In this section collocation is applied to the integral equation
formulation of the Helmholtz equation given in the previous section.
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 Fortran subroutine ABSEMGEN
is outlined.
In this section it is shown how collocation is applied to derive the
discrete form of the integral equations. The boundary and shell
are divided into planar triangular elements. The boundary S is divided into
nS elements DS1, DS2, ..., DSnS, the
shell G is divided into nG elements
DG1, DG2, ... DGnG
and the boundary functions and shell functions are approximated
by a constant on each element. Let p1, p2, ..., pnS
and q1, q2, ..., qNG be the collocation
point with pi Î DSi for i=1, 2,..., nS
and qi Î DGi for i=1, 2, ..., nG
and each lying at the centroid of the respective element.
It is helpful to introduce the following notation. Let the matrices
LSS, LSG, LGS,
and LGG be defined as follows:
[LSS]ij = { Lk h }DSj (pi) (i=1,2,...,nS),(j=1,2,...,nS) ,
[LSG]ij = { Lk h }DGj (pi) (i=1,2,...,nS), (j=1,2,...,nG) ,
[LGS]ij = { Lk h }DSj (qi) (i=1,2,...,nG), (j=1,2,...,nS) ,
[LGG]ij = { Lk h }DGj (qi) (i=1,2,...,nG), (j=1,2,...,nG)
where h is the unit function. The other integral operators
can be discretised in a similar way. This allows us to
construct the following linear systems of approximations which
are the discrete analogues of equations (15), (16), and (17).
where gG = [g1, g2, ..., gnG]T with gj = g(pj)
and fG = [f1, f2, ..., fnG]T
with fj = f(pj) .
Let the vibratory properties of the shell be modelled by the nM most
fundamental mode shapes. Let the nG ×nM matrix of mode shapes
UGM be defined as follows:
In the physical system of interest, the velocity of the surface of the
bodies, the modal forcing on the shells and the dynamic properties of the
shell are assumed known. From approximation (19), (20)
and (21) we may now construct the
following linear system for computing approximations
[^(j)]S, [^(dG)],
and [^(V)]M
to
jS, dS, and VM
é ê ê ê
ê ê ë
a(MSS -
1
2
ISS ) + bNSS
aMSG + bNSG
0SM
NGS
NGG
- UGM
0MS
i rwUTMG
UTGM DMM
ù ú ú ú
ú ú û
é ê ê ê
ê ê ë
^
j
S
^
d
G
^
V
M
ù ú ú ú
ú ú û
=
é ê ê ê
ê ê ë
( aLSS + b( MSS +
1
2
ISS )vS
MtGSvS
UMGTUGMFM
ù ú ú ú
ú ú û
.
(27)
In subroutine ABSEMGEN the integrals corresponding to the discrete
integral operators were computed using Gaussian quadrature on a triangle
[15] when regular. In the case when the integrals are singular,
techniques similar to those given in [6] and [8] are
used. See reference [8] for more details. In this work,
the complex numbers a and b were chosen so that
a = || NSS || and
b = i || MSS - [1/2] ISS || which is
consistent with the method advocated in reference [18].
For the computation of the solution in the exterior domain and the
sound power and radiation ratio, the corresponding discrete analogues
of equations (18), (11), (12) and (13)
are implemented.
In outline, subroutine ABSEMGEN accepts the frequencies of interest,
a description of the shape and position of the bodies and shells, the
dynamic properties of the shells, the distribution of vibration on
the bodies, the distribution of the direct forcing on the shells and
the points in the exterior where the sound pressure is sought.
As output, the subroutine gives, for each chosen frequency,
the surface intensity pattern, the sound power, the radiation efficiency
and the sound pressure at the selected exterior points.