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

264 Waves
r/a
5
4
3
2
1
0 0 2 4 6 8
arise on the boundary at in®nity.y This is discussed by Bettess. 40 The terms can be
evaluated, but they are not symmetrical (or hermitian), and therefore impose a
change of solution technique. An alternative, which eliminates the terms at in®nity,
was proposed by Astley et al. 48 In this a `geometrical factor' is included in the
weighting function, which then takes the form
N i…r; † r i
r
3
e ik…r ÿ r i†
…8:43†
It has been shown that this form of weighting functions gives very good results. Such
wave envelope in®nite elements have been further developed by Coyette, Cremers and
Fyfe. 49;50 These elements have incorporated a more general mapping than that in the
original Zienkiewicz et al. mapped in®nite wave element. Cremers and Fyfe allow the
mapping to vary in the local and directions.
8.14 Accuracy of in®nite elements
The use of a complex conjugate weighting in the wave envelope in®nite elements means
that the original variational statement, Eq. (8.22), must be changed to allow the use of
the di€erent weighting function. This gives rise to a number of issues relating to the
nature of the weighted residual statement and the existence of various terms. These
issues were touched on by Bettess, 40 but have been subsequently subjected to more
detailed study. Gerdes and Demkowitz, 55;56 analysed the wave envelope elements,
and subsequently the wave in®nite elements. 57 Some of this work is restricted to
z/a
Incident mode number = 1
Reduced frequency ka == 11
Spiral mode number m φ = 8
Plane wavelength
Conventional FEM
Wave envelope FEM
Fig. 8.9 Computed acoustical pressure contours for a hyperbolic duct … 0 ˆ 708, ka ˆ 11, m ˆ 8†. Conventional
and wave envelope element solutions, Astley. 20
y Some writers, particularly mathematicians, prefer to call the usual wave in®nite elements, unconjugated
in®nite elements, and the Astley type wave envelope in®nite elements, conjugated in®nite elements.