FOURTH WORKSHOP ON THIN STRUCTURES Naples

Transcript

FOURTH WORKSHOP ON THIN STRUCTURES
Naples, September 8-10, 2016
Eremo SS. Salvatore, via dell’Eremo 87 (Camaldoli), 80131 Napoli, Italy
http://www.convegni.unicas.it/WTS2016
SCIENTIFIC COMMITTEE
Antonio Gaudiello
Università degli Studi di Cassino e del Lazio Meridionale, Italy
François Murat
Université Pierre et Marie Curie Paris VI, France
ORGANIZING COMMITTEE
Giuseppe Cardone
Università degli Studi del Sannio, Italy
Umberto De Maio
Università degli Studi di Napoli ”Federico II”, Italy
Luisa Faella
Giuliano Gargiulo
Antonio Gaudiello
François Murat
Carmen Perugia
Elvira Zappale
Università degli Studi di Salerno, Italy
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SPONSORS
Banca Popolare del Cassinate
Consulat Général de France à Naples
Dipartimento di Ingegneria (Università degli Studi del Sannio)
Dipartimento di Ingegneria Industriale (Università degli Studi di Salerno)
Dipartimento di Matematica e Applicazioni ”Renato Caccioppoli” (Università degli Studi di Napoli ”Federico II”)
Dipartimento di Scienze e Tecnologie (Università degli Studi del Sannio)
Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (Istituto Nazionale di Alta Matematica ”F. Severi”)
Laboratoire Jacques-Louis Lions (Université Pierre et Marie Curie Paris VI)
UniCredit
Università degli Studi di Cassino e del Lazio Meridionale
Università degli Studi di Napoli ”Federico II”
Università degli Studi di Salerno
Università degli Studi del Sannio
Università Italo Francese / Université Franco Italienne
OPENING SESSION
Prof. Giovanni Betta (Rettore dell’Università degli Studi di Cassino e
del Lazio Meridionale)
Prof. Vittorio Coti Zelati (Vice Presidente dell’Unione Matematica Italiana)
Dott. Luigi de Magistris (Sindaco del Comune di Napoli)
Prof. Nicola Fusco (Member of the Executive Committee of the European
Mathematical Society)
Prof. Gioconda Moscariello (Direttore del Dipartimento di Matematica e Applicazioni ”R. Caccioppoli” dell’Università degli Studi di Napoli
”Federico II”)
Prof. Carlo Sbordone (Segretario Generale della Società Nazionale di
Scienze, Lettere e Arti in Napoli)
M. Jean-Paul Seytre (Consul Général de France à Naples et Directeur de
l’Institut français de Naples)
Prof. Guido Trombetti (Ex Rettore dell’Università degli Studi di Napoli
”Federico II”)
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SPEAKERS AND TALKS
Emilio Acerbi (Università degli Studi di Parma, Italy)
Relaxation of second order one-dimensional energies
As a preliminary step to the case of surfaces, we study the relaxation of
plastic and elastic energies depending on curvature and defined on regular
curves in high codimension. Through an extension to suitable currents,
the notions of generalized length and generalized curvature (in the plastic
case) or p-curvature (in the elastic one) are given, through which an easily
readable representation formula is provided.
Yves Achdou (Université Paris-Diderot, France)
Hamilton-Jacobi equations on networks. Dimension reduction and homogenization
Hamilton-Jacobi (HJ) equations on heterogeneous structures (comprising networks) have recently received an increased interest.
Recent results on control problems on networks and related first order
HJ equations will be reviewed. Then a dimension reduction problem will
be considered: a sequence of thin domains whose thickness tends to zero,
converges to a network. A state constrained optimal control problem is
set in such domains and the aim is to pass to the limit as the thickness
parameter tends to 0. An effective transmission condition at the crosspoints
of the network is found at the limit. The needed correctors problems, set in
unbounded domains, had not been tackled in the literature devoted to HJ
equations.
If there remains time, we will discuss a rather similar situation, i.e.
regional optimal control problems in Rd divided into two unbounded subdomains, with an interface obtained as the graph of a smooth periodic real
valued function: the dynamics and running costs are discontinuous across
the interface. Homogenization will be performed when the period and the
amplitude of the interface oscillations are small and of the same order. An
effective transmission condition on a flat interface will be found at the limit.
