Abstract This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand? effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a fluid-structure interaction problem, we aim at generalizing previous works towards a more reliable description of the physics involved. In particular, we provide several insights on how the introduction of an elastic structure influences the bifurcating behavior. We have addressed the computational burden by developing a reduced order branch-wise algorithm based on a monolithic proper orthogonal decomposition. We compared different constitutive relations for the solid, and we observed that a nonlinear hyper-elastic law delays the bifurcation w.r.t. the standard model, while the same effect is even magnified when considering linear elastic solid.

VL - n/a SN - 0271-2091 UR - https://doi.org/10.1002/fld.5118 IS - n/a JO - International Journal for Numerical Methods in Fluids ER - TY - UNPB T1 - Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations Y1 - 2022 A1 - Martin W. Hess A1 - Gianluigi Rozza ER - TY - JOUR T1 - The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2022 A1 - Davide Papapicco A1 - Nicola Demo A1 - Michele Girfoglio A1 - Giovanni Stabile A1 - Gianluigi Rozza KW - Advection KW - Computational complexity KW - Deep neural network KW - Deep neural networks KW - Linear subspace KW - Multiphase simulations KW - Non linear KW - Nonlinear hyperbolic equation KW - Partial differential equations KW - Phase space methods KW - Pre-processing KW - Principal component analysis KW - reduced order modeling KW - Reduced order modelling KW - Reduced-order model KW - Shifted-POD AB -Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate the Kolmogorov N−width decay thereby obtaining smaller linear subspaces with improved accuracy. These methods however must rely on the knowledge of the characteristic speeds in phase space of the solution, limiting their range of applicability to problems with explicit functional form for the advection field. In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework by implementing a deep-learning architecture. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation. © 2022 Elsevier B.V.

VL - 392 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85124488633&doi=10.1016%2fj.cma.2022.114687&partnerID=40&md5=12f82dcaba04c4a7c44f8e5b20101997 ER - TY - JOUR T1 - A POD-Galerkin reduced order model for the Navier–Stokes equations in stream function-vorticity formulation Y1 - 2022 A1 - Michele Girfoglio A1 - Annalisa Quaini A1 - Gianluigi Rozza KW - Galerkin projection KW - Navier–Stokes equations KW - Proper orthogonal decomposition KW - Reduced order model KW - Stream function-vorticity formulation AB -We develop a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for the efficient numerical simulation of the parametric Navier–Stokes equations in the stream function-vorticity formulation. Unlike previous works, we choose different reduced coefficients for the vorticity and stream function fields. In addition, for parametric studies we use a global POD basis space obtained from a database of time dependent full order snapshots related to sample points in the parameter space. We test the performance of our ROM strategy with the well-known vortex merger benchmark and a more complex case study featuring the geometry of the North Atlantic Ocean. Accuracy and efficiency are assessed for both time reconstruction and physical parametrization.

SN - 0045-7930 UR - https://www.sciencedirect.com/science/article/pii/S0045793022001645 JO - Computers & Fluids ER - TY - ABST T1 - Projection based semi–implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid–Structure Interaction problems Y1 - 2022 A1 - Monica Nonino A1 - Francesco Ballarin A1 - Gianluigi Rozza A1 - Yvon Maday AB -The goal of this manuscript is to present a partitioned Model Order Reduction method that is based on a semi-implicit projection scheme to solve multiphysics problems. We implement a Reduced Order Method based on a Proper Orthogonal Decomposition, with the aim of addressing both time-dependent and time-dependent, parametrized Fluid-Structure Interaction problems, where the fluid is incompressible and the structure is thick and two dimensional.

ER - TY - ABST T1 - A Proper Orthogonal Decomposition approach for parameters reduction of Single Shot Detector networks Y1 - 2022 A1 - Laura Meneghetti A1 - Nicola Demo A1 - Gianluigi Rozza KW - Computer Vision and Pattern Recognition (cs.CV) KW - FOS: Computer and information sciences KW - FOS: Mathematics KW - Machine Learning (cs.LG) KW - Numerical Analysis (math.NA) UR - https://arxiv.org/abs/2207.13551 ER - TY - UNPB T1 - An artificial neural network approach to bifurcating phenomena in computational fluid dynamics Y1 - 2021 A1 - Federico Pichi A1 - Francesco Ballarin A1 - Gianluigi Rozza A1 - Jan S Hesthaven ER - TY - JOUR T1 - ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis JF - Software Impacts Y1 - 2021 A1 - Francesco Romor A1 - Marco Tezzele A1 - Gianluigi Rozza VL - 10 ER - TY - UNPB T1 - A CERTIFIED REDUCED BASIS Method FOR LINEAR PARAMETRIZED PARABOLIC OPTIMAL CONTROL PROBLEMS IN SPACE-TIME FORMULATION Y1 - 2021 A1 - Maria Strazzullo A1 - Francesco Ballarin A1 - Gianluigi Rozza ER - TY - JOUR T1 - On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis JF - Computers & Fluids Y1 - 2021 A1 - Mahmoud Gadalla A1 - Marta Cianferra A1 - Marco Tezzele A1 - Giovanni Stabile A1 - Andrea Mola A1 - Gianluigi Rozza KW - Dynamic mode decomposition KW - Ffowcs Williams and Hawkings KW - Hydroacoustics KW - Large eddy simulation KW - Model reduction KW - Proper orthogonal decomposition AB -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.

VL - 216 UR - https://www.sciencedirect.com/science/article/pii/S0045793020303893 ER - TY - UNPB T1 - Consistency of the full and reduced order models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows Y1 - 2021 A1 - Maria Strazzullo A1 - Michele Girfoglio A1 - Francesco Ballarin A1 - T. Iliescu A1 - Gianluigi Rozza ER - TY - UNPB T1 - A data-driven partitioned approach for the resolution of time-dependent optimal control problems with dynamic mode decomposition Y1 - 2021 A1 - Eleonora Donadini A1 - Maria Strazzullo A1 - Marco Tezzele A1 - Gianluigi Rozza ER - TY - ABST T1 - A Dimensionality Reduction Approach for Convolutional Neural Networks Y1 - 2021 A1 - Laura Meneghetti A1 - Nicola Demo A1 - Gianluigi Rozza ER - TY - CONF T1 - Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation T2 - Numerical Mathematics and Advanced Applications ENUMATH 2019 Y1 - 2021 A1 - Nirav Shah A1 - Martin W. Hess A1 - Gianluigi Rozza ED - Vermolen, Fred J. ED - Vuik, Cornelis AB -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.

JF - Numerical Mathematics and Advanced Applications ENUMATH 2019 PB - Springer International Publishing CY - Cham SN - 978-3-030-55874-1 ER - TY - JOUR T1 - A dynamic mode decomposition extension for the forecasting of parametric dynamical systems JF - arXiv preprint arXiv:2110.09155 Y1 - 2021 A1 - Francesco Andreuzzi A1 - Nicola Demo A1 - Gianluigi Rozza ER - TY - JOUR T1 - Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method JF - Advances in Computational Mathematics Y1 - 2021 A1 - Moreno Pintore A1 - Federico Pichi A1 - Martin W. Hess A1 - Gianluigi Rozza A1 - Claudio Canuto AB -The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

VL - 47 ER - TY - JOUR T1 - An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques JF - Bolletino dell Unione Matematica Italiana Y1 - 2021 A1 - Nicola Demo A1 - Giulio Ortali A1 - Gianluca Gustin A1 - Gianluigi Rozza A1 - Gianpiero Lavini AB -This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretization of a two-phase (water and air) fluid. Since the fluid dynamics model can result very expensive—especially dealing with complex industrial geometries—we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique. Thanks to the quick approximation, a genetic optimization algorithm becomes feasible to converge towards the optimal shape.

