01746nas a2200217 4500008004100000020001400041245012200055210006900177260001600246520096600262653003001228653003001258653004101288653002501329653001801354100002701372700001901399700001701418700002101435856007201456 2021 eng d a0898-122100aA Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems0 aReduced Order Cut Finite Element method for geometrically parame c2021/08/12/3 a
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.
10aCut Finite Element Method10aNavier–Stokes equations10aParameter–dependent shape geometry10aReduced Order Models10aUnfitted mesh1 aKaratzas, Efthymios, N1 aNonino, Monica1 aBallarin, F.1 aRozza, Gianluigi uhttps://www.sciencedirect.com/science/article/pii/S089812212100279000618nas a2200169 4500008004100000020002200041245016600063210006900229260001300298300001400311490000800325100002200333700001800355700001700373700002100390856003700411 2021 eng d a978-3-030-55873-400aReduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences0 aReduced Order Methods for Parametrized Nonlinear and Time Depend bSpringer a841–8500 v1391 aStrazzullo, Maria1 aZainib, Zakia1 aBallarin, F.1 aRozza, Gianluigi uhttps://arxiv.org/abs/1912.0788601664nas a2200181 4500008004100000020002200041245016600063210006900229260005200298520089600350100002201246700001801268700001701286700002101303700002201324700001901346856011701365 2021 eng d a978-3-030-55874-100aReduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences0 aReduced Order Methods for Parametrized Nonlinear and Time Depend aChambSpringer International Publishingc2021//3 aWe introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.
1 aStrazzullo, Maria1 aZainib, Zakia1 aBallarin, F.1 aRozza, Gianluigi1 aVermolen, Fred, J1 aVuik, Cornelis uhttps://www.springerprofessional.de/en/reduced-order-methods-for-parametrized-non-linear-and-time-depen/1912267600588nas a2200169 4500008004100000245013500041210006900176260001000245300001600255490000700271100001700278700002300295700002200318700002100340700002000361856003700381 2021 eng d00aReduced order models for the incompressible Navier-Stokes equations on collocated grids using a `discretize-then-project' approach0 aReduced order models for the incompressible NavierStokes equatio bWiley a2694–27220 v931 aStar, Kelbij1 aSanderse, Benjamin1 aStabile, Giovanni1 aRozza, Gianluigi1 aDegroote, Joris uhttps://doi.org/10.1002/fld.499401531nas a2200145 4500008004100000245011100041210006900152300001200221490000700233520104500240100002101285700002101306700002101327856003701348 2020 eng d00aReduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature0 aReduced Basis Model Order Reduction for NavierStokes equations i a119-1260 v343 aWe consider the Navier-Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. 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 valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.
1 aHess, Martin, W.1 aQuaini, Annalisa1 aRozza, Gianluigi uhttps://arxiv.org/abs/1901.0370801653nas a2200145 4500008004100000245011300041210007100154300001200225490000700237520104600244100002101290700002101311700002101332856015401353 2020 eng d00aReduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature0 aReduced basis model order reduction for Navier–Stokes equations a119-1260 v343 aWe consider the Navier–Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced-order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced-order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. 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 valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.
1 aHess, Martin, W.1 aQuaini, Annalisa1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85085233294&doi=10.1080%2f10618562.2019.1645328&partnerID=40&md5=e2ed8f24c66376cdc8b5485aa400efb001868nas a2200181 4500008004100000245012100041210006900162260003800231520122900269100002701498700002201525700001901547700002401566700002101590700001601611700002201627856003701649 2020 eng d00aA Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries0 aReduced Order Approach for the Embedded Shifted Boundary FEM and bSpringer International Publishing3 aA 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.
1 aKaratzas, Efthymios, N1 aStabile, Giovanni1 aAtallah, Nabib1 aScovazzi, Guglielmo1 aRozza, Gianluigi1 aFehr, Jörg1 aHaasdonk, Bernard uhttps://arxiv.org/abs/1807.0775301572nas 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=7b15836ae65fa28dcfe8733788d7730c02309nas a2200313 4500008004100000020001400041245013100055210006900186260001500255300001000270490000800280520128800288653003401576653002201610653001701632653002101649653002601670653001701696653003301713653003601746653002601782100001801808700001701826700002401843700002001867700002501887700002101912856006201933 2020 eng d a2040-793900aReduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation0 aReduced order methods for parametric optimal flow control in cor c2020/05/27 ae33670 vn/a3 aAbstract Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease. In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step toward matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are: (a) lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (b) high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition-Galerkin.
