In this work, Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD) methodologies are applied to hydroacoustic dataset computed using Large Eddy Simulation (LES) coupled with Ffowcs Williams and Hawkings (FWH) analogy. First, a low-dimensional description of the flow fields is presented with modal decomposition analysis. Sensitivity towards the DMD and POD bases truncation rank is discussed, and extensive dataset is provided to demonstrate the ability of both algorithms to reconstruct the flow fields with all the spatial and temporal frequencies necessary to support accurate noise evaluation. Results show that while DMD is capable to capture finer coherent structures in the wake region for the same amount of employed modes, reconstructed flow fields using POD exhibit smaller magnitudes of global spatiotemporal errors compared with DMD counterparts. Second, a separate set of DMD and POD modes generated using half the snapshots is employed into two data-driven reduced models respectively, based on DMD mid cast and POD with Interpolation (PODI). In that regard, results confirm that the predictive character of both reduced approaches on the flow fields is sufficiently accurate, with a relative superiority of PODI results over DMD ones. This infers that, discrepancies induced due to interpolation errors in PODI is relatively low compared with errors induced by integration and linear regression operations in DMD, for the present setup. Finally, a post processing analysis on the evaluation of FWH acoustic signals utilizing reduced fluid dynamic fields as input demonstrates that both DMD and PODI data-driven reduced models are efficient and sufficiently accurate in predicting acoustic noises.

10aDynamic mode decomposition10aFfowcs Williams and Hawkings10aHydroacoustics10aLarge eddy simulation10aModel reduction10aProper orthogonal decomposition1 aGadalla, Mahmoud1 aCianferra, Marta1 aTezzele, Marco1 aStabile, Giovanni1 aMola, Andrea1 aRozza, Gianluigi uhttps://www.sciencedirect.com/science/article/pii/S004579302030389301664nas a2200169 4500008004100000022001400041245011000055210006900165300000800234490000600242520112900248100001701377700001901394700001701413700002101430856004301451 2021 eng d a2077-131200aHull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing0 aHull Shape Design Optimization with Parameter Space and Model Re a1850 v93 aIn the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.

1 aDemo, Nicola1 aTezzele, Marco1 aMola, Andrea1 aRozza, Gianluigi uhttps://www.mdpi.com/2077-1312/9/2/18501497nas a2200169 4500008004100000245010500041210006900146520085900215100002101074700001601095700001701111700001901128700002301147700002201170700001701192856011801209 2020 eng d00aAdvances in reduced order methods for parametric industrial problems in computational fluid dynamics0 aAdvances in reduced order methods for parametric industrial prob3 aReduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.

1 aRozza, Gianluigi1 aMalik, M.H.1 aDemo, Nicola1 aTezzele, Marco1 aGirfoglio, Michele1 aStabile, Giovanni1 aMola, Andrea uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395686&partnerID=40&md5=fb0b1a3cfdfd35a104db9921bc9be67501462nas a2200145 4500008004100000245009400041210006900135520097900204100001901183700001701202700002201219700001701241700002101258856003701279 2020 eng d00aEnhancing CFD predictions in shape design problems by model and parameter space reduction0 aEnhancing CFD predictions in shape design problems by model and 3 aIn this work we present an advanced computational pipeline for the approximation and prediction of the lift coefficient of a parametrized airfoil profile. The non-intrusive reduced order method is based on dynamic mode decomposition (DMD) and it is coupled with dynamic active subspaces (DyAS) to enhance the future state prediction of the target function and reduce the parameter space dimensionality. The pipeline is based on high-fidelity simulations carried out by the application of finite volume method for turbulent flows, and automatic mesh morphing through radial basis functions interpolation technique. The proposed pipeline is able to save 1/3 of the overall computational resources thanks to the application of DMD. Moreover exploiting DyAS and performing the regression on a lower dimensional space results in the reduction of the relative error in the approximation of the time-varying lift coefficient by a factor 2 with respect to using only the DMD.

1 aTezzele, Marco1 aDemo, Nicola1 aStabile, Giovanni1 aMola, Andrea1 aRozza, Gianluigi uhttps://arxiv.org/abs/2001.0523701572nas a2200181 4500008004100000245008200041210006900123300001200192490000800204520090200212100002201114700001701136700001901153700002301172700002201195700002101217856015201238 2020 eng d00aReduced order isogeometric analysis approach for pdes in parametrized domains0 aReduced order isogeometric analysis approach for pdes in paramet a153-1700 v1373 aIn this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.

1 aGarotta, Fabrizio1 aDemo, Nicola1 aTezzele, Marco1 aCarraturo, Massimo1 aReali, Alessandro1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85089615035&doi=10.1007%2f978-3-030-48721-8_7&partnerID=40&md5=7b15836ae65fa28dcfe8733788d7730c01427nas a2200121 4500008004100000245013900041210006900180520096200249100001701211700001901228700002101247856003701268 2020 eng d00aA supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems0 asupervised learning approach involving active subspaces for an e3 aIn 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.

1 aDemo, Nicola1 aTezzele, Marco1 aRozza, Gianluigi uhttps://arxiv.org/abs/2006.0728202116nas a2200133 4500008004100000245013800041210006900179520154200248100001701790700001901807700001701826700002101843856011801864 2019 eng d00aA complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems0 acomplete datadriven framework for the efficient solution of para3 aIn the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in [31, 24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in [7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry — assuming the topology is inaltered by the deformation —, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water.

