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 1349 prodotti
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70/2021 - 16/11/2021
Beirao da Veiga, L.; Canuto, C.; Nochetto, R.H.; Vacca, G.; Verani, M.
Adaptive VEM: Stabilization-Free A Posteriori Error Analysis | Abstract | | In the present paper we initiate the challenging task of building a mathematically sound theory for Adaptive Virtual Element Methods (AVEMs). Among the realm of polygonal meshes, we restrict our analysis to triangular meshes with hanging nodes in 2d -- the simplest meshes with a systematic refinement procedure that preserves shape regularity and optimal complexity. A major challenge in the a posteriori error analysis of AVEMs is the presence of the stabilization term, which is of the same order as the residual-type error estimator but prevents the equivalence of the latter with the energy error.
Under the assumption that any chain of recursively created hanging nodes has uniformly bounded length, we show that the stabilization term can be made arbitrarily small relative to the error estimator provided the stabilization parameter of the scheme is sufficiently large. This quantitative estimate leads to stabilization-free upper and lower a posteriori bounds for the energy error. This novel and crucial property of VEMs hinges on the largest subspace of continuous piecewise linear functions and the delicate interplay between its coarser scales and the finer ones of the VEM space. Our results apply to H^1-conforming (lowest order) VEMs of any kind, including the classical and enhanced VEMs. |
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69/2021 - 06/11/2021
Antonietti, P.F.; Caldana, M.; Dede', L.
Accelerating Algebraic Multigrid Methods via Artificial Neural Networks | Abstract | | We present a novel Deep Learning-based algorithm to accelerate - through the use of Artificial Neural Networks (ANNs) - the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from Finite Element discretizations of Partial Differential Equations. We show that ANNs can be be successfully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algorithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN,
we consider the iterative solution via the AMG method of the algebraic system of equations stemming from Finite Element discretizations of a two-dimensional elliptic equation with a highly heterogeneous diffusion coefficient. We train (off-line) our ANN with a rich data-set and present an in-depth analysis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme. |
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68/2021 - 03/11/2021
Regazzoni, F.; Salvador, M.; Dede', L.; Quarteroni, A.
A machine learning method for real-time numerical simulations of cardiac electromechanics | Abstract | | We propose a machine learning-based method to build a system of differential equations that approximates the dynamics of 3D electromechanical models for the human heart, accounting for the dependence on a set of parameters. Specifically, our method permits to create a reduced-order model (ROM), written as a system of Ordinary Differential Equations (ODEs) wherein the forcing term, given by the right-hand side, consists of an Artificial Neural Network (ANN), that possibly depends on a set of parameters associated with the electromechanical model to be surrogated. This method is non-intrusive, as it only requires a collection of pressure and volume transients obtained from the full-order model (FOM) of cardiac electromechanics. Once trained, the ANN-based ROM can be coupled with hemodynamic models for the blood circulation external to the heart, in the same manner as the original electromechanical model, but at a dramatically lower computational cost. Indeed, our method allows for real-time numerical simulations of the cardiac function. Our results show that the ANN-based ROM is accurate with respect to the FOM (relative error between $10^{-3}$ and $10^{-2}$ for biomarkers of clinical interest), while requiring very small training datasets (30-40 samples). We demonstrate the effectiveness of the proposed method on two relevant contexts in cardiac modeling. First, we employ the ANN-based ROM to perform a global sensitivity analysis on both the electromechanical and hemodynamic models. Second, we perform a Bayesian estimation of two parameters starting from noisy measurements of two scalar outputs. In both these cases, replacing the FOM of cardiac electromechanics with the ANN-based ROM makes it possible to perform in a few hours of computational time all the numerical simulations that would be otherwise unaffordable, because of their overwhelming computational cost, if carried out with the FOM. As a matter of fact, our ANN-based ROM is able to speedup the numerical simulations by more than three orders of magnitude. |
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67/2021 - 03/11/2021
Salvador, M.; Regazzoni, F.; Pagani, S.; Dede', L.; Trayanova, N.; Quarteroni, A.
The role of mechano-electric feedbacks and hemodynamic coupling in scar-related ventricular tachycardia | Abstract | | Mechano-electric feedbacks (MEFs), which model how mechanical stimuli are transduced into electrical signals, have received sparse investigation by considering electromechanical simulations in simplified scenarios. In this paper, we study the effects of different MEFs modeling choices for myocardial deformation and nonselective stretch-activated channels (SACs) in the monodomain equation. We perform numerical simulations during ventricular tachycardia (VT) by employing a biophysically detailed and anatomically accurate 3D electromechanical model for the left ventricle (LV) coupled with a 0D closed-loop model of the cardiocirculatory system. We model the electromechanical substrate responsible for scar-related VT with a distribution of infarct and peri-infarct zones. Our mathematical framework takes into account the hemodynamic effects of VT due to myocardial impairment and allows for the classification of their hemodynamic nature, which can be either stable or unstable. By combining electrophysiological, mechanical and hemodynamic models, we observe that all MEFs may alter the propagation of the action potential and the morphology of the VT. In particular, we notice that the presence of myocardial deformation in the monodomain equation may change the VT basis cycle length and the conduction velocity but do not affect the hemodynamic nature of the VT. Finally, nonselective SACs may affect wavefront stability, by possibly turning a hemodynamically stable VT into a hemodynamically unstable one and vice versa. |
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66/2021 - 03/11/2021
Antonietti, P.F.; Botti, M.; Mazzieri, I.
