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09:45 to 10:45 |
A. K. Nandakumaran (IISc, Bengaluru, India) |
General Introduction to Homogenization Multi scales arise in many physical and industrial problems. Many industrial constructions include very complicated structures. Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. This study is basically developed from material science in the creation of composite materials though the present application is much far and wide. It has applications in {\it composite media, porous domains, laminar structures, domains with rapidly oscillating boundaries}, to name a few. The PDE problems posed on such complicated domains lead to a type of asymptotic analysis known as homogenization. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium.
There are various methods developed in the last 50 years to understand the mathematical homogenization theory and some them are; Asymptotic Expansion, Energy Method, Compensated Compactness, Two-scale and multi-scale convergence, Gamma Convergence, Bloch Wave Analysis, Method of Unfolding etc. We give a quick introduction on the topic and briefly discuss the aim of the discussion meeting and topics to be discussed in the next two weeks.
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10:45 to 11:45 |
Patrizia Donato (University of Rouen, France) |
Introduction to Sobolev Spaces and Weak Solutions of PDEs The aim of this welcome lecture is to give a general overview of variational problems, in order to prepare the audience to the following talks. A short introduction to variational PdE’s will be followed by the definition of a class of Sobolev spaces and its properties. then we introduce the variational formulation and give some existence and uniqueness result for the model equation in the divergence form. The case of homogeneus Dirichlet boundary conditions and periodic boundary conditions Dirichlet are treated. This last case allows to introduce the corrector functions appearing in homogenization.
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11:45 to 12:15 |
Break |
Tea/coffee |
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12:15 to 13:15 |
Editha Jose (University of Phillippines, Los Banos) |
Multiscale Expansion Method for Periodic Homogenization (Lecture 1) Homogenization is a way of seeking the average properties of a material out of its components. In this lecture, we focus on the homogenization of periodic structures by considering an asymptotic analysis of the PDEs describing a certain property of a material. The limiting process is done by taking the period to approach zero and obtain a homogeneous model with homogenized coefficients that depend on the coefficients of the components. In this particular lecture, we discuss the multiscale expansion method which is a heuristic one but allows one to formally homogenize a great variety of equations posed in a periodic domain.
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13:15 to 14:30 |
Break |
Lunch |
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14:30 to 16:00 |
Editha Jose (University of Phillippines, Los Banos) |
Multiscale Expansion Method for Periodic Homogenization (Lecture 2) Homogenization is a way of seeking the average properties of a material out of its components. In this lecture, we focus on the homogenization of periodic structures by considering an asymptotic analysis of the PDEs describing a certain property of a material. The limiting process is done by taking the period to approach zero and obtain a homogeneous model with homogenized coefficients that depend on the coefficients of the components. In this particular lecture, we discuss the multiscale expansion method which is a heuristic one but allows one to formally homogenize a great variety of equations posed in a periodic domain.
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16:00 to 16:30 |
Break |
Tea/coffee |
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16:30 to 18:00 |
Tutorials/Discussion |
Tutorials/Discussion/Lecture |
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