Project Overview
Neutral Inclusions and Cloaking
Department(s)
Mathematics
Abstract
Many people remembers (reading or watching) Harry Potter using his amazing invisibiilty cloak during his many adventures at Hogwarts. The cool property of the cloak was that it rendered the wearer invisible. Cloaking and invisibility are old staples of popular fiction, especially of science fiction. But with the laws of physics in the real world, is this possible, even in theory? Not so long ago, physicists and mathematicians found that the answer to this question is “yes.” The key: to engineer a “metamaterial” with special microstructure that bends electromagnetic waves in a quantifiable and controllable way.
In 2006, John Pendry, David Schurig, and David Smith published an idea for a cloak that would render an object in two or more dimensions invisible to probing by electromagnetic waves at a fixed frequency by surrounding it with a specially designed metamaterial. Back in 2003, Greenleaf, Lassas, and Uhlmann had already described essentially the same notion
when studying an inverse problem for electrical impedance tomography
The goal of cloaking is to render an object invisible to detection from electromagnetic energy by surrounding the object with a specially engineered “metamaterial” that redirects electromagnetic waves around the object. In this project, we will first, understand the mathematical ideas behind this phenomenon. We will understand how to cloak an object against detection from impedance tomography and though the mathematical ideas, apply the knowledge to much more general forms of imaging.
A neutral inclusion, when inserted in a matrix containing a uniform applied electric field, does not disturb the outside field. Back in 1953, Mansfield was the first to observe that reinforced holes, “neutral holes”, could be cut out of a uniformly stressed plate without disturbing the surrounding stress field in the plate Mansfield. The well known Hashin coated sphere onstruction is an example of a neutral coated inclusion for the conductivity problem. The problem of determining nonlinear neutral inclusions in (electrical or ther-
mal) conductivity will also be considered in this project. We will work to construct neutral inclusions from nonlinear materials.
Student Qualifications
Calculus I,II,III
Linear Algebra,
Differential Equations
Preference: Students that have also taken Real Analysis I and/or Partial Differential Equaitons
Number of Student Researchers
2 students
Project Length
8 weeks