These are joint works with Nicoletta Tchou (IRMAR, Université de
Rennes) and partially with Salomé Oudet when she was a Ph.D. student
at IRMAR.
José M. Arrieta (Universidad Complutense de Madrid, Spain)
Thin domains with a locally periodic highly oscillatory boundary
We consider a two dimensional thin domain where the boundary has a
highly oscillatory behavior but the oscillations are not purely periodic. For
instance, we may consider the case where the thin domain is of the type
R = {(x, y) : 0 < x < 1; 0 < y < G(x, x/)} where the function G(x, ·)
is periodic of period L(x), for some function L(·). Observe that we are
allowing that the period and amplitude of the oscillations varies in space.
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We will analyze the homogenized limit as the thickness of the domain goes
to 0. We are interested in understanding how the varying amplitude and
period appear in the homogenized limit problem.
This is a joint work with Manuel Villanueva-Pesquera (UCM, Madrid).
Gang Bao (Zhejiang University, Hangzhou, P. R. China)
Multiscale scattering and inverse scattering problems
Scattering and inverse scattering problems arise in diverse areas of
industrial and military applications, such as nondestructive testing, seismic
imaging, submarine detections, near-field and nano optical imaging, and
medical imaging. A model problem in wave propagation is concerned with
a plane wave incident on a medium enclosed by a bounded domain. Given
the incident field, the direct problem is to determine the scattered field for
the known scatterer. The inverse problem is to determine the scatterer from
the boundary measurements of near field currents densities. Although this
is a classical problem in mathematical physics, mathematical issues and numerical solution of the inverse problems remain to be challenging since the
problems are nonlinear, large-scale, and most of all ill-posed! The severe
ill-posedness has thus far limited in many ways the scope of inverse problem
methods in practical applications. In this talk, the speaker will report recent progress in mathematical analysis and computational studies of the inverse boundary value problems. A novel stable continuation approach based
on the uncertainty principle will be presented. By using multi-frequency
or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse problems. New stability results and
techniques for the inverse problems will be presented. The speaker will also
discuss some multiscale problems in related modeling and design applications.
Peter Bella (Max Planck Institute for Mathematics in the Sciences, Leipzig,
Germany)
The coarsening of folds in hanging drapes
In this talk I will discuss shape of a hanging drape - a thin elastic sheet,
pulled down by the force of gravity, with fine-scale folding at the top that
achieves approximately uniform confinement. This example of energy-driven
pattern formation in a thin elastic sheet is of particular interest because the
length scale of folding varies with height. I will focus on how the minimum
elastic energy depends on the physical parameters, and will explain that selfsimilar coarsening achieves the optimal scaling law in a certain parameter
regime, and that constructions (involving lateral spreading of the sheet) do
better in other regions of parameter space.
This is based on a joint work with Robert V. Kohn.
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Andrea Braides (Università di Roma ”Tor Vergata”, Italy)
Interfaces in discrete thin films
In the simplest lattice setting a discrete thin film can be thought as a superposition of atomistic interactions between sites placed on a finite number
of layers of planar lattices. We consider ferromagnetic lattice spin systems,
which can be described macroscopically by interfacial energies between two
magnetic phases. For the latter, dimension-reduction problems have been
previously considered by Braides and Fonseca. The simpler discrete setting
allows to follow that approach and to consider additional phenomena, such
as
1) dependence on the number of layers;
2) modeling of quasicrystalline and aperiodic effects;
3) random thin films with possible percolation phenomena.
I will also give examples of antiferromagnetic thin films, where the
number of parameters may vary in dependence of the number of layers
Works in collaboration with R. Alicandro, M. Cicalese, M. Ruf, and
M. Solci.
Gilles Carbou (Université de Pau et des Pays de l’Adour, France)
Walls’ dynamics in thin ferromagnetic nanotubes
Ferromagnetic nanotubes are proposed as an alternative to ferromagnetic
nanowires for data-storage applications. In this talk we construct a twodimensional model of such devices and we establish the stability of moving
walls in the Walker regime when the tube is subject to a small magnetic
field.