VL - 14 ER - TY - UNPB T1 - AN EXTENDED PHYSICS INFORMED NEURAL NETWORK FOR PRELIMINARY ANALYSIS OF PARAMETRIC OPTIMAL CONTROL PROBLEMS Y1 - 2021 A1 - Nicola Demo A1 - Maria Strazzullo A1 - Gianluigi Rozza ER - TY - JOUR T1 - Hierarchical model reduction techniques for flow modeling in a parametrized setting JF - Multiscale Modeling and Simulation Y1 - 2021 A1 - Matteo Zancanaro A1 - F. Ballarin A1 - Simona Perotto A1 - Gianluigi Rozza AB -In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.

VL - 19 ER - TY - JOUR T1 - Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing JF - Journal of Marine Science and Engineering Y1 - 2021 A1 - Nicola Demo A1 - Marco Tezzele A1 - Andrea Mola A1 - Gianluigi Rozza AB -In 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.

VL - 9 UR - https://www.mdpi.com/2077-1312/9/2/185 ER - TY - JOUR T1 - Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters JF - Fluids Y1 - 2021 A1 - Matteo Zancanaro A1 - Markus Mrosek A1 - Giovanni Stabile A1 - Carsten Othmer A1 - Gianluigi Rozza PB - MDPI AG VL - 6 UR - https://doi.org/10.3390/fluids6080296 ER - TY - UNPB T1 - A local approach to parameter space reduction for regression and classification tasks Y1 - 2021 A1 - Francesco Romor A1 - Marco Tezzele A1 - Gianluigi Rozza JF - arXiv preprint arXiv:2107.10867 N1 - Submitted ER - TY - JOUR T1 - A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems JF - Fluids Y1 - 2021 A1 - Monica Nonino A1 - F. Ballarin A1 - Gianluigi Rozza AB -The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid–Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek–Hron benchmark test case, with a fluid Reynolds number Re=100.

VL - 6 UR - https://www.mdpi.com/2311-5521/6/6/229 ER - TY - CONF T1 - Multi-fidelity data fusion for the approximation of scalar functions with low intrinsic dimensionality through active subspaces T2 - Proceedings in Applied Mathematics & Mechanics Y1 - 2021 A1 - Francesco Romor A1 - Marco Tezzele A1 - Gianluigi Rozza JF - Proceedings in Applied Mathematics & Mechanics PB - Wiley Online Library VL - 20 ER - TY - JOUR T1 - Multi-fidelity data fusion through parameter space reduction with applications to automotive engineering JF - arXiv preprint arXiv:2110.14396 Y1 - 2021 A1 - Francesco Romor A1 - Marco Tezzele A1 - Markus Mrosek A1 - Carsten Othmer A1 - Gianluigi Rozza ER - TY - ABST T1 - The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations Y1 - 2021 A1 - Davide Papapicco A1 - Nicola Demo A1 - Michele Girfoglio A1 - Giovanni Stabile A1 - Gianluigi Rozza ER - TY - JOUR T1 - Non-intrusive data-driven ROM framework for hemodynamics problems JF - Acta Mechanica Sinica Y1 - 2021 A1 - Michele Girfoglio A1 - Leonardo Scandurra A1 - Francesco Ballarin A1 - Giuseppe Infantino A1 - Francesca Nicolò A1 - Andrea Montalto A1 - Gianluigi Rozza A1 - Roberto Scrofani A1 - Marina Comisso A1 - Francesco Musumeci VL - 37 ER - TY - JOUR T1 - A novel iterative penalty method to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamics problems JF - Communications in Computational Physics Y1 - 2021 A1 - Kelbij Star A1 - Giovanni Stabile A1 - Francesco Belloni A1 - Gianluigi Rozza A1 - Joris Degroote AB - A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamic problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the control function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the speedup ratio between the reduced order models and the full order model is of the order 1000 for the lid driven cavity case and of the order 100 for the Y-junction test case. PB - Global Science Press VL - 30 ER - TY - JOUR T1 - A numerical approach for heat flux estimation in thin slabs continuous casting molds using data assimilation JF - International Journal for Numerical Methods in Engineering Y1 - 2021 A1 - Umberto Emil Morelli A1 - Patricia Barral A1 - Peregrina Quintela A1 - Gianluigi Rozza A1 - Giovanni Stabile PB - Wiley VL - 122 UR - https://doi.org/10.1002/nme.6713 ER - TY - JOUR T1 - A POD-Galerkin reduced order model for a LES filtering approach JF - Journal of Computational Physics Y1 - 2021 A1 - Michele Girfoglio A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for an implementation of the Leray model that combines a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented.

VL - 436 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85102138957&doi=10.1016%2fj.jcp.2021.110260&partnerID=40&md5=73115708267e80754f343561c26f4744 ER - TY - JOUR T1 - A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step JF - Applied Mathematical Modelling Y1 - 2021 A1 - Kelbij Star A1 - Giovanni Stabile A1 - Gianluigi Rozza A1 - Joris Degroote AB -A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 105 times faster than the RANS simulations that are performed on eight cores.

VL - 89 ER - TY - JOUR T1 - PyGeM: Python Geometrical Morphing JF - Software Impacts Y1 - 2021 A1 - Marco Tezzele A1 - Nicola Demo A1 - Andrea Mola A1 - Gianluigi Rozza KW - Free form deformation KW - Geometrical morphing KW - Inverse distance weighting KW - Python KW - Radial basis functions interpolation AB - PyGeM is an open source Python package which allows to easily parametrize and deform 3D object described by CAD files or 3D meshes. It implements several morphing techniques such as free form deformation, radial basis function interpolation, and inverse distance weighting. Due to its versatility in dealing with different file formats it is particularly suited for researchers and practitioners both in academia and in industry interested in computational engineering simulations and optimization studies. VL - 7 ER - TY - JOUR T1 - A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems JF - Computer & Mathematics With Applications Y1 - 2021 A1 - Efthymios N Karatzas A1 - Monica Nonino A1 - F. Ballarin A1 - Gianluigi Rozza KW - Cut Finite Element Method KW - Navier–Stokes equations KW - Parameter–dependent shape geometry KW - Reduced Order Models KW - Unfitted mesh AB -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.

SN - 0898-1221 UR - https://www.sciencedirect.com/science/article/pii/S0898122121002790 JO - Computers & Mathematics with Applications ER - TY - CONF T1 - Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences T2 - Numerical Mathematics and Advanced Applications ENUMATH 2019 Y1 - 2021 A1 - Maria Strazzullo A1 - Zakia Zainib A1 - F. Ballarin A1 - Gianluigi Rozza JF - Numerical Mathematics and Advanced Applications ENUMATH 2019 PB - Springer VL - 139 SN - 978-3-030-55873-4 UR - https://arxiv.org/abs/1912.07886 ER - TY - CONF T1 - Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences T2 - Numerical Mathematics and Advanced Applications ENUMATH 2019 Y1 - 2021 A1 - Maria Strazzullo A1 - Zakia Zainib A1 - F. Ballarin A1 - Gianluigi Rozza ED - Fred J Vermolen ED - Cornelis Vuik AB -We 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.

