arXiv subjects

Gianluigi Rozza

Publications and source records attributed to Gianluigi Rozza.

At least 91 records · Page 5Linked to original sources

arXiv2024

A brief review of Reduced Order Models using intrusive and non-intrusive techniques

Reduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief overview about such a topic. We discuss both an intrusive framework based on a Galerkin projection technique and non-intrusive approaches, including Physics Informed Neural Networks (PINN), purely Data-Driven Neural Networks (DDNN), Radial Basis Functions (RBF), Dynamic Mode Decomposition (DMD) and Gaussian Process Regression (GPR). We also briefly mention geometrical parametrization and dimensionality reduction methods like Active Subspaces (AS). Then we present some results related to academic test cases as well as a preliminary investigation related to an industrial application.

math.NA↗

arXiv2024

A segregated reduced order model of a pressure-based solver for turbulent compressible flows

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 of a properly trained neural network. By exploiting the aforementioned expedients it is possible to predict accurate solutions with respect to the full-order problems characterized by high Reynolds numbers and elevated Mach numbers.

physics.flu-dyn↗

arXiv2024

Modal Analysis of the Wake Shed Behind a Horizontal Axis Wind Turbine with Flexible Blades

The proper orthogonal decomposition has been applied on a full-scale horizontal-axis wind turbine to shed light on the wake characteristics behind the wind turbine. In reality, the blade tip experiences high deflections even at the rated conditions which definitely alter the wake flow field, and in the case of a wind farm, may complicate the inlet conditions of the downstream wind turbine. The turbine under consideration is the full-scale model of the National Renewable Energy Laboratory 5MW onshore wind turbine which is accompanied by several simulation complexities including turbulence, mesh motion and fluid-structure interaction. Results indicated an almost similar modal behaviour for the rigid and flexible turbines at the wake region. In addition, more flow structures in terms of local vortices and fluctuating velocity fields take place at the far wake region. The flow structures due to the wake shed from the tower tend to move towards the center and merge with that of the nacelle leading to an integral vortical structure 2.5 diameter away from the rotor. Also, it is concluded that the exclusion of the tower leads to missing a major part of the wake structures, especially at far-wake positions.

math.NA↗

arXiv2024

Computational study of numerical flux schemes for mesoscale atmospheric flows in a Finite Volume framework

We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two two classical benchmarks, namely the smooth rising thermal bubble and the density current.

math.NA↗

arXiv2024

Stabilized POD Reduced Order Models for convection-dominated incompressible flows

We present a comparative computational study of two stabilized Reduced Order Models (ROMs) for the simulation of convection-dominated incompressible flow (Reynolds number of the order of a few thousands). Representative solutions in the parameter space, which includes either time only or time and Reynolds number, are computed with a Finite Volume method and used to generate a reduced basis via Proper Orthogonal Decomposition (POD). Galerkin projection of the Navier-Stokes equations onto the reduced space is used to compute the ROM solution. To ensure computational efficiency, the number of POD modes is truncated and ROM solution accuracy is recovered through two stabilization methods: i) adding a global constant artificial viscosity to the reduced dimensional model, and ii) adding a different value of artificial viscosity for the different POD modes. We test the stabilized ROMs for fluid flow in an idealized medical device consisting of a conical convergent, a narrow throat, and a sudden expansion. Both stabilization methods significantly improve the ROM solution accuracy over a standard (non-stabilized) POD-Galerkin model.

physics.flu-dyn↗

arXiv2024

Linear and nonlinear filtering for a two-layer quasi-geostrophic ocean model

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300).

math.NA↗

arXiv2024

A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

math.NA↗

arXiv2024

A LSTM-enhanced surrogate model to simulate the dynamics of particle-laden fluid systems