Cesare Davini (Università degli Studi di Udine, Italy)
Composite thin-walled beams by Γ-convergence
The behavior of thin-walled beams does not fit the De Saint-Venant’s
theory of beams and a multitude of ad hoc models have been proposed
throughout the years, starting with that of Vlasov. In two joint papers with
R. Paroni and L. Freddi we have considered a beam whose cross-section is
a tubular neighborhood of a simple curve γ for the two instances that the
curve is either open or closed. We assumed that the wall thickness scales
with a parameter δε and the length of γ with ε, with δε /ε → 0. Starting
from fully anisotropic inhomogeneous three dimensional linear elasticity, we
derived the Γ-limit problem for the case in which the ratio between ε2 and
δε remains bounded. The approach provides two asymptotic models that
encompass, in particular, Vlasov’s theory. In a more recent paper we have
also tried to establish a bridge between those mathematical results and their
implementation to a real problem. My purpose here is to illustrate all this.
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Rita Ferreira (King Abdullah University of Sciences and Technology,
Thuwal, Saudi Arabia)
Lower-dimensional models for multi-domains involving bending moments
In this talk, we address a dimension-reduction problem in the context
of nonlinear elasticity where the applied external surface forces induce a
bending moment. The underlying domain is a multi-domain in R3 consisting
of a thin tube-shaped domain placed upon a thin plate-shaped domain. The
problem involves two small parameters, the radius of the cross section of
the tube-shaped domain and the thickness of the plate-shaped domain. The
limiting problem as these two parameters converge to zero is characterized;
in particular, when this limiting problem is coupled, the limiting junction
condition is also characterized.
Filippo Gazzola (Politecnico di Milano, Italy)
On the variation of longitudinal and torsional frequencies in a partially
hinged rectangular plate
We consider a thin and partially hinged rectangular plate and we analyze its normal modes. There are two families of modes, longitudinal and
torsional. We study the variation of the corresponding eigenvalues under domain deformations. We investigate the possibility of finding a shape functional able to quantify the torsional instability of the plate, namely how prone
is the plate to transform longitudinal oscillations into torsional ones. This
functional should obey several rules coming from both theoretical and practical evidences, in particular for plates modeling the deck of a suspension
bridge. We show that a simple functional obeying all the required rules
does not exist and that the functionals available in literature are not reliable.
This is a joint work with Elvise Berchio and Davide Buoso.
Carolin Kreisbeck (Universiteit Utrecht, The Netherlands)
Heterogeneous thin films: combining dimension reduction and homogenization of functionals with differential constraints
Working with variational principles subject to linear PDE constraints
conveyed by a constant-rank operator A allows us to treat a number of
problems in continuum mechanics and electromagnetism in a unified way.
The topic of this talk is the rigorous derivation of lower dimensional, effective
limit models for thin films with periodic heterogeneities in this general framework. We analyze the asymptotic behavior of a multiscale problem given
by a sequence of integral functionals with two characteristic length scales,
namely the film thickness and the period of the oscillating microstructures,
by means of Γ-convergence. On a technical level, this requires a subtile
merging of homogenization tools, such as multiscale convergence methods,
with dimension reduction techniques for functionals on A-free vector fields.
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One observes that the results depend critically on the relative magnitude
between the two scales. Interestingly, this even regards the fundamental
question of locality of the limit model.
Michel Lenczner (Université de Technologie de Belfort-Montbéliard
& FEMTO-ST, Besançon, France)
A framework for computer-aided derivation of families of multiscale models
In the recent years, we have introduced a theoretical framework for
computer-aided derivation of a family of multi-scale models. It relies on
asymptotic methods and term rewriting techniques issued from theoretical
computing science. A multi-scale model derivation is characterized by the
features taken into account in the asymptotic analysis. Its formulation consists in a derivation of a reference proof associated to a reference model, and
in the combination of a set of extensions to be applied to this proof until it
takes into account the wanted features. The overall method will be presented
and illustrated on a thin structure, namely an array of micromirrors.
Giovanni Leoni (Carnegie Mellon University, Pittsburgh, USA)
A model for dislocations in epitaxially strained elastic films
We present a variational model for nucleation of dislocations in
epitaxially strained films. This is joint work with Nicola Fusco, Irene Fonseca, and Massimiliano Morini.