JF - Numerical Mathematics and Advanced Applications ENUMATH 2019 PB - Springer International Publishing CY - Cham SN - 978-3-030-55874-1 UR - https://www.springerprofessional.de/en/reduced-order-methods-for-parametrized-non-linear-and-time-depen/19122676 ER - TY - JOUR T1 - Reduced order models for the incompressible Navier-Stokes equations on collocated grids using a `discretize-then-project' approach JF - International Journal for Numerical Methods in Fluids Y1 - 2021 A1 - Kelbij Star A1 - Benjamin Sanderse A1 - Giovanni Stabile A1 - Gianluigi Rozza A1 - Joris Degroote PB - Wiley VL - 93 UR - https://doi.org/10.1002/fld.4994 ER - TY - JOUR T1 - A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems JF - SIAM Journal on Scientific Computing Y1 - 2021 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza AB -In 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.

VL - 43 UR - https://arxiv.org/abs/2006.07282 IS - 3 ER - TY - JOUR T1 - A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences JF - Computers and Mathematics with Applications Y1 - 2021 A1 - G. Carere A1 - Maria Strazzullo A1 - Francesco Ballarin A1 - Gianluigi Rozza A1 - R. Stevenson VL - 102 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85117948561&doi=10.1016%2fj.camwa.2021.10.020&partnerID=40&md5=cb57d59a6975a35315b2cf5d0e3a6001 ER - TY - CONF T1 - Advances in reduced order methods for parametric industrial problems in computational fluid dynamics T2 - Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018 Y1 - 2020 A1 - Gianluigi Rozza A1 - M.H. Malik A1 - Nicola Demo A1 - Marco Tezzele A1 - Michele Girfoglio A1 - Giovanni Stabile A1 - Andrea Mola AB -Reduced 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.

JF - Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395686&partnerID=40&md5=fb0b1a3cfdfd35a104db9921bc9be675 ER - TY - CHAP T1 - Basic ideas and tools for projection-based model reduction of parametric partial differential equations T2 - Model Order Reduction, Volume 2 Snapshot-Based Methods and Algorithms Y1 - 2020 A1 - Gianluigi Rozza A1 - Martin W. Hess A1 - Giovanni Stabile A1 - Marco Tezzele A1 - F. Ballarin JF - Model Order Reduction, Volume 2 Snapshot-Based Methods and Algorithms PB - De Gruyter CY - Berlin, Boston SN - 9783110671490 UR - https://www.degruyter.com/view/book/9783110671490/10.1515/9783110671490-001.xml ER - TY - JOUR T1 - Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height JF - Computers and Mathematics with Applications Y1 - 2020 A1 - F. Ballarin A1 - Rebollo, T.C. A1 - E.D. Ávila A1 - Marmol, M.G. A1 - Gianluigi Rozza AB -In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

VL - 80 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85085843368&doi=10.1016%2fj.camwa.2020.05.013&partnerID=40&md5=7c6596865ec89651319c7dd97159dd77 ER - TY - JOUR T1 - Data-driven POD-Galerkin reduced order model for turbulent flows JF - Journal of Computational Physics Y1 - 2020 A1 - Saddam Hijazi A1 - Giovanni Stabile A1 - Andrea Mola A1 - Gianluigi Rozza AB -In 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 $Re=O(10^5)$.

VL - 416 UR - https://arxiv.org/abs/1907.09909 ER - TY - JOUR T1 - Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method JF - Advances in Computational Mathematics Y1 - 2020 A1 - Moreno Pintore A1 - Federico Pichi A1 - Martin W. Hess A1 - Gianluigi Rozza A1 - Claudio Canuto AB -The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

UR - https://arxiv.org/abs/1912.06089 ER - TY - JOUR T1 - Efficient Geometrical parametrization for finite-volume based reduced order methods JF - International Journal for Numerical Methods in Engineering Y1 - 2020 A1 - Giovanni Stabile A1 - Matteo Zancanaro A1 - Gianluigi Rozza AB -In this work, we present an approach for the efficient treatment of parametrized geometries in the context of POD-Galerkin reduced order methods based on Finite Volume full order approximations. On the contrary to what is normally done in the framework of finite element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the Discrete Empirical Interpolation Method (D-EIM) to handle together non-affinity of the parametrization and non-linearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a Radial Basis Function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the non-orthogonal correction. In the second numerical example the methodology is tested on a geometrically parametrized incompressible Navier–Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level

VL - 121 UR - https://arxiv.org/abs/1901.06373 ER - TY - CONF T1 - The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows T2 - Lecture Notes in Computational Science and Engineering Y1 - 2020 A1 - Saddam Hijazi A1 - Shafqat Ali A1 - Giovanni Stabile A1 - F. Ballarin A1 - Gianluigi Rozza AB -We present in this double contribution 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.

JF - Lecture Notes in Computational Science and Engineering PB - Springer International Publishing CY - Cham SN - 978-3-030-30705-9 ER - TY - JOUR T1 - Enhancing CFD predictions in shape design problems by model and parameter space reduction JF - Advanced Modeling and Simulation in Engineering Sciences Y1 - 2020 A1 - Marco Tezzele A1 - Nicola Demo A1 - Giovanni Stabile A1 - Andrea Mola A1 - Gianluigi Rozza AB -In 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.

VL - 7 UR - https://arxiv.org/abs/2001.05237 IS - 40 ER - TY - JOUR T1 - A hybrid reduced order method for modelling turbulent heat transfer problems JF - Computers & Fluids Y1 - 2020 A1 - Sokratia Georgaka A1 - Giovanni Stabile A1 - Kelbij Star A1 - Gianluigi Rozza A1 - Michael J. Bluck AB -A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a k-ω SST URANS full order model, is tested against the full order solver in a 3D T-junction pipe with parametric velocity inlet boundary conditions.

VL - 208 UR - https://arxiv.org/abs/1906.08725 ER - TY - BOOK T1 - Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method Y1 - 2020 A1 - Francesco Romor A1 - Marco Tezzele A1 - Lario, Andrea A1 - Gianluigi Rozza ER - TY - ABST T1 - MicroROM: An Efficient and Accurate Reduced Order Method to Solve Many-Query Problems in Micro-Motility Y1 - 2020 A1 - Nicola Giuliani A1 - Martin W. Hess A1 - Antonio DeSimone A1 - Gianluigi Rozza KW - FOS: Mathematics KW - Numerical Analysis (math.NA) UR - https://arxiv.org/abs/2006.13836 ER - TY - CHAP T1 - Non-intrusive Polynomial Chaos Method Applied to Full-Order and Reduced Problems in Computational Fluid Dynamics: A Comparison and Perspectives T2 - Quantification of Uncertainty: Improving Efficiency and Technology: QUIET selected contributions Y1 - 2020 A1 - Saddam Hijazi A1 - Giovanni Stabile A1 - Andrea Mola A1 - Gianluigi Rozza AB -In this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones.

