4  Derivation of the Numerical Method

In this section collocation is applied to the integral equation formulation of the Helmholtz equation (13). The resulting discrete equations are then coupled with the equations that govern the motion of the structure. A linear system that models the coupled acoustic-structure interaction problem results. The corresponding eigenvalue problem has the form (1). Quadratic polynomial interpolation (with respect to k) of the matrix coefficients is then applied to derive a method for computing the mode shapes and resonant frequencies of the coupled system.

4.1  Application of Collocation

In this section it is shown how collocation is applied to derive the discrete form of the integral equations. The boundary S is divided into N elements DS1, DS2, ... DSN and the boundary functions are approximated by a constant on each element. Thus the integral equation (13) is reduced to the following
B(k) j » C(k) v
(15)
where B(k) and C(k) are N ×N matrices with
[B(k)]ij = a æ
è 		{Mk h }DSj (pi) - 1
2
 dij ö
ø 		+ b{Nk h }DSj (pi) ,
(16)

[C(k)]ij = a{Lk h }DSj (pi) + b æ
è 		{Mkt h }DSj (pi)  + 1
2
 dij ö
ø 		 .
(17)
with j = [ j1, j2, ..., jN]T with jj = j(pj) (j=1,2, ..., N), v = [ v1, v2, ..., vN]T with vj = [(j)/(n)](pj) (j=1,2, ..., N) and h is the unit function.
The component of the matrices are defined in terms of integrals. In most cases the integrals are regular and can be approximated satisfactorily using Gaussian quadrature. However, for the diagonal components of the matrices B(k) and C(k) the corresponding integrals are singular and special techniques need to be employed. See [5,16,17] for the background to this. In this work, the complex numbers a and b were chosen so that a = ||[{Nk h }DSj (pi) ]|| and b = i ||[ ( {Mk h }DSj (pi) - [1/2] dij ) ] ||, this is consistent with the method advocated in reference [18].

4.2  Discretising the Vibratory Properties

The following equation is immediate from (11):
g = f - i rk c j
(18)
where g = [g1, g2, ..., gN]T with gj = g(pj) , where f = [f1, f2, ..., fN]T with fj = f(pj) . Let the vibratory properties of the structure be modelled by the M most fundamental mode shapes. Let the N ×M matrix of mode shapes U be defined as follows:
[U]ij = uj(pi) . npi   for   i=1,2,...,N   and   j=1,2,...,M  .
(19)
We may then write g » U G , f » U F , and v » U V where F = [F1, F2, ..., FM]T, G = [G1, G2, ..., GM]T and V = [V1, V2, ..., VM]T.

4.3  Discrete Formulation of Coupled Problem

Equation (18) can now be written in the form
U G » U F - i rk c j .
(20)
Let D(k) be the M ×M diagonal matrix defined as follows:
D(k) = diag { l1 (k), l2 (k), ..., lM (k) } .
(21)
Thus we may write
G = D(k) V = - i wD(k) W = - i k c D(k) W .
(22)
We can now construct the discrete representation of the coupled problem which follows from (15), (20) and (22).:

é
ê
ê
ë
B(k)
i k c C(k) U
i rkc UT
- i k c UT U D(k)
 ù
ú
ú
û 		é
ê
ê
ë
j
W
 ù
ú
ú
û 		» é
ê
ê
ë
0
UT U F
 ù
ú
ú
û 		.
(23)
The resonant frequencies and mode shapes can now be determined by finding the non-trivial solutions to the non-linear eigenvalue problem of the form (1) with
A(k) = é
ê
ê
ë
B(k)
i k c C(k) U
i rk c UT
- i k c U UT D(k)
 ù
ú
ú
û 		 .
(24)

4.4  Solution of the Non-Linear Eigenvalue Problem.

In the derivation of a method for solving (1), the matrix A(k) is approximated by a quadratic polynomial in k
A(k) » A[0] + k A[1] + k2 A[2]  .
(25)
Thus we may replace (1) with the following eigenvalue problem
[A[0] + k A[1] + k2 A[2]]x = 0  ,
(26)
the solutions (k*, x*) of which approximate the solutions of (1). The solutions of (26) are the same as those of the following generalised linear eigenvalue problem:
é
ê
ê
ë
A[0]
A[1]
0
I
 ù
ú
ú
û 		é
ê
ê
ë
x
k x
 ù
ú
ú
û 		= k é
ê
ê
ë
0
-A[2]
I
0
 ù
ú
ú
û 		é
ê
ê
ë
x
k x
 ù
ú
ú
û 		.
(27)
Equation (27) is amenable to solution by the QZ algorithm [19], which may be invoked, for example, via NAG routine F02GJF [20]. Methods of this type are considered in references [9,10,21-23]. A Fortran subroutine that computes eigenvalues of interpolated matrices is listed in reference [9].
The generalised eigenvalue problem (27) will generally have 2(N + M) solutions. In general, many of the solutions to (1) will result as a consequence of the approximation (25) and are not solutions of the underlying matrix eigenvalue problem (1). However, since we expect the resonant frequencies of the acoustic-structure problem to have the property |Re(k*)| >> |Im(k*)| and we determine that Re(k*) > 0, then the spurious solutions of (27) can generally be filtered out.
There is a wide variety of methods for deriving an approximation (25). In this paper the method employed involves computing A(k) at the three Chebyshev (¥ norm) interpolation points for any selected range [kA, kB]. The subroutine listed in reference [9] was used to solve the linear eigenvalue problem (27).