Marta Lewicka (University of Pittsburgh, USA)
Prestrained elasticity: curvature constraints and differential geometry with
low regularity
A prestrained elastic body is modeled by an open three dimensional
domain and an ideal Riemannian metric. This metric, induced by some
mechanism such as growth or plasticity, is the main driving force of the
elastic deformations which determine the shape of the body. Namely, the
body wants to realize the distances between its constitutive cell elements
which are set by the metric, but this realization is taking place in a flat
three dimensional space, and hence is impossible if the curvature tensor of
the metric is nonzero. In the static theory, we seek the closest realization by
an elastic variational principle, seeking the best possible immersion of the
given three dimensional Riemannian manifold into the three dimensional
flat space.
In the context of thin structures, one can rigorously derive the variational
principles governing prestrained films using the machinery of Γ-convergence.
A particular aspect of the resulting residual theories is the emergence of
isometry constraints on elastic immersions of two dimensional Riemannian
manifolds into the 3d space, which lack the usual smoothness properties
of classical differential geometry. Some of these constraints are manifested
as a Monge-Ampere type constraints. We will discuss their derivation and
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show how the Nash-Kuiper convex integration approach, combined with the
new insights gained from the prestrained elasticity, result in the rigidity
and flexibility of the two dimensional Monge-Ampere equation at Hölder
regularity.
Olivier Pantz (Université de Nice Sophia-Antipolis, France)
Vesicles: modeling, justification, in silico experimentations
Vesicles are closed mechanical two dimensional structures made of
phospholipids that self-assemble in an aqueous environment. They are the
basic mechanical structure of every mammal cells, notably red blood cells.
In the seventies, Canham and Helfrich proposed to model those
structures as elastic shells whose energy only depends on the mean and
Gaussian curvatures. This approach has drawn a lot of attention : several
justifications have been proposed as well as numerical methods. The initial
model have also been enriched, what some authors claim to be necessary to
fit some experimental observations.
Interestingly, there is no genuine consensus on what the ”correct”
modeling is.
In this presentation, we propose to model the structure as a three dimensional structure. Letting the thickness of the vesicle go to zero, we derive
several models of vesicles. Finally, we show that the three-dimensional initial modeling can be used to perform numerical simulations. We intend in
a nearby future to compare our in silico experiments to actual observations,
in order to determine the more accurate model.
Olivier Pironneau (Université Pierre et Marie Curie Paris VI, France)
Analysis of a coupled fluid-structure model with applications to hemodynamics
We propose and analyse a simplified fluid-structure coupled model for
flows with compliant walls. The wall reaction to the fluid is modelled by
a small displacement visco-elastic shell where the tangential stress components and displacements are neglected. We show that within this small
displacement approximation a transpiration condition can be used which
does not require an update of the geometry at each time step, for pipe flow
at least. Such simplifications lead to a model which is well posed and for
which a semi-implicit time discretization can be shown to converge. We
present some numerical results and a comparison with a standard test case
taken from hemodynamics. The model is more stable and less computer
demanding than full models with moving mesh. We apply the model to a
3D arterial flow with a stent.
This is joint work with Tomás Chacón (Universidad de Sevilla, Spain
and Laboratoire Jacques-Louis Lions, France), François Murat (Laboratoire
Jacques-Louis Lions, France), and Vivette Girault (Laboratoire JacquesLouis Lions, France).
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Valeriy V. Slastikov (University of Bristol, UK)
Thin ferromagnetic films with periodic roughness
We investigate the behavior of thin ferromagnetic films with periodic
surface roughness and show that in the leading order roughness gives a local
contribution to the micromagnetic energy that might significantly change
the magnetization behavior in the ferromagnet.
Franco Tomarelli (Politecnico di Milano, Italy)
Some variational problems for Föppl-von Kármán plates
Some variational problems for a Föppl-von Kármán plate subject to equilibrated shear load are studied. The existence of equilibrium globally minimizing the energy is shown under the assumption that out-of-plane displacement fulfils homogeneous Dirichlet condition on the whole boundary. When
the Dirichlet condition is prescribed only in a subset of the boundary the energy may be unbounded from below; in such case, for simple
geometry with prescribed thickness, the energy is scaled by taking ε times
the out-of-plane displacement in order to show an asymptotic expansion of
the functional coherent with the appearance of wrinkling patterns.
