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

absorbed or damped in such a way that it does not return into the computational
domain. See the papers by Monk and Collino, Hayder and Driscoll in reference 30.
8.9 Linking to exterior solutions
A general methodology for linking ®nite elements to exterior solutions was proposed
by Zienkiewicz et al., 33;34 following various ad hoc developments, and this is also
discussed in Volume 1, particularly in Chapter 13. The exterior solution can take
any form, and those chie¯y used are (a) exterior series solutions and (b) exterior
boundary integrals, although others are possible. The two main innovators in these
cases were Berkho€, 10;35 for coupling to boundary integrals, and Chen and Mei 36;37
for coupling to exterior series solutions. Although the methods proposed are quite
di€erent, it is useful to cast them in the same general form. More details of this
procedure are given in reference 33. Basically the energy functional given in
Eq. (8.23) is again used. If the functions used in the exterior automatically satisfy
the wave equation, then the contribution on the boundary reduces to a line integral
of the form
ˆ 1
…
2 ÿ
@
dÿ …8:25†
@n
It can be shown 15;33;34 that if the free parameters in the interior and exterior are b and
a respectively, the coupled equations can be written
K K T
" #
a f 0
‡ ˆ …8:26†
K K b 0 0
where
K ji ˆ 1
2
…
…
‰…PNj†Ni ‡ Nj…PNi†Š dÿ and Kji ˆ ‰…PNj†…Ni† dÿ …8:27†
ÿ
In the above P is an operator giving the normal derivative, i.e. P @=@n, N is the
®nite element shape function, N is the exterior shape function, and K corresponds
to the normal ®nite element matrix. The approach described above can be used
with any suitable form of exterior solution, as we will see. All the nodes on the
boundary become coupled.
8.9.1 Linking to boundary integrals
Berkho€ 10;35 adopted the simple expedient of identifying the nodal values of velocity
potential obtained using the boundary integral, with the ®nite element nodal values.
This leads to a rather clumsy set of equations, part symmetrical, real and banded, and
part unsymmetrical, complex and dense. The direct boundary integral method for the
Helmholtz equation in the exterior leads to a matrix set of equations
A~g ˆ B @~g
…8:28†
@n
ÿ
Linking to exterior solutions 255