10acoronary artery bypass grafts10adata assimilation10aflow control10aGalerkin methods10ahemodynamics modeling10aOptimization10apatient-specific simulations10aProper orthogonal decomposition10areduced order methods1 aZainib, Zakia1 aBallarin, F.1 aFremes, Stephen, E.1 aTriverio, Piero1 aJiménez-Juan, Laura1 aRozza, Gianluigi uhttps://onlinelibrary.wiley.com/doi/10.1002/cnm.3367?af=R01537nas a2200121 4500008004100000245011600041210006900157520098400226100002001210700002101230700002101251856014301272 2020 eng d00aA reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation0 areduced order modeling technique to study bifurcating phenomena 3 aWe propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a reduced order modeling (ROM) technique, suitably supplemented with a hyperreduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called the Gross{Pitaevskii equation, as one or two physical parameters are varied. In the two-parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard full order method.
1 aPichi, Federico1 aQuaini, Annalisa1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803&doi=10.1137%2f20M1313106&partnerID=40&md5=47d6012d10854c2f9a04b9737f87059201419nas a2200121 4500008004100000245010700041210006900148520098100217100002001198700002101218700002101239856003701260 2020 eng d00aA Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation0 aReduced Order technique to study bifurcating phenomena applicati3 aWe propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.
1 aPichi, Federico1 aQuaini, Annalisa1 aRozza, Gianluigi uhttps://arxiv.org/abs/1907.0708201444nas a2200157 4500008004100000245010200041210006900143490000800212520080500220100002701025700002201052700001701074700002401091700002101115856015001136 2020 eng d00aA reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations0 areducedorder shifted boundary method for parametrized incompress0 v3703 aWe 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.
1 aKaratzas, Efthymios, N1 aStabile, Giovanni1 aNouveau, Leo1 aScovazzi, Guglielmo1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85087886522&doi=10.1016%2fj.cma.2020.113273&partnerID=40&md5=d864e4808190b682ecb1c8b27cda72d802191nas a2200169 4500008004100000245015000041210006900191300001200260490000800272520148000280100002701760700002201787700001701809700002401826700002101850856015001871 2019 eng d00aA reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow0 areduced basis approach for PDEs on parametrized geometries based a568-5870 v3473 aWe 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.
1 aKaratzas, Efthymios, N1 aStabile, Giovanni1 aNouveau, Leo1 aScovazzi, Guglielmo1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85060107322&doi=10.1016%2fj.cma.2018.12.040&partnerID=40&md5=1a3234f0cb000c91494d946428f8ebef01370nas a2200133 4500008004100000245010900041210006900150300001400219490000700233520091800240100002001158700002101178856003701199 2019 eng d00aReduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations0 aReduced basis approaches for parametrized bifurcation problems h a112–1350 v813 aThis work focuses on the computationally efficient 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 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. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode. journal = Journal of Scientific Computing
1 aPichi, Federico1 aRozza, Gianluigi uhttps://arxiv.org/abs/1804.0201401440nas a2200133 4500008004100000245010900041210006900150300001200219490000700231520087600238100002001114700002101134856015101155 2019 eng d00aReduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations0 aReduced Basis Approaches for Parametrized Bifurcation Problems h a112-1350 v813 aThis work focuses on the computationally efficient 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 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. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.
1 aPichi, Federico1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85068973907&doi=10.1007%2fs10915-019-01003-3&partnerID=40&md5=a09af83ce45183d6965cdb79d87a919b01735nas a2200157 4500008004100000245007200041210006900113300001400182490000700196520114500203100002201348700001701370700001801387700002101405856015101426 2019 eng d00aA reduced order variational multiscale approach for turbulent flows0 areduced order variational multiscale approach for turbulent flow a2349-23680 v453 aThe purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. 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 levels. 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.
1 aStabile, Giovanni1 aBallarin, F.1 aZuccarino, G.1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85068076665&doi=10.1007%2fs10444-019-09712-x&partnerID=40&md5=af0142e6d13bbc2e88c6f31750aef6ad02258nas a2200145 4500008004100000245013400041210006900175300001200244490000700256520163800263100001801901700002001919700002101939856015201960 2018 eng d00aReduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings0 aReduced Basis Approximation and A Posteriori Error Estimation Ap a203-2470 v153 aIn this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely 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.