1 aDemo, Nicola1 aTezzele, Marco1 aMola, Andrea1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85075342565&partnerID=40&md5=d76b8a1290053e7a84fb8801c0e6bb3d02563nas a2200169 4500008004100000245009100041210006900132520193100201100001702132700001902149700002102168700002502189700001902214700002102233700002102254856011802275 2019 eng d00aEfficient reduction in shape parameter space dimension for ship propeller blade design0 aEfficient reduction in shape parameter space dimension for ship 3 aIn this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

1 aMola, Andrea1 aTezzele, Marco1 aGadalla, Mahmoud1 aValdenazzi, Federica1 aGrassi, Davide1 aPadovan, Roberta1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395143&partnerID=40&md5=b6aa0fcedc2f88e78c295d0f437824d001894nas a2200145 4500008004100000245010300041210006900144300001200213490000800225520130700233100001701540700001901557700002101576856015101597 2019 eng d00aA non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces0 anonintrusive approach for the reconstruction of POD modal coeffi a873-8810 v3473 aReduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions—computed for properly chosen parameters, using a full-order model—in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI)—a data-driven reduced order method—with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.

1 aDemo, Nicola1 aTezzele, Marco1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85075379471&doi=10.1016%2fj.crme.2019.11.012&partnerID=40&md5=dcb27af39dc14dc8c3a4a5f681f7d84b02446nas a2200121 4500008004100000245014200041210006900183520189700252100001902149700001702168700002102185856011802206 2019 eng d00aShape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces0 aShape optimization through proper orthogonal decomposition with 3 aWe 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.

1 aTezzele, Marco1 aDemo, Nicola1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85075390244&partnerID=40&md5=3e1f2e9a2539d34594caff13766c94b800587nas a2200133 4500008004100000245012400041210006900165260001300234300001400247100001900261700002400280700002100304856012800325 2018 eng d00aCombined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods0 aCombined parameter and model reduction of cardiovascular problem bSpringer a185–2071 aTezzele, Marco1 aBallarin, Francesco1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/combined-parameter-and-model-reduction-cardiovascular-problems-means-active-subspaces02307nas a2200169 4500008004100000245011900041210006900160260000800229300000700237490000600244520167500250100001901925700002501944700001701969700002101986856013002007 2018 eng d00aDimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems0 aDimension reduction in heterogeneous parametric spaces with appl cSep a250 v53 aWe 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.

1 aTezzele, Marco1 aSalmoiraghi, Filippo1 aMola, Andrea1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/dimension-reduction-heterogeneous-parametric-spaces-application-naval-engineering-shape02869nas a2200241 4500008004100000022002200041245016200063210006900225260007400294520193000368653002102298653002802319653003102347653003202378653002602410653003002436653002602466100001702492700001902509700001702528700002102545856006102566 2018 eng d a978-1-880653-87-600aAn efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment0 aefficient shape parametrisation by freeform deformation enhanced aSapporo, JapanbInternational Society of Offshore and Polar Engineers3 aIn 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 which is a common benchmark in ship hydrodynamics simulations.10aActive subspaces10aBoundary element method10aDynamic mode decomposition10aFluid structure interaction10aFree form deformation10aFully nonlinear potential10aNumerical towing tank1 aDemo, Nicola1 aTezzele, Marco1 aMola, Andrea1 aRozza, Gianluigi uhttps://www.onepetro.org/conference-paper/ISOPE-I-18-48100373nas a2200133 4500008004100000245003700041210003600078300000800114490000600122100001700128700001900145700002100164856005400185 2018 eng d00aEZyRB: Easy Reduced Basis method0 aEZyRB Easy Reduced Basis method a6610 v31 aDemo, Nicola1 aTezzele, Marco1 aRozza, Gianluigi uhttps://joss.theoj.org/papers/10.21105/joss.0066101777nas a2200157 4500008004100000245013300041210006900174260003000243520120300273100001901476700001701495700002101512700001701533700002101550856004801571 2018 eng d00aModel Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics0 aModel Order Reduction by means of Active Subspaces and Dynamic M aTrieste, ItalybIOS Press3 aWe 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 full order 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.1 aTezzele, Marco1 aDemo, Nicola1 aGadalla, Mahmoud1 aMola, Andrea1 aRozza, Gianluigi uhttp://ebooks.iospress.nl/publication/4927000402nas a2200133 4500008004100000245004500041210004400086300000800130490000600138100001700144700001900161700002100180856006700201 2018 eng d00aPyDMD: Python Dynamic Mode Decomposition0 aPyDMD Python Dynamic Mode Decomposition a5300 v31 aDemo, Nicola1 aTezzele, Marco1 aRozza, Gianluigi uhttps://joss.theoj.org/papers/734e4326edd5062c6e8ee98d03df9e1d01912nas a2200157 4500008004100000245009800041210006900139260003000208520136800238100001701606700001901623700002101642700002201663700002101685856004801706 2018 eng d00aShape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition0 aShape Optimization by means of Proper Orthogonal Decomposition a aTrieste, ItalybIOS Press3 aShape 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.1 aDemo, Nicola1 aTezzele, Marco1 aGustin, Gianluca1 aLavini, Gianpiero1 aRozza, Gianluigi uhttp://ebooks.iospress.nl/publication/4922902112nas a2200217 4500008004100000245018600041210006900227260003600296520123100332100002501563700002401588700002001612700001701632700001901649700002101668700002101689700002101710700001701731700001601748856013001764 2016 en d00aAdvances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives0 aAdvances in geometrical parametrization and reduced order models aCrete, GreecebECCOMASc06/20163 aSeveral 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.

1 aSalmoiraghi, Filippo1 aBallarin, Francesco1 aCorsi, Giovanni1 aMola, Andrea1 aTezzele, Marco1 aRozza, Gianluigi1 aPapadrakakis, M.1 aPapadopoulos, V.1 aStefanou, G.1 aPlevris, V. uhttps://www.math.sissa.it/publication/advances-geometrical-parametrization-and-reduced-order-models-and-methods-computational