Franco Brezzi (Univ. di Pavia)
Generalised cochaines and higher order MFD
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Abstract: Classical k-cochains describe the value of k-forms at the corresponding simplexes
of a chain. In the Finite Element (FE) terminology this corresponds to nodal values
(for k=0), line integrals on edges (k=1), fluxes through faces (k=n-1), or element
averages (k=n), where n is the dimension and we assume to be given a decmposition
of the computational domain.
Typically, they correspond to lowest order accuracy, ending up in first
order approximation schemes in the context of Mimetic Finite Differences (MFD).
In order to produce higher order schemes (as in classical FE schemes) one has
to introduce more general approximations, mixing different types of degrees
of freedom.
The talk will first recall classical MFD schemes and classical cochains, and
then discuss the genaralization on several examples coming from PDE problems.
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Luis Caffarelli (Univ. of Austin, Texas)
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Abstract: In my lectures I was planning to make a review of recent developments and
open problems involving non local diffusions: divergence and non divergence,
energy and viscosity methods, smooth approximations, boundary data, geometric problems.
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Charlie Elliott (Univ. of Warwick)
Applications of the transport formula to surface partial differential equations
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Abstract: In this talk we show how the transport formula for moving hypersurfaces may be used to
derive conservation laws on moving surfaces, derive gradient flows for surfaces and derive and analyse
a finite element method for advection diffusion equations on moving surfaces.
We will motivate with some physical examples.
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Charbel Farhat (Stanford University)
A high-fidelity computational framework for the analysis of failure driven by multi-phase fluid-structure interaction
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Abstract:  (25 KB)
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Luca Formaggia (Politecnico di Milano)
Coupling models for haemodynamic simulations
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Abstract: In this presentation we will give an overview of techniques for the
numerical simulation of haemodynamics in arteries based on the
coupling of heterogenous PDE models. In particular, the coupling of
three dimensional fluid structure interaction models with simplified
models based on a system of hyperbolic equations.
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Jean-Frédéric Gerbeau (INRIA, Rocquencourt)
Direct and inverse modeling in hemodynamics
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Abstract: (100 KB)
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Claude LeBris (ENPC, Marne la Valleé)
Mechanics for the 21st century: an interdisciplinary science
Multiscale problems involving randomness: Basics, and Some recent
theoretical and numerical developments
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Pierre-Louis Lions (Collège de France)
On Mean Field Games
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Abstract: This talk will be a general presentation of Mean Field Games (MFG in short),
a new class of mathematical models and problems introduced and studied in
collaboration with Jean-Michel Lasry. Roughly speaking, MFG are mathematical
models that aim to describe the behavior of a very large number of "agents"
who optimize their decisions while taking into account and interacting with
the other agents. The derivation of MFG, which can be justified rigorously
from Nash equilibria for N players games, letting N go to infinity, leads to
new nonlinear systems involving ordinary differential equations or partial
differential equations. Many classical systems are particular cases of MFG
like, for example, compressible Euler equations, Hartree equations, porous
media equations, semilinear elliptic equations, Hamilton-Jacobi-Bellman
equations, Vlasov-Boltzmann models... In this talk we shall explain in a
very simple example how MFG models are derived and present some overview of
the theory, its connections with many other fields and its applications.
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Mohamed Masmoudi (Univ. P. Sabatier, Toulouse)
The topological gradient method: from optimal
design to image processing
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Abstract: To find an optimal domain is equivalent to identify its characteristic function.
At first sight this problem is not differentiable. The two classical ways to make it differentiable are:
- The relaxation technique, which allows the “characteristic function” to take all possible values in
the interval [0,1];
- The level set approach where the characteristic function is replaced by a regular level set function
which is positive inside the unknown domain and negative in its complementary.
As a third method, we propose the topological gradient. It gives the variation of a cost function when
we switch the characteristic function from one to zero or from zero to one in a small area.
We focus our interest on the application of the topological gradient approach to edge detection, which
is the basis of our image processing algorithms. We will consider the classical thermal diffusion
technique and improve it by modeling the edges by highly insolating cracks.
We solve major image processing problems in O(n log(n)) operations.
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Fabio Nobile (Politecnico di Milano)
Partitioned algorithms for fluid-structure interaction problems in hemodynamics
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Abstract: Simulating the interaction between blood and arterial wall in
large arteries is a challenging problem due, on the one hand, to the
complex dynamics of the arterial tissue and, on the other hand, to the
substantial energy exchange between the fluid and the structure in
each cardiac beat.
We will consider partitioned procedures that solve at each time step
the two subproblems iteratively.
We review the concept of added mass of the fluid on the structure, and
its crucial role in the performance of partitioned procedures. Simple
partitioned procedure are totally ineffective (if not divergent) in
presence of a large added mass as it is the case in hemodynamic applications.
The added mass effect can be significantly reduced by a proper choice
of the quantities exchanged between the fluid and the structure at
each iteration. We present a class of algorithms based on Robin
transmission conditions, i.e. linear combinations of stresses and
velocities, and show how they can be tuned to reduce the added mass
effect.
