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 1249 prodotti
-
101/2024 - 29/11/2024
Bonetti, S.; Corti, M.
Unified discontinuous Galerkin analysis of a thermo/poro-viscoelasticity model | Abstract | | We present and analyze a discontinuous Galerkin method for the numerical modeling of a Kelvin-Voigt thermo/poro-viscoelastic problem. We present the derivation of the model, and we develop a stability analysis in the continuous setting that holds both for the full inertial and quasi-static problems and that is robust with respect to most of the physical parameters of the problem. For spatial discretization, we propose an arbitrary-order weighted symmetric interior penalty scheme that supports general polytopal grids and is robust with respect to strong heterogeneities in the model coefficients. For the semi-discrete problem, we prove the extension of the stability result demonstrated in the continuous setting. A wide set of numerical simulations is presented to assess the convergence and robustness properties of the proposed method. Moreover, we test the scheme with literature and physically sound test cases for proof-of-concept applications in the geophysical context. |
-
102/2024 - 29/11/2024
Bucelli, M.
The lifex library version 2.0 | Abstract | | This article presents updates to lifex [Africa, SoftwareX (2022)], a C++ library for high-performance finite element simulations of multiphysics, multiscale and multidomain problems. In this release, we introduce an additional intergrid transfer method for non-matching multiphysics coupling on the same domain, significantly optimize nearest-neighbor point searches and interface coupling utilities, extend the support for 2D and mixed-dimensional problems, and provide improved facilities for input/output and simulation serialization and restart. These advancements also propagate to the previously released modules of lifex specifically designed for cardiac modeling and simulation, namely lifex-fiber [Africa et al., BMC Bioinformatics (2023)], lifex-ep [Africa et al., BMC Bioinformatics (2023)] and lifex-cfd [Africa et al., Computer Physics Communications (2024)]. The changes introduced in this release aim at consolidating lifex's position as a valuable and versatile tool for the simulation of multiphysics systems. |
-
100/2024 - 27/11/2024
Farenga, N.; Fresca, S.; Brivio, S.; Manzoni, A.
On latent dynamics learning in nonlinear reduced order modeling | Abstract | | In this work, we present the novel mathematical framework of latent dynamics models (LDMs) for reduced order modeling of parameterized nonlinear time-dependent PDEs. Our framework casts this latter task as a nonlinear dimensionality reduction problem, while constraining the latent state to evolve accordingly to an (unknown) dynamical system, namely a latent vector ordinary differential equation (ODE). A time-continuous setting is employed to derive error and stability estimates for the LDM approximation of the full order model (FOM) solution. We analyze the impact of using an explicit Runge-Kutta scheme in the time-discrete setting, and further explore the learnable setting where deep neural networks approximate the discrete LDM components, while providing a bounded approximation error with respect to the FOM. Moreover, we extend the concept of parameterized Neural ODE - recently proposed as a possible way to build data-driven dynamical systems with varying input parameters - to be a convolutional architecture, where the input parameters information is injected by means of an affine modulation mechanism, while designing a convolutional autoencoder neural network able to retain spatial-coherence, thus enhancing interpretability at the latent level. Numerical experiments, including the Burgers’ and the advection-reaction-diffusion equations, demonstrate the framework’s ability to obtain, in a multi-query context, a time-continuous approximation of the FOM solution, thus being able to query the LDM approximation at any given time instance while retaining a prescribed level of accuracy. Our findings highlight the remarkable potential of the proposed LDMs, representing a mathematically rigorous framework to enhance the accuracy and approximation capabilities of reduced order modeling for time-dependent parameterized PDEs. |
-
99/2024 - 23/11/2024
Ragni, A.; Masci, C.; Paganoni, A. M.
Analysis of Higher Education Dropouts Dynamics through Multilevel Functional Decomposition of Recurrent Events in Counting Processes | Abstract | | This paper analyzes the dynamics of higher education dropouts through an innovative approach that integrates recurrent events modeling and point process theory with functional data analysis. We propose a novel methodology that extends existing frameworks to accommodate hierarchical data structures, demonstrating its potential through a simulation study. Using administrative data from student careers at Politecnico di Milano, we explore dropout patterns during the first year across different bachelor's degree programs and schools.
