MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1268 products
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93/2021 - 12/27/2021
Parolini, N.; Dede', L; Ardenghi, G.; Quarteroni, A.
Modelling the COVID-19 epidemic and the vaccination campaign in Italy by the SUIHTER model | Abstract | | Several epidemiological models have been proposed to study the evolution of COVID-19 pandemic. In this paper, we propose an extension of the SUIHTER model, first introduced in [Parolini et al, Proc R. Soc. A., 2021] to analyse the COVID-19 spreading in Italy, which accounts for the vaccination campaign and the presence of new variants when they become dominant. In particular, the specific features of the variants (e.g. their increased transmission rate) and vaccines (e.g. their efficacy to prevent transmission, hospitalization and death) are modeled, based on clinical evidence. The new model is validated comparing its near-future forecast capabilities with other epidemiological models and exploring different scenario analyses. |
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92/2021 - 12/27/2021
Antonietti, P.F.; Manzini, G.; Scacchi, S.; Verani, M.
On arbitrarily regular conforming virtual element methods for elliptic partial differential equations | Abstract | | The Virtual Element Method (VEM) is a very effective framework to
design numerical approximations with high global regularity to the
solutions of elliptic partial differential equations.
In this paper, we review the construction of such approximations
for an elliptic problem of order $p_1$ using conforming, finite
dimensional subspaces of $H^2{p_2}(Omega)$, where $p_1$ and
$p_2$ are two integer numbers such that $p_2geqp_1geq1$
and $OmegainR^2$ is the computational domain.
An abstract convergence result is presented in a suitably defined
energy norm.
The space formulation and major aspects such as the choice and
unisolvence of the degrees of freedom are discussed, also
providing specific examples corresponding to various practical
cases of high global regularity.
Finally, the construction of the ``enhanced'' formulation of the
virtual element spaces is also discussed in details with a proof
that the dimension of the ``regular'' and ``enhanced'' spaces is
the same and that the virtual element functions in both spaces can
be described by the same choice of the degrees of freedom. |
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91/2021 - 12/16/2021
Arnone, E.; Sangalli, L.M.; Vicini, A.
Smoothing spatio-temporal data with complex missing data patterns | Abstract | | We consider spatio-temporal data and functional data with spatial dependence, characterized by complicated missing data patterns. We propose a new method capable to efficiently handle these data structures, including the case where data are missing over large portions of the spatio-temporal domain. The method is based on regression with partial differential equation regularization. The proposed model can accurately deal with data scattered over domains with irregular shapes and can accurately estimate fields exhibiting complicated local features. We demonstrate the consistency and asymptotic normality of the estimators. Moreover, we illustrate the good performances of the method in simulations studies, considering different missing data scenarios, from sparse data to more challenging scenarios where the data are missing over large portions of the spatial and temporal domains and the missing data are clustered in space and/or in time. The proposed method is
compared to competing techniques, considering predictive accuracy and uncertainty quantification measures. Finally, we show an application to the analysis of lake surface water temperature data, that further illustrates the ability of the method to handle data featuring complicated patterns of missingness and highlights its potentiality for environmental
studies. |
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90/2021 - 12/16/2021
Hernandez, V.M.; Paolucci, R.; Mazzieri, I.
3D numerical modeling of ground motion in the Valley of Mexico: a case study from the Mw3.2 earthquake of July 17, 2019 | Abstract | | In this study a 3D physics-based numerical approach, based on the spectral element numerical code SPEED (http://speed.mox.polimi.it), is used to simulate seismic wave propagation due to a local earthquake in the Mexico City basin. The availability of detailed geological, geophysical, geotechnical, and seismological data allowed for the creation of a large-scale (60 km x 60 km) heterogeneous 3D numerical model of the Mexico City area, dimensioned to accurately propagate frequencies up to 1.3 Hz. Results of numerical simulations are validated against the ground motion recordings of the July 17, 2019, Mw3.2 earthquake, which produced peak ground acceleration (PGA) exceeding 0.3g about 1 km away of the epicenter. Results show that for the hill and transition zones of the Valley of Mexico there is a good agreement with records. For the lake zone, the simulated decay trend of the PGV with epicentral distance was reasonably close to the observations, both for the horizontal and vertical components, but synthetics present in general shorter duration with respect to records, probably due to insufficient accuracy of considered values of the quality factor. In spite of these limitations, the simulations proved to be suitable to provide a comprehensive picture of seismic wave propagation in the lake zone of Mexico City, including the onset of long-duration quasi-monochromatic ground motion with strong amplification between 0.5 and 0.6 Hz. The numerical results also suggest that higher-mode surface waves dominate the wavefield in the lake zone of Mexico City, as evident from the measured phase velocities and vertical displacements along vertical arrays. Based on these positive outcomes, we conclude that this numerical model may be used for the simulation of ground motions during larger magnitude earthquakes, for example in view of generation of seismic damage scenarios in Mexico City. |
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89/2021 - 12/16/2021
Boulakia, M.; Grandmont, C.; Lespagnol, F.; Zunino, P.