The numerical treatment of fluid-particle systems is a very challenging problem because of the complex coupling phenomena occurring between the two phases. Although accurate mathematical modelling is available to address this kind of application, the computational cost of the numerical simulations is very expensive. The use of the most modern high-performance computing infrastructures could help to mitigate such an issue but not completely fix it. In this work, we develop a non-intrusive data-driven reduced order model (ROM) for Computational Fluid Dynamics (CFD) - Discrete Element Method (DEM) simulations. The ROM is built using the proper orthogonal decomposition (POD) for the computation of the reduced basis space and the Long Short-Term Memory (LSTM) network for the computation of the reduced coefficients. We are interested in dealing both with system identification and prediction. The most relevant novelties rely on (i) a filtering procedure of the full-order snapshots to reduce the dimensionality of the reduced problem and (ii) a preliminary treatment of the particle phase. The accuracy of our ROM approach is assessed against the classic Goldschmidt fluidized bed benchmark problem. Finally, we also provide some insights about the efficiency of our ROM approach.

math.NA↗

arXiv2024

A local approach to parameter space reduction for regression and classification tasks

Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models' design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), which explores the synergies of active subspaces with supervised clustering techniques in order to carry out a more efficient dimension reduction in the parameter space. The clustering is performed without losing the input-output relations by introducing a distance metric induced by the global active subspace. We present two possible clustering algorithms: K-medoids and a hierarchical top-down approach, which is able to impose a variety of subdivision criteria specifically tailored for parameter space reduction tasks. This method is particularly useful for the community working on surrogate modelling. Frequently, the parameter space presents subdomains where the objective function of interest varies less on average along different directions. So, it could be approximated more accurately if restricted to those subdomains and studied separately. We tested the new method over several numerical experiments of increasing complexity, we show how to deal with vectorial outputs, and how to classify the different regions with respect to the local active subspace dimension. Employing this classification technique as a preprocessing step in the parameter space, or output space in case of vectorial outputs, brings remarkable results for the purpose of surrogate modelling.

stat.ML↗

arXiv2024

Enhancing non-intrusive Reduced Order Models with space-dependent aggregation methods

In this manuscript, we combine non-intrusive reduced order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM. The prediction of the mixed formulation is given by a convex linear combination of the predictions of some previously-trained ROMs, where we assign to each model a space-dependent weight. The ROMs taken into account to build the mixed model exploit different reduction techniques, such as Proper Orthogonal Decomposition (POD) and AutoEncoders (AE), and/or different approximation techniques, namely a Radial Basis Function Interpolation (RBF), a Gaussian Process Regression (GPR) or a feed-forward Artificial Neural Network (ANN). The contribution of each model is retained with higher weights in the regions where the model performs best, and, vice versa, with smaller weights where the model has a lower accuracy with respect to the other models. Finally, a regression technique, namely a Random Forest, is exploited to evaluate the weights for unseen conditions. The performance of the aggregated model is evaluated on two different test cases: the 2D flow past a NACA 4412 airfoil, with an angle of attack of 5 degrees, having as parameter the Reynolds number varying between 1e5 and 1e6 and a transonic flow over a NACA 0012 airfoil, considering as parameter the angle of attack. In both cases, the mixed-ROM has provided improved accuracy with respect to each individual ROM technique.

math.NA↗

arXiv2024

Optimized Bayesian Framework for Inverse Heat Transfer Problems Using Reduced Order Methods

A stochastic inverse heat transfer problem is formulated to infer the transient heat flux, treated as an unknown Neumann boundary condition. Therefore, an Ensemble-based Simultaneous Input and State Filtering as a Data Assimilation technique is utilized for simultaneous temperature distribution prediction and heat flux estimation. This approach is incorporated with Radial Basis Functions not only to lessen the size of unknown inputs but also to mitigate the computational burden of this technique. The procedure applies to the specific case of a mold used in Continuous Casting machinery, and it is based on the sequential availability of temperature provided by thermocouples inside the mold. Our research represents a significant contribution to achieving probabilistic boundary condition estimation in real-time handling with noisy measurements and errors in the model. We additionally demonstrate the procedure's dependence on some hyperparameters that are not documented in the existing literature. Accurate real-time prediction of the heat flux is imperative for the smooth operation of Continuous Casting machinery at the boundary region where the Continuous Casting mold and the molten steel meet which is not also physically measurable. Thus, this paves the way for efficient real-time monitoring and control, which is critical for preventing caster shutdowns.

