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 1347 products
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23/2014 - 06/13/2014
Ieva, F., Paganoni, A.M., Tarabelloni, N.
Covariance Based Unsupervised Classification in Functional Data Analysis | Abstract | | In this paper we propose a new algorithm to perform unsupervised classification of multivariate and functional data when the difference between the two populations lies in their covariances, rather than in their means. The algorithm relies on a proper quantification of distance between the estimated covariance operators of the populations, and identifies as clusters those groups maximising the distance between their induced covariances. The naive implementation of such an algorithm is computationally forbidding, so we propose an heuristic formulation with a much lighter complexity and we study its convergence properties, along with its computational cost. We also propose to use an enhanced estimator for the estimation of discrete covariances of functional data, namely a linear shrinkage estimator, in order to improve the precision of the classification. We establish the effectiveness of our algorithm through applications to both synthetic data and a real dataset coming from a biomedical context, showing also how the use of shrinkage estimation may lead to substantially better results. |
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22/2014 - 06/10/2014
Arioli, G.
Insegnare Matematica con Mathematica | Abstract | | n questo articolo vogliamo discutere le possibilità offerte da soft- ware di calcolo simbolico, in particolare parliamo di Mathematica, nell’insegnamento della Matematica nelle scuole primarie e secondarie. Dopo un breve excursus sulla ricerca recente in didattica della matematica e sull’importanza del problem solving nell’insegnamento della matematica, illustriamo come Mathematica possa essere un ottimo strumento didattico, e introduciamo tre applet create con Mathematica per illustrare le possibilità di questo strumento. |
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21/2014 - 06/09/2014
Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S
The benefits of anisotropic mesh adaptation for brittle fractures under plane-strain conditions | Abstract | | We develop a reliable a posteriori anisotropic first order estimator for the numerical simulation of the Francfort and Marigo model of brittle fracture, after its approximation by means of the Ambrosio-Tortorelli variational model. We show that an adaptive algorithm based on this estimator reproduces all the previously obtained well-known benchmarks on fracture development with particular attention to the fracture directionality. Additionally, we explain why our method, based on an extremely careful tuning of the anisotropic adaptation, has the potential of outperforming significantly in terms of numerical complexity the ones used to achieve similar degrees of accuracy in previous studies. |
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20/2014 - 05/26/2014
Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S.
Anisotropic mesh adaptation for crack detection in brittle materials | Abstract | | The quasistatic brittle fracture model proposed by G. Francfort and J.-J. Marigo can be Gamma-approximated at each time evolution step by the Ambrosio-Tortorelli functional. In this paper, we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field. Secondly, we build on this variational model an adapted discretization to numerically compute the time-evolving minimizing solution. We present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure. The main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack, but only far away from the crack tip, while they show a highly isotropic behavior in a neighborhood of the crack tip instead. As a consequence of these properties, the resulting discretizations follow very closely the propagation of the fracture, which is not significantly influenced by the discretization itself, delivering a physically sound prediction of the crack path, with a reasonable computational effort. In fact, we provide numerical tests which assess the balance between accuracy and complexity of the algorithm. We compare our results with isotropic mesh adaptation and we highlight the remarkable improvements both in terms of accuracy and computational cost with respect to simulations in the pertinent most recent literature. |
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19/2014 - 05/22/2014
Bonaventura, L.; Ferretti, R.
Semi-Lagrangian methods for parabolic problems in divergence form | Abstract | | Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection-diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches. |
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18/2014 - 05/21/2014
Tumolo, G.; Bonaventura, L.
An accurate and efficient numerical framework for adaptive numerical weather prediction | Abstract | | We present an accurate and efficient discretization approach for the adaptive discretization of typical model equations employed in numerical weather prediction. A semi-Lagrangian approach is combined with the TR-BDF2 semi-implicit time discretization method and with a spatial discretization based on adaptive discontinuous finite elements. The resulting method has full second order accuracy in time and can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element, in order to balance accuracy and computational cost. The p-adaptivity approach employed does not require remeshing, therefore it is especially suitable for applications, such as numerical weather prediction, in which a large number of physical quantities are associated with a given mesh. Furthermore, although the proposed method can be implemented on arbitrary unstructured and nonconforming meshes, even its application on simple Cartesian meshes in spherical coordinates can cure effectively the pole problem by reducing the polynomial degree used in the polar elements. Numerical simulations of classical benchmarks for the shallow water and for the fully compressible Euler equations validate the method and demonstrate its capability to achieve accurate results also at large Courant numbers, with time steps up to 100 times larger than those of typical explicit discretizations of the same problems, while reducing the computational cost thanks to the adaptivity algorithm. |
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17/2014 - 05/06/2014
Discacciati, M.; Gervasio, P.; Quarteroni, A.
Interface Control Domain Decomposition (ICDD) Method for Stokes-Darcy coupling | Abstract | | We propose the ICDD method to solve the coupling between Stokes and Darcy equations. According to this approach, the problem is formulated as an optimal control problem whose control variables are the traces of the velocity and the pressure on the internal boundaries of the subdomains that provide an overlapping decomposition of the original computational domain. A theoretical analysis is carried out, an efficient solution algorithm is proposed, and several numerical tests are implemented. Our results show the accuracy of the ICDD methos, its computational efficiency and robustness with respect to the different parameters involved (grid-size, polynomial degrees, permeability of the porous domain, thickness of the overlapping region). The ICDD approach is more versatile and easier to implement than the model based on the Beavers, Joseph and Saffman coupling conditions. |
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16/2014 - 05/05/2014
Dede, L.; Jaggli, C.; Quarteroni, A.
Isogeometric numerical dispersion analysis for elastic wave propagation | Abstract | | In this paper, we carry out a numerical dispersion analysis for the linear elastodynamics equations approximated by means of NURBS-based Isogeometric Analysis in the framwork of the Galerkin method; specifically, we consider the analysis of harminic plane waves in an isotropic and homogeneous elastic medium. We compare and discuss the errors associated to the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considerin B-splines and NURBS basis functions of different regularity, namely globally C^0- and C^(p-1)- continuous, being p the polynomial degree. We conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface. |
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