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 1239 products
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36/2017 - 07/09/2017
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 - 07/09/2017
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 - 07/05/2017
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 - 07/03/2017
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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32/2017 - 06/29/2017
Riccobelli, D.; Ciarletta, P.
Shape transitions in a soft incompressible sphere with residual stresses | Abstract | | Residual stresses may appear in elastic bodies due to the formation of misfits in the micro-structure, driven by plastic deformations, thermal or growth processes. They are especially widespread in living matter, resulting from the dynamic remodelling processes aiming at optimizing the overall structural response to environmental physical forces.
From a mechanical viewpoint, residual stresses are classically modelled through the introduction of a virtual incompatible configuration that maps the natural state of the body. In this work, we instead employ an alternative approach based on a strain energy function that constitutively depends on both the deformation gradient and the residual stress tensor. In particular, our objective is to study the morphological stability of an incompressible sphere, made of a neo-Hookean material and subjected to given distributions of residual stresses.
The boundary value elastic problem is studied with analytic and numerical tools. Firstly, we perform a linear stability analysis on the pre-stressed sphere using the method of incremental deformations. The marginal stability conditions are given as a function of a control parameter, being the dimensionless variable that represents the characteristic intensity of the residual stresses. Secondly, we perform finite element simulations using a mixed formulation in order to investigate the post-buckling morphology in the fully nonlinear regime.
Considering different initial distributions of the residual stresses, we find that different morphological transitions are all localized around the material domain where the hoop residual stress reaches its maximum compressive value. The loss of spherical symmetry is found to be controlled by the mechanical and geometrical properties of the sphere, as well as on the spatial distribution of the residual stress.
The results provide useful guidelines in order to design morphable soft spheres, for example by controlling the residual stresses through active deformations. They finally open a pathway for the non-disruptive characterization of residual stresses in soft tissues, such as solid tumours. |
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31/2017 - 06/27/2017
Schiltz, F.; Masci, C.; Agasisti, T.; Horn, D.
Using Machine Learning to Model Interaction Effects in Education: a Graphical Approach. | Abstract | | Educational systems can be characterized by a complex structure: students, classes and teachers, schools and principals, and providers of education. The added value of schools is likely influenced by all these levels and, especially, by interactions between them. We illustrate the ability of Machine Learning (ML) methods (Regression Trees, Random Forests and Boosting) to model this complex ‘education production function’ using Hungarian data. We find that, in contrast to ML methods, classical regression approaches fail to identify relevant nonlinear interactions such as the role of school principals to accommodate district size policies. We visualize nonlinear interaction effects in a way that can be easily interpreted. |
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30/2017 - 06/20/2017
Bacchelli, V.; Micheletti, S.; Perotto, S.; Pierotti, D.
Parameter identification for the linear wave equation with Robin boundary condition | Abstract | | We consider an initial-boundary value problem for the classical linear wave
equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both
the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization. |
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29/2017 - 06/20/2017
Antonietti, P.F.; Mascotto, L.; Verani, M.
A multigrid algorithm for the $p$--version of the Virtual Element Method | Abstract | | We present a multigrid algorithm for the solution of the linear systems of equations stemming from the p?version of the Virtual Element discretization of a two-dimensional Poisson problem. The sequence of coarse spaces are constructed decreasing progressively the polynomial approximation degree of the Virtual Element space, as in standard p-multigrid schemes. The construction of the interspace operators relies on auxiliary Virtual Element spaces, where it is possible to compute higher order polynomial projectors. We prove that the multigrid scheme is uniformly convergent, provided the number of smoothing steps is chosen sufficiently large. We also demonstrate that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom that can be employed to accelerate the convergence of classical Krylov-based iterative schemes. Numerical experiments validate the theoretical results. |
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