Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1268 prodotti
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14/2022 - 17/03/2022
Zappon, E.; Manzoni, A.; Quarteroni A.
Efficient and certified solution of parametrized one-way coupled problems through DEIM-based data projection across non-conforming interfaces | Abstract | | One of the major challenges of coupled problems is to manage nonconforming meshes at the interface between two models and/or domains, due to different numerical schemes or domains discretizations employed. Moreover, very often complex submodels depend on (e.g., physical or geometrical) parameters. Understanding how outputs of interest are affected by parameter variations thus plays a key role to gain useful insights on the problem's physics; however, expensive repeated solutions of the problem using high-fidelity, full-order models are often unaffordable. In this paper, we propose a parametric reduced order modeling (ROM) technique for parametrized one-way coupled problems made by a first independent model, the master model, and a second model, the slave model, that depends on the master model through Dirichlet interface conditions. We combine a reduced basis (RB) method, applied to each subproblems, with the discretized empirical interpolation method (DEIM) to efficiently interpolate or project Dirichlet data across conforming and non-conforming meshes at the domains interface, building a low-dimensional representation of the overall coupled problem. The proposed technique is then numerically verified by considering a series of test cases involving both steady and unsteady problems, and deriving a-posteriori error estimates on the solution of the coupled problem in both cases. This work arises from the need to solve staggered cardiac electrophysiological models and represents the first step towards the setting of ROM techniques for the more general two-way Dirichlet-Neumann coupled problems solved with domain decomposition sub-structuring methods, when interface non-conformity is involved. |
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13/2022 - 26/02/2022
Grasselli, M.; Parolini, N.; Poiatti, A.; Verani, M.
Non-isothermal non-Newtonian fluids: the stationary case | Abstract | | The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suit-able power law depending on p in (1,2) (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases.Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments. |
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12/2022 - 25/02/2022
Antonietti, P.F.; Dassi, F.; Manuzzi, E.
Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods | Abstract | | We propose two new strategies based on Machine Learning techniques to handle polyhedral grid refinement, to be possibly employed within an adaptive framework. The first one employs the k-means clustering algorithm to partition the points of the polyhedron to be refined. This strategy is a variation of the well known Centroidal Voronoi Tessellation. The second one employs Convolutional Neural Networks to classify the “shape” of an element so that “ad-hoc” refinement criteria can be defined. This strategy can be used to enhance existing refinement strategies, including the k-means strategy, at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polyhedral elements, namely the Virtual Element Method (VEM) and the Polygonal Discontinuous Galerkin (PolyDG) method. We demonstrate that these strategies do preserve the structure and the quality of the underlaying grids, reducing the overall computational cost and mesh complexity. |
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11/2022 - 22/02/2022
Sampaoli, S.; Agosti, A.; Pozzi, G.; Ciarletta,P.
A toy model of misfolded protein aggregation and neural damage propagation in neudegenerative diseases | Abstract | | Neurodegenerative diseases (NDs) result from the transformation and accumulation of misfolded proteins within the nervous system. They have common features, like the chronic nature and the progressive destruction of neurons in specific areas of the brain. Several mathematical models have been proposed to investigate the biological processes underlying NDs, focusing on the kinetics of polymerization and fragmentation at the microscale and on the spread of neural damage at a macroscopic level. The aim of this work is to bridge the gap between microscopic and macroscopic approaches proposing a toy partial differential model able to take into account both for the short-time dynamics of the misfolded proteins aggregating in plaques and the long-term evolution of tissue damage. Using the theoretical framework of mixtures theory, we considered the brain as a biphasic material made of misfolded protein aggregates and of healthy tissue. The resulting Cahn-Hilliard type equation for the misfolded proteins contains a growth term depending on the local availability of precursor proteins, that follow a reaction-diffusion equation. The misfolded proteins also posses a chemotactic mass flux driven by gradients of neural damage, that is caused by local accumulation of misfolded protein and that evolves slowly according to an Allen-Cahn equation. The partial differential model is solved numerically using the finite element method in a simple two-dimensional domain, evaluating the effects of the mobility of the misfolded protein and the diffusion of the neural damage. We considered both isotropic and anisotropic mobility coefficients, highlighting that the spreading front of the neural damage follows the direction of the largest eigenvalue of the mobility tensor. In both cases, we computed two biomarkers for quantifying the aggregation in plaques and the evolution of neural damage, that are in qualitative agreement with the characteristic Jack curves for many NDs. |
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10/2022 - 22/02/2022
Fresca, S.; Manzoni, A.
Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models | Abstract | | Simulating fluid flows in different virtual scenarios is of key importance in engineering applications. However, high-fidelity, full-order models relying, e.g., on the finite element method, are unaffordable whenever fluid flows must be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on proper orthogonal decomposition (POD) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. However, they might require expensive hyper-reduction strategies for handling parameterized nonlinear terms, and enriched reduced spaces (or Petrov-Galerkin projections) if a mixed velocity-pressure formulation is considered, possibly hampering the evaluation of reliable solutions in real-time. Dealing with fluid-structure interactions entails even higher difficulties. The proposed deep learning (DL)-based ROMs overcome all these limitations by learning in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics. To do so, they rely on deep neural networks, after performing a former dimensionality reduction through POD enhancing their training times substantially. The resulting POD-DL ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid-structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm. |
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09/2022 - 22/02/2022
Corti, M.; Zingaro, A.; Dede', L.; Quarteroni, A.
Impact of Atrial Fibrillation on Left Atrium Haemodynamics: A Computational Fluid Dynamics Study | Abstract | | We analyze left atrium haemodynamics, highlighting differences among healthy individuals and patients affected by atrial fibrillation. The computational study is based on patient-specific geometries of the left atria to simulate blood flow dynamics. We devise a novel procedure aimed at recovering the boundary conditions for the 3D haemodynamics simulations, particularly useful in absence of specific ones provided by clinical measurements. With this aim, we introduce a parametric definition of the atria displacement, and we employ a closed-loop lumped parameter model of the whole cardiocirculatory system conveniently tuned on the basis of the patient characteristics. We evaluate a number of fluid dynamics indicators for the atrial haemodynamics, validating our numerical results in terms of several clinical measurements; we investigate the impact of geometrical and clinical features on the risk of thrombosis. To analyse the correlation of thrombus formation with atrial fibrillation, coherently with the medical evidence, we propose a novel indicator, which we call age stasis and that arises from the combination of Eulerian and Lagrangian quantities. This indicator identifies regions where the slow flow cannot rinse the chamber properly, accumulating stale blood particles and creating optimal conditions for clot formation.
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08/2022 - 02/02/2022
Gobat, G.; Opreni, A.; Fresca, S.; Manzoni, A.; Frangi, A.
Reduced order modeling of nonlinear microstructures through Proper Orthogonal Decomposition | Abstract | | We apply the Proper Orthogonal Decomposition (POD) method for the efficient simulation of several scenarios undergone by Micro-Electro Mechanical-Systems, involving nonlinearites of geometric and electrostatic nature. The former type of nonlinearity, associated to the large displacements of the devices, leads to polynomial terms up to cubic order that are reduced through exact projection onto a low dimensional subspace spanned by the Proper Orthogonal Modes (POMs). On the contrary, electrostatic nonlinearities are modeled resorting to precomputed manifolds in terms of the amplitudes of the electrically active POMs. We extensively test the reliability of the assumed linear trial space in challenging applications focusing on resonators, micromirrors and arches also displaying internal resonances. We discuss several options to generate the matrix of snapshots using both classical time marching schemes and more advanced Harmonic Balance (HB) approaches. Furthermore, we propose a comparison between the periodic orbits computed with POD and the invariant manifold approximated with Direct Normal Form approaches, further stressing the reliability of the technique and its remarkable predictive capabilities, e.g., in terms of estimation of the frequency response function of selected output quantities of interest |
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07/2022 - 02/02/2022
Sinigaglia, C.; Quadrelli, D.E.; Manzoni, A.; Braghin, F.
Fast active thermal cloaking through PDE-constrained optimization and reduced-order modeling | Abstract | | In this paper we show how to efficiently achieve thermal cloaking from a computational standpoint in several virtual scenarios by controlling a distribution of active heat sources. We frame this problem in the setting of PDE-constrained optimization, where the reference field is the solution of the time-dependent heat equation in the absence of the object to cloak. The optimal control problem then aims at actuating the space-time control field so that the thermal field outside the obstacle is indistinguishable from the reference field. In particular, we consider multiple scenarios where material’s thermal diffusivity, source intensity and obstacle’s temperature are allowed to vary within a user-defined range. To tackle the thermal cloaking problem in a rapid and reliable way, we rely on a parametrized reduced order model built through the reduced basis method, thus entailing huge computational speedups compared to high-fidelity, full-order model exploiting the finite element method while dealing both with complex target shapes and disconnected control domains. |
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