Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

262 Waves
to using such coordinate systems:
. The volume integrals separate into radial and angular parts which can be carried
out independently. This leads to economies in computation.
. The radial integration is identical for every such in®nite element, so that the only
integration which needs to be carried out for every in®nite element is along the
®nite element interface.
. The radial integration is the only part containing the wave number.
. The ellipsoidal coordinates can be used to enclose a large variety of di€erent
geometries in the ®nite element interior, while still retaining a guarantee of
convergence in 3D.
The angular shape functions are written in the conventional polynomial form. The
radial shape functions take the form
Xm
ÿ ikr
Ni ˆ e
j ˆ 1
h ij
…kr† j
…8:40†
The coe cients h ij are given from the condition of circumferential compatibility
between adjacent in®nite elements. There is e€ectively no di€erence between this
radial behaviour and that originally proposed in the mapped in®nite wave elements
by Bettess et al. 45;46 The di€erence in the in®nite element methodology lies in the
fact that the radial variable, r, is now in ellipsoidal coordinates. Burnett and
Holford 51ÿ53 give the necessary detailed information for the element integrations
and the programming of these in®nite elements, together with some results. The
analytical expressions are too long to include here. The elements have been used on
submarine ¯uid±structure interaction problems, and substantial e ciencies over
the use of boundary integral models for the scattered waves have been claimed. In
one case Burnett states that the ®nite and in®nite element model ran for 7 hours on
a workstation. His projected time for the corresponding boundary element model
was about 3000 hours, the in®nite elements giving a dramatic improvement!
The Burnett elements have been tested up to very short-wave cases, up to ka ˆ 100
for an elastic sphere di€raction problem, which is shown in Fig. 8.8.
8.13 Wave envelope in®nite elements
Astley introduced a new type of ®nite element, in which the weighting function is the
complex conjugate of the shape function. 19;20 The great simpli®cation which this
introduces is that the oscillatory function exp…ikr† cancels after being multiplied by
exp…ÿikr†, and the remaining terms are all polynomials, which can be integrated
using standard techniques, like Gauss±Legendre integration (see Volume 1, Chapter
9). This type of element was originally large (i.e. many wavelengths in extent), but not
in®nite. Figure 8.9 shows an example from acoustics, that of acoustical pressure in a
hyperbolic duct. Good results were obtained despite using a relatively coarse mesh.
Astley's shape function was of the form
N i…r; † r i
r eÿik…r ÿ r i†
…8:41†