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Ricardo H. Nochetto (Univ. of Maryland, College Park)
Geometric Flows I
Abstract: Electrowetting Electrowetting on dielectric (EWOD)
refers to a parallel-plate micro-device that moves fluid droplets
through electrically actuated surface tension effects. These devices
have found applications in biomedical lab-on-a-chip systems, autofocus
cell phone lenses, colored oil pixels, controlled micro-fluidic
transport, and others. We present two models that account for several
physical effects, such as contact line pinning, along with analysis and
computation. We first model the fluid dynamics using Hele-Shaw type
equations (in 2d) with contact line pinning being described as a
variational inequality on the liquid-gas interface. We analyze this
approach, present simulations, and compare them to experimental videos
of EWOD driven droplets exhibiting pinching and merging events. This is
joint work with S. Walker, A. Bonito, and B. Shapiro. We next discuss a
diffuse interface 3d model coupling electrostatics with incompressible
two-phase flows with different densities and moving contact lines. The
latter are governed by a generalized Navier boundary condition. We
propose a fully discrete FEM with an energy law that mimics the
continuous problem. Numerical simulations show realistic motion of drops
including pinching. This is joint work with S. Walker and A. Salgado.
Geometric Flows II
Abstract: Biomembranes and Fluids We present two models for
biomembranes. The first one is purely geometric since the equilibrium
shapes are the minimizers of the Willmore (or bending) energy under area
and volume constraints. We present a novel method based on ideas from
shape differential calculus. The second model incorporates the effect of
the inside (bulk) viscous incompressible fluid and leads to more
physical dynamics. We use a parametric approach, which gives rise to
fourth order highly nonlinear PDEs on surfaces and involves large domain
deformations. We discretize these PDEs in space with an adaptive finite
element method (AFEM), with either piecewise linear or quadratic
polynomials, and a semi-implicit time stepping scheme. We employ the
Taylor-Hood element for the Navier-Stokes equations together with
iso-parametric elements, the latter being crucial for the correct
approximation of curvature, within a low order ALE framework. We discuss
several computational tools such as space-time adaptivity and mesh
smoothing. This work is joint with A. Bonito and M.S. Pauletti. We also
present time-discrete ALE formulations of any order for
advection-diffusion equations based on dG discretization. The
variational structure of dG is crucial for the validity of Reynolds'
transport theorem, which leads to stability and optimal a priori error
estimates including Runge-Kutta-Radau methods of any order. This is
joint work with A. Bonito and I. Kyza.
Geometric Flows III
Abstract: Biomembranes and Director Fields We introduce a
nonlinear model for the evolution of biomembranes driven by the
L2-gradient flow of a novel elasticity functional describing the
interaction of a director field on a membrane with its curvature. Such
an interaction gives rise to a spontaneous curvature. In the linearized
setting of a graph we present a practical finite element method (FEM),
and prove stability and convergence. The FEM deals with the length
constraint of director fields while decreasing energy, an idea
originally proposed by F. Alouges for finite differences. We extend this
approach to the nonlinear model on closed surfaces and introduce a
parametric FEM. We present numerical experiments, for both linear and
nonlinear models, which agree well with the expected behavior in simple
situations and show that defects on the director field may have a
dramatic effect on membrane shape. This is joint work with S. Bartels
and G. Dolzmann. We also discuss a method to execute refinement,
coarsening, and smoothing of meshes on manifolds with incomplete
information about their geometry and yet preserve position and curvature
accuracy. This is a new paradigm in adaptivity and is important for
parametric FEM. We show an application to shape optimization. This is
joint work with A. Bonito and M.S. Pauletti.
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Alfio Quarteroni (EPFL, Lausanne and Politecnico di Milano)
Opening Lecture:
I modelli matematici nella medicina, nell'ambiente, nella tecnologia e nello sport
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Abstract: La matematica oggi permea ogni ambito del sapere. Usiamo, spesso inconsapevolmente, algoritmi
matematici quando inviamo immagini dai nostri telefoni cellulari, o quando i motori di ricerca ci dispensano
risposte a qualsivoglia tipo di richiesta, pescando fra decine di miliardi di pagine del web in tempi
infinitesimali.
I modelli matematici sono rappresentazioni sintetiche ma funzionali di un sistema complesso
(fisico, biologico, economico o sociale): preservandone le caratteristiche fondamentali, essi consentono
la simulazione di processi di interesse reale che si incontrano nelle scienze, nell’ingegneria, nella
medicina e nell’economia.
I modelli che governano la fisica dell’atmosfera vengono usati quotidianamente per formulare previsioni
meteorologiche su scala continentale, regionale o locale. Modelli vengono usati per l’analisi di rischio
sismico, la valutazione d’impatto di inondazioni o esondazioni, la simulazione di processi di inquinamento
atmosferico o idrico.
In ambito industriale, grazie ai modelli si possono descrivere le diverse fasi di un processo produttivo,
dal design al controllo e all'ottimizzazione, consentendo l'analisi rapida di diversi scenari, riducendo in
modo significativo gli investimenti in tempo e denaro.
