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 Î D) . |
| (5.9) |
If we write the discrete form of this equation for a selected set of
points q1, q2, ..., qm Î D then the
following linear system of equations can be obtained:
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):
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| (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
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| (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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