The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.

1 aShah, Nirav1 aHess, Martin, W.1 aRozza, Gianluigi1 aVermolen, Fred, J.1 aVuik, Cornelis uhttps://www.math.sissa.it/publication/discontinuous-galerkin-model-order-reduction-geometrically-parametrized-stokes-000498nas a2200109 4500008004100000245009500041210006900136100002500205700001700230700002100247856012000268 2021 eng d00aA dynamic mode decomposition extension for the forecasting of parametric dynamical systems0 adynamic mode decomposition extension for the forecasting of para1 aAndreuzzi, Francesco1 aDemo, Nicola1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/dynamic-mode-decomposition-extension-forecasting-parametric-dynamical-systems01178nas a2200157 4500008004100000245006900041210006700110300001100177490000800188520070800196100001900904700002200923700001700945700002100962856003700983 2020 eng d00aData-driven POD-Galerkin reduced order model for turbulent flows0 aDatadriven PODGalerkin reduced order model for turbulent flows a1095130 v4163 aIn this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to $Re=O(10^5)$.

1 aHijazi, Saddam1 aStabile, Giovanni1 aMola, Andrea1 aRozza, Gianluigi uhttps://arxiv.org/abs/1907.0990902307nas 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-shape