We focus on steady and unsteady Navier–Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1], [2], [3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.

%B Computer & Mathematics With Applications %8 2021/08/12/ %@ 0898-1221 %G eng %U https://www.sciencedirect.com/science/article/pii/S0898122121002790 %! Computers & Mathematics with Applications %0 Journal Article %J Computers and Mathematics with Applications %D 2020 %T Projection-based reduced order models for a cut finite element method in parametrized domains %A Efthymios N Karatzas %A F. Ballarin %A Gianluigi Rozza %XThis work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.

%B Computers and Mathematics with Applications %V 79 %P 833-851 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85070900852&doi=10.1016%2fj.camwa.2019.08.003&partnerID=40&md5=2d222ab9c7832955d155655d3c93e1b1 %R 10.1016/j.camwa.2019.08.003 %0 Conference Paper %B IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 %D 2020 %T A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries %A Efthymios N Karatzas %A Giovanni Stabile %A Nabib Atallah %A Guglielmo Scovazzi %A Gianluigi Rozza %E Fehr, Jörg %E Bernard Haasdonk %XA 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.

%B IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 %I Springer International Publishing %G eng %U https://arxiv.org/abs/1807.07753 %R 10.1007/978-3-030-21013-7_8 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2020 %T A reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations %A Efthymios N Karatzas %A Giovanni Stabile %A Leo Nouveau %A Guglielmo Scovazzi %A Gianluigi Rozza %XWe investigate a projection-based reduced order model of the steady incompressible Navier–Stokes equations for moderate Reynolds numbers. In particular, we construct an “embedded” reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.

%B Computer Methods in Applied Mechanics and Engineering %V 370 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85087886522&doi=10.1016%2fj.cma.2020.113273&partnerID=40&md5=d864e4808190b682ecb1c8b27cda72d8 %R 10.1016/j.cma.2020.113273 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2019 %T A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow %A Efthymios N Karatzas %A Giovanni Stabile %A Leo Nouveau %A Guglielmo Scovazzi %A Gianluigi Rozza %XWe propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to deal with 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. This unfitted boundary method permits to avoid remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is 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.

%B Computer Methods in Applied Mechanics and Engineering %V 347 %P 568-587 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85060107322&doi=10.1016%2fj.cma.2018.12.040&partnerID=40&md5=1a3234f0cb000c91494d946428f8ebef %R 10.1016/j.cma.2018.12.040