JF - Quantification of Uncertainty: Improving Efficiency and Technology: QUIET selected contributions PB - Springer International Publishing CY - Cham SN - 978-3-030-48721-8 UR - https://doi.org/10.1007/978-3-030-48721-8_10 ER - TY - UNPB T1 - POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations Y1 - 2020 A1 - Maria Strazzullo A1 - F. Ballarin A1 - Gianluigi Rozza AB -In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

ER - TY - JOUR T1 - POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation JF - Journal of Scientific Computing Y1 - 2020 A1 - Maria Strazzullo A1 - F. Ballarin A1 - Gianluigi Rozza AB -In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

VL - 83 ER - TY - JOUR T1 - POD–Galerkin reduced order methods for combined Navier–Stokes transport equations based on a hybrid FV-FE solver JF - Computers and Mathematics with Applications Y1 - 2020 A1 - S. Busto A1 - Giovanni Stabile A1 - Gianluigi Rozza A1 - M.E. Vázquez-Cendón AB -The purpose of this work is to introduce a novel POD–Galerkin strategy for the semi-implicit hybrid high order finite volume/finite element solver introduced in Bermúdez et al. (2014) and Busto et al. (2018). The interest is into the incompressible Navier–Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

VL - 79 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068068567&doi=10.1016%2fj.camwa.2019.06.026&partnerID=40&md5=a8dcce1b53b8ee872d174bbc4c20caa3 ER - TY - JOUR T1 - Projection-based reduced order models for a cut finite element method in parametrized domains JF - Computers and Mathematics with Applications Y1 - 2020 A1 - Efthymios N Karatzas A1 - F. Ballarin A1 - Gianluigi Rozza AB -This 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.

VL - 79 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85070900852&doi=10.1016%2fj.camwa.2019.08.003&partnerID=40&md5=2d222ab9c7832955d155655d3c93e1b1 ER - TY - JOUR T1 - Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature JF - International Journal of Computational Fluid Dynamics Y1 - 2020 A1 - Martin W. Hess A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We 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.

VL - 34 UR - https://arxiv.org/abs/1901.03708 ER - TY - JOUR T1 - Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature JF - International Journal of Computational Fluid Dynamics Y1 - 2020 A1 - Martin W. Hess A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We 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.

VL - 34 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85085233294&doi=10.1080%2f10618562.2019.1645328&partnerID=40&md5=e2ed8f24c66376cdc8b5485aa400efb0 ER - TY - CONF T1 - A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries T2 - IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 Y1 - 2020 A1 - Efthymios N Karatzas A1 - Giovanni Stabile A1 - Nabib Atallah A1 - Guglielmo Scovazzi A1 - Gianluigi Rozza ED - Fehr, Jörg ED - Bernard Haasdonk AB -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.

JF - IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 PB - Springer International Publishing UR - https://arxiv.org/abs/1807.07753 ER - TY - JOUR T1 - Reduced order isogeometric analysis approach for pdes in parametrized domains JF - Lecture Notes in Computational Science and Engineering Y1 - 2020 A1 - Fabrizio Garotta A1 - Nicola Demo A1 - Marco Tezzele A1 - Massimo Carraturo A1 - Alessandro Reali A1 - Gianluigi Rozza AB -In 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.

VL - 137 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85089615035&doi=10.1007%2f978-3-030-48721-8_7&partnerID=40&md5=7b15836ae65fa28dcfe8733788d7730c ER - TY - JOUR T1 - Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation JF - International Journal for Numerical Methods in Biomedical EngineeringInternational Journal for Numerical Methods in Biomedical EngineeringInt J Numer Meth Biomed Engng Y1 - 2020 A1 - Zakia Zainib A1 - F. Ballarin A1 - Stephen E. Fremes A1 - Piero Triverio A1 - Laura Jiménez-Juan A1 - Gianluigi Rozza KW - coronary artery bypass grafts KW - data assimilation KW - flow control KW - Galerkin methods KW - hemodynamics modeling KW - Optimization KW - patient-specific simulations KW - Proper orthogonal decomposition KW - reduced order methods AB -Abstract 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.

VL - n/a SN - 2040-7939 UR - https://onlinelibrary.wiley.com/doi/10.1002/cnm.3367?af=R IS - n/a JO - International Journal for Numerical Methods in Biomedical Engineering ER - TY - JOUR T1 - A reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation JF - SIAM Journal on Scientific Computing Y1 - 2020 A1 - Federico Pichi A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We 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.

UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803&doi=10.1137%2f20M1313106&partnerID=40&md5=47d6012d10854c2f9a04b9737f870592 ER - TY - JOUR T1 - A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation JF - SIAM Journal on Scientific Computing Y1 - 2020 A1 - Federico Pichi A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We 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.

UR - https://arxiv.org/abs/1907.07082 ER - TY - JOUR T1 - A reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2020 A1 - Efthymios N Karatzas A1 - Giovanni Stabile A1 - Leo Nouveau A1 - Guglielmo Scovazzi A1 - Gianluigi Rozza AB -We 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.

VL - 370 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85087886522&doi=10.1016%2fj.cma.2020.113273&partnerID=40&md5=d864e4808190b682ecb1c8b27cda72d8 ER - TY - JOUR T1 - Special Issue on Reduced Order Models in CFD JF - International Journal of Computational Fluid Dynamics Y1 - 2020 A1 - Simona Perotto A1 - Gianluigi Rozza VL - 34 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084258805&doi=10.1080%2f10618562.2020.1756497&partnerID=40&md5=d9316aad9ba95f244e07379318ebbcba ER - TY - JOUR T1 - A spectral element reduced basis method for navier–stokes equations with geometric variations JF - Lecture Notes in Computational Science and Engineering Y1 - 2020 A1 - Martin W. Hess A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization. It is shown that an expansion in element-wise local degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

VL - 134 ER - TY - JOUR T1 - Stabilized reduced basis methods for parametrized steady Stokes and Navier–Stokes equations JF - Computers and Mathematics with Applications Y1 - 2020 A1 - Shafqat Ali A1 - F. Ballarin A1 - Gianluigi Rozza AB -It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf–sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf–sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf–sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi–Pitkaranta, Franca–Hughes, streamline upwind Petrov–Galerkin, Galerkin Least Square. In the spirit of offline–online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline–online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf–sup stability is still preserved at the reduced order level.

VL - 80 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85083340115&doi=10.1016%2fj.camwa.2020.03.019&partnerID=40&md5=7ace96eee080701acb04d8155008dd7d ER - TY - JOUR T1 - BladeX: Python Blade Morphing JF - The Journal of Open Source Software Y1 - 2019 A1 - Mahmoud Gadalla A1 - Marco Tezzele A1 - Andrea Mola A1 - Gianluigi Rozza VL - 4 ER - TY - CONF T1 - A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems T2 - VIII International Conference on Computational Methods in Marine Engineering Y1 - 2019 A1 - Nicola Demo A1 - Marco Tezzele A1 - Andrea Mola A1 - Gianluigi Rozza AB -In 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.

JF - VIII International Conference on Computational Methods in Marine Engineering UR - https://arxiv.org/abs/1905.05982 ER - TY - CONF T1 - A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems T2 - 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 Y1 - 2019 A1 - Nicola Demo A1 - Marco Tezzele A1 - Andrea Mola A1 - Gianluigi Rozza AB -In 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.

JF - 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075342565&partnerID=40&md5=d76b8a1290053e7a84fb8801c0e6bb3d ER - TY - CONF T1 - Efficient reduction in shape parameter space dimension for ship propeller blade design T2 - 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 Y1 - 2019 A1 - Andrea Mola A1 - Marco Tezzele A1 - Mahmoud Gadalla A1 - Valdenazzi, Federica A1 - Grassi, Davide A1 - Padovan, Roberta A1 - Gianluigi Rozza AB -In 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.

