Within the frame of potential theory and the assumption of weak nonlinearity of wave motion, a numerical method is developed for the third-order triple-frequency wave loads on fixed axisymmetric bodies in monochromatic incident waves. Applying the numerical code, numerical computations were carried out for surge and heave forces and pitch moments on a uniform cylinder, truncated cylinders and a hemisphere. Examinations were made of the contribution to third-order forces and moments from potentials at each wave order, and the relation of third-order forces and moments to wave number and drafts of cylinders.

A Fast Melipole Mechod (FMM) is developed as nurnerical approach to the reduction of tile computauorai cost and requirement memory capacity for a large in solving large-scale problems. In this paper it is applied to the boundary integral equation method (BIEM) for current diffraction from arbitrary 3D bodies. The boundary integral equation is dis-crotized by higher order elements, the FMM is applied to avoid the matrix/vector product, and the resulting algebraic e-quatiun is solved by the Generalized Conjugate Residual method (GCR). Numerical examination shows that the FMM is more efficient than the direct evaluation method in computational cost and storage of computers.

A horizontal two-dimensional finite element model is developed in order to estimate the process of scour around a large-scale cylinder due to waves. The present model differs from previous models in the sense that the wave model is based on an elliptic mild slope equation and the sediment transport induced by the steady streaming is considered. The current induced by the gradient of radiation stress is considered and calculated using a depth integrated shallow water equation. The contributions of the Lagrangian drift velocity to the scour is also considered in this model. The model is validated against a few cases where experimental data are available. The comparison of the calculation results with the experimental data indicates that the present numerical model predicts the scour around a large cylinder reasonably well. The effects of Keulegan-Capenter (KC) number, the grain size of sediments and the model scale on scour around a large cylinder are also investigated.