math.NA↗

arXiv2024

A comparative computational study of different formulations of the compressible Euler equations for mesoscale atmospheric flows in a finite volume framework

We consider three conservative forms of the mildly compressible Euler equations, called CE1, CE2 and CE3, with the goal of understanding which leads to the most accurate and robust pressure-based solver in a finite volume environment. Forms CE1 and CE2 are both written in density, momentum, and specific enthalpy, but employ two different treatments of the buoyancy and pressure gradient terms: for CE1 it is the standard pressure splitting implemented in open-source finite volume solvers (e.g., OpenFOAM), while for CE2 it is the typical pressure splitting found in computational atmospheric studies. Form CE3 is written in density, momentum, and potential temperature, with the buoyancy and pressure terms addressed as in CE2. For each formulation, we adopt a computationally efficient splitting approach. The three formulations are thoroughly assessed and compared through six benchmark tests involving dry air flow over a flat terrain or orography. We found that all three models are able to provide accurate results for the tests with a flat terrain, although the solvers based on the CE2 and CE3 forms are more robust. As for the mountain tests, CE1 solutions become unstable, while the CE2 and CE3 models provide results in very good agreement with data in the literature, the CE3 model being the most accurate. Hence, the CE3 model is the most accurate, reliable, and robust for the simulation of mesoscale atmospheric flows when using a pressure-based approach and space discretization by a finite volume method.

physics.flu-dyn↗

arXiv2024

An optimisation-based domain-decomposition reduced order model for parameter-dependent non-stationary fluid dynamics problems

In this work, we address parametric non-stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the optimisation-based domain decomposition approach, we derive an optimal control problem, for which we present a convergence analysis in the case of non-stationary incompressible Navier-Stokes equations. We discretize the problem with the finite element method and we compare different model order reduction techniques: POD-Galerkin and a non-intrusive neural network procedure. We show that the classical POD-Galerkin is more robust and accurate also in transient areas, while the neural network can obtain simulations very quickly though being less precise in the presence of discontinuities in time or parameter domain. We test the proposed methodologies on two fluid dynamics benchmarks with physical parameters and time dependency: the non-stationary backward-facing step and lid-driven cavity flow.

math.NA↗

arXiv2024

A Predictive Surrogate Model for Heat Transfer of an Impinging Jet on a Concave Surface

This paper aims to comprehensively investigate the efficacy of various Model Order Reduction (MOR) and deep learning techniques in predicting heat transfer in a pulsed jet impinging on a concave surface. Expanding on the previous experimental and numerical research involving pulsed circular jets, this investigation extends to evaluate Predictive Surrogate Models (PSM) for heat transfer across various jet characteristics. To this end, this work introduces two predictive approaches, one employing a Fast Fourier Transformation augmented Artificial Neural Network (FFT-ANN) for predicting the average Nusselt number under constant-frequency scenarios. Moreover, the investigation introduces the Proper Orthogonal Decomposition and Long Short-Term Memory (POD-LSTM) approach for random-frequency impingement jets. The POD-LSTM method proves to be a robust solution for predicting the local heat transfer rate under random-frequency impingement scenarios, capturing both the trend and value of temporal modes. The comparison of these approaches highlights the versatility and efficacy of advanced machine learning techniques in modelling complex heat transfer phenomena.

