INTER SCHOOL
Gamma-convergence and homogenization in continuum mechanics
SCHEDULE
October 31 – November 4, 2016
- Introduction to Homogenization and G-convergence, 15 h,
by Chiara Zanini, Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino.
chiara.zanini@polito.it
Abstract: Periodic structures are employed to describe composite materials, which appear in many fields like Mechanics, Physics, Chemistry and Engineering. The typical situation is that the physical parameters (conductivity, elasticity coefficients,…) are discontinuous and oscillating between the different values characterizing each of the components. When the components are intimately mixed, these parameters oscillate very rapidly and the microscopic structure becomes complicated.
Homogenization is a mathematical tool which, roughly speaking connects the length scales associated with the microscopic and the macroscopic phenomena. Indeed, the idea is to get a good approximation of the macroscopic behavior of such a heterogeneous material by means of a homogeneous material, whose overall response is close to that of the composite (periodic) material. In other words, starting from Partial Differential Equations of Physics describing a heterogeneous material with a fine periodic structure, Homogenization deals with the asymptotic analysis when the parameter describing the fineness of the microscopic structure tends to zero.
The course provides an introduction to Homogenization and G-convergence, and will be mainly focused on:
- Homogenization of second order linear elliptic operators
- Homogenization of monotone operators
- G-convergence, H-convergence
- Gamma-convergence methods for statics and evolution, 15h,
by Dr. rer. nat. Marita Thomas, Weierstrass Institute, Berlin (Germany),
marita.thomas@wias-berlin.de
Abstract: Physical and geometrical aspects in material modeling often lead to parameter-dependent families of variational problems. For instance, to determine the state of a deformable solid with changing material properties along a sharp interface, one may instead decide to analyze the respective minimum problem where the material discontinuity is replaced by a smooth transition zone located in a thin neighborhood of the interface. But how does this family of smooth problems relate to the original problem? What is the asymptotic behavior of their minima and minimizers? Going back on De Giorgi in the 1970s, the framework of Gamma-convergence provides a general and flexible tool to describe the asymptotic behavior of minimum problems for families of functionals in the calculus of variations. In particular, it provides exactly the criteria needed to ensure that cluster points of minimizers of the approximating functional are minimizers of the limit functional.
In the first part of the course, we will study general properties of Gamma-convergence. In this way, we will be able to handle static problems in continuum mechanics.
But what can be said about the asymptotics of evolution problems? In the second part of the course we will investigate how to refine the ideas of static Gamma-convergence in order to describe the asymptotics of different kinds of evolution problems in terms of evolutionary Gamma-convergence. Particular attention will be paid to the asymptotics of gradient flows and their generalizations, which play an important role in the treatment of dissipative processes in deformable solids, such as plasticity, phase transformation or separation processes, damage, or fracture. Based on this, we will also discuss how to handle asymptotics in settings that couple dissipative processes with the dynamics of the deformation. Again, in all these settings the key task will be to introduce suitable notions of solution and to establish general criteria which ensure that the notion of solution is preserved in asymptotics, so that cluster points of solutions of the approximating problems define a solution of the limit problem.