Reduced models for the Poisson problem in perforated domains | Abstract | | We develop a fictitious domain method to approximate a Dirichlet problem on a domain with small circular holes (simply called a perforated domain). To address the case of many small inclusions or exclusions, we propose a reduced model based on the projection of the homogeneous Dirichlet boundary constraint on a finite dimensional approximation space. We analyze the existence of the solution of this reduced problem and prove its convergence towards the limit problem without holes. We next obtain an estimate of the gap between the solution of the reduced model and the solution of the full initial model with small holes, the convergence rate depending on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes. The approximation properties of the finite element method are analyzed by a-priori estimates and confirmed by numerical experiments. elliptic differential equations, small inclusions, asymptotic analysis, approximated numerical method |
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88/2021 - 12/16/2021
Kuchta, M.; Laurino, F.; Mardal, K.A.; Zunino, P.
Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers | Abstract | | Coupled partial differential equations defined on domains with different dimensionality are usually called mixed dimensional PDEs. We address mixed dimensional PDEs on three-dimensional (3D) and one-dimensional domains, giving rise to a 3D-1D coupled problem.
Such problem poses several challenges from the standpoint of existence of solutions and numerical approximation. For the coupling conditions across dimensions, we consider the combination of essential and natural conditions, basically the combination of Dirichlet and Neumann conditions. To ensure a meaningful formulation of such conditions, we use the Lagrange multiplier method, suitably adapted to the mixed dimensional case. The well posedness of the resulting saddle point problem is analyzed. Then, we address the numerical approximation of the problem in the framework of the finite element method. The discretization of the Lagrange multiplier space is the main challenge. Several options are proposed, analyzed and compared, with the purpose to determine a good balance between the mathematical properties of the discrete problem and flexibility of implementation of the numerical scheme. The results are supported by evidence based on numerical experiments. |
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87/2021 - 12/16/2021
Both, J.W.; Barnafi, N.A.; Radu, F.A.; Zunino, P.; Quarteroni, A.
Iterative splitting schemes for a soft material poromechanics model | Abstract | | We address numerical solvers for a poromechanics model particularly adapted for soft materials, as it generally respects thermodynamics principles and energy balance. Considering the multi-physics nature of the problem, which involves solid and fluid species, interacting on the basis of mass balance and momentum conservation, we decide to adopt a solution strategy of the discrete problem based on iterative splitting schemes. As the model is similar (but not equivalent to) the Biot poromechanics problem, we follow the abundant literature for solvers of the latter equations, developing two approaches that resemble the well known undrained and fixed-stress splits for the Biot model. A thorough convergence analysis of the proposed schemes is performed. In particular, the undrained-like split is developed and analyzed in the framework of generalized gradient flows, whereas the fixed-stress-like split is understood as block-diagonal $L^2$-type stabilization and analyzed by means of a relative stability analysis. In addition, the application of Anderson acceleration is suggested, improving the robustness of the split schemes. Finally, we test these methods on different benchmark tests, and we also compare their performance with respect to a monolithic approach. Together with the theoretical analysis, the numerical examples provide guidelines to appropriately choose what split scheme shall be used to address realistic applications of the soft material poromechanics model. |
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86/2021 - 12/16/2021
Possenti, L.; Cicchetti, A.; Rosati, R.; Cerroni, D.; Costantino, M.L.; Rancati, T.; Zunino, P.
A Mesoscale Computational Model for Microvascular Oxygen Transfer | Abstract | | We address a mathematical model for oxygen transfer in the microcirculation. The model includes blood flow and hematocrit transport coupled with the interstitial flow, oxygen transport in the blood and the tissue, including capillary-tissue exchange effects. Moreover, the model is suited to handle arbitrarily complex vascular geometries. The purpose of this study is the validation of the model with respect to classical solutions and the further demonstration of its adequacy to describe the heterogeneity of oxygenation in the tissue microenvironment. Finally, we discuss the importance of these effects in the treatment of cancer using radiotherapy. |
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