6 Boundary Element Method Solution through Frequency Interpolation
The concept of frequency interpolation in the BEM was first introduced
with a view to speeding up the numerical solution of boundary value
problems [32-33]. The use of
frequency interpolation as a means to solving the eigenvalue problem
was first considered in Kirkup and Amini [].
A similar technique for the modal analysis of coupled fluid-structure
problems is considered in Kirkup and Amini [34].
The method employed for solving (23)
requires that in an interval [kA, kB] of values of k
the matrix Ak is approximated by a matrix polynomial in k
Ak » A[0] + k A[1] + ... + kmA[N]forkreal.
(39)
The non-linear eigenvalue problem (23) can be replaced
with the following eigenvalue problem
[A[0] + k A[1] + ... + kmA[N]]
~
j
= 0.
(40)
The solutions of (40) are the same as those of the following
generalised linear eigenvalue problem
é ê ê ê
ê ê ê ë
A[0]
A[1]
A[2]
.
.
A[N-2]
A[N-1]
0
I
0
.
.
0
0
:
:
:
:
:
:
:
0
0
0
.
.
I
0
0
0
0
.
.
0
I
ù ú ú ú
ú ú ú û
é ê ê ê ê ê
ê ê ê ê ë
~
j
~
k
~
j
:
~
k
m-2
~
j
~
k
m-1
~
j
ù ú ú ú ú ú
ú ú ú ú û
(41)
=
~
k
é ê ê ê
ê ê ê ë
0
0
0
.
.
0
-A[N]
I
0
0
.
.
0
0
:
:
:
:
:
:
:
0
0
0
.
.
0
0
0
0
0
.
.
I
0
ù ú ú ú
ú ú ú û
é ê ê ê ê ê
ê ê ê ê ë
~
j
~
k
~
j
:
~
k
N-2
~
j
~
k
N-1
~
j
ù ú ú ú ú ú
ú ú ú ú û
.
Equation (41) is amenable to solution by the QZ algorithm.
Methods for solving problems of the form (40)
are considered in Kublanovskaya [26,35-38].
On solution, the eigenvalues [k\tilde] are the approximations
to the eigenfrequencies k* of the Helmholtz problem. The approximation
[(j)\tilde](p)
to the eigenfunctions j*(p) (p Î D ÈS)
can be recovered through the
substitution of the approximations into (18),
~
j
(p) = -
n å
i=1
~
j
i
{M[k\tilde] zi }(p) (p Î D) and
(42)
~
j
(p) = -
1
c(p)
n å
i=1
~
j
i
{M[k\tilde] zi }(p) (p Î S) .
The full set
of solutions will contain approximations to the true eigenvalues of the
underlying Helmholtz problem. Spurious solutions are
produced as a result of the collocation method and approximation (39),
but they can be easily excluded.
In this paper we consider interpolants with N=2 only. In this
case equation (41) takes the form
é ê
ê ë
A[0]
A[1]
0
I
ù ú
ú û
é ê ê
ê ë
~
j
~
k
~
j
ù ú ú
ú û
=
~
k
é ê
ê ë
0
-A[2]
I
0
ù ú
ú û
é ê ê
ê ë
~
j
~
k
~
j
ù ú ú
ú û
.
(43)
where Ak is computed at the three Chebyshev (¥ norm)
interpolation points for any selected range [kA, kB]. The
coefficient matrices
A[0], A[1]
and A[2] in (39) are obtained through
Newton's divided differences. The generalised eigenvalue problem
(41) is then solved through invoking NAG routine F02GJF [39].
Further detail on the development of the method is
given in Kirkup and Amini [18].