8_10},
    preprint = {https://arxiv.org/abs/1711.10829}
    }

  2. F. Ballarin, A. D’Amario, S. Perotto, and G. Rozza, “A POD-Selective Inverse Distance Weighting method for fast parametrized shape morphing”, 2017.
    [BibTeX] [Abstract] [Download preprint]
    Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a model reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.

    @unpublished{BallarinDAmarioPerottoRozza2017,
    author = {Ballarin, Francesco and D'Amario, Alessandro and Perotto, Simona and Rozza, Gianluigi},
    title = {A POD-Selective Inverse Distance Weighting method for fast parametrized shape morphing},
    year = {2017},
    preprint = {https://arxiv.org/abs/1710.09243},
    abstract = {Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a model reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.}
    }

  3. D. Devaud and G. Rozza, “Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation”, in Spectral and High Order Methods for Partial Differential Equations), Springer, vol. 119, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA) is a growing research theme in scientific computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. For a general overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate that it is also possible for some class of problems to deal with ane geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2] sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work we show a certication of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation.

    @inbook{DevaudRozza2017,
    chapter = {Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation},
    booktitle = {Spectral and High Order Methods for Partial Differential Equations},
    volume = {119},
    year = {2017},
    publisher = {Springer},
    abstract = {In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA) is a growing research theme in scientific computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. For a general overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate that it is also possible for some class of problems to deal with ane geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2] sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work we show a certication of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation.},
    author = {Devaud, Denis and Gianluigi Rozza},
    preprint = {https://arxiv.org/abs/1710.06148},
    doi = {10.1007/978-3-319-65870-4_3}
    }

  4. G. Pitton, A. Quaini, and G. Rozza, “Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology”, Journal of Computational Physics, 344, p. pp. 557, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier-Stokes equations for a Newtonian and viscous fluid in contraction-expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

    @article{PittonQuainiRozza2017,
    title = {Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology},
    journal = {Journal of Computational Physics},
    volume = {344},
    year = {2017},
    month = {09/2017},
    pages = {557},
    chapter = {534},
    abstract = {We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier-Stokes equations for a Newtonian and viscous fluid in contraction-expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.},
    doi = {10.1016/j.jcp.2017.05.010},
    preprint = {https://arxiv.org/abs/1708.09718},
    author = {Giuseppe Pitton and Annalisa Quaini and Gianluigi Rozza}
    }

  5. G. Pitton and G. Rozza, “On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics”, Journal of Scientific Computing, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

    @article{PittonRozza2017,
    title = {On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics},
    journal = {Journal of Scientific Computing},
    year = {2017},
    abstract = {In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.},
    doi = {10.1007/s10915-017-0419-6},
    author = {Giuseppe Pitton and Gianluigi Rozza},
    preprint = {https://arxiv.org/abs/1801.00923}
    }

  6. G. Stabile, S. N. Hijazi, S. Lorenzi, A. Mola, and G. Rozza, “Pod-Galerkin Reduced Order Methods for CFD Using Finite Volume Discretisation: Vortex Shedding Around a Circular Cylinder”, Communication in Applied Industrial Mathematics, 8(1), p. pp. 210–236, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

    @article{StabileHijaziLorenziMolaRozza2017,
    title = {Pod-Galerkin Reduced Order Methods for CFD Using Finite Volume Discretisation: Vortex Shedding Around a Circular Cylinder},
    journal = {Communication in Applied Industrial Mathematics},
    year = {2017},
    volume = {8},
    number = {1},
    year = {2017},
    month = {dec},
    pages = {210--236},
    abstract = {Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.},
    preprint = {https://arxiv.org/abs/1701.03424},
    author = {Stabile, Giovanni and Hijazi, Saddam NY and Lorenzi, Stefano and Andrea Mola and Gianluigi Rozza},
    doi = {10.1515/caim-2017-0011}
    }

  7. M. Tezzele, F. Ballarin, and G. Rozza, “Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods”, 2017.
    [BibTeX] [Abstract] [Download preprint]
    In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.

    @unpublished{TezzeleBallarinRozza2017,
    author = {Tezzele, Marco and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods},
    year = {2017},
    preprint = {https://arxiv.org/abs/1711.10884},
    abstract = {In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.}
    }

  8. D. Torlo, F. Ballarin, and G. Rozza, “Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs”, 2017.
    [BibTeX] [Abstract] [Download preprint]
    In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline-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 fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

    @unpublished{TorloBallarinRozza2017,
    author = {Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs},
    year = {2017},
    preprint = {https://arxiv.org/abs/1711.11275},
    abstract = {In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline-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 fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.}
    }

2016

  1. F. Salmoiraghi, F. Ballarin, G. Corsi, A. Mola, M. Tezzele, and G. Rozza, “Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives”, in Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, 2016.
    [BibTeX] [Abstract] [View on publisher website]
    Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.

    @INPROCEEDINGS{SalmoiraghiBallarinCorsiMolaTezzeleRozza2016,
    author = {Salmoiraghi, F. and Ballarin, F. and Corsi, G. and Mola, A. and Tezzele, M. and Rozza, G.},
    title = {Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives},
    booktitle = {Proceedings of the {ECCOMAS} {Congress} 2016, {VII} {E}uropean {C}onference on {C}omputational {M}ethods in {A}pplied {S}ciences and {E}ngineering},
    year = {2016},
    editor = {Papadrakakis, M. and Papadopoulos, V. and Stefanou, G. and Plevris, V.},
    abstract = {Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.},
    url = {http://www.eccomas.org/cvdata/cntr1/spc7/dtos/img/mdia/eccomas-2016-vol-1.pdf}
    }