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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29/2015 - 13/06/2015
Antonietti, P.F.; Cangiani, A.; Collis, J.; Dong, Z.; Georgoulis, E.H.; Giani, S.; Houston, P.
Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains | Abstract | | The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp–version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp–DGFEM approximation of both second–order elliptic and first–order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements. |
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28/2015 - 13/06/2015
Taffetani, M.; Ciarletta, P.
Beading instability in soft cylindrical gels with capillary energy: weakly non-linear analysis and numerical simulations | Abstract | | Soft cylindrical gels can develop a long-wavelength peristaltic pattern driven by a competition between surface tension and bulk elastic energy. In contrast to the Rayleigh-Plateau instability for viscous fluids, the macroscopic shape in soft solids evolves toward a stable beading, which strongly differs from the buckling arising in compressed elastic cylinders.
This work proposes a novel theoretical and numerical approach for studying the onset and the non-linear development of the elastocapillary beading in soft cylinders, made of neo-Hookean hyperelastic material with capillary energy at the free surface, subjected to axial stretch. Both a theoretical study, deriving the linear and the weakly non-linear stability analyses for the problem, and numerical simulations, investigating the fully non-linear evolution of the beaded morphology, are performed. The theoretical results prove that an axial elongation can not only favour the onset of beading, but also determine the nature of the elastic bifurcation. The fully non-linear phase diagrams of the beading are also derived from finite element numerical simulations, showing two peculiar morphological transitions when varying either the axial stretch or the material properties of the gel. Since the bifurcation is found to be subcritical for very slender cylinders, an imperfection sensitivity analysis is finally performed. In this case, it is shown that a surface sinusoidal imperfection can resonate with the corresponding marginally stable solution, thus selecting the emerging beading wavelength.
In conclusion, the results of this study provide novel guidelines for controlling the beaded morphology in different experimental conditions, with important applications in micro-fabrication techniques, such as electrospun fibres. |
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27/2015 - 25/05/2015
Marron, J.S.; Ramsay, J.O.; Sangalli, L.M.; Srivastava, A.
Functional Data Analysis of Amplitude and Phase Variation | Abstract | | The abundance of functional observations in scientific endeavors has led to a significant development in tools for functional data analysis (FDA). This kind of data comes with several challenges: infinite dimensionality
of function spaces, observation noise, and so on. However, there is another interesting phenomena that creates problems in FDA. The functional data often comes with lateral displacements/deformations in curves, a phenomenon which is different from the height or amplitude variability and is termed phase variation. The presence of phase variability artificially often inflates data variance, blurs underlying data structures and distorts principal components. While the separation and/or removal of phase from amplitude data is desirable, this is a difficult problem. In particular, a commonly-used alignment procedure, based on minimizing the L2 norm between functions, does not provide satisfactory results. In this paper we motivate the importance of dealing with the phase variability and summarize several current ideas for separating phase and amplitude components. These approaches differ in: (1) the definition and mathematical representation of phase variability, (2) the objective functions that are used in functional data alignment, and (3) the algorithmic tools for solving estimation/optimization problems. We use simple examples to illustrate various approaches and to provide useful contrast between them. |
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26/2015 - 25/05/2015
Tagliabue, A.; Dede', L.; Quarteroni, A.
Nitsche’s Method for Parabolic Partial Differential Equations with Mixed Time Varying Boundary Conditions | Abstract | | We investigate a finite element approximation of an initial boundary value problem associated with parabolic Partial Differential Equations endowed with mixed time varying boundary conditions, switching from essential to natural and viceversa. The switching occurs both in time and in different portions of the boundary. For this problem, we apply and extend the Nitsche’s method presented in [Juntunen and Stenberg,Mathematics of Computation, 2009] to the case of mixed time varying boundary conditions. After proving existence and numerical stability of the full discrete numerical solution obtained by using the ?-method for time discretization, we present and discuss a numerical test that compares our method to a standard approach based on remeshing and projection procedures. |
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25/2015 - 25/05/2015
Del Pra, M.; Fumagalli, A.; Scotti, A.
