PML in Three Dimensions
In three dimensions, the linearized Euler equations may be written in a matrix form
as
d u 9 u „ 9 u
+ A + В
dt dx dy
where
p 
M 
1 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
1 
0 

u 
0 
M 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 

v 
, a = 
0 
0 
M 
0 
0 
, В = 
0 
0 
0 
0 
1 
, c = 
0 
0 
0 
0 
0 
w 
0 
0 
0 
M 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 

p_ 
0 
1 
0 
0 
M 
0 
0 
1 
0 
0 
0 
0 
0 
1 
0 
A straightforward application of the method of Section 9.5.1 yields the following PML equations:



u + (ay + az )ql + ayazq2 u + (az + ax )ql + azaxq2
U + (ax + ay )ql + axayQ2
(ax + ay + az)U + (ayaz + azax + axay )4l + axayaz02
The auxiliary variables ql and q2 are given by
Notice that the absorption coefficients ax, a and az are arbitrary functions of x, y, and z, respectively. Again, as in the twodimensional case, the auxiliary variables, q1 and q2 need to be computed only in the PML regions.
The PML absorbing boundary condition is especially useful in a ducted environment. It is very effective for simulating a longduct termination. Figure 9.13 shows a computational domain inside a circular duct with rigid walls. In cylindrical coordinates, (x, r, ф), the linearized Euler equations may be written as
where
p 
M 
1 
0 
0 
0 
0 
0 
1 
0 
0 

u 
0 
M 
0 
0 
1 
0 
0 
0 
0 
0 

U = 
v 
, 
a 
= 
0 
0 
M 
0 
0 
, 
В= 
0 
0 
0 
0 
1 

w 
0 
0 
0 
M 
0 
0 
0 
0 
0 
0 

p _ 
0 
1 
0 
0 
M 
0 
0 
1 
0 
0 

‘ 0 
0 
0 
1 
0 ‘ 
‘ 0 
0 
1 0 
0 

0 
0 
0 
0 
0 
0 
0 
о о 
0 

C= 
0 
0 
0 
0 
0 
, d 
= 
0 
0 
О О 
0 

0 
0 
0 
0 
1 
0 
0 
О О 
0 

0 
0 
0 
1 
0 
0 
0 
1 0 
0 

The corresponding PML equation is
where
In applying PML to a ducted domain, the wall boundary condition for the PML is the same as that for the Euler domain. For instance, for the rigid wall circular duct problem of Figure 9.13, the same rigid wall boundary condition, namely, v = 0, at r = D/2 is to be used in the PML regions.
In implementing PML, the damping coefficients, say, ctx(x), cty(y), are often taken as smooth functions. A good practice is to set these functions equal to zero at the interface with the Euler domain. It is also a good practice to let ctx(x) increase smoothly to a constant level in the main part of the PML. At the termination of the PML, the computed variables are usually exponentially small so that reflection is not a concern. However, it is a good practice to impose a standard radiation boundary condition that is of minimal cost to the computation.