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
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31/2015 - 26/06/2015
Pini, A.; Vantini, S.; Colombo, D.; Colosimo, B. M.; Previtali, B.
Domain-selective functional ANOVA for process analysis via signal data: the remote monitoring in laser welding | Abstract | | In many application domains, process monitoring and process optimization have to deal with functional responses, also known as profile data.
In these scenarios, a relevant industrial problem consists in discovering which specific parts of the functional response is mostly affected by the process changes. As a matter of fact, knowledge of the specific locations where the curve is more sensitive to process changes can bring several advantages. It can be exploited to design specific monitoring devices directly focusing on the functional data pertaining to the selected intervals. Secondly, the dimensional reduction can eventually bring to an increase of the power to detect process changes.
This paper proposes a methodology to inferentially select the parts of the output functions that are more informative in terms of the underlying factors. The procedure is based on a non-parametric domain-selective ANOVA for functional data, which results in the selection of the intervals of the domain presenting statistically significant effects of each factor. To illustrate its potential in industrial applications, the proposed procedure is applied to a case study on remote laser welding, where the main aim is monitoring the gap between the welded plates through the observation of the emission spectra of the welded material. |
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30/2015 - 13/06/2015
Pini, A.; Vantini, S.
Interval-wise testing for functional data | Abstract | | We propose an inferential procedure for functional data, able to select the intervals of the domain imputable of rejecting a functional null hypothesis.
The procedure is based on three different steps: (i) a functional test is performed on any interval of the domain; (ii) an unadjusted and an adjusted p-value function are defined from the results of the previous tests; (iii) the significant intervals of the domain are selected by thresholding the unadjusted or the adjusted p-value functions, depending on the desired type of control of the family-wise error rate (i.e., point-wise or interval-wise, respectively). In detail, we prove that the newly defined unadjusted p-value function provides a control of the point-wise error rate (i.e., given any point of the domain where the null hypothesis is not violated - in an L2 sense to be suitably defined - the probability of wrongly selecting it as significant is controlled) and that it is point-wise consistent (i.e., given any point of the domain where the null hypothesis is violated - in an L2 sense to be suitably defined - the probability of selecting it as significant goes to one as the sample size goes to infinity). Similarly, we prove that the newly defined adjusted p-value function provides instead a control of the interval-wise error rate (i.e., given any interval of the domain where the null hypothesis is almost-everywhere not violated the probability of wrongly selecting it as significant is controlled) and that it is interval-wise consistent (i.e., given any interval of the domain where the null hypothesis is almost-everywhere violated the probability of selecting it as significant goes to one as the sample size goes to infinity).
The procedure is also applied - together to other two state-of-the-art procedures - to the analysis of of the Canadian daily temperatures, to test for pairwise differences between four climatic regions. In detail, we show how the new procedure hereby proposed is able to give a new deeper and useful insight on the possible rejection of the null hypothesis that consists in the selection of the periods of the years presenting significant differences between each couple of regions. |
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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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