9 Potential use of the Boundary and Shell Element Method
The Helmholtz equation, which governs the acoustic field exterior
to an isolated shield or shell, can be re-formulated as an integral equation
in a way similar to that for a closed surface, see Ben Mariem and Hamdi (1987)
or Warham (1988). The resulting equation is termed a shell
integral equation.
The situation we consider in this paper consists of a closed boundary
(the engine block) and at least one thin shield. This can also be
reformulated as an integral equation termed a boundary and
shell integral equation (BSIE)
which is a hybrid of a boundary integral
equation and a shell integral equation.
Discretisation of this integral equation allows us to derive the
boundary and shell element
method. The BSEM is clearly a generalisation of the BEM and hence
the background to the BEM in section 4.3 is also relevant to the BSEM.
For the particular BSEM applied in this paper, the BSIE was based on
the generalisation of the integral equation of
Burton and Miller (1971). In order to derive the
BSEM, the surfaces are approximated by a set of triangles and the surface
functions are approximated by a constant on each triangle. Further detail
on the method is given in Kirkup (1991).
The test problems that are considered all have a 10cm cube of which
one face is vibrating uniformly, the vibration of the other
faces is zero. A square plate of side 10cm is placed 10cm from the
vibrating face. Air is the acoustic medium and the shield and cube are
assumed to be perfectly reflecting.
The test problem is illustrated in figure 9.
Figure 9. Illustration of the test problem.
The radiation ratios of the system with or without the shield
were computed at 10Hz, 20Hz, ..., 5000Hz.
The radiation ratio curve for the system is given in figure 10.
The graph shows that the shield has a major effect on the
radiation ratio of the vibrating cube. The radiation ratios
for the shielded and unshielded cube are similar when the
frequency is less than around 1300Hz. However, the shield
makes a significant reduction in radiation ratio at around 1800Hz,
3000Hz and 4500Hz and significantly increases the radiation ratio
at around 2200Hz and 3700Hz.
Further results are given in Kirkup, Henwood and Tyrrell (1991)
and Kirkup (1991).
Figure 10. Radiation ratio curves for the shielded and unshielded cubes.