Computation of the Adjoint Green’s Function for a Conical Surface

A relatively simple way to compute the homogenous solution is to convert Eqs. (14.66) to (14.69) into time-dependent equations by replacing – io by Jt and add a factor of e-imt to the right side of boundary condition (14.70). The resulting problem may then be time marched to a periodic state, which is the nonhomoge­neous adjoint solution. For this purpose, it is advantageous to switch to an oblique Cartesian coordinate system (f, n) as shown in Figure 14.9. The oblique Cartesian coordinates and the cylindrical coordinates are related by

x

n = r – x tan S, f = or r = n + f sin S, x = f cos S. (14.78)

cos S

With respect to the oblique Cartesian coordinates, the adjoint equations (14.66) to (14.69) in the time domain may be rewritten in the following form:

where

‘ u(a) ‘ um vm

«0 sec 8 0 0 0

/ 0 0 0 0 0 0 0 0 0 – sec8 0 0

„-i—t

A

w(a) wm p(a) pm 0 0 0 «0 tan 8 0 0 0 0 Bm = 0 0 0 -a0 0 C = ’ Cm 0 0 0 0 0 0 0 0 0 0 i«0m tan 8 -1 0 0 0 -1 im 0 /

The boundary condition on the conical surface is

ц = 0, pm = – j( —f sin вх sin 8 ) e-i—(fcos 5 cos 0i+t). (14.80)

a0 /

To avoid the reflection of outgoing disturbances back into the computational domain, it is recommended to install a perfectly matched layer (PML) around the computational domain as shown in Figure 14.9. It is straightforward to develop a set of PML equations. A split-variable version of these equations is

Um Um1 + Um2	(14.81a)
9 Um1 + A dUm + a U 0 at + A" df + a U"‘ = 0	(14.81b)
d u 1 m V1 + + fsin 8 CmUm + aUm2 = 0. оц n + f sin 8	(14.81c)

9 Um2 d t

+ B

In Eq. (14.81), a is the damping function. Figure 14.10 shows a distribution of a(z) in the f – ц plane. For best results, it is recommended that a profile of a(z) resembling that in Figure 14.11 be used.

The solution of Eq. (14.79) together with PML boundary condition (14.81) and nonhomogeneous boundary condition (14.80) may be computed by using the 7-point stencil dispersion-relation-preserving (DRP) scheme. Boundary condition (14.80) may be enforced by the ghost point method. This boundary condition is responsible for introducing the correct disturbances into the computational domain. Since a time periodic solution is sought, a zero initial condition is the simplest way to get the solution started. The solution is to be marched in time until a time periodic solution is reached. Note: the adjoint Green’s function is spatially oscillatory because boundary condition (14.80) is an oscillatory function. The period of oscillation of boundary condition (14.80) may be used to estimate the spatial resolution required for computing the adjoint Green’s function. Experience suggests that it is prudent to turn the nonhomogeneous boundary term on gradually. This may be done by multiplying the right side of boundary condition (14.80) by a factor (1 – e^t/T) and taking T to be a number of oscillatory periods long. The reason for this is that, after discretization, the response of the finite difference system to a sudden imposition of boundary condition is sometimes not the same as that for a partial differential system.