On mathematical and numerical modelling of multiphysics wave propagation with polygonal Discontinuous Galerkin methods | Abstract | | In this work we present discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic,
poro-elastic, and poro-elasto-acoustic materials. Wave ropagation is modeled by using either the elastodyanmics equation, in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poro-elastic one.
The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed on the interface between the domains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods. |
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65/2021 - 03/11/2021
Mazzieri, I.; Muhr, M.; Stupazzini, M.; Wohlmuth, B.
Elasto-acoustic modelling and simulation for the seismic response of structures: The case of the Tahtali dam in the 2020 Izmir earthquake | Abstract | | As a mean to assess the risk dam structures are exposed to during earthquakes, we employ an abstract mathematical, three dimensional, elasto-acoustic coupled wave-propagation model taking into account (i) the dam structure itself, embedded into (ii) its surrounding topography, (iii) different material soil layers, (iv) the seismic source as well as (v) the reservoir lake filled with water treated as an acoustic medium. As a case study for extensive numerical simulations we consider the magnitude 7 seismic event of the 30th of October 2020 taking place in the Icarian Sea (Greece) and the Tahtali dam around 30km from there (Turkey). A challenging task is to resolve the multiple length scales that are present due to the huge differences in size between the dam building structure and the area of interest, considered for the propagation of the earthquake. Interfaces between structures and highly non-conforming meshes on different scales are resolved by means of a discontinuous Galerkin approach. The seismic source is modeled using inversion data about the real fault plane. Ultimately, we perform a real data driven, multi-scale, full source-to-site, physics based simulation based on the discontinuous Galerkin spectral element method, which allows to precisely validate the ground motion experienced along the Tahtali dam, comparing the synthetic seismograms against actually observed ones. A comparison with a more classical computational method, using a plane wave with data from a deconvolved seismogram reading as an input, is discussed. |
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64/2021 - 22/10/2021
Clarotto, L; Allard, D.; Menafoglio, A.
A new class of alpha-transformations for the spatial analysis of Compositional Data | Abstract | | Georeferenced compositional data are prominent in many scientic fields and in spatial statistics. This work addresses the problem of proposing models and methods to analyze and predict, through kriging, this type of data. To this purpose, a novel class of transformations, named the Isometric alpha-transformation (alpha-IT), is proposed, which encompasses the traditional Isometric Log-Ratio (ILR) transformation. It is shown that the ILR is the limit case of the alpha-IT as alpha tends to 0 and that alpha = 1 corresponds to a linear transformation of the data. Unlike the ILR, the proposed transformation accepts 0s in the compositions when alpha > 0. Maximum likelihood estimation of the parameter alpha is established. Prediction using kriging on alpha-IT transformed data is validated on synthetic spatial compositional data, using prdiction scores computed either in the geometry induced by the alpha-IT, or in the simplex. Application to land cover data shows that the relative superiority of the various approaches w.r.t. a prediction objective depends on whether the compositions contained any zero component. When all components are positive, the limit cases (ILR or linear transformations) are optimal for none of the considered metrics. An intermediate geometry, corresponding to the alpha-IT with maximum likelihood estimate, better describes the dataset in a geostatistical setting. When the amount of compositions with 0s is not negligible, some side-effects of the transformation gets amplied as alpha decreases, entailing poor kriging performances both within the alpha-IT geometry and for metrics in the simplex. |
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63/2021 - 22/10/2021
Rosafalco, L.; Torzoni, M.; Manzoni, A.; Mariani, S.; Corigliano, A.
Online structural health monitoring by model order reduction and deep learning algorithms | Abstract | | Within a structural health monitoring (SHM) framework, we propose a simulation-based classification strategy to move towards online damage localization. The procedure combines parametric Model Order Reduction (MOR) techniques and Fully Convolutional Networks (FCNs) to analyze raw vibration measurements recorded on the monitored structure. First, a dataset of possible structural responses under varying operational conditions is built through a physics-based model, allowing for a finite set of predefined damage scenarios. Then, the dataset is used for the offline training of the FCN. Because of the extremely large number of model evaluations required by the dataset construction, MOR techniques are employed to reduce the computational burden. The trained classifier is shown to be able to map unseen vibrational recordings, e.g. collected on-the-fly from sensors placed on the structure, to the actual damage state, thus providing information concerning the presence and also the location of damage. The proposed strategy has been validated by means of two case studies, concerning a 2D portal frame and a 3D portal frame railway bridge; MOR techniques have allowed us to respectively speed up the analyses about 30 and 420 times. For both the case studies, after training the classifier has attained an accuracy greater than 85%. |
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