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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

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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