In this section the eigenvalue problem for the same domain as in the
previous section is considered.
The boundary conditions (in the form of section 2.6) are such that
j(p) = 0 for 0 < p1 < 1
and v(p)=0 on the remainder of the boundary. The shell condition
is such that n(p)=V(p)=0 for all points on the shell.
6.1 Direct and Indirect Non-linear Eigenvalue Problems
From (19) and (20) it follows that
the approximation to the eigenvalues and eigenfunctions on the boundary
and shell are given by finding the solution of the non-linear eigenvalue problem
é ê ê ê
ê ê ê ë
(Mk,SS +
1
2
ISS )
- Lk,SS
0S G
- Mk,S G
DaSS
DbSS
0S G
0S G
Mk,GS
- Lk,GS
IGG
-Mk,GG
Nk,GS
-Mtk,GS
0GG
-Nk,GG
ù ú ú ú
ú ú ú û
é ê ê ê ê
ê ê ê ë
~
j
S
~
v
S
~
F
G
~
d
G
ù ú ú ú ú
ú ú ú û
=
é ê ê ê
ê ê ë
0S
0S
0G
0G
ù ú ú ú
ú ú û
(21)
for the direct method and
é ê ê ê ê
ê ê ê ë
ISS
0SS
- Lk,SS
0S G
- Mk,S G
0SS
ISS
-(Mtk,SS +
1
2
ISS)
0S G
- Nk,S G
DaSS
DbSS
0SS
0S G
0S G
0GS
0GS
- Lk,GS
IGG
- Mk,GG
0GS
0GS
- Mtk,GS
0GG
- Nk,GG
ù ú ú ú ú
ú ú ú û
é ê ê ê ê ê
ê ê ê ê ë
~
j
S
~
v
S
~
s
S
~
F
G
~
d
G
ù ú ú ú ú ú
ú ú ú ú û
=
é ê ê ê
ê ê ê ë
0S
0S
0S
0G
0G
ù ú ú ú
ú ú ú û
(22)
for the indirect method. The approximation to the eigenfunction in the domain
can the be obtained through the substitution of the results from the
eigenvectors into the discrete form of equation (5) for the
direct method and the discrete form of (10) for the indirect method.
The eigenvalue problems (21) and (22) both have the form
Ak
~
m
= 0
(23)
where each component of Ak is a continuously differentiable complex-valued
function of k. Non-linear eigenvalue problems of this form are
considered in [11-13].
In this paper the eigenvalue problem is solved using the method
introduced in Kirkup and Amini [8]. This method involves
approximating Ak by a matrix polynomial
in an interval [kA, kB] of real values of k,
Ak » A[0] + k A[1] + ... + kmA[m]forkreal.
(24)
The non-linear eigenvalue problem (23) can be replaced
with the following eigenvalue problem
[A[0] + k A[1] + ... + kmA[m]]m = 0.
(25)
The solutions of (25) 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[m-2]
A[m-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 ë
m
k m
:
km-2 m
km-1 m
ù ú ú ú
ú ú ú û
(26)
= k
é ê ê ê
ê ê ê ë
0
0
0
.
.
0
-A[m]
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 ë
m
k m
:
km-2 m
km-1 m
ù ú ú ú
ú ú ú û
.
Equation (26) is amenable to solution by the QZ algorithm
[14], which may be invoked, for example, via NAG routine
F02GJF [15].
Methods for solving problems of the form (25)
are considered in references
[12,13,15].
On solution, the eigenvalues k* include the approximations
to the eigenfrequencies of the Helmholtz problem. Spurious
solutions to (26) are also obtained but these can be easily
excluded, for further details on this see Kirkup and Amini [8].
The approximation
to the eigenfunctions in D can be recovered through the
substitution of the relevant sub-vectors of the computed eigenvectors into
the discrete form of equation (5) for the direct method or
(10) for the indirect method.
In this paper the method employed for deriving the polynomial approximation
(24)
involves computing Ak at the m+1 Chebyshev (¥ norm)
interpolation points for any selected range [kA, kB]. The
coefficient matrices
A[0], A[1], ..., A[m] in (24) are obtained through
Newton's divided differences. The generalised eigenvalue problem
(26) is then solved through invoking NAG routine F02GJF.