RISM (Riemann International School of Mathematics)
organizes the meeting
September 25 - 30, 2011
Verbania (Italy) Hotel Il Chiostro

Franco Brezzi (Univ. di Pavia)
Generalised cochaines and higher order MFD
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.

Luis Caffarelli (Univ. of Austin, Texas)
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.

Charlie Elliott (Univ. of Warwick)
Applications of the transport formula to surface partial differential equations
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.

Charbel Farhat (Stanford University)
A high-fidelity computational framework for the analysis of failure driven by multi-phase fluid-structure interaction
Abstract: pdf format (25 KB)

Luca Formaggia (Politecnico di Milano)
Coupling models for haemodynamic simulations
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.

Jean-Frédéric Gerbeau (INRIA, Rocquencourt)
Direct and inverse modeling in hemodynamics
Abstract: pdf format (100 KB)

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

Pierre-Louis Lions (Collège de France)
On Mean Field Games
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.

Mohamed Masmoudi (Univ. P. Sabatier, Toulouse)
The topological gradient method: from optimal design to image processing
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.

Fabio Nobile (Politecnico di Milano)
Partitioned algorithms for fluid-structure interaction problems in hemodynamics
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.

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.

Alfio Quarteroni (EPFL, Lausanne and Politecnico di Milano)
Opening Lecture:
I modelli matematici nella medicina, nell'ambiente, nella tecnologia e nello sport
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.
Complexity Reduction in Partial Differential Equations
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.

Sandro Salsa (Politecnico di Milano)
Two phase Stefan-type problems: recent results and open question
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.

Giuseppe Savaré (Univ. di Pavia)
From gradient flows to rate-independent problems: a variational approach
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)

J.A. Sethian (Dept. of Mathematics University of California, Berkeley)
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.

Juergen Sprekels (WIAS, Berlin)
Mathematical challenges in the industrial growth of semiconductor bulk single crystals
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.

Endre Süli (Univ. of Oxford)
Existence and equilibration of global weak solutions to Navier-Stokes-Fokker--Planck systems
Abstract: pdf format (39 KB)

Stéphane Zaleski (Univ. Pierre et Marie Curie)
Volume of Fluid and Height Function methods for Direct Numerical Simulations of droplets and bubbles
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.