In questa presentazione, di carattere divulgativo, verrà formulato un approccio generale alla modellistica
matematica e verranno presentati svariati esempi di applicazione alla medicina, all'ambiente, all'innovazione
tecnologica e allo sport da competizione.
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Talk:
Complexity Reduction in Partial Differential Equations
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Abstract: Mathematical models of complex physical problems can be based on heterogeneous partial differential
equations (PDEs), i.e. on boundary-value problems of different kind in different subregions of the computational
domain. This is for instance the case of problems in multiphysics.
In different circumstances, especially in control and optimization problems for parametrized PDEs, reduced
order models such as the reduced basis method can be used to alleviate the computational complexity.
After introducing some illustrative examples, in this presentation a variety of numerical approaches will be
proposed, then a few representative applications to blood flow modeling, sports design, and technology
innovation will be addressed.
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Sandro Salsa (Politecnico di Milano)
Two phase Stefan-type problems: recent results and open question
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Abstract: We consider a class of free boundary problems which includes the classical two phase Stefan
problem and their solutions in the viscosity sense. We describe recent results on existence,
uniqueness, regularity and some questions and generalizations that remains unsolved.
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Giuseppe Savaré (Univ. di Pavia)
From gradient flows to rate-independent problems: a variational approach
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Abstract: Energetic solutions of rate-independent evolution problems can be obtained by solving a recursive minimization scheme which involves a functional governing the evolution perturbed by a suitable dissipation term. Rate-independence is guaranteed by the 1-homogeneity of the dissipation, which therefore has a linear growth and provides just a BV estimate in time for the approximating solutions.
Thus jumps can typically occur during the evolution, at least when the functional is not convex. These jumps can be characterized by a global stability condition which reflects the global character of each minimization step of the approximation scheme.
In order to obtain a localized condition (which should also be easier to solve numerically), various viscosity-type corrections have been proposed. The simplest ones consist to add a (asymptotically small) quadratic perturbation to the dissipation term.
In this talk we discuss some characterizations of the limit BV solutions obtained by this kind of viscosity approximation schemes, their link with the variational theory of Gradient Flows and with the original energetic formulation.
(In collaboration with A. Mielke and R. Rossi)
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J.A. Sethian (Dept. of Mathematics University of California, Berkeley)
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Abstract: Many scientific and engineering problems involve interconnected moving interfaces separating
different regions, including dry foams, crystal grain growth and multi-cellular structures in man-made
and biological materials. materials. Producing consistent and well-posed mathematical models that capture
the motion of these interfaces, especially at degeneracies, such as triple points and triple lines where
multiple interfaces meet, is challenging. We introduce an efficient and robust mathematical and
computational methodology for computing the solution to two and three-dimensional multi-interface
problems involving complex junctions and topological changes in an evolving general multiphase system.
We demonstrate the method on a collection of problems, including geometric coarsening flows under
curvature and incompressible flow coupled to multi-fluid interface problems. Finally, we compute the
dynamics of unstable foams, such as soap bubbles, evolving under the combined effects of gas-fluid
interactions, thin-film lamella drainage, and topological bursting.
This work is joint with R. Saye of UC Berkeley.
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Juergen Sprekels (WIAS, Berlin)
Mathematical challenges in the industrial growth of semiconductor bulk
single crystals
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Abstract: Semiconductor bulk single crystals form the material basis for modern
technologies. Their prooduction is therefore of utmost technological and
economical importance. The modeling of the production processes leads to
challenging mathematical problems: free boundaries, strongly coupled
nonlinear PDE systems, jumping material coefficients and boundary
conditions, radiative heat transfer, multiple scales in time and space,
and nonsmooth geometries, to name only a few. In this talk, we discuss
some "real-life" production processes and the corresponding models, and
we report on how the inherent mathematical difficulties have been dealt
with at WIAS Berlin in industrial cooperations.
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Endre Süli (Univ. of Oxford)
Existence and equilibration of global weak solutions to Navier-Stokes-Fokker--Planck systems
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Abstract: (39 KB)
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Stéphane Zaleski (Univ. Pierre et Marie Curie)
Volume of Fluid and Height Function methods for Direct Numerical Simulations of droplets and bubbles
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Abstract: We describe issues arising in Direct Numerical Simulations of droplets and bubbles through the example
of computations performed using the Gerris code. Gerris is an open-source software implementing finite
volume solvers on an octree adaptive grid together with a piecewise linear volume of fluid interface
tracking method.
Height Function methods are used for the computation of interface curvature and surface tension.
The parallelisation of Gerris is achieved by domain decomposition. We show examples of the capabilities
of Gerris on several types of problems. The impact of a droplet on a layer of the same liquid results in
the formation of a thin air layer trapped between the droplet and the liquid layer that the adaptive
refinement allows to capture. It is followed by the jetting of a thin corolla emerging from below the
impacting droplet. The jet atomization problem is another extremely challenging computational problem,
in which a large number of small scales are generated.
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