points which are to be used for the integral with respect to (x, y). For these integrals the Gauss quadrature is applied, with 6 × 6 = 36 points used as integration points within one panel.
When the field point is located near or within the integration panel, a special treatment is required for singularities due to functions of 1/R and in R included in the Green function.
In this paper, analytical integrations are performed for the integrals of 1/R and ln R multiplied by the cubic B-spine functions. The analytical procedure adopted is the same as that for higher-order polynomial distributions of sources described in Newman7).
4. FIRST-ORDER PRESSURE FORCE
A solution of (17), together with (16), determines the pressure on the bottom of the structure.
The analysis in the radiation problem including specified elastic modes is an easy extension from the rigid-body problem; hence the forces acting in the i-th direction due to the j-th mode of motion can be defined as follows:
We note that the integrals needed for evaluating (22) and (24) are the same as in (20).
It is important to validate the results of numerical computations. The most reliable method may be to verify numerical convergence of pressure forces as the number of panels increases; however, this is laborious particularly when the computation time is long.
Another method, which is possibly easier, is to check numerically various hydrodynamic relations which are theoretically proven. In this paper two different relations are considered: One is the energyconservation principle associated with the damping force, and the other is the Haskind relation between the radiation and diffraction problems.
Omitting derivations, final results can be expressed in the following form:
is defined as the Kochin function.
Equation (25) must be equal to the same coefficient in (22), and likewise (26) must be equal to (24).
5. RESULTS AND DISCUSSIONS
Computations were performed for a rectangular plate of L/B = 5.0 in the incident regular wave of β = 30 degrees. Since our interest is placed on short wavelength region, the finite-depth effect may be negligible and thus was taken as infinity.
The discretization of the structure into panels is made such that the region of x > 0 and 71 > 0 is subdivided
into NX in the x-axis and NY in the 71-axis with ratio NX/NY 5.0, meaning that each panel is a square.
The numbers of mode functions defined in Table 1 were taken equal to 25, and all computations were implemented, using a scalar workstation of HP 9000 series/model 735.
The developed computer program is not fully optimized. Nevertheless, as shown in Table 2, the average computation time for one wavelength, including numerical check of the Haskind relation and the energconservation principle, was found rather fast.
Table 2 Computation time vs. numbers of panels
Convergence of numerical results can be seen in Figs. 3 and 4, where Fig. 3 shows the added mass in heave mode as a rigid body ( ζj= u0 (x)v0(y) in Table 1) and Fig.4 the damping coefficient in the same heave mode. The abscissa is the length ratio L/A.
It is clearly seen that the results approach smoothly converged values as the number of panels increases. Rough estimate indicates that reliable results may be obtained, provided NX > 0.8 *L/λ is satisfied.
It is noteworthy that the computed results are very smooth at least for computed points of L/A and no phenomena of irregular frequencies can be seen.
Another thing to be emphasized is that all the results plotted in Figs. 3 and 4 satisfy quite accurately the Haskind relation and the energy-conservation principle, even when the results are away from converged values. One example for that is shown in Table 3, which shows the relative error in the energyconservation principle, computed for L/λ 45 with NX = 40 and NY = 8. Surprisingly the error is within 0.03% for all specified mode shapes. (Results for higher mode shapes are omitted due to space limit.)
