**Seminari e Convegni**

**Archiviata**

# 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****counterpart**

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****tube structures**

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.