28 Febbraio, 2013 17:45

Seminario Matematico e Fisico di Milano

THE ATMOSPHERE-ICES-OCEAN SYSTEMS. ITS COMPLEXITY, ITS MODELING

Workshop: Mathematics in a Complex World
on the occasion of 150th year of Politecnico di Milano

ROGER TEMAM, Indiana University Bloomington

Politecnico di Milano

27 Febbraio, 2013 12:00

Seminari PhD di Equazioni Differenziali

Semigroups of set-valued maps and the equations of the atmosphere

MICHELE COTI ZELATI, Indiana University, Bloomington

Aula seminari III piano

abstract

abstract

From the point of view of longterm dynamics, we study multivalued and
single-valued semigroups of operators acting on complete metric
spaces. We provide necessary
and sufficient conditions for the existence of the global attractor under minimal
requirements in terms of continuity of the semigroup. As
an application, we consider the multivalued semigroup generated by the
equation ruling
the evolution of the specific humidity in a system of moist air, and we
prove the existence
of a regular global attractor.

26 Febbraio, 2013 15:00

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Elastic energy of a convex body

Antoine Henrot, Institut Elie Cartan

Aula seminari VI piano

abstract

abstract

Following L. Euler, we define the elastic energy E(K) of a regular compact set K in the plane
as 1/2 times the integral over the boundary of K of the square of the boundary curvature. We will denote by $A(K)$ the area of $K$ and
$P(K)$ its perimeter. In this talk, we prove that for any convex set K the quotient
A(K)E(K)/P(K) is larger than or equal to pi/2, with equality only for the disk. We deduce that the disk
minimizes the elastic energy with an area constraint.
We will also consider analogous tridimensional problems involving the
Willmore (or the Helfrich) energy linked to the modelling of vesicles.

26 Febbraio, 2013 10:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

An extension of the Crouzeix-Raviart and Raviart-Thomas spaces to general meshes with application to quasi-incompressible linear elasticity and Stokes flow

Daniele di Pietro, Université Montpellie

Aula Seminari al III PIANO Dipartimento di Matematica ,Politecnico di Milano

abstract

abstract

In this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of
the standard Crouzeix-Raviart space, namely the continuity of mean values at interfaces and the existence of an interpolator which
preserves the mean value of the gradient inside each element. The construction borrows ideas from both Cell Centered Galerkin and Hybrid
Finite Volume methods. More specifically, the discrete function space is defined from cell and face unknowns by introducing a suitable piecewise
affine reconstruction on a pyramidal subdivision of the original mesh.

This subdivision is fictitious in the sense that the original mesh is the only one that needs to be manipulated by the end-user. Two applications are considered in which the discrete space plays an important role, namely (i) the design of a locking-free primal (as opposed to mixed) method for quasi-incompressible linear elasticity on
general polygonal meshes; (ii) the design of an inf-sup stable method for the Stokes equations on general polygonal or polyhedral meshes.
In both cases, the relation between the proposed method and classical finite volume as well as finite element methods on standard meshes is
investigated. Finally, it is shown how similar ideas can be exploited to mimic key properties of the lowest-order Raviart-Thomas space on
general polygonal or polyhedral meshes.

contatto:
luca.formaggia@polimi.it; marco.verani@polimi.it

22 Febbraio, 2013 11:00

Seminario Matematico e Fisico di Milano

SOLUZIONI DI ENERGIA INFINITA DI EQUAZIONI ELLITTICHE CON DATI NON REGOLARI.

LUCIO BOCCARDO, Dipartimento di Matematica Guido Castelnuovo, Università Roma La Sapienza

Dipartimento di Matematica del Politecnico di Milano, Aula Consiglio 7° piano

In this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of
the standard Crouzeix-Raviart space, namely the continuity of mean values at interfaces and the existence of an interpolator which
preserves the mean value of the gradient inside each element. The construction borrows ideas from both Cell Centered Galerkin and Hybrid
Finite Volume methods. More specifically, the discrete function space is defined from cell and face unknowns by introducing a suitable piecewise
affine reconstruction on a pyramidal subdivision of the original mesh.

This subdivision is fictitious in the sense that the original mesh is the only one that needs to be manipulated by the end-user. Two applications are considered in which the discrete space plays an important role, namely (i) the design of a locking-free primal (as opposed to mixed) method for quasi-incompressible linear elasticity on
general polygonal meshes; (ii) the design of an inf-sup stable method for the Stokes equations on general polygonal or polyhedral meshes.
In both cases, the relation between the proposed method and classical finite volume as well as finite element methods on standard meshes is
investigated. Finally, it is shown how similar ideas can be exploited to mimic key properties of the lowest-order Raviart-Thomas space on
general polygonal or polyhedral meshes.

contatto:
luca.formaggia@polimi.it; marco.verani@polimi.it

18 Febbraio, 2013 11:15 in punto

Seminario Matematico e Fisico di Milano

Evaluation of Cardiovascular Mechanics: from Imaging-Based Numerical Assessment to Clinical Applications

Jan O. Mangual, Post Doctoral Whitaker Scholar Università di Firenze

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

It is well established that ventricular function is of utmost importance in maintaining hemodynamic stability and cardiac performance in its entirety. A handful of methods for evaluating flow in the cardiovascular system in vivo
have been studied, yet little information has been obtained on the adequate relation of the flow pattern to proper cardiac performance. To this end, a novel method that combines both non invasive imaging and computational fluid
dynamics has been developed. This integrated approach has been designed and developed with the scope of obtaining more accurate visualization and characterization of the human ventricles than can be obtained from traditional imaging alone.
Imaging-based numerical simulations offer, not only qualitative information, but a quantitative method for characterizing cardiac fluid and tissue mechanics of the ventricles. To name a few, the fluid residence time distribution,
pressure gradient and principal strain, can be attained with the hopes of proving their relevance and importance in analyzing the progression of pathologic conditions and cardiac remodeling. An important aspect of this work
is the lack of cardiac diagnostic markers obtained in normal subjects and the progression of mechanic adjustments leading to heart failure and remodeling. With this in mind a fundamental factor of the work presented herein is in the
evaluation of the mechanical properties in healthy subjects compared with patients that have undergone remodeling as a result of heart failure.

