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 1239 prodotti
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71/2023 - 28/09/2023
Conni, G.; Piccardo, S.; Perotto, S.; Porta, G.M.; Icardi, M.
HiPhome: HIgh order Projection-based HOMogEnisation for advection diffusion reaction problems | Abstract | | We propose a new model reduction technique for multiscale scalar transport problems that exhibit dominant axial dynamics.
To this aim, we rely on the separation of variables to combine a Hierarchical Model (HiMod) reduction with a two-scale asymptotic expansion. We extend the two-scale asymptotic expansion to an arbitrary order and exploit the high-order correctors to define the HiMod modal basis which approximates the transverse dynamics of the flow, while we adopt a finite element discretisation to model the leading stream.
The resulting method, which is named HiPhome (HIgh-order Projection-based HOMogEnisation), is successfully assessed both in steady and unsteady advection-diffusion-reaction settings. The numerical results confirm the very good performance of HiPhome which improves the accuracy and the convergence rate of HiMod and extends the reliability of the standard homogenised solution to transient and pre-asymptotic regimes. |
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70/2023 - 25/09/2023
Ragni, A.; Ippolito, D.; Masci, C.
Assessing the Impact of Hybrid Teaching on Students' Academic Performance via Multilevel Propensity Score-based techniques | Abstract | | This study employs multilevel propensity score techniques in an innovative analysis pipeline to assess the impact of hybrid teaching – a blend of face-to-face and online learning – on student performance within engineering programs at Politecnico di Milano. By analyzing students' credits earned and grade point average, the investigation compares outcomes of students engaged in hybrid teaching against those solely in face-to-face instruction that precedes the Covid-19 pandemic.
Tailored multilevel models for earned credits and grade point averages are fitted onto meticulously constructed dataframes, effectively minimizing potential biases stemming from variables such as gender, age at career initiation, previous academic track, admission test scores, and student origins across the two groups.
The methodology accounts for variations across distinct educational programs and investigates disparities among them.
Our findings suggest marginal overall disparities in student performance, indicating, on average, a subtle inclination toward a modest rise in earned credits and a slight decrease in grade point averages among those exposed to hybrid teaching. The use of multilevel models to analyze data within the same institution revealed that the impact of hybrid teaching on students' performances can vary significantly across different engineering programs, providing valuable insights into its effectiveness in diverse educational contexts. |
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69/2023 - 20/09/2023
Ferro, N.; Micheletti, S.; Parolini, N.; Perotto, S.; Verani, M.; Antonietti, P. F.
Level set-fitted polytopal meshes with application to structural topology optimization | Abstract | | We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method. |
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68/2023 - 14/09/2023
Vitullo, P.; Colombo, A.; Franco, N.R.; Manzoni, A.; Zunino, P.
Nonlinear model order reduction for problems with microstructure using mesh informed neural networks | Abstract | | Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment. |
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66/2023 - 05/09/2023
Fresca, S.; Gobat, G.; Fedeli, P.; Frangi, A.; Manzoni, A.
Deep learning-based reduced order models for the real-time simulation of the nonlinear dynamics of microstructures | Abstract | | We propose a non-intrusive Deep Learning-based Reduced Order Model (DL-ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity snapshots are used to generate a POD-Galerkin ROM which is subsequently exploited to generate the data, covering the whole parameter range, used in the training phase of the DL-ROM. A convolutional autoencoder is employed to map the system response onto a low-dimensional representation and, in parallel, to model the reduced nonlinear trial manifold. The system dynamics on the manifold is described by means of a deep feedforward neural network that is trained together with the autoencoder. The strategy is benchmarked against high fidelity solutions on a clamped-clamped beam and on a real micromirror with softening response and multiplicity of solutions. By comparing the different computational costs, we discuss the impressive gain in performance and show that the DL-ROM truly represents a real-time tool which can be profitably and efficiently employed in complex system-level simulation procedures for design and optimisation purposes. |
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67/2023 - 05/09/2023
Conti, P.; Guo, M.; Manzoni, A.; Hesthaven, J.S.
Multi-fidelity surrogate modeling using long short-term memory networks | Abstract | | When evaluating quantities of interest that depend on the solutions to differential equations, we inevitably face the trade-off between accuracy and efficiency. Especially for parametrized, time-dependent problems in engineering computations, it is often the case that acceptable computational budgets limit the availability of high-fidelity, accurate simulation data. Multi-fidelity surrogate modeling has emerged as an effective strategy to overcome this difficulty. Its key idea is to leverage many low-fidelity simulation data, less accurate but much faster to compute, to improve the approximations with limited high-fidelity data. In this work, we introduce a novel data-driven framework of multi-fidelity surrogate modeling for parametrized, time-dependent problems using long short-term memory (LSTM) networks, to enhance output predictions both for unseen parameter values and forward in time simultaneously -- a task known to be particularly challenging for data-driven models. We demonstrate the wide applicability of the proposed approaches in a variety of engineering problems with high- and low-fidelity data generated through fine versus coarse meshes, small versus large time steps, or finite element full-order versus deep learning reduced-order models. Numerical results show that the proposed multi-fidelity LSTM networks not only improve single-fidelity regression significantly, but also outperform the multi-fidelity models based on feed-forward neural networks. |
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65/2023 - 03/09/2023
Gatti, F.; de Falco, C.; Perotto, S.; Formaggia, L.; Pastor, M.
A scalable well-balanced numerical scheme for the modelling of two-phase shallow granular landslide consolidation | Abstract | | We introduce a new method to efficiently solve a variant of the Pitman-Le two-phase depth-integrated system of equations, for the simulation of a fast landslide consolidation process. In particular, in order to cope with the loss of hyperbolicity typical of this system, we generalize Pelanti’s proposition for the Pitman-Le model to the case of a non-null excess pore water pressure configuration. The variant of the Pitman-Le model is numerically solved by relying on the approach the authors set to discretize the corresponding single-phase model, jointly with a fictitious inter-phase drag force which avoids arising the spurious numerical oscillations induced by the loss of hyperbolicity. To verify the reliability of the proposed simulation tool, we first assess the accuracy and efficiency of the new method in ideal scenarios. In particular, we investigate the well-balancing property and provide some relevant scaling results for the parallel implementation of the method. Successively, we challenge the procedure on real configurations from the available literature. |
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64/2023 - 28/08/2023
Heltai, L.; Zunino, P.
Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains | Abstract | | Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues -- just to mention a few examples -- can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the inf-sup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples. |
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