Well posedness of fully coupled fracture/bulk Darcy flow with XFEM | Abstract | | In this work we consider the coupled problem of Darcy’s flow in a fracture and the surrounding porous medium. The fracture is represented as a (d ? 1)-dimensional interface and it is non-matching with the computational grid thanks to a suitable XFEM enrichment of the mixed finite element spaces. In the existing literature well posedness has been proven for the discrete problem in the hypothesis of given solution in the fracture. This works provides theoretical results on the stability and convergence of the discrete, fully coupled problem, yielding sharp conditions on the fracture geometry and on the computational grid to ensure that the inf-sup conditions is satisfied by the enriched spaces, as confirmed by numerical experiments.
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24/2015 - 15/05/2015
Bonaventura, L:
Local Exponential Methods: a domain decomposition approach to exponential time integration of PDE. | Abstract | | A local approach to the time integration of PDEs by exponential methods is
proposed, motivated by theoretical estimates by A.Iserles on the decay of off-diagonal terms in the exponentials of sparse matrices. An overlapping domain decomposition technique is outlined, that allows to replace the computation of a global exponential matrix by a number of independent and easily parallelizable local problems. Advantages and potential
problems of the proposed technique are discussed. Numerical experiments on simple, yet relevant model problems show that the resulting method allows to increase computational efficiency with respect to standard implementations of exponential methods. |
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23/2015 - 08/05/2015
Paolucci, R.; Mazzieri, I.; Smerzini, C.
Anatomy of strong ground motion: near-source records and 3D physics-based numerical simulations of the Mw 6.0 May 29 2012 Po Plain earthquake, Italy | Abstract | | Stimulated by the recent advances in computational tools for the simulation of seismic wave propagation problems in realistic geologic environments, this paper presents a 3D physics-based numerical study on the prediction of earthquake ground motion in the Po Plain, with reference to the Mw 6.0 May 29 2012 earthquake.
To respond to the validation objectives aimed at reproducing with a reasonable accuracy some of the most peculiar features of the near-source strong motion records and of the damage distribution, this study required a sequence of investigations, starting from the analysis of a nearly unprecedented set of near-source records, to the calibration of an improved kinematic seismic source model, up to the development of a 3D numerical model of the portion of the Po Plain interested by the earthquake, including the irregular buried morphology, with sediment thickness varying from few tens of m to some km. The spatial resolution of the numerical model is suitable to propagate up to about 1.5 Hz. Numerical simulations were performed using the open-source high-performance code SPEED, based on the Discontinuous Galerkin Spectral Elements (DGSE) method.
The 3D numerical model coupled with the updated slip distribution along the rupturing fault proved successful to reproduce with reasonable accuracy, measured through quantitative goodness-of-fit criteria, the most relevant features of the observed ground motion both at the near- and far-field scales. These include: (i) the large fault normal velocity peaks at the near-source stations driven by up-dip directivity effects; (ii) the small-scale variability at short distance from the source, resulting in the out-of-phase motion at stations separated by only 3 km distance; (iii) the propagation of prominent trains of surface waves, especially in the Northern direction, induced by the irregular buried morphology in the near-source area; (iv) the map of earthquake-induced ground uplift with maximum values of about 10 cm, in substantial agreement with satellite measurements; and (v) the two-lobed pattern of the peak ground velocity map, well correlated with the distribution of macroseismic intensity. |
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22/2015 - 05/05/2015
Bonaventura, L.; Ferretti, R.
Flux form Semi-Lagrangian methods for parabolic problems | Abstract | | A semi-Lagrangian method for parabolic problems is proposed,
that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations.
A basic consistency and convergence analysis are proposed. Numerical examples
validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection--diffusion and nonlinear parabolic problems. |
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