Igor Velčić (University of Zagreb, Croatia)
Homogenization of thin structures in nonlinear elasticity
We will give the results on the models of thin plates and rods in nonlinear
elasticity by doing simultaneous homogenization and dimensional reduction.
We will derive the models by means of Γ-convergence and will focus on the
models in bending and von Kármán regime. In the case of bending plate
we are able to obtain the models only under periodicity assumption and
assuming some special relation between the periodicity of the material and
thickness of the body. In the von Kármán regime of rods and plates and in
the bending regime of rods we are able to obtain the models in the general
non-periodic setting.
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POSTER SESSION
Eric Bonnetier (Université de Grenoble-Alpes, France)
Pointwise bounds on the gradients in between the inhomogenities of a composite medium and the Neumann-Poincaré operator
Davide Buoso (Politecnico di Torino, Italy)
Eigenvalues of free plates
Khaled Chacouche (Universitá degli Studi di Cassino e del Lazio Meridionale, Italy & Université Paris Est Créteil Val de Marne, France)
Ferromagnetic nanowire of infinite length and ferromagnetic thin film of
infinite diameter
Francesco Ferraresso (Università degli Studi di Padova, Italy)
The dumbbell problem for a free plate
Fatima Goffi (University of Sciences and Technology Houari Boumediène,
Algiers, Algeria)
Scattering of an electromagnetic wave by a perfectly conducting obstacle
coated with two thin layers
Hiromichi Itou (Tokyo University of Science, Japan)
On asymptotic behavior of the displacement field near a tip of thin obstacles
in linearized elasticity
Alexander G. Kolpakov (Siberian State University of Telecommunication
and Informatics, Novosibirsk, Russia)
Complete asymptotic decomposition in the problem of joined elastic plates
Piotr Kozarzewski (University of Warsaw, Poland)
Optimal design for thin structures in generalized Sobolev spaces
Heiner Olbermann (Leipzig University, Germany)
Energy scaling law for a single disclination in a thin elastic sheet
Ravi Prakash (University of Concepción, Chile)
Asymptotic analysis of Neumann periodic optimal control problem
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LIST OF PARTICIPANTS
Emilio Acerbi
Università degli Studi di Parma, Italy
[email protected]
Yves Achdou
Université Paris-Diderot, France
[email protected]
Angela Alberico
CNR., Napoli, Italy
[email protected]
Angelo Alvino
[email protected]
José M. Arrieta
Universidad Complutense de Madrid, Spain
[email protected]
Gang Bao
Zhejiang University, Hangzhou, P.R. China
[email protected]
Annamaria Barbagallo
[email protected]
Raffaele Barretta
[email protected]
Peter Bella
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
[email protected]
Francesca Betta
Università degli Studi di Napoli ”Parthenope”, Italy
[email protected]
Eric Bonnetier
Université de Grenoble-Alpes, France
[email protected]
Pierre Bousquet
Institut de Mathématiques de Toulouse, France
[email protected]
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Andrea Braides
Università di Roma ”Tor Vergata”, Italy
[email protected]
Barbara Brandolini
[email protected]
Lorenzo Brasco
Università degli Studi di Ferrara, Italy
[email protected]
Davide Buoso
Politecnico di Torino, Italy
[email protected]
Luciano Carbone
[email protected]
Gilles Carbou
Université de Pau et des Pays de l’Adour, France
[email protected]
Giuseppe Cardone
Università degli Studi del Sannio, Benevento, Italy
[email protected]
Raffaele Carlone
[email protected]
Khaled Chacouche
Université Paris Est Créteil Val de Marne, Créteil, France & Università degli
Studi di Cassino e del Lazio Meridionale, Italy
[email protected]
Bidhan Chandra Sardar
Indian Institute of Science, Bangalore, India
[email protected]
Francesco Chiacchio
[email protected]
Antonio Corbo Esposito
[email protected]
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Vittorio Coti Zelati
[email protected]
Emma D’Aniello
Seconda Università degli Studi di Napoli, Caserta, Italy
[email protected]
Cesare Davini