math.NA↗

arXiv2024

PyDMD: A Python package for robust dynamic mode decomposition

The dynamic mode decomposition (DMD) is a simple and powerful data-driven modeling technique that is capable of revealing coherent spatiotemporal patterns from data. The method's linear algebra-based formulation additionally allows for a variety of optimizations and extensions that make the algorithm practical and viable for real-world data analysis. As a result, DMD has grown to become a leading method for dynamical system analysis across multiple scientific disciplines. PyDMD is a Python package that implements DMD and several of its major variants. In this work, we expand the PyDMD package to include a number of cutting-edge DMD methods and tools specifically designed to handle dynamics that are noisy, multiscale, parameterized, prohibitively high-dimensional, or even strongly nonlinear. We provide a complete overview of the features available in PyDMD as of version 1.0, along with a brief overview of the theory behind the DMD algorithm, information for developers, tips regarding practical DMD usage, and introductory coding examples. All code is available at https://github.com/PyDMD/PyDMD .

stat.CO↗

arXiv2024

Physics Informed Neural Network Framework for Unsteady Discretized Reduced Order System

This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent reduced-order system. In this work, first, the governing equations are discretized using a finite difference scheme (whereas, any other discretization technique can be adopted), then projected on a reduced or latent space using the Proper Orthogonal Decomposition (POD)-Galerkin approach and next, the residual arising from discretized reduced order equation is considered as an additional loss penalty term alongside the data-driven loss term using different variants of deep learning method such as Artificial neural network (ANN), Long Short-Term Memory based neural network (LSTM). The LSTM neural network has been proven to be very effective for time-dependent problems in a purely data-driven environment. The current work demonstrates the LSTM network's potential over ANN networks in physics-informed neural networks (PINN) as well. The potential of using discretized governing equations instead of continuous form lies in the flexibility of input to the PINN. Different sizes of data ranging from small, medium to big datasets are used to assess the potential of discretized-physics-informed neural networks when there is very sparse or no data available. The proposed methods are applied to a pitch-plunge airfoil motion governed by rigid-body dynamics and a one-dimensional viscous Burgers' equation. The current work also demonstrates the prediction capability of various discretized-physics-informed neural networks outside the domain where the data is available or governing equation-based residuals are minimized.

math.NA↗

arXiv2024

A Shape Optimization Pipeline for Marine Propellers by means of Reduced Order Modeling Techniques

In this paper, we propose a shape optimization pipeline for propeller blades, applied to naval applications. The geometrical features of a blade are exploited to parametrize it, allowing to obtain deformed blades by perturbating their parameters. The optimization is performed using a genetic algorithm that exploits the computational speed-up of reduced order models to maximize the efficiency of a given propeller. A standard offline-online procedure is exploited to construct the reduced-order model. In an expensive offline phase, the full order model, which reproduces an open water test, is set up in the open-source software OpenFOAM and the same full order setting is used to run the CFD simulations for all the deformed propellers. The collected high-fidelity snapshots and the deformed parameters are used in the online stage to build the non-intrusive reduced-order model. This paper provides a proof of concept of the pipeline proposed, where the optimized propeller improves the efficiency of the original propeller.

math.OC↗

arXiv2024

A Dynamic Mode Decomposition Extension for the Forecasting of Parametric Dynamical Systems

Dynamic mode decomposition (DMD) has recently become a popular tool for the non-intrusive analysis of dynamical systems. Exploiting Proper Orthogonal Decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial basis evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context. Initially all the snapshots -- for different parameters and different time instants -- are projected to a reduced space; then DMD, or one of its variants, is employed to approximate reduced snapshots for future time instants. Exploiting the low dimension of the reduced space the predicted reduced snapshots are then combined using regression techniques, thus enabling the possibility to approximate any untested parametric configuration in future. This paper depicts in detail the algorithmic core of this method; we also present and discuss three test cases for our algorithm: a simple dynamical system with a linear parameter dependency, a heat problem with nonlinear parameter dependency and a fluid dynamics problem with nonlinear parameter dependency.

math.NA↗