Papers by Gianluigi Rozza
arXiv (Cornell University), Sep 20, 2019
In this work we recast parametrized time dependent optimal control problems governed by partial d... more In this work we recast parametrized time dependent optimal control problems governed by partial differential equations in a saddle point formulation 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 hand, 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 experiments the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.
arXiv (Cornell University), Jan 18, 2019
In this work, we present an approach for the efficient treatment of parametrized geometries in th... more 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.
Fluids, Aug 25, 2021
We present a stabilized POD-Galerkin reduced order method (ROM) for a Leray model. For the implem... more We present a stabilized POD-Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space. We show that the evolve and filtered velocity spaces have to be enriched with the supremizer solutions related to both evolve and filter pressure fields in order to obtain stable and accurate solutions with the supremizer enrichment method. We test our ROM approach on a 2D unsteady flow past a cylinder at Reynolds number 0 ≤ Re ≤ 100. We find that both stabilization strategies produce comparable errors in the reconstruction of the lift and drag coefficients, with the pressure Poisson equation method being more computationally efficient.
Journal of Scientific Computing, Jan 7, 2023
In this work the development of a machine learning-based Reduced Order Model (ROM) for the invest... more In this work the development of a machine learning-based Reduced Order Model (ROM) for the investigation of hemodynamics in a patient-specific configuration of Coronary Artery Bypass Graft (CABG) is proposed. The computational domain is referred to left branches of coronary arteries when a stenosis of the Left Main Coronary Artery (LMCA) occurs. The method extracts a reduced basis space from a collection of high-fidelity solutions via a Proper Orthogonal Decomposition (POD) algorithm and employs Artificial Neural Networks (ANNs) for the computation of the modal coefficients. The Full Order Model (FOM) is represented by the incompressible Navier-Stokes equations discretized using a Finite Volume (FV) technique. Both physical and geometrical parametrization are taken into account, the former one related to the inlet flow rate and the latter one related to the stenosis severity. With respect to the previous works focused on the development of a ROM framework for the evaluation of coronary artery disease, the novelties of our study include the use of the FV method in a patient-specific configuration, the use of a data-driven ROM technique and the mesh deformation strategy based on a Free Form Deformation (FFD) technique. The performance of our ROM approach is analyzed in terms of the error between full order and reduced order solutions as well as the speedup achieved at the online stage.
arXiv (Cornell University), Mar 2, 2020
A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds aver... more 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 10 5 times faster than the RANS simulations that are performed on eight cores.
arXiv (Cornell University), Jun 30, 2022
We consider an optimal flow control problem in a patient-specific coronary artery bypass graft wi... more We consider an optimal flow control problem in a patient-specific coronary artery bypass graft with the aim of matching the blood flow velocity with given measurements as the Reynolds number varies in a physiological range. Blood flow is modelled with the steady incompressible Navier-Stokes equations. The geometry consists in a stenosed left anterior descending artery where a single bypass is performed with the right internal thoracic artery. The control variable is the unknown value of the normal stress at the outlet boundary, which is need for a correct set-up of the outlet boundary condition. For the numerical solution of the parametric optimal flow control problem, we develop a data-driven reduced order method that combines proper orthogonal decomposition (POD) with neural networks. We present numerical results showing that our data-driven approach leads to a substantial speed-up with respect to a more classical POD-Galerkin strategy proposed in [59], while having comparable accuracy.
arXiv (Cornell University), Oct 18, 2018
The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volu... more The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volume/finite element solver introduced in [9] and . 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.
arXiv (Cornell University), Nov 4, 2019
Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent par... more Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease (CAD). 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 towards matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are, (i). lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (ii). 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 (POD)-Galerkin.
Computers & Fluids, Sep 1, 2018
In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier-S... more 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.
arXiv (Cornell University), Dec 17, 2019
We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and ... more 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 (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.
arXiv (Cornell University), Dec 2, 2019
A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamics problems w... more A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamics problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the lifting 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 reduced order models are 270-308 times faster than the full order models for the lid driven cavity test case and 13-24 times for the Y-junction test case.