Specifically, we employ Cox-based recurrent event models, treating dropouts as repeated occurrences within both programs and schools. Additionally, we apply functional modeling of recurrent events and multilevel principal component analysis to disentangle latent effects associated with degree programs and schools, identifying critical periods of dropout risk and providing valuable insights for institutions seeking to implement strategies aimed at reducing dropout rates. |
-
98/2024 - 14/11/2024
Castiglione, C.; Arnone, E.; Bernardi, M.; Farcomeni, A.; Sangalli, L.M.
PDE-regularised spatial quantile regression | Abstract | | We consider the problem of estimating the conditional quantiles of an unknown distribution from data gathered on a spatial domain. We propose a spatial quantile regression model with differential regularisation. The penalisation involves a partial differential equation defined over the considered spatial domain, that can display a complex geometry. Such regularisation permits, on one hand, to model complex anisotropy and non-stationarity patterns, possibly on the basis of problem-specific knowledge, and, on the other hand, to comply with the complex conformation of the spatial domain. We define an innovative functional Expectation-Maximisation algorithm, to estimate the unknown quantile surface. We moreover describe a suitable discretisation of the estimation problem, and investigate the theoretical properties of the resulting estimator. The performance of the proposed method is assessed by simulation studies, comparing with state-of-the-art techniques for spatial quantile regression. Finally, the considered model is applied to two real data analyses, the first concerning rainfall measurements in Switzerland and the second concerning sea surface conductivity data in the Gulf of Mexico. |
-
97/2024 - 12/11/2024
Ferro, N.; Mezzadri, F.; Carbonaro, D.; Galligani, E.; Gallo, D.; Morbiducci, U.; Chiastra, C.; Perotto, S.
Designing novel vascular stents with enhanced mechanical behavior through topology optimization of existing devices | Abstract | | A variety of different vascular stent designs are currently available on the market, featuring different geometries, manufacturing materials, and physical characteristics. Here, we propose a framework for designing innovative stents that replicate and enhance the mechanical properties of existing devices. The framework includes a SIMP-based topology optimization formulation, assisted by the homogenization theory to constrain the mechanical response, along with a minimum length scale requirement to ensure manufacturability to the designed devices. The optimization problem, discretized on a sequence of computational meshes anisotropically adapted, generates a 2D stent unit cell, which can be automatically converted into a 3D digital version of the device. This virtual prototype is validated through in silico testing via a radial crimping simulation to assess the stent insertion into the catheter, prior to implantation. The results prove that the proposed framework can identify stent designs that are competitive with respect to existing devices in terms of relevant clinical requirements, such as foreshortening, radial stiffness and surface contact area. |
-
94/2024 - 11/11/2024
Franco, N.R.; Fresca, S.; Tombari, F.; Manzoni, A.
Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks | Abstract | | Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution of parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency. This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs in the presence of geometrical variability. In particular, we propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme where a GNN architecture is used to efficiently evolve the system. With respect to the majority of surrogate models, the proposed approach stands out for its ability of tackling problems with parameter dependent spatial domains, while simultaneously generalizing to different geometries and mesh resolutions. We assess the effectiveness of the proposed approach through a series of numerical experiments, involving both two- and three-dimensional problems, showing that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios. We also assess, from a numerical standpoint, the importance of using GNNs, rather than classical dense deep neural networks, for the proposed framework. |
-
93/2024 - 11/11/2024
Conti, P.; Kneifl, J.; Manzoni, A.; Frangi, A.; Fehr, J.; Brunton, S.L.; Kutz, J.N.
VENI, VINDy, VICI - a variational reduced-order modeling framework with uncertainty quantification | Abstract | | The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when governing equations are unknown or partially known, or when access to full order solvers is restricted, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified.
Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by VINDy. Once trained offline, the identified model can be queried for new parameter instances and/or new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the Rossler system with different noise intensities and sources. Then the performance of the overall method – named VENI, VINDy, VICI – is tested on PDE benchmarks including structural mechanics and fluid dynamics. |
|
