Partial Differential Equations - Modelling and ... - ResearchGate

∂H
∂t
= −∇ × E,
∂E
∂t
Electromagnetic Scattering 97
= ∇ × H. (2)
Here, E and H denote the electric and magnetic field intensity respectively,
dΩ denotes an element of surface area, in the direction normal to the surface,
and dΓ is an element of contour length, in the tangent direction to the contour.
Consideration will be restricted to the solution of two-dimensional problems,
involving TE polarized waves. In this case, relative to a Cartesian x, y, z coordinate
system, the field intensity vectors E =(E x ,E y , 0) and H =(0, 0,H z )
are functions of t, x and y only.
The scattering simulations that will be undertaken will involve the interaction
between a known incident field, generated by a source located in the
far field, and a scatterer, surrounded by free space. It will be assumed that the
scatterer is a perfect electrical conductor (PEC) and that the incident field is
a plane single frequency wave. For such simulations, it is convenient to split
the total electric and magnetic fields as
E = E inc + E scat , H = H inc + H scat , (3)
where the subscripts inc and scat refer to the incident and scattered wave components
respectively. The problem is then formulated in terms of the scattered
fields. The boundary condition at the surface of the scatterer is the requirement
that the tangential component of the total electric field should be zero.
The infinite solution domain must be truncated to enable a numerical simulation
and the condition that must be imposed at the truncated far field
boundary is that the scattered field should only consist of outgoing waves.
This requirement is imposed by surrounding the computational domain with
an artificial perfectly matched layer (PML) [Ber94, BP97].
3 A Finite Element Method
An explicit finite element time domain (FETD) method, for implementation
on a general unstructured mesh of triangles, can be developed by initially
writing the equations (2) in the form
∂U
∂t = −∂Fk ∂x k
= −A k ∂U
∂x k
, (4)
where k takes the values 1 and 2 and the summation convention is employed.
Here x 1 = x, x 2 = y and
⎡ ⎤
⎡
⎤
H z
0 −(k − 1) (2 − k)
U = ⎣E x
⎦ , A k = ⎣ (k − 1) 0 0 ⎦ . (5)
E y −(2 − k) 0 0
This equation is discretised using the explicit TG2 algorithm [DH03]. In this
method, the solution is advanced over a time step, ∆t, in a two-stage process.