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• #### Appendix-List of Participants 1077-1084

can absorb waves efficiently in some cases, but are not always effective in general. Even when they are effective, they have to be placed sufficiently far away from the body. This means that the computational domain is usually big, which will require a large number of elements. On the other hand, when the required memory exceeds the physical memory of the computer, only a small percentage of CPU will be used. This makes the calculation extremely inefficient. Wu and Eatock Taylor [5] therefore adopted domain decomposition. The required memory will then depend on the sizes of the subdomains which can always be subdivided if necessary. The continuity across the subdomain is achieved through iteration.

In this work, we shall use the three dimensional finite element method to consider the problem of a vertical cylinder in a wave tank. The methodology is first verified using the analytical solution for a two dimensional wave maker. The case may seem simple enough, but it is found that care is needed in dealing with cross waves. The computer code is also verified by the linearized analytical solution for three dimensional standing waves. The relative merits of various domain decomposition schemes are discussed. Results for the vertical cylinders in the tank are provided.

##### MATHEMATICAL FORMULATION AND NUMERICAL TECHNIQUE

We consider the problem of a vertical cylinder in a wave tank as shown in figure 1. (x,y,z) denotes a Cartesian co-ordinate system with x axis pointing in the longitudinal direction of the tank and z upwards. The origin of the

Fig. 1 The layout of the wave tank

system is located on the mean position of the free surface and the centre of the cylinder. B, L and d in the figure indicate the width, length and depth of the tank, respectively.

Based on the usual assumptions of ideal flow, the velocity potential satisfies Laplace's equation:

2=0 (1)

in the fluid domain Ω. The condition on the piston wave maker can be written as

(2)

where U(t) is the velocity of the wave maker. On the fixed boundary the condition is:

(3)

where n is the normal of the surface pointing out of the fluid domain. On the free surface z=η(x,y,t), the kinematic and dynamic conditions can be written as

(4)

(5)

where g is the gravitational acceleration. These are then combined with the initial conditions which usually assume that the wave elevation and the potential on the free surface are zero.

When the wave generated by the wavemaker encounters the cylinder, it will be diffracted. The reflected wave will travel back towards the wave maker. The transmitted wave, on the other hand, will propagate towards the other end of the tank. As the time step increases, the waves reflected by the wavemaker and the far end of the tank will arrive at the cylinder. This will distort the wave loading on the cylinder. Several approaches have been proposed to absorb the reflected wave (e.g. [1]) and the transmitted wave (e.g. [4]). They are effective in some cases but they all have their limitations. Here we do not intend to investigate the effectiveness of various wave absorption schemes at the far end. Instead we simply use a relatively long tank and the computation is

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