JF - 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395143&partnerID=40&md5=b6aa0fcedc2f88e78c295d0f437824d0 ER - TY - JOUR T1 - A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization JF - Computers & Fluids Y1 - 2019 A1 - Michele Girfoglio A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

VL - 187 UR - https://arxiv.org/abs/1901.05251 ER - TY - JOUR T1 - A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization JF - Computers and Fluids Y1 - 2019 A1 - Michele Girfoglio A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in EFR algorithm. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

VL - 187 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85065471890&doi=10.1016%2fj.compfluid.2019.05.001&partnerID=40&md5=c982371b5b5d4b5664a676902aaa60f4 ER - TY - JOUR T1 - A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2019 A1 - Martin W. Hess A1 - Alla, Alessandro A1 - Annalisa Quaini A1 - Gianluigi Rozza A1 - Max Gunzburger AB -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.

VL - 351 UR - https://arxiv.org/abs/1807.08851 ER - TY - JOUR T1 - A localized reduced-order modeling approach for PDEs with bifurcating solutions JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2019 A1 - Martin W. Hess A1 - Alla, Alessandro A1 - Annalisa Quaini A1 - Gianluigi Rozza A1 - Max Gunzburger AB -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. Although ROMs have been successfully used in many settings, ROMs built specifically for the efficient treatment of PDEs having solutions that bifurcate as the values of input parameters change have not received much attention. 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 not respect the often large differences in the PDE solutions corresponding to different subregions. In this work, we develop and test a new ROM approach specifically aimed at bifurcation problems. 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.

VL - 351 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85064313505&doi=10.1016%2fj.cma.2019.03.050&partnerID=40&md5=8b095034b9e539995facc7ce7bafa9e9 ER - TY - JOUR T1 - A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces JF - Comptes Rendus - Mecanique Y1 - 2019 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza AB -Reduced 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.

VL - 347 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075379471&doi=10.1016%2fj.crme.2019.11.012&partnerID=40&md5=dcb27af39dc14dc8c3a4a5f681f7d84b ER - TY - JOUR T1 - Parametric POD-Galerkin Model Order Reduction for Unsteady-State Heat Transfer Problems JF - Communications in Computational Physics Y1 - 2019 A1 - Sokratia Georgaka A1 - Giovanni Stabile A1 - Gianluigi Rozza A1 - Michael J. Bluck AB -A parametric reduced order model based on proper orthogonal decom- position with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power plants. Thermal mixing of different temperature coolants in T-junction pipes leads to tem- perature fluctuations and this could potentially cause thermal fatigue in the pipe walls. The novelty of this paper is the development of a parametric ROM considering the three dimensional, incompressible, unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume approximation. Two different paramet- ric cases are presented in this paper: parametrization of the inlet temperatures and parametrization of the kinematic viscosity. Different training spaces are considered and the results are compared against the full order model.

VL - 27 UR - https://arxiv.org/abs/1808.05175 ER - TY - CONF T1 - POD-Galerkin Reduced Order Model of the Boussinesq Approximation for Buoyancy-Driven Enclosed Flows T2 - International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2019 Y1 - 2019 A1 - Kelbij Star A1 - Giovanni Stabile A1 - Sokratia Georgaka A1 - Francesco Belloni A1 - Gianluigi Rozza A1 - Joris Degroote JF - International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2019 SN - 9780894487699 ER - TY - JOUR T1 - A POD-selective inverse distance weighting method for fast parametrized shape morphing JF - International Journal for Numerical Methods in Engineering Y1 - 2019 A1 - F. Ballarin A1 - A. D'Amario A1 - Simona Perotto A1 - Gianluigi Rozza AB -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 significantly compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion that automatically selects a subset of the original set of control points. Then, we combine this new approach with a dimensionality 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.

VL - 117 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85056396233&doi=10.1002%2fnme.5982&partnerID=40&md5=6aabcbdc9a0da25e36575a0ebfac034f ER - TY - JOUR T1 - A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2019 A1 - Efthymios N Karatzas A1 - Giovanni Stabile A1 - Leo Nouveau A1 - Guglielmo Scovazzi A1 - Gianluigi Rozza AB -We 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.

VL - 347 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85060107322&doi=10.1016%2fj.cma.2018.12.040&partnerID=40&md5=1a3234f0cb000c91494d946428f8ebef ER - TY - JOUR T1 - Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations JF - Journal of Scientific Computing Y1 - 2019 A1 - Federico Pichi A1 - Gianluigi Rozza AB -This 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.

VL - 81 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068973907&doi=10.1007%2fs10915-019-01003-3&partnerID=40&md5=a09af83ce45183d6965cdb79d87a919b ER - TY - JOUR T1 - Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations Y1 - 2019 A1 - Federico Pichi A1 - Gianluigi Rozza AB -This 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

VL - 81 UR - https://arxiv.org/abs/1804.02014 ER - TY - JOUR T1 - A reduced order variational multiscale approach for turbulent flows JF - Advances in Computational Mathematics Y1 - 2019 A1 - Giovanni Stabile A1 - F. Ballarin A1 - G. Zuccarino A1 - Gianluigi Rozza AB -The 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.

VL - 45 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068076665&doi=10.1007%2fs10444-019-09712-x&partnerID=40&md5=af0142e6d13bbc2e88c6f31750aef6ad ER - TY - CONF T1 - Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces T2 - 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 Y1 - 2019 A1 - Marco Tezzele A1 - Nicola Demo A1 - Gianluigi Rozza AB -We 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.

JF - 8th International Conference on Computational Methods in Marine Engineering, MARINE 2019 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075390244&partnerID=40&md5=3e1f2e9a2539d34594caff13766c94b8 ER - TY - JOUR T1 - A spectral element reduced basis method in parametric CFD JF - Lecture Notes in Computational Science and Engineering Y1 - 2019 A1 - Martin W. Hess A1 - Gianluigi Rozza AB -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 solutions 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 (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) 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.

VL - 126 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85060005503&doi=10.1007%2f978-3-319-96415-7_64&partnerID=40&md5=d1a900db8ddb92cd818d797ec212a4c6 ER - TY - CHAP T1 - A Spectral Element Reduced Basis Method in Parametric CFD T2 - Numerical Mathematics and Advanced Applications - ENUMATH 2017 Y1 - 2019 A1 - Martin W. Hess A1 - Gianluigi Rozza ED - Radu, Florin Adrian ED - Kumar, Kundan ED - Berre, Inga ED - Nordbotten, Jan Martin ED - Pop, Iuliu Sorin AB -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.

JF - Numerical Mathematics and Advanced Applications - ENUMATH 2017 PB - Springer International Publishing VL - 126 UR - https://arxiv.org/abs/1712.06432 ER - TY - JOUR T1 - A Weighted POD Method for Elliptic PDEs with Random Inputs JF - Journal of Scientific Computing Y1 - 2019 A1 - L.Venturi A1 - F. Ballarin A1 - Gianluigi Rozza AB -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 assess 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 high dimensional problems.

VL - 81 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85053798049&doi=10.1007%2fs10915-018-0830-7&partnerID=40&md5=5cad501b6ef1955da55868807079ee5d ER - TY - JOUR T1 - Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs JF - PoliTO Springer Series Y1 - 2019 A1 - L. Venturi A1 - D. Torlo A1 - F. Ballarin A1 - Gianluigi Rozza AB -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.

UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084009379&doi=10.1007%2f978-3-030-04870-9_2&partnerID=40&md5=446bcc1f331167bbba67bc00fb170150 ER - TY - JOUR T1 - Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models JF - Journal of Scientific Computing Y1 - 2018 A1 - Immanuel Martini A1 - Bernard Haasdonk A1 - Gianluigi Rozza VL - 74 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85017156114&doi=10.1007%2fs10915-017-0430-y&partnerID=40&md5=023ef0bb95713f4442d1fa374c92a964 ER - TY - CHAP T1 - Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods T2 - Mathematical and Numerical Modeling of the Cardiovascular System and Applications Y1 - 2018 A1 - Marco Tezzele A1 - F. Ballarin A1 - Gianluigi Rozza JF - Mathematical and Numerical Modeling of the Cardiovascular System and Applications PB - Springer ER - TY - CHAP T1 - Computational methods in cardiovascular mechanics T2 - Cardiovascular Mechanics Y1 - 2018 A1 - Auricchio, Ferdinando A1 - Conti, Michele A1 - Lefieux, Adrian A1 - Morganti, Simone A1 - Alessandro Reali A1 - Gianluigi Rozza A1 - Veneziani, Alessandro ED - Michel F. Labrosse AB -The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

JF - Cardiovascular Mechanics PB - CRC Press UR - https://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201%2Fb21917-5 ER - TY - JOUR T1 - Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems JF - Advanced Modeling and Simulation in Engineering Sciences Y1 - 2018 A1 - Marco Tezzele A1 - Filippo Salmoiraghi A1 - Andrea Mola A1 - Gianluigi Rozza AB -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.

VL - 5 ER - TY - Generic T1 - An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment T2 - The 28th International Ocean and Polar Engineering Conference Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Andrea Mola A1 - Gianluigi Rozza KW - Active subspaces KW - Boundary element method KW - Dynamic mode decomposition KW - Fluid structure interaction KW - Free form deformation KW - Fully nonlinear potential KW - Numerical towing tank AB - 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 which is a common benchmark in ship hydrodynamics simulations. JF - The 28th International Ocean and Polar Engineering Conference PB - International Society of Offshore and Polar Engineers CY - Sapporo, Japan UR - https://www.onepetro.org/conference-paper/ISOPE-I-18-481 ER - TY - JOUR T1 - EZyRB: Easy Reduced Basis method JF - The Journal of Open Source Software Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza VL - 3 UR - https://joss.theoj.org/papers/10.21105/joss.00661 ER - TY - JOUR T1 - Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations JF - Computers and Fluids Y1 - 2018 A1 - Giovanni Stabile A1 - Gianluigi Rozza AB -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.

VL - 173 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85043366603&doi=10.1016%2fj.compfluid.2018.01.035&partnerID=40&md5=c15435ea3b632e55450da19ba2bb6125 ER - TY - JOUR T1 - Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation JF - International Journal of Computational Fluid Dynamics Y1 - 2018 A1 - Filippo Salmoiraghi A1 - Scardigli, Angela A1 - Telib, Haysam A1 - Gianluigi Rozza AB -In this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.

PB - Taylor & Francis VL - 32 ER - TY - CONF T1 - Model Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics T2 - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research Y1 - 2018 A1 - Marco Tezzele A1 - Nicola Demo A1 - Mahmoud Gadalla A1 - Andrea Mola A1 - Gianluigi Rozza AB - 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 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. JF - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research PB - IOS Press CY - Trieste, Italy UR - http://ebooks.iospress.nl/publication/49270 ER - TY - JOUR T1 - Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering JF - SIAM Journal on Scientific Computing Y1 - 2018 A1 - Maria Strazzullo A1 - F. Ballarin A1 - Mosetti, R. A1 - Gianluigi Rozza VL - 40 UR - https://doi.org/10.1137/17M1150591 ER - TY - JOUR T1 - PyDMD: Python Dynamic Mode Decomposition JF - The Journal of Open Source Software Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza VL - 3 UR - https://joss.theoj.org/papers/734e4326edd5062c6e8ee98d03df9e1d ER - TY - JOUR T1 - Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings JF - SEMA SIMAI Springer Series Y1 - 2018 A1 - D.B.P. Huynh A1 - Federico Pichi A1 - Gianluigi Rozza AB -In 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.

VL - 15 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85055036627&doi=10.1007%2f978-3-319-94676-4_8&partnerID=40&md5=e9c07038e7bcc6668ec702c0653410dc ER - TY - CHAP T1 - Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings T2 - Numerical Methods for PDEs Y1 - 2018 A1 - Huynh, D. B. P. A1 - Federico Pichi A1 - Gianluigi Rozza JF - Numerical Methods for PDEs VL - 15 UR - https://link.springer.com/chapter/10.1007/978-3-319-94676-4_8 ER - TY - CONF T1 - Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition T2 - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluca Gustin A1 - Gianpiero Lavini A1 - Gianluigi Rozza AB - 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. JF - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research PB - IOS Press CY - Trieste, Italy UR - http://ebooks.iospress.nl/publication/49229 ER - TY - CONF T1 - SRTP 2.0 - The evolution of the safe return to port concept T2 - Technology and Science for the Ships of the Future - Proceedings of NAV 2018: 19th International Conference on Ship and Maritime Research Y1 - 2018 A1 - D. Cangelosi A1 - A. Bonvicini A1 - M. Nardo A1 - Andrea Mola A1 - A. Marchese A1 - Marco Tezzele A1 - Gianluigi Rozza AB -In 2010 IMO (International Maritime Organisation) introduced new rules in SOLAS with the aim of intrinsically increase the safety of passenger ships. This requirement is achieved by providing safe areas for passengers and essential services for allowing ship to Safely Return to Port (SRtP). The entry into force of these rules has changed the way to design passenger ships. In this respect big effort in the research has been done by industry to address design issues related to the impact on failure analysis of the complex interactions among systems. Today the research activity is working to bring operational matters in the design stage. This change of research focus was necessary because human factor and the way to operate the ship itself after a casualty on board may have a big impact in the design of the ship/systems. Also the management of the passengers after a casualty is becoming a major topic for safety. This paper presents the state of the art of Italian knowledge in the field of system engineering applied to passenger ship address to safety improvement and design reliability. An overview of present tools and methodologies will be offered together with future focuses in the research activity.

JF - Technology and Science for the Ships of the Future - Proceedings of NAV 2018: 19th International Conference on Ship and Maritime Research ER - TY - JOUR T1 - Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs JF - SIAM-ASA Journal on Uncertainty Quantification Y1 - 2018 A1 - D. Torlo A1 - F. Ballarin A1 - Gianluigi Rozza AB -In this work, we propose viable and eficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov-Galerkin (SUPG) in our case) in the ofine-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-fdelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

VL - 6 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85058246502&doi=10.1137%2f17M1163517&partnerID=40&md5=6c54e2f0eb727cb85060e988486b8ac8 ER - TY - JOUR T1 - On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics JF - Journal of Scientific Computing Y1 - 2017 A1 - Giuseppe Pitton A1 - Gianluigi Rozza AB -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.

ER - TY - CHAP T1 - Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation T2 - Spectral and High Order Methods for Partial Differential Equations Y1 - 2017 A1 - Denis Devaud A1 - Gianluigi Rozza AB -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 scientic 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. Section 2 recalls the abstract setting for parametrized PDEs, Section 3

recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

JF - Spectral and High Order Methods for Partial Differential Equations PB - Springer CY - Heildeberg VL - 119 SN - 978-3-319-65869-8 ER - TY - JOUR T1 - On a certified smagorinsky reduced basis turbulence model JF - SIAM Journal on Numerical Analysis Y1 - 2017 A1 - Rebollo, T.C. A1 - E.D. Ávila A1 - Marmol, M.G. A1 - F. Ballarin A1 - Gianluigi Rozza VL - 55 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85039928218&doi=10.1137%2f17M1118233&partnerID=40&md5=221d9cd2bcc74121fcef93efd9d3d76c ER - TY - JOUR T1 - Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology JF - Journal of Computational Physics Y1 - 2017 A1 - Giuseppe Pitton A1 - Annalisa Quaini A1 - Gianluigi Rozza KW - Parametrized Navier–Stokes equations KW - Reduced basis method KW - Stability of flows KW - Symmetry breaking bifurcation AB -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.

