Section: New Results
Discontinuous Galerkin methods for Maxwell's equations
DGTD- method based on
hierarchical polynomial interpolation
Participants : Loula Fezoui, Joseph Charles, Stéphane Lanteri.
The DGTD (Discontinuous Galerkin Time Domain) method originally
proposed by the team for the solution of the time domain Maxwell's
equations [14] relies on an arbitrary high order
polynomial interpolation of the component of the electromagnetic
field, and its computer implementation makes use of nodal (Lagrange)
basis expansions on simplicial elements. The resulting method is
often denoted by DGTD-
DGTD- method
on multi-element meshes
Participants : Clément Durochat, Stéphane Lanteri, Claire Scheid, Mark Loriot [Distene, Pôle Teratec, Bruyères-le-Chatel] .
In this work, we study a multi-element DGTD method formulated on a
hybrid mesh which combines a structured (orthogonal) quadrangulation
of the regular zones of the computational domain with an unstructured
triangulation for the discretization of the irregularly shaped
objects. The general objective is to enhance the flexibility and the
efficiency of DGTD methods for large-scale time domain electromagnetic
wave propagation problems with regards to the discretization process
of complex propagation scenes. As a first step, we have designed and
analyzed a DGTD-
DGTD- method for dispersive materials
Participants : Claire Scheid, Maciej Klemm [Communication Systems & Networks Laboratory, Centre for Communications Research, University of Bristol, UK] , Stéphane Lanteri.
This work is undertaken in the context of a collaboration with the
Communication Systems & Networks Laboratory, Centre for
Communications Research, University of Bristol (UK). This laboratory
is studying imaging modalities based on microwaves with applications
to dynamic imaging of the brain activity (Dynamic Microwave Imaging)
on one hand, and to cancerology (imaging of breast tumors) on the
other hand. The design of imaging systems for these applications is
extensively based on computer simulation, in particular to assess the
performances of the antenna arrays which are at the heart of these
systems. In practice, one has to model the propagation of
electromagnetic waves emitted from complex sources and which propagate
and interact with biological tissues. In relation with these issues,
we study the extension of the DGTD-
DGFD- method for the frequency
domain Maxwell equations
Participants : Victorita Dolean, Mohamed El Bouajaji, Stéphane Lanteri, Ronan Perrussel [Laplace Laboratory, INP/ENSEEIHT/UPS, Toulouse] .
For certain types of problems, a time harmonic evolution can be
assumed leading to the formulation of the frequency domain Maxwell
equations, and solving these equations may be more efficient than
considering the time domain variant. We are studying a high order
Discontinuous Galerkin Frequency Domain (DGFD-
Hybridized DGFD- method
Participants : Stéphane Lanteri, Liang Li, Ronan Perrussel [Laplace Laboratory, INP/ENSEEIHT/UPS, Toulouse] .
One major drawback of DG methods is their intrinsic cost due to the very large number of globally coupled degrees of freedom as compared to classical high order conforming finite element methods. Different attempts have been made in the recent past to improve this situation and one promising strategy has been recently proposed by Cockburn et al. [40] in the form of so-called hybridizable DG formulations. The distinctive feature of these methods is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. This work is concerned with the study of such Hybridizable Discontinuous Galkerkin (HDG) methods for the solution of the system of Maxwell equations in the time domain when the time integration relies on an implicit scheme, or in the frequency domain. As a first setp, HDGTD and HDGFD [33] methods have been developed for the solution of the 2D propagation problems.
Exact transparent condition
in a DGFD- method
Participants : Mohamed El Bouajaji, Nabil Gmati [ENIT-LAMSIN, Tunisia] , Stéphane Lanteri, Jamil Salhi [ENIT-LAMSIN, Tunisia] .
In the numerical treatment of propagation problems theoretically posed in unbounded domains, an artificial boundary is introduced on which an absorbing condition is imposed. For the frequency domain Maxwell equations, one generally use the Silver-Müller condition which is a first order approximation of the exact radiation condition. Then, the accuracy of the numerical treatment greatly depends on the position of the artificial boundary with regards to the scattering object. In this work, we have conducted a preliminary study aiming at improving this situation by using an exact transparent condition in place of the Silver-Müller condition. Promising results have been obtained in the 2D case and call for an extension of this work to the more challenging 3D case.