Project Overview
Homogenization Theory for Partial Differential Equations
Department(s)
Mathematics
Abstract
The project will be about the mathematical theory of homogenization. In rough terms, homogenization is a rigorous version of what is known as averaging. The goal of homogenization is to extract homogeneous effective parameters from disordered or heterogeneous media.
Homogenization has first been developed for periodic structures. Indeed, in many fields of science and technology one has to solve boundary value problems in periodic media. Quite often the size of the period is small compared to the size of a sample of the medium, and an asymptotic analysis, as the size of the period goes to zero, is called for. Starting from a microscopic description of a problem, we seek a macroscopic, or effective, description. This process of making an asymptotic analysis and seeking an averaged formulation is called homogenization. Students learn the basic background of the theory of homogenization and apply it to problems arising from materials science.
Homogenization offers a rigorous approach to modeling the macroscopic behavior of highly heterogeneous multiscale media with periodic microstructures of size 0 < ε « 1, assuming scale separation in the material. By employing asymptotic analysis, homogenization extracts homogeneous effective parameters from heterogeneous media, providing a direct link between the material’s constituents and its effective properties; hence, it bridges the gap between microscopic and macroscopic descriptions and accurately captures the interactions between different phases.
The primary benefit of homogenization theory lies in its ability to reduce computational complexity and provide insights into the overall behavior of multiscale systems. From the point of
view of applications, this method enables the design of new materials with desired characteristics/tailored properties, offering advantages over phenomenological models that lack straightforward control over effective properties. Because of that, this approach has found wide-ranging applications in various fields, enabling us to tackle problems that would otherwise be intractable due to the presence of multiple scales, reducing computational costs, and improving the efficiency of simulations used in the design and testing of new materials.
The method of two-scale asymptotic expansions is presented, and its mathematical justification will be studied.
Student Qualifications
MATH 161,162,162.
MATH 214.
MATH 308.
Not required but desirable: MATH 377, MATH 408.
Number of Student Researchers
2 students
Project Length
8 weeks
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