This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

JF - Encyclopedia of Computational Mechanics Second Edition PB - John Wiley & Sons ER - TY - JOUR T1 - Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts JF - Biomechanics and Modeling in Mechanobiology Y1 - 2017 A1 - F. Ballarin A1 - Elena Faggiano A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza A1 - Sonia Ippolito A1 - Roberto Scrofani VL - 16 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851&doi=10.1007%2fs10237-017-0893-7&partnerID=40&md5=c388f20bd5de14187bad9ed7d9affbd0 ER - TY - JOUR T1 - POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder JF - Communications in Applied and Industrial Mathematics Y1 - 2017 A1 - Giovanni Stabile A1 - Saddam Hijazi A1 - Andrea Mola A1 - Stefano Lorenzi A1 - Gianluigi Rozza AB -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.

VL - 8 ER - TY - JOUR T1 - Reduced Basis Methods for Uncertainty Quantification JF - SIAM/ASA Journal on Uncertainty Quantification Y1 - 2017 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB -In 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.

JF - Model Reduction of Parametrized Systems PB - Springer International Publishing ER - TY - CONF T1 - Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives T2 - Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, Y1 - 2016 A1 - Filippo Salmoiraghi A1 - F. Ballarin A1 - Giovanni Corsi A1 - Andrea Mola A1 - Marco Tezzele A1 - Gianluigi Rozza ED - Papadrakakis, M. ED - Papadopoulos, V. ED - Stefanou, G. ED - Plevris, V. AB -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.

JF - Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, PB - ECCOMAS CY - Crete, Greece U1 - 35466 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - RPRT T1 - A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts Y1 - 2016 A1 - F. Ballarin A1 - Elena Faggiano A1 - Andrea Manzoni A1 - Gianluigi Rozza A1 - Alfio Quarteroni A1 - Sonia Ippolito A1 - Roberto Scrofani A1 - Carlo Antona AB - A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases. PB - Submitted UR - http://urania.sissa.it/xmlui/handle/1963/35240 U1 - 35545 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - RPRT T1 - Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes Y1 - 2016 A1 - Filippo Salmoiraghi A1 - F. Ballarin A1 - Luca Heltai A1 - Gianluigi Rozza AB - In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model. PB - Springer, AMOS Advanced Modelling and Simulation in Engineering Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35199 U1 - 35493 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - CHAP T1 - Model Order Reduction: a survey T2 - Wiley Encyclopedia of Computational Mechanics, 2016 Y1 - 2016 A1 - Francisco Chinesta A1 - Antonio Huerta A1 - Gianluigi Rozza A1 - Karen Willcox JF - Wiley Encyclopedia of Computational Mechanics, 2016 PB - Wiley UR - http://urania.sissa.it/xmlui/handle/1963/35194 U1 - 35470 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel JF - Annals of Nuclear Energy, 87, 2 (2016): pp. 198-208 Y1 - 2016 A1 - Alberto Sartori A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB - In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well. PB - Elsevier VL - 87 UR - http://urania.sissa.it/xmlui/handle/1963/35191 U1 - 35471 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations Y1 - 2016 A1 - Stefano Lorenzi A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB - Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition – Reduced Order Model (POD – ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods. PB - Computer Methods in Applied Mechanics and Engineering, Elsevier U1 - 35502 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems JF - International Journal Numerical Methods for Fluids Y1 - 2016 A1 - F. Ballarin A1 - Gianluigi Rozza AB - In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD)–Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline-online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic POD–Galerkin method for the online computation of the global structural displacement, fluid velocity and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced order method and its computational performances PB - Wiley U1 - 35465 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods JF - NERS-14-1062; ASME J of Nuclear Rad Sci, 2, 2 (2016) 021019 Y1 - 2016 A1 - Alberto Sartori A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB - This 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. PB - ASME VL - 2 UR - http://urania.sissa.it/xmlui/handle/1963/35192 IS - 2 N1 - 8 pages U1 - 35473 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Reduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods JF - Communications in Computational Physics Y1 - 2016 A1 - Alberto Sartori A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB -In 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.

PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/34963 IS - in press U1 - 35188 U2 - Mathematics ER - TY - JOUR T1 - Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries JF - Computers and Mathematics with Applications Y1 - 2016 A1 - Laura Iapichino A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - The 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. PB - Elsevier VL - 71 IS - 1 U1 - 35187 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - BOOK T1 - Certified Reduced Basis Methods for Parametrized Partial Differential Equations T2 - Springer Briefs in Mathematics Y1 - 2015 A1 - Jan S Hesthaven A1 - Gianluigi Rozza A1 - Benjamin Stamm KW - a posteriori error bounds KW - empirical interpolation KW - parametrized partial differential equations KW - reduced basis methods, greedy algorithms AB -This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

JF - Springer Briefs in Mathematics PB - Springer CY - Switzerland SN - 978-3-319-22469-5 ER - TY - RPRT T1 - Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization Y1 - 2015 A1 - F. Ballarin A1 - Elena Faggiano A1 - Sonia Ippolito A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza A1 - Roberto Scrofani AB - In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD–Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to perform sensitivity analysis studies, so far out of reach. UR - http://urania.sissa.it/xmlui/handle/1963/34623 U1 - 34824 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Model order reduction of parameterized systems (MoRePaS): Preface to the special issue of advances in computational mathematics JF - Advances in Computational Mathematics Y1 - 2015 A1 - Peter Benner A1 - Mario Ohlberger A1 - Anthony Patera A1 - Gianluigi Rozza A1 - Sorensen, D.C. A1 - Karsten Urban VL - 41 ER - TY - JOUR T1 - Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations JF - Numerische Mathematik, (2015), 36 p. Article in Press Y1 - 2015 A1 - Gianluigi Rozza A1 - Peng Chen A1 - Alfio Quarteroni AB - In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34491 U1 - 34680 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system JF - Advances in Computational Mathematics Y1 - 2015 A1 - Immanuel Martini A1 - Gianluigi Rozza A1 - Bernard Haasdonk KW - Domain decomposition KW - Error estimation KW - Non-coercive problem KW - Porous medium equation KW - Reduced basis method KW - Stokes flow AB -The 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.

VL - special issue for MoRePaS 2012 IS - in press ER - TY - JOUR T1 - Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number JF - Lecture Notes in Computational Science and Engineering Y1 - 2015 A1 - Pacciarini, P. A1 - Gianluigi Rozza AB -In 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.

VL - 103 ER - TY - JOUR T1 - Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations JF - Computers and Mathematics with Applications Y1 - 2015 A1 - Federico Negri A1 - Andrea Manzoni A1 - Gianluigi Rozza AB -This 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.

