Chapter 6
Absorbing Boundary Conditions
As see from the previous chapter, exact 2D radiation boundary conditions can be established on a circle by
using cylindrical harmonics, or on a arbitrary surface using the surface integral equation. However, these
exact radiation boundary conditions make part of the the impedance matrix (the part that relates to the
surface unknowns) dense, thus significantly increasing the computational cost. This dense matrix is due to
nonlocal radiation of sources on the outer boundary.
Therefore, it is desirable to have a radiation boundary condition that can retain the sparsity of the
impedance matrix of the FEM. Such approximate radiation boundary conditions are usually known as
the absorbing boundary conditions (ABCs). An absorbing boundary condition approximates the radiation
boundary condition required for an unbounded problem so that negligible erroneous reflections occur at
the computational edge when an unbounded domain is truncated by the ABC. In the previous chapter, we
already learn the exact radiation boundary conditions, either on a circular outer boundary or the integral
equation at an arbitrary outer boundary, although they make the impedance matrix dense for the part
associated with boundary unknowns.
In this Chapter, the BaylissTurkel ABC, a local approximate ABC, and the perfectly matched layer
(PML) ABC are presented to overcome this difficulty. The PML boundary conditions can be designed to
yield no reflections at the boundary interface, but to make it usable for the FEM and other partial differential
equation methods (such as the FDTD), one must truncate the PML medium to a finite thickness; thus,
there will be reflections from the outer boundary. From this viewpoint, the PML ABC is also approximate.
However, such approximate ABC will preserve the sparseness of the FEM system matrix.
6.1
The BaylissTurkel ABC
The BaylissTurkel ABC is based on the farzone radiation condition. In 2D, Sommerfeld radiation condition
for the scattered field
E
s
z
at
ρ
→ ∞
is
lim
ρ
→∞
∂E
s
z
∂ρ
=

jk
b
E
s
z
(6.1)
in a background medium with the wavenumber
k
b
. This is a local condition, but it holds only at
ρ
→ ∞
.
For a finite distance
ρ
, this condition is only approximate. Since the definition of the far field zone is with
respect to the wavelength, it is more convenient to define a normalized length
˜
ρ
=
k
b
ρ
(6.2)
so that the radiation condition can be rewritten as
lim
˜
ρ
→∞
∂E
s
z
(˜
ρ, φ
)
∂
˜
ρ
=

jE
s
z
(6.3)
105
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CHAPTER 6.
ABSORBING BOUNDARY CONDITIONS
For a finite value of ˜
ρ
, this is only approximately valid, and will give rise to some approximation errors.
Below we will use this normalized length ˜
ρ
to obtain a better approximation than (6.2) for a finite distance
from the scatterer.
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 Spring '09
 QingLiu
 PML, Q. H. Liu

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