Solvers for 2D electromagnetic problems
Principle ideas of our methodology
- Efficient simulation of thin dialectic coating layers (know-how).
- Special approximation of the Sommerfeld radiation condition.
Formulation of the problem.
Let E(x,y,z),H(x,y,z) be the electromagnetic field generated
by a time-harmonic plane electromagnetic wave Ei,Hi
in a presence of a coated infinite cylindrical obstacle.
The coated obstacle is the union of an ideal conductor D and a dielectric coating of
variable thickness.
The incident wave is assumed to be E-polarized (H-polarized). In this case the Maxwell
equations are reduced to the 2D Helmholtz wave equation in the xy-plane with the Dirichlet
(the Neumann) boundary condition on the conductor surface. The dielectric layer
is assumed to be thin with respect to the obstacle size.
Scattering by a coated ellipse.
Let the obstacle be an ellipse with semi-axes 2 and 1 coated by a dielectric
of thickness 0.1 with dielectric permeability 1 and permettivity 100.
The incident wave of length 1 is propagated in the direction shown on the
left picture below. The problem parameters are such that the Leontovich
boundary condition can not be applied, i.e. the problem can not be reduced
to a homogeneous problem in vacuum with an absorbing boundary conditions
on the obstacle surface.
The acoustic thickness of the covering is 1 which leads to numerous reflections
from the dielectric-vacuum boundary and to accumulation of the energy inside
the coating. Suppose that the incident wave has an unit amplitude, then the
amplitude of the scattered wave achieves 25. The next pictures shows isolines
of the total field.
Flexibility of the developed software.
The flexibility of the program package developed for solving
scattering problems is demonstrated with two examples.
In the first example the coating has the same material properties
as the surrounding media. This allows us to use the same software to
solve homogeneous scattering problems. The figure below shows
strong back scattering signals from the plane cockpit and tail.
In the second example we use specially designed coating layers
to either absorb radar waves or reduced back scattering signals.
The test obstacle is the NACA 0012 profile with a tail dielectric comb
made of a material with a complex permeability. A relatively small
comb (see figure below) with carefully chosen parameters may reduce
the back scattering signal by a few ten decibels.
References.
- G.Abdoulaev, Y.Achdou, Y.Kuznetsov, K.Lipnikov, J.Periaux and O.Pironneau.
Finite element methods with nonmatching grids and applications.
Proceedings of the Conference on Applied Mathematics
and Computer Science, 28-29 October 1996, Moscow, Russia,
French-Russian A.M.Liapunov Institute, Moscow State University,
pp.65-81.
- Y.Kuznetsov and K.Lipnikov.
On the application of fictitious domain and domain
decomposition methods for scattering problems on Cray Y-MP C98.
Report No.9557, University of Nijmegen, The Netherlands,
December 1995.