Contatto: davide.ambrosi@polimi.it

15 Febbraio, 2013 14:00

Seminari PhD di Equazioni Differenziali

Stability issues in the theory of compressible fluids

EDUARD FEIREISL, Institute of Mathematics, Academy of Sciences of the Czech Republic, Praha

Aula Seminari III piano

abstract

abstract

We discuss several issues related to stability of the families of solutions to
the Navier-Stokes equations describing the motion of a compressible, viscous
and/or heat conducting fluid. We introduce the concept of relative entropy
and use it to address several problems:
• Weak strong uniqueness principle
• Regularity criteria for weak solutions
• Singular limits of rotating fluids

11 Febbraio, 2013 11:00 in punto

Seminari PhD di Equazioni Differenziali

Stokes and Navier-Stokes equations under slip or leak boundary conditions of friction type

Takahito Kashiwabara, University of Tokyo

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

We consider the incompressible flow with some
nonlinear boundary conditions, called the slip (resp. leak) boundary condition of friction type.
They basically mean that: if the tangential (resp. normal) component of the fluid stress remains under the given threshold, there occurs no tangential (resp. normal) flow i.e. slip (resp. leak); otherwise slip (resp. leak) is allowed to take place on the boundary.
In other words, they are nonlinear intermediates between Dirichlet and Neumann boundary conditions. We can also regard them as fluid-dynamical versions of the Coulomb-type friction condition known in elasticity.

This talk presents, from a theoretical viewpoint, numerical and analytical studies for those boundary conditions. First we notice that a weak form of this problem is written by a variational inequality of the second kind. There appears an L1 norm defined on the boundary, which reflects the nonlinearity of the boundary conditions.
Then, in the numerical part, we propose a discretization by FEM for the stationary Stokes case, discussing error analysis and solution algorithm.

The key point is the introduction of numerical-integration formulas for the L1 norm.
This enables us to preserve in a discrete sense some nice properties known in the continuous (i.e. PDE) level.
For example, the continuous variational inequality problem is known to admit an equivalent saddle point formulation, and so does our discrete variational inequality.
Finally, in the analytical part, the nonstationary Navier-Stokes case is addressed, where we prove the existence and uniqueness of a strong solution. The technique is rather standard and classical; we regularize the L1-norm term to reduce the variational inequality into an equation, construct Galerkin approximations, derive a priori estimates of Ladyzhenskaya type, and pass to the limit.

What is not standard is a treatment of the compatibility condition at the initial condition, where we need to perturb the initial velocity.
Another remark is that in the case of leak boundary condition, we cannot obtain a global-in-time solution without assuming some smallness on data, even in the 2D case.

8 Febbraio, 2013 09:30 in punto

Seminari PhD di Equazioni Differenziali

Discretizations of two-phase Darcy flows in heterogeneous porous media with discontinuous capillary pressures

Konstantin Brenner and Roland Masson, University of Nice Sophia Antipolis and INRIA

SALA CONSIGLIO VII PIANO Dipartimento di Matematica Politecnico di Milano

abstract

abstract

The talk deals with two finite volume discretizations of two-phase Darcy flows in heterogeneous porous media. More precisely we assume that the flow takes place in a domain made of two different rocks so that the capillary pressure is discontinuous across the interface between the rocks. As a consequence the solution itself is discontinuous across the interface with nonlinear transmission conditions. The discretization of such problems is an important issue to modelize capillary barriers arising for example in CO2 storage or basin modelling.

The model is first discretized by means of a numerical scheme which reduces to a standard two point flux finite volume scheme on each sub-domain assuming the orthogonality of the mesh. This first approach is shown to lead to a convergent scheme towards a weak solution of the continuous problem.

contatto:luca.formaggia@polimi.it

In order to deal with anisotropic media or more general meshes, the previous approach is then extended using the vertex approximate gradient scheme (VAG). The proposed extension of the VAG scheme is based on a specific choice of the
primary unknowns at the vertices that allows for discontinuous saturations at the interface between different rock types and respect the continuity of the phase pressures. Numerical
tests exhibit the efficiency of both schemes on various 2D and 3D problems.

contatto:
luca.formaggia@polimi.it, edie. miglio@polimi.it

8 Febbraio, 2013 14:00

Seminari PhD di Equazioni Differenziali

Un modello geometrico per la localizzazione di sorgenti acustiche a partire dalle TDOA.

ROBERTO NOTARI, Dip. di Matematica

Aula seminari III piano

abstract

abstract

Viene presentato un modello per la localizzazione di sorgenti sonore a partire dalle differenze dei tempi di ricezione del segnale da un insieme di sensori sincronizzati la cui posizione e' nota. Assumendo di avere a disposizione solo tre sensori, studiamo il modello con tecniche geometriche e di algebra multilineare, arrivando in particolare ad individuare le condizioni per l'unicità della localizzazione e le regioni potenzialmente sensibili agli errori sui dati misurati.