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 1251 prodotti
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41/2017 - 26/07/2017
Beretta, E.; Micheletti, S.; Perotto, S.; Santacesaria, M.
Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT | Abstract | | In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions. |
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39/2017 - 13/07/2017
Ciarletta, P.
Matched asymptotic solution for crease nucleation in soft solids | Abstract | | A soft solid subjected to a finite compression can suddenly develop sharp self-contacting folds at its free surface, also known as creases. This singular instability is of utmost importance in material science, since it can be positively used to fabricate objects with adaptive surface morphology at different length-scales. Creasing is physically different from other instabilities in elastic materials, like buckling or wrinkling. Indeed, it is a scale-free, fully nonlinear phenomenon displaying similar features as phase-transformations, but lacking an energy barrier. Despite recent experimental and numerical advances, the theoretical understanding of crease nucleation remains elusive, yet crucial for driving further progress in engineering applications.
This work solves the quest for a theoretical explanation of crease nucleation. Creasing is proved to occur after a global bifurcation allowing the co-existence of an affine outer deformation and an inner discontinuous solution with localised self-contact at the free surface. The most fundamental insight is the theoretical prediction of the crease nucleation threshold, in excellent agreement with experiments and numerical simulations. A matched asymptotic approximation is also provided within the intermediate region between the two co-existing inner and outer solutions. The near-field incremental problem becomes singular because of the surface self-contact, acting like the point-wise disturbance in the Oseen's correction for the 2D Stokes problem of the flow past a circle. Using Green's functions in the half-space, analytic expressions of the matching solution and the relative range of validity are derived, perfectly fitting the results of numerical simulations without any adjusting parameter. |
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38/2017 - 13/07/2017
Bonaventura, L.; Fernandez Nieto, E.; Garres Diaz, J.; Narbona Reina, G.;
Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization | Abstract | | We propose an extension of the discretization approaches for mul-
tilayer shallow water models, aimed at making them more
exible and
ecient for realistic applications to coastal
ows. A novel discretiza-
tion approach is proposed, in which the number of vertical layers and
their distribution are allowed to change in di erent regions of the com-
putational domain. Furthermore, semi-implicit schemes are employed
for the time discretization, leading to a signi cant eciency improve-
ment for subcritical regimes. We show that, in the typical regimes in
which the application of multilayer shallow water models is justi ed,
the resulting discretization does not introduce any major spurious fea-
ture and allows again to reduce substantially the computational cost
in areas with complex bathymetry. As an example of the potential of
the proposed technique, an application to a sediment transport prob-
lem is presented, showing a remarkable improvement with respect to
standard discretization approaches. |
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37/2017 - 13/07/2017
Formaggia, L.; Vergara, C.; Zonca, S.
Unfitted Extended Finite Elements for composite grids | Abstract | | We consider an Extended Finite Elements method to handle the case
of composite independent grids that lead to un tted meshes. We detail
the corresponding discrete formulation for the Poisson problem with dis-
continuous coecients. We also provide some technical details for the 3D
implementation. Finally, we provide some numerical examples with the aim
of showing the e ectiveness of the proposed formulation. |
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36/2017 - 09/07/2017
Koeppl, T.; Vidotto, E.; Wohlmuth, B.; Zunino, P.
Mathematical modelling, analysis and numerical approximation of second order elliptic problems with inclusions | Abstract | | Many biological and geological systems can be modelled as porous media with small inclusions. Vascularized tissue, roots embedded in soil or fractured rocks are examples of such systems. In these applicatons, tissue, soil or rocks are considered to be porous media, while blood vessels, roots or fractures form small inclusions. To model flow processes in thin inclusions, one-dimensional (1D) models of Darcy- or Poiseuille type have been used, whereas Darcy-equations of higher dimension have been considered for the flow processes within the porous matrix. A coupling between flow in the porous matrix and the inclusions can be achieved by setting suitable source terms for the corresponding models, where the source term of the higher-dimensional model is concentrated on the centre lines of the inclusions.
