by Leonidas Mindrinos (Computational Science Center, University of Vienna, Austria)
In many applications, like material parameter investigation and nondestructive testing, arises the inverse problem of reconstructing the position, the shape and / or the properties of a cavity or an inclusion from given measurements. A good knowledge of the direct problem is necessary before studying the inverse problem.
We consider the scattering of obliquely incident electromagnetic waves by an infinitely long penetrable cylinder. The mathematical model for the scattering process is based on Maxwell’s equations with different parameters (permittivity and permeability) in and outside the cylinder. The fields and their derivatives are coupled at the boundary considering transmission conditions. Then, the direct problem is to compute the scattered electric and magnetic fields and their far-field patterns (far from the medium), given the system of PDEs, the boundary conditions and specific radiation conditions.
Figure: The norms of the electric fields (left column) and the magnetic fields (right column) for different boundary curves, parameters, and polar angles of the incident wave (π/2 top row and π/9 bottom row).
If we restrict ourselves to a homogeneous cylinder, oriented for instance parallel to the z-axis and oblique incidence, the three-dimensional problem can be reduced to two two-dimensional problems for the Helmholtz equation in the x-y plane but with more involved boundary conditions. We have shown uniqueness and existence of solution for this problem by formulating an equivalent system of boundary integral equations in a Sobolev space setting. The operators however are singular and special treatment is needed for the numerical scheme. We considered quadrature rules and applied a collocation method to obtain a linear system. The numerical results where also tested with analytic fields derived from fundamental solutions.
Joint work with Drossos Gintides (National Technical University of Athens, Greece).
For further information please see the published paper.