Università degli Studi di Udine, Italy
[email protected]
Francesco Della Pietra
[email protected]
Umberto De Maio
[email protected]
Giuseppina Di Blasio
[email protected]
Giovanni Di Fratta
Ecole Polytechnique, Palaiseau, France
[email protected]
Luigi D’Onofrio
[email protected]
Tiziana Durante
Università degli Studi di Salerno, Fisciano, Italy
[email protected]
Luisa Faella
[email protected]
Fernando Farroni
[email protected]
Filomena Feo
[email protected]
Vincenzo Ferone
[email protected]
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Francesco Ferraresso
Università degli Studi di Padova, Italy
[email protected]
Rita Ferreira
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
[email protected]
Alberto Fiorenza
[email protected]
Giuseppe Floridia
[email protected]
Maria Rosaria Formica
[email protected]
Lorenzo Freddi
Università degli Studi di Udine
[email protected]
Nicola Fusco
[email protected]
Giuliano Gargiulo
[email protected]
Antonio Gaudiello
[email protected]
Filippo Gazzola
Politecnico di Milano, Italy
[email protected]
Daniela Giachetti
Sapienza Università di Roma, Italy
[email protected]
Flavia Giannetti
[email protected]
Raffaella Giova
[email protected]
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Fatima Goffi
University of Sciences and Technology Houari Boumediène, Algiers, Algeria
[email protected]
Luigi Greco
[email protected]
Michael Hauck
Fraunhofer ITWM, TU Kaiserslautern, Germany
[email protected]
Hiromichi Itou
Tokyo University of Science, Japan
[email protected]
Alexander G. Kolpakov
Siberian State University of Telecommunication and Informatics, Novosibirsk, Russia
[email protected]
Piotr Kozarzewski
University of Warsaw, Poland
[email protected]
Carolin Kreisbeck
Universiteit Utrecht, The Netherlands
[email protected]
Michel Lenczner
Université de Technologie de Belfort-Montbéliard & FEMTO-ST, Besançon,
France
[email protected]
Giovanni Leoni
Carnegie Mellon University, Pittsburgh, USA
[email protected]
Marta Lewicka
University of Pittsburgh, USA
[email protected]
Francesco Maiolino
[email protected]
Fabrizio Marignetti
[email protected]
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Anna Mercaldo
[email protected]
Sara Monsurrò
[email protected]
Gioconda Moscariello
[email protected]
François Murat
[email protected]
Carlo Nitsch
[email protected]
Heiner Olbermann
Leipzig University, Germany
[email protected]
Olivier Pantz
Université de Nice Sophia-Antipolis, France
[email protected]
Roberto Paroni
Università degli Studi di Sassari, Alghero, Italy
[email protected]
Antonia Passarelli di Napoli
[email protected]
Carmen Perugia
[email protected]
Olivier Pironneau
[email protected]
Giovanni Pisante
[email protected]
Maria Rosaria Posteraro
[email protected]
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Ravi Prakash
University of Concepción, Chile
[email protected]
Annie Raoult
Université Paris Descartes, France
[email protected]
Giovanni Romano
[email protected]
Giacomo Russo
[email protected]
Carlo Sbordone
[email protected]
Giovanni Scilla
[email protected]
Roberta Schiattarella
[email protected]
Ali Sili
Université de Toulon et du Var, La Garde, France
[email protected]
Valeriy V. Slastikov
University of Bristol, UK
[email protected]
Bianca Stroffolini
[email protected]
Nicoletta Tchou
Université de Rennes 1, France
[email protected]
Franco Tomarelli
Politecnico di Milano, Italy
[email protected]
Gerardo Toraldo
[email protected]
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Cristina Trombetti
[email protected]
Guido Trombetti
[email protected]
Igor Velčić
University of Zagreb, Croatia
[email protected]
Anna Verde
[email protected]
Renato Verde
ITC ”Minzoni”, Giugliano in Campania, Italy
[email protected]
Antonio Vitolo
[email protected]
Roberta Volpicelli
[email protected]
Stephan Wackerle
Fraunhofer ITWM, TU Kaiserslautern, Germany
[email protected]
Elvira Zappale
[email protected]
Gabriella Zecca
[email protected]
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FOURTH WORKSHOP ON THIN STRUCTURES Naples

Transcript

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