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OCR for page 725
Summary of the Group Discussion on Boundary Integral Method
for Radiation/Diffraction Problems
Chairman: O. M. Faltinsen
Norwegian Institute of Technology
Trondheim, Norway
Co-cha~rman: M. Takaki
Hiroshima University
Hiroshima, Japan
More than 40 participants took part in this
group discussions and actively discussed about
the following topics.
1. Singularities due to the body
Zhao showed the behaviour of the potential
near the corner of a rectangular cylinder by
using a lower order panel method based on
Green's second identity. The method assumes
the velocity potential and its normal deriva-
tive are constant over each element. By com-
paring with analytical solutions it was
demonstrated that the velocity potential at
the element closest to the corner will always
be wrong.
2. Free-surface intersection. 1 ine integral
Zhao showed the behaviour of the wave
elevation near the intersection between the
free-surface and the body surface in the case
of a plate suddenly starts to move with a con-
stant velocity. The solution is based on
linear free-surface condition. The analytical
solution by Roberts shows the wave amplitude
is f inite everywhere. However, the wave
elevation near the intersection point between
the free-surface and the body has infinite
number of oscillations. That means we cannot
numerically solve the problem by using a
finite number of elements and assuming the
velocity potential to be constant over each
element.
Regarding the latter problem, Cointe,
Paul oski, and Takagi commented that the 1 i near
theory is inconsistent, we should treat it as
a nonlinear problem.
Kyouzuka presented 2-D second order forces
by the perturbation theory. Singularities oc-
curred in the 2nd order theory at the inter-
section between the free-surface and the body.
In order to overcome this problem, it is use-
ful to integrate analytically the potential on
a few free-surface elements which are close to
the body.
Higo showed that the line integral did not
effect significantly hydrodynami c f orces on a
vertical circular cylinder. Ohkusu commented
that the line integral effects the wave field
near the body much more than the forces on a
body. therefore the effect of the line in-
tegral should be checked by the values of wave
f ield.
Kashiwagi investigated the validity of a
linear solution for an oscillating and moving
surface piercing body. The solution is based
on the classical free-surface condition. A
singular solution occurs at the intersection
between the free-surface and the body surface.
The multiple expansion method is useful for
overcoming that problem. It was found that
additional contribution to the rate of energy
flux arise from the solution around an inter-
section point.
3. Radiation condition
Takagi and Naito investigated the linear-
and a nonlinear radiation condition due to 2-D
REM (Boundary Elemental Method). They ex-
plained that the idea of the active wave ab-
sorber could be used for the radiation condi-
tion for BEM. Cointe pointed out that we
should not use the word wave radiation condi-
tion. It is better to call it wave absorption
condition.
725
OCR for page 726
4. Calculation of velocity on the body and the
ink-terms
Kinnas told about calculation of velocity
near the body by using a lower order panel
method. There will always be numerical
problems in calculating the velocity at the
body boundary. He recommended that we should
_~ ~ ~ . Zhao mentioned
that a similar problem occurred in the Gal-
culation of the ink-terms.
use a higher order scheme
5. Yerif ication procedures
The following items were addressed by the
chairman.
a. Convergence by increasing panel numbers.
This does not always occur, for instance
sharp corners.
b. Importance of analytical results.
c. How to qualify errors.
near
Ohkusu talked about a 3-D panel method with
forward speed. He carried out numerical cal-
culations on a submerged spheroid to exclude
the problem of the 1 ine integral. A conver-
gence by 3-D panel method becomes better by
increasing number of panels. 1200 panels on a
submerged spheroid are good enough. So we
have already reached to the confident result
from a practical point of view. However. as
the body is close to a free-surface, the ac-
curacy of results becomes worse. This means
the calculations for surface piercing bodies
create probl ems.
Validations of numerical results about 3-D
panel method have still the following
problems:
a. How many panels do we have to use?
b. How should we treat the wave component
with a shorter wavelength?
c.
Are there any singularities at the inter
section between the free-surface and the
body surface?
Lee commented that using establ ished rela-
tions is better than checking the convergence
by increasing the number of panels. This
means we should check the mass conservation
and the body boundary condition etc.
6. Iterative solution of large equation system
Hermans told that there are a lot of
references in the 1 iterature about iterative
solvers. For different problems we should use
different iterative solvers. It is difficult
to know which one is the best for a given
numerical problem.
726
Representative terms from entire chapter:
radiation condition