In this paper, we investigate an alternative coupling scheme. Here, the source term lives on the boundary of the inclusions. By doing so, we lift the dimension by one and thus increase the regularity of the solution. We show that this model can be derived from a full-dimensional model and the occurring modelling errors are estimated. Furthermore, we prove the well-posedness of the variational formulation and discuss the convergence behaviour of standard finite element methods with respect to this model. Our theoretical results are confirmed by numerical tests. Finally, we demonstrate how the new coupling concept can be used to simulate stationary flow through a capillary network embedded in a biological tissue. |
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35/2017 - 09/07/2017
Piercesare Secchi
On the role of statistics in the era of big data: a call for a debate | Abstract | | While discussing the plenary talk of Dunson (2016) at the 48th Scienti c Meet-
ing of the Italian Statistical Society, I formulated a few general questions on the
role of statistics in the era of big data which stimulated an interesting debate.
They are reported here with the aim of engaging a larger audience on an issue
which promises to change radically our discipline and, more generally, science
as we know it. But is it so? |
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34/2017 - 05/07/2017
Agosti, A.
Analysis of a Discontinuous Galerkin Finite Element discretization of a degenerate Cahn-Hilliard equation with a single-well potential | Abstract | | This work concerns the construction and the convergence analysis of a Discontinuous Galerkin Finite
Element approximation of a Cahn-Hilliard type equation with degenerate mobility and single-well singular
potential of Lennard-Jones type. This equation has been introduced in literature as a diffuse interface
model for the evolution of solid tumors. Differently from the Cahn-Hilliard equation analyzed in the
literature, in this model the singularity of the potential does not compensate the degeneracy of the
mobility at zero by constraining the solution to be strictly positive. In previous works a finite element
approximation with continuous elements of the problem has been developed by the author and co-
authors. In the latter case, the positivity of the solution is enforced through a discrete variational
inequality, which is solved only on active nodes of the triangulation where the degenerate operator
can be inverted. Moreover, a lumping approximation of the L2 scalar product is introduced in the
formulation in order to select the solutions with a moving support with finite speed of velocity from the
unphysical solutions with fixed support. As a consequence of this approximation, the order of convergence
of the method is lowered down with respect to the case of the classical Cahn-Hilliard equation with
constant mobility. In the present discretization with discontinuous elements, the concept of active nodes
is delocalized to the concept of active elements of the triangulation and no lumping approximation of the
mass products is needed to select the physical solutions. The well posedness of the discrete formulation
is shown, together with the convergence to the weak solution. Different algorithms to solve the discrete
variational inequality, based on iterative solvers of the associated complementarity system, are derived
and implemented. Simulation results in two space dimensions are reported in order to test the validity
of the proposed algorithms, in which the dynamics of the spinodal decomposition and the evolution
behaviour in the coarsening regime are studied. Similar results to the ones obtained in standard phase
ordering dynamics are found, which highlight nucleation and pattern formation phenomena and the
evolution of single domains to steady state with constant curvature. Since the present formulation does
not depend on the particular form of the potential, but it’s based on the fact that the singularity set
of the potential and the degeneracy set of the mobility do not coincide, it can be applied also to the
degenerate CH equation with smooth potential.
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33/2017 - 03/07/2017
Fumagalli, I.
A free-boundary problem with moving contact points | Abstract | | This paper concerns the theoretical and numerical analysis of a free boundary problem for the Laplace equation, with a curvature condition on the free boundary. This boundary is described as the graph of a function, and contact angles are imposed at the moving contact points. The equations are set in the framework of classical Sobolev Banach spaces, and existence and uniqueness of the solution are proved via a fixed-point iteration, exploiting a suitably defined lifting operator from the free boundary. The free-boundary function and the bulk solution are approximated by piecewise linear finite elements, and the well-posedness and convergence of the discrete problem are proved. This proof hinges upon a stability result for the Riesz projection onto the discrete space, which is separately proven and has an interest per se. |
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