VL - 69 ER - TY - JOUR T1 - Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations Y1 - 2015 A1 - F. Ballarin A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number. PB - Wiley UR - http://urania.sissa.it/xmlui/handle/1963/34701 U1 - 34915 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Comparison between reduced basis and stochastic collocation methods for elliptic problems Y1 - 2014 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O (1) to moderate dimensions O (10) and to high dimensions O (100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34727 U1 - 34916 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics JF - Annals of Nuclear Energy Y1 - 2014 A1 - Alberto Sartori A1 - Davide Baroli A1 - Antonio Cammi A1 - Davide Chiesa A1 - Lelio Luzzi A1 - Roberto R. Ponciroli A1 - Ezio Previtali A1 - Marco E. Ricotti A1 - Gianluigi Rozza A1 - Monica Sisti AB -In this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

PB - Elsevier VL - 71 UR - http://urania.sissa.it/xmlui/handle/1963/35039 U1 - 35270 U2 - Physics U4 - 1 ER - TY - JOUR T1 - Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems JF - International Journal of Computational Fluid Dynamics Y1 - 2014 A1 - Forti, D. A1 - Gianluigi Rozza AB - We present some recent advances and improvements in shape parametrisation techniques of interfaces for reduced-order modelling with special attention to fluid–structure interaction problems and the management of structural deformations, namely, to represent them into a low-dimensional space (by control points). This allows to reduce the computational effort, and to significantly simplify the (geometrical) deformation procedure, leading to more efficient and fast reduced-order modelling applications in this kind of problems. We propose an efficient methodology to select the geometrical control points for the radial basis functions based on a modal greedy algorithm to improve the computational efficiency in view of more complex fluid–structure applications in several fields. The examples provided deal with aeronautics and wind engineering. VL - 28 ER - TY - CHAP T1 - Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications T2 - Separated representations and PGD-based model reduction : fundamentals and applications Y1 - 2014 A1 - Gianluigi Rozza KW - reduced basis method, linear elasticity, heat transfer, error bounds, parametrized PDEs AB -In this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive PDEs. We discuss all the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present an application of the RB method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.

JF - Separated representations and PGD-based model reduction : fundamentals and applications T3 - CISM International Centre for Mechanical Sciences PB - Springer CY - Wien VL - 554 ER - TY - JOUR T1 - An improvement on geometrical parameterizations by transfinite maps JF - Comptes Rendus Mathematique Y1 - 2014 A1 - Jäggli, C. A1 - Laura Iapichino A1 - Gianluigi Rozza AB - We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries. VL - 352 ER - TY - JOUR T1 - Model Order Reduction in Fluid Dynamics: Challenges and Perspectives Y1 - 2014 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references. PB - Springer U1 - 34923 U2 - Mathematics U4 - 1 ER - TY - CONF T1 - Reduced basis method for the Stokes equations in decomposable domains using greedy optimization T2 - ECMI 2014 proceedings Y1 - 2014 A1 - Laura Iapichino A1 - Alfio Quarteroni A1 - Gianluigi Rozza A1 - Volkwein, Stefan JF - ECMI 2014 proceedings ER - TY - BOOK T1 - Reduced Order Methods for Modeling and Computational Reduction T2 - MS&A Y1 - 2014 A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis AB -This 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.

JF - MS&A PB - Springer CY - Milano VL - 9 ER - TY - Generic T1 - A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics T2 - 22nd International Conference on Nuclear Engineering ICONE22 Y1 - 2014 A1 - Alberto Sartori A1 - Davide Baroli A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB -

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.

JF - 22nd International Conference on Nuclear Engineering ICONE22 PB - American Society of Mechanical Engineers (ASME) CY - Prague, Czech Republic SN - 978-079184595-0 UR - http://urania.sissa.it/xmlui/handle/1963/35123 N1 - 2014 22nd International Conference on Nuclear Engineering, ICONE 2014; Prague; Czech Republic; 7 July 2014 through 11 July 2014; Code 109131; U1 - 35360 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows Y1 - 2014 A1 - F. Ballarin A1 - Andrea Manzoni A1 - Gianluigi Rozza A1 - Sandro Salsa AB - Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34698 U1 - 34914 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Stabilized reduced basis method for parametrized advection-diffusion PDEs JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2014 A1 - Pacciarini, P. A1 - Gianluigi Rozza AB -In this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the ``double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

VL - 274 ER - TY - CONF T1 - Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts T2 - 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 Y1 - 2014 A1 - Pacciarini, P. A1 - Gianluigi Rozza AB -Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

JF - 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 UR - https://infoscience.epfl.ch/record/203327/files/ECCOMAS_PP_GR.pdf ER - TY - JOUR T1 - A weighted empirical interpolation method: A priori convergence analysis and applications Y1 - 2014 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method. PB - EDP Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35021 U1 - 35253 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices JF - Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598 Y1 - 2013 A1 - Denis Devaud A1 - Andrea Manzoni A1 - Gianluigi Rozza KW - Partial differential equations AB -We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.

PB - Elsevier UR - http://hdl.handle.net/1963/7389 U1 - 7434 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Free Form Deformation Techniques Applied to 3D Shape Optimization Problems JF - Communications in Applied and Industrial Mathematics Y1 - 2013 A1 - Anwar Koshakji A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation. ER - TY - JOUR T1 - Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants JF - Numerische Mathematik, 2013 Y1 - 2013 A1 - Gianluigi Rozza A1 - Phuong Huynh A1 - Andrea Manzoni KW - parametrized Stokes equations AB - In 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. PB - Springer UR - http://hdl.handle.net/1963/6339 U1 - 6269 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - CHAP T1 - Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA) T2 - CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches Y1 - 2013 A1 - Denis Devaud A1 - Gianluigi Rozza JF - CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches VL - 48 ER - TY - JOUR T1 - Reduced basis method for parametrized elliptic optimal control problems JF - SIAM Journal on Scientific Computing Y1 - 2013 A1 - Federico Negri A1 - Gianluigi Rozza A1 - Andrea Manzoni A1 - Alfio Quarteroni AB - We 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. VL - 35 ER - TY - RPRT T1 - A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics Y1 - 2013 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza PB - SISSA U1 - 6571 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - RPRT T1 - A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows Y1 - 2013 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza PB - SISSA U1 - 6555 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - RPRT T1 - Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics Y1 - 2013 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Gianluigi Rozza PB - SISSA U1 - 6554 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - JOUR T1 - Stochastic optimal robin boundary control problems of advection-dominated elliptic equations JF - SIAM Journal on Numerical Analysis Y1 - 2013 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. VL - 51 ER - TY - JOUR T1 - A weighted reduced basis method for elliptic partial differential equations with random input data JF - SIAM Journal on Numerical Analysis Y1 - 2013 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems. VL - 51 ER - TY - JOUR T1 - Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty JF - Mathematical Modelling and Numerical Analysis, in press, 2012-13 Y1 - 2012 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - shape optimization AB - We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework. PB - Cambridge University Press UR - http://hdl.handle.net/1963/6337 U1 - 6267 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - CHAP T1 - Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs T2 - Springer, Indam Series, Vol. 4, 2012 Y1 - 2012 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - solution manifold AB - The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates. JF - Springer, Indam Series, Vol. 4, 2012 PB - Springer UR - http://hdl.handle.net/1963/6340 U1 - 6270 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - Generic T1 - Reduction strategies for PDE-constrained oprimization problems in Haemodynamics T2 - European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 Y1 - 2012 A1 - Gianluigi Rozza A1 - Andrea Manzoni A1 - Federico Negri KW - inverse problems AB - Solving 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. JF - European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 UR - http://hdl.handle.net/1963/6338 U1 - 6268 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - JOUR T1 - Simulation-based uncertainty quantification of human arterial network hemodynamics JF - International Journal Numerical Methods Biomedical Engineering Y1 - 2012 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction AB - This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time. PB - Wiley U1 - 6467 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER -