MEETING ON ANALYSIS AND MODELING OF MULTI-SCALE PROBLEMS
Micol Amar (University of Roma “La Sapienza”)
Homogenization of a heat conduction problem involving tangential operators
We present a model for the heat conduction in a composite having a microscopic structure
arranged in a periodic array made by two phases separated by a thermally active membrane.
The thermal behavior of the membrane is described by a parabolic equation involving the
Laplace-Beltrami operator. Such interface equation furnishes the contact temperature of the
two diffusive phases in terms of the jump of the heat fluxes at the interface.
We obtain the macroscopic behavior of the material via an homogenization procedure based
on the unfolding technique, providing the equation satisfied by the effective temperature.
We are also able to prove an error estimate on the rate of convergence of the sequence of
approximating solutions to the homogenized solution.
These results are part of a joint research with R. Gianni.
Andrea Braides (University of Roma "Tor Vergata")
A simple discrete model for damage exhibiting infinitely-many phases and a continuum
We give a variational interpretation of a model by Novak and Truskinovsky of a discrete
systems where microscopic fracture produces macroscopic damage. Depending on the
arrangement at a discrete level the overall response corresponds to different damaged
materials. We give a formal continuum counterpart depending on a small parameter
mimicking the discrete dimension. The homogenized description as this parameter tends to
zero departs from the previous one in certain regimes. Work in collaboration with A.Causin,
M.Solci and L.Truskinovsky.
Paolo Cermelli (Univerrsity of Torino)
The limit of the cut functional on dense graph sequences
The cut functional on a finite graph is a measure of the total number of edges connecting
different communities, and can be used to find optimal splittings of the graph into highly
connected components. It also arises as a spin functional with simple pairwise interactions
between the nodes. Given that large graphs have increasing importance in applications, it is
important to understand the structure of the cut functional when the order of the graph goes
to infinity, as well as its relation with its finite counterpart. In this work we exploit the spinfunctional
analogy, the theory of limits of graph sequences and Gamma-convergence to
explore the limit functional on dense graph sequences, in order to elucidate the structure of
the interfaces in the large graph limit.
Grigory Panasenko ( Univ Lyon, UJM-Saint-Etienne, CNRS, Institute Camille Jordan UMR
5208, SFR MODMAD FED 4169, F-42023, Saint-Etienne, FRANCE)
Asymptotic reduction and hybrid dimension models for the flows in domains containing thin
Thin structures are some finite unions of thin rectangles (in 2D settings) or cylinders (in 3D
settings) depending on small parameter ɛ << 1 that is, the ratio of the thickness of the rectangle
(cylinder) to its length. We consider thin structures and multistructures which consist of several
“massive” domains independent of ɛ connected by thin structures. Viscous flows in such structures
are modeled by steady or non-steady Stokes or Navier-Stokes equations stated in thin structures or
multistructures with the no-slip boundary condition at the lateral boundary of the cylinders and
with the inflow and outflow conditions with the given velocity on some part of the boundary.
For thin structures an asymptotic expansion of the solution is constructed and justified. It has a form
of a Poiseuille (or Womersley) flow within thin cylinders at some distance from the bases while the
boundary layers near the ends of the cylinders decay exponentially. The algorithm of construction
of the expansion deals with a special Reynolds type problem on the graph for the pressure. This
structure of the expansion allows to reduce the dimension within the cylinders at the distance of
order ɛ |ln ɛ| from the bases of the cylinders and derive the junction conditions between models of
different dimensions. This approach is extended for multistructures (Stokes equations).
Finally, we discuss the possibility of asymptotic derivation of boundary conditions describing the
elasticity of the wall and of non-Newtonian equations for the fluid motion.