1 aHuynh, D.B.P.1 aPichi, Federico1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85055036627&doi=10.1007%2f978-3-319-94676-4_8&partnerID=40&md5=e9c07038e7bcc6668ec702c0653410dc00501nas a2200121 4500008004100000245013400041210006900175490000700244100002100251700002000272700002100292856006600313 2018 eng d00aReduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings0 aReduced Basis Approximation and A Posteriori Error Estimation Ap0 v151 aHuynh, D., B. P.1 aPichi, Federico1 aRozza, Gianluigi uhttps://link.springer.com/chapter/10.1007/978-3-319-94676-4_802504nas a2200157 4500008004100000245005700041210005700098260001200155300000800167490000600175520201600181100001502197700002202212700002102234856009102255 2017 eng d00aReduced Basis Methods for Uncertainty Quantification0 aReduced Basis Methods for Uncertainty Quantification c08/2017 a8690 v53 aIn this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.
Read More: http://epubs.siam.org/doi/abs/10.1137/151004550
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.
1 aBallarin, F.1 aRozza, Gianluigi1 aMaday, Yvon1 aBenner, Peter1 aOhlberger, Mario1 aPatera, Anthony1 aRozza, Gianluigi1 aUrban, Karsten uhttps://www.math.sissa.it/node/1294801691nas a2200169 4500008004100000245008700041210006900128260001800197300000600215490000600221520116500227100002101392700001901413700001701432700002101449856005101470 2016 en d00aA Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods0 aReduced Basis Approach for Modeling the Movement of Nuclear Reac bASMEc02/2016 a80 v23 aThis work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased.1 aSartori, Alberto1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3519201826nas a2200145 4500008004100000245012700041210006900168260001600237520129800253100002101551700001901572700001701591700002101608856005101629 2016 en d00aReduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods0 aReduced basis approaches in timedependent noncoercive settings f bSISSAc20163 aIn this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.
1 aSartori, Alberto1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3496301905nas a2200157 4500008004100000245012000041210006900161260002200230300000800252490000700260520128900267100002101556700002201577700002101599856012701620 2016 en d00aReduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries0 aReduced basis method and domain decomposition for elliptic probl bElsevierc01/2016 a4300 v713 aThe aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed.1 aIapichino, Laura1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-method-and-domain-decomposition-elliptic-problems-networks-and-complex02042nas a2200217 4500008004100000022001400041245010200055210006900157490003500226520122900261653002501490653002101515653002501536653002701561653002501588653001601613100002201629700002101651700002201672856013001694 2015 eng d a1019-716800aReduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system0 aReduced basis approximation and aposteriori error estimation for0 vspecial issue for MoRePaS 20123 aThe coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.
10aDomain decomposition10aError estimation10aNon-coercive problem10aPorous medium equation10aReduced basis method10aStokes flow1 aMartini, Immanuel1 aRozza, Gianluigi1 aHaasdonk, Bernard uhttps://www.math.sissa.it/publication/reduced-basis-approximation-and-posteriori-error-estimation-coupled-stokes-darcy-system01086nas a2200133 4500008004100000245009800041210006900139300001400208490000800222520055000230100001900780700002100799856013200820 2015 eng d00aReduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number0 aReduced basis approximation of parametrized advectiondiffusion P a419–4260 v1033 aIn this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.
1 aPacciarini, P.1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approximation-parametrized-advection-diffusion-pdes-high-p%C3%A9clet-number01235nas a2200145 4500008004100000245010300041210006900144300001400213490000700227520066400234100002000898700002000918700002100938856013000959 2015 eng d00aReduced basis approximation of parametrized optimal flow control problems for the Stokes equations0 aReduced basis approximation of parametrized optimal flow control a319–3360 v693 aThis paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.
1 aNegri, Federico1 aManzoni, Andrea1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approximation-parametrized-optimal-flow-control-problems-stokes-equations00566nas a2200133 4500008004100000245010000041210006900141300001000210100002100220700002200241700002100263700002100284856012700305 2014 eng d00aReduced basis method for the Stokes equations in decomposable domains using greedy optimization0 aReduced basis method for the Stokes equations in decomposable do a1–71 aIapichino, Laura1 aQuarteroni, Alfio1 aRozza, Gianluigi1 aVolkwein, Stefan uhttps://www.math.sissa.it/publication/reduced-basis-method-stokes-equations-decomposable-domains-using-greedy-optimization01878nam a2200181 4500008004100000022002200041245006700063210006700130250000600197260002100203300000800224490000600232520123600238653007801474100002201552700002101574856010101595 2014 eng d a978-3-319-02089-100aReduced Order Methods for Modeling and Computational Reduction0 aReduced Order Methods for Modeling and Computational Reduction a1 aMilanobSpringer a3340 v93 aThis monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.
Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.
This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.
10areduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-order-methods-modeling-and-computational-reduction01688nas a2200193 4500008004100000020002000041245009500061210006900156250004400225260008500269300002800354520096400382100002101346700001901367700001901386700001701405700002101422856005101443 2014 en d a978-079184595-000aA reduced order model for multi-group time-dependent parametrized reactor spatial kinetics0 areduced order model for multigroup timedependent parametrized re aAmerican Society Mechanical Engineering aPrague, Czech RepublicbAmerican Society of Mechanical Engineers (ASME)c07/2014 aV005T17A048-V005T17A0483 a
In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity "truth" finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter. In order to model a continuous movement of the rod, a piecewise affine transformation based on subdomain division has been implemented. The proposed ROM is capable to efficiently reproduce the neutron flux distribution allowing to take into account the spatial effects induced by the movement of the control rod with a computational speed-up of 30000 times, with respect to the "truth" model.
1 aSartori, Alberto1 aBaroli, Davide1 aCammi, Antonio1 aLuzzi, Lelio1 aRozza, Gianluigi uhttp://urania.sissa.it/xmlui/handle/1963/3512302183nas a2200145 4500008004100000245015300041210006900194260001300263520163200276653003401908100002101942700001801963700002001981856003602001 2013 en d00aReduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants0 aReduced basis approximation and a posteriori error estimation fo bSpringer3 aIn this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle.10aparametrized Stokes equations1 aRozza, Gianluigi1 aHuynh, Phuong1 aManzoni, Andrea uhttp://hdl.handle.net/1963/633900531nas a2200121 4500008004100000245011700041210006900158300001100227490000700238100001800245700002100263856012500284 2013 eng d00aReduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)0 aReduced Basis Approximation for the StructuralAcoustic Design ba a98-1150 v481 aDevaud, Denis1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-basis-approximation-structural-acoustic-design-based-energy-finite-element01701nas a2200157 4500008004100000245007600041210006900117300001800186490000700204520113900211100002001350700002101370700002001391700002201411856011001433 2013 eng d00aReduced basis method for parametrized elliptic optimal control problems0 aReduced basis method for parametrized elliptic optimal control p aA2316–A23400 v353 aWe propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.1 aNegri, Federico1 aRozza, Gianluigi1 aManzoni, Andrea1 aQuarteroni, Alfio uhttps://www.math.sissa.it/publication/reduced-basis-method-parametrized-elliptic-optimal-control-problems00548nas a2200133 4500008004100000245009200041210006900133260001000202100001800212700002000230700002200250700002100272856012100293 2013 en d00aA Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics0 aReduced Computational and Geometrical Framework for Inverse Prob bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-computational-and-geometrical-framework-inverse-problems-haemodynamics00568nas a2200133 4500008004100000245010500041210006900146260001000215100001800225700002000243700002200263700002100285856012800306 2013 en d00aA reduced-order strategy for solving inverse Bayesian identification problems in physiological flows0 areducedorder strategy for solving inverse Bayesian identificatio bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduced-order-strategy-solving-inverse-bayesian-identification-problems-physiological00490nas a2200121 4500008004100000245007900041210006900120260001000189100001800199700002000217700002100237856011000258 2013 en d00aReduction Strategies for Shape Dependent Inverse Problems in Haemodynamics0 aReduction Strategies for Shape Dependent Inverse Problems in Hae bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/reduction-strategies-shape-dependent-inverse-problems-haemodynamics01651nas a2200133 4500008004100000245008400041210006900125520120500194653002101399100002101420700002001441700002001461856003601481 2012 en d00aReduction strategies for PDE-constrained oprimization problems in Haemodynamics0 aReduction strategies for PDEconstrained oprimization problems in3 aSolving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation.10ainverse problems1 aRozza, Gianluigi1 aManzoni, Andrea1 aNegri, Federico uhttp://hdl.handle.net/1963/6338