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

244 Waves
the familiar form
where
…
M ˆ
K ÿ ! 2 ÿ
M
N T …
1
N d K ˆ
g
D ˆ
h 0
0 h
~g ˆ 0 …8:9†
B T DB d …8:10†
It is thus an eigenvalue problem as discussed in Chapter 17 of Volume 1. The K and M
matrices are analagous to structure sti€ness and mass matrices. The eigenvalues will
give the natural frequencies of oscillation of the water in the basin and the eigenvectors
give the mode shapes of the water surface. Such an analysis was ®rst carried
out using ®nite elements by Taylor et al. 12 and the results are shown as Fig. 17.5 of
Volume 1. There are analytical solutions for harbours of regular shape and constant
depth. 1;3 The reader should ®nd it easy to modify the standard element routine given
in Volume 1, Chapter 20, to generate the wave equation `sti€ness' and `mass'
matrices. In the corresponding acoustic problems, the eigenvalues give the natural
resonant frequencies and the eigenvectors give the modes of vibration. The model
described above will give good results for harbour and basin resonance problems,
and other problems governed by the Helmholtz equation. In modelling the Helmholtz
equation, it is necessary to retain a mesh which is su ciently ®ne to ensure an accurate
solution. A `rule of thumb', which has been used for some time, is that there should be
10 nodes per wavelength. This has been accepted as giving results of acceptable engineering
accuracy for many wave problems. However, recently more accurate error
analysis of the Helmholtz equation has been carried out. 13;14 In wave problems it is
not su cient to use a ®ne mesh only in zones of interest. The entire domain must
be discretized to a suitable element density. There are essentially two types of error:
. The wave shape may not be a good representation of the true wave, that is the local
elevations or pressures may be wrong.
. The wave length may be in error.
This second case causes a poor representation of the wave in one part of the problem
to cause errors in another part of the problem. This e€ect, where errors build up
across the model, is called a pollution error. It has been implicitly understood since
the early days of modelling of the Helmholtz equation, as can be seen from the
uniform size of ®nite element used in meshes.
BabusÏ ka et al. 13;14 show some results for various ®nite element models, using di€erent
element types, and the error as a function of element size, h, and wave number, k.
The sharper error results show that the simple rule of thumb given above is not always
adequate. Since the wave number, k, and the wavelength, , are related by k ˆ 2 = ,
the condition of 10 nodes per wavelength can be written as kh 0:6. But keeping to
this limit is not su cient. The pollution error grows as k 3 h 2 . BabusÏ ka et al. propose
a posteriori error indicators to asssess the pollution error. See the cited references and
Chapter 14, Volume 1, for further discussion of these matters.