5.2  The Schenck Method

The Schenck method [75] represented an important step forward in the boundary element solution of the exterior Helmholtz equation. The method takes advantage of the complementary equation of equations (5.2) and (5.3) for points in the interior;
{ Mk f}S (p) = { Lk v }S (p)    (p D) .
(5.9)
If we write the discrete form of this equation for a selected set of points q1, q2, ..., qm D then the following linear system of equations can be obtained:

M
 

k 
f
L
 

k 
v
(5.10)
where [`L]k is an m ×n matrices with

[[`L]k]ij = {Lk [e*] }DS*j(qi)  and the other matrix is defined similarly.

The matrix approximation (5.10) is regarded as a further set of m approximations that relate f and v. Schenck suggested that instead of basing the numerical method on the equation related to (5.7) alone, the surface solution is found from the following matrix-vector approximation that is constructed from both the approximation (5.7) and (5.10):







Mk - 1
2
I

M
 

k 






f




Lk

L
 

k 





v ,
(5.11)
where the matrices in (5.11) have n columns and n+m rows.

A least squares method may be employed to find an approximation to the unknown boundary function. For example for the Neumann problem this is equivalent to finding the solution of the matrix-vector equation






(Mk - 1
2
I)T
(
M
 

k 
)T











Mk - 1
2
I

M
 

k 






^
f
 
=




(Mk - 1
2
I)T
(
M
 

k 
)T










Lk

L
 

k 





v ,
(5.12)
which is arrived at through pre-multiplying both sides of the system (5.11) by the transpose of the matrix on the left hand side. Since the transposed matrix has the dimension n ×(n+m) then the matrices on both sides of (5.12) are n ×n when multiplied out and the equation is simply a system of n equations in n unknowns that can be solved by standard methods.

For further details on the Schenck method, often also termed the CHIEF method, the reader is referred to Seybert et al [76], [77], [78] and Juhl [37], for example. The Schenck method has the potential of greatly extending the range of wavenumbers over which a solution of the exterior Helmholtz equation can be achieved when compared to the elementary methods. However, the Schenck method suffers from the difficulty that the number of interior points and their positions is not clear. At higher wavenumbers, more and more interior points are required to maintain accuracy on the one hand but the size of the matrices must correspondingly increase on the other, signalling a loss of efficiency.

The Schenck method remains popular, particularly since it avoids the necessity of discretising the Nk operator, which is generally required by the improved formulations that are considered in the next Section.


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