arXiv (Cornell University), Nov 24, 2020
In this contribution we propose reduced order methods to fast and reliably solve parametrized opt... more In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control can be used in order to fill the gap between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by viscous Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not losing accuracy. Keywords. Reduced order modelling, optimal control problems, time dependent nonlinear partial differential equations, Lagrangian approach. This contribution is rooted in control systems and controllability theory for partial differential equation. A control problem is a system on which you can act through suitable external variables said controls . The controllability theory answers to the need of steering a system towards a desired configuration. Is it always possible? Under which conditions can I reach an exact prescribed profile for my system? The problem is quite fascinating and of utmost usefulness in many applications. Even if linear partial differential control systems have many complex aspects and features to be analyzed both theoretically and numerically , our main focus will concern nonlinearity in fluid dynamics. In this setting, the problem becomes more challenging and, besides the growing complexity, the need of a control tool increases. The case of nonlinear partial differential equations is much more complicated to handle. The control theory for fluid models, e.g. Navier-Stokes equations, prospered in the eighties thanks to the research of J. L. Lions. The main idea of his production relied on the role which nonlinearity plays as a control itself, giving the possibility or preventing the achievement of peculiar motion behaviours . This intuition paves the way to a wide range of literature which addresses the problem . However, in many applied contexts, it is clear that not all the systems are controllable and, furthermore, it is not possible to prove the existence of controls which give the exact desired solution profile one wants to reach. This is the reason why the controllabilty theory expands towards optimization and optimal control theory. Namely, the new objective is to find a way to reach the most similar configuration with respect to the desired one, satisfying the underling partial differential equation
arXiv (Cornell University), May 19, 2022
This article provides a reduced-order modelling framework for turbulent compressible flows discre... more This article provides a reduced-order modelling framework for turbulent compressible flows discretized by the use of finite volume approaches. The basic idea behind this work is the construction of a reduced-order model capable of providing closely accurate solutions with respect to the high fidelity flow fields. Full-order solutions are often obtained through the use of segregated solvers (solution variables are solved one after another), employing slightly modified conservation laws so that they can be decoupled and then solved one at a time. Classical reduction architectures, on the contrary, rely on the Galerkin projection of a complete Navier-Stokes system to be projected all at once, causing a mild discrepancy with the high order solutions. This article relies on segregated reduced-order algorithms for the resolution of turbulent and compressible flows in the context of physical and geometrical parameters. At the full-order level turbulence is modeled using an eddy viscosity approach. Since there is a variety of different turbulence models for the approximation of this supplementary viscosity, one of the aims of this work is to provide a reduced-order model which is independent on this selection. This goal is reached by the application of hybrid methods where Navier-Stokes equations are projected in a standard way while the viscosity field is approximated by the use of data-driven interpolation methods or by the evaluation *
SIAM Journal on Scientific Computing, 2020
We propose a computationally efficient framework to treat nonlinear partial differential equation... more 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.
Computers & Fluids, Aug 1, 2020
A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition wi... more 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 flow 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.
arXiv (Cornell University), Mar 5, 2021
In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection ... more In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours . We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.
Computers & mathematics with applications, Feb 1, 2020
This work presents a reduced order modeling technique built on a high fidelity embedded mesh fini... more 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.
arXiv (Cornell University), Mar 21, 2020
In this work we propose reduced order methods as a reliable strategy to efficiently solve paramet... more 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.
arXiv (Cornell University), Nov 24, 2020
In this contribution we propose reduced order methods to fast and reliably solve parametrized opt... more In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control can be used in order to fill the gap between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by viscous Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not losing accuracy. Keywords. Reduced order modelling, optimal control problems, time dependent nonlinear partial differential equations, Lagrangian approach. This contribution is rooted in control systems and controllability theory for partial differential equation. A control problem is a system on which you can act through suitable external variables said controls . The controllability theory answers to the need of steering a system towards a desired configuration. Is it always possible? Under which conditions can I reach an exact prescribed profile for my system? The problem is quite fascinating and of utmost usefulness in many applications. Even if linear partial differential control systems have many complex aspects and features to be analyzed both theoretically and numerically , our main focus will concern nonlinearity in fluid dynamics. In this setting, the problem becomes more challenging and, besides the growing complexity, the need of a control tool increases. The case of nonlinear partial differential equations is much more complicated to handle. The control theory for fluid models, e.g. Navier-Stokes equations, prospered in the eighties thanks to the research of J. L. Lions. The main idea of his production relied on the role which nonlinearity plays as a control itself, giving the possibility or preventing the achievement of peculiar motion behaviours . This intuition paves the way to a wide range of literature which addresses the problem . However, in many applied contexts, it is clear that not all the systems are controllable and, furthermore, it is not possible to prove the existence of controls which give the exact desired solution profile one wants to reach. This is the reason why the controllabilty theory expands towards optimization and optimal control theory. Namely, the new objective is to find a way to reach the most similar configuration with respect to the desired one, satisfying the underling partial differential equation
Society for Industrial and Applied Mathematics eBooks, 2022
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Papers by Gianluigi Rozza