Transparent Boundaries

If we are interested in the propagation of waves around a small
region of interest, like e.g. an obstacle inside a very big
("unbounded") domain, one way to bring such problems to the
computer and solve them numerically is to cut that unbounded
domain to a bounded domain. But to have a well-posed problem we
have to prescribe boundary conditions on the so-called artificial
boundary, which are not inherent in our original problem. This is
a classical problem which is not only connected to wave
phenomena.


Sonia Fliss is interested in so-called transparent boundary
conditions. These are the boundary conditions on the artificial
boundaries with just the right properties. There are several
classical methods like perfectly matched layers (PML) around the
region of interest. They are built to absorb incoming waves
(complex stretching of space variable). But unfortunately this
does not work for non-homogeneous media.


Traditionally, also boundary integral equations were used to
construct transparent boundary conditions. But in general, this
is not possible for anisotropic media (or heterogenous media,
e.g. having periodic properties).


The main idea in the work of Sonia Fliss is quite simple: She
surrounds the region of interest with half spaces (three or
more). Then, the solutions in each of these half spaces are
determined by Fourier transform (or Floquet waves for periodic
media, respectively). The difficulty is that in the overlap of
the different half spaces the representations of the solutions
have to coincide.


Sonia Fliss proposes a method which ensures that this is true
(eventually under certain compatibility conditions). The chosen
number of half spaces does not change the method very much. The
idea is charmingly simple, but the proof that these solutions
exist and have the right properties is more involved. She is
still working on making the proofs more easy to understand and
apply.


It is a fun fact, that complex media were the starting point for
the idea, and only afterwards it became clear that it also works
perfectly well for homogeneous (i.e. much less complex) media.
One might consider this to be very theoretical result, but they
lead to numerical simulations which match our expectations and
are quite impressive and impossible without knowing the right
transparent boundary conditions.


Sonia Fliss is still very fascinated by the many open theoretical
questions. At the moment she is working at Ecole Nationale
Supérieure des Techniques avancées (ENSTA) near Paris as Maitre
de conférence.
Literature and additional material

C. Besse, J. Coatleven, S. Fliss, I. Lacroix-Violet, K.
Ramdani: Transparent boundary conditions for locally perturbed
infinite hexagonal periodic media, arXiv preprint
arXiv:1205.5345, 2012.

S. Fliss, P. Joly: Exact boundary conditions for
time-harmonic wave propagation in locally perturbed periodic
media, Applied Numerical Mathematics 59.9: 2155-2178, 2009.

L. Bourgeois, S. Fliss: On the identification of defects in a
periodic waveguide from far field data, Inverse Problems 30.9:
095004, 2014.