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Twenty-First Symposium on NAVAL HYDRODYNAMICS
oped to overcome these difficulties. Most of the higher-order methods are based on piecewise polynomial approximations of the geometry and potential on each panel, usually restricted to linear or quadratic representations using local polynomials of first- or second-degree, respectively. In this paper we describe a different higher-order approach, where B-spline basis functions are used to represent the geometry and potential. This offers the possibility of a more continuous representation, with greater geometrical flexibility and numerical efficiency. Following a preliminary investigation of this technique in two dimensions (Hsin et al., ), we have developed a three-dimensional panel program which will be referred to as ‘HIPAN'. A more detailed description of the method is contained in the thesis of Maniar .
HIPAN was developed initially to solve linearized radiation-diffraction problems in the frequency domain. Thus the body is fixed, or performing small oscillatory motions about a fixed mean position, and plane progressive waves are incident from a prescribed direction. The fluid is either of constant finite depth, or infinitely deep. The body may be floating on the surface or submerged. For this case the free-surface Green function can be used effectively. The analogous low-order panel method WAMIT, which is described by Lee , can be used to test the accuracy and relative computational efficiency of HIPAN. The efficiency of HIPAN is most apparent for relatively complicated body shapes. This will be illustrated by considering a large array of floating cylinders similar to a floating bridge. In that particular problem we find not only that substantially larger arrays can be analyzed, but also that the results display very interesting features closely related to the occurrence of trapped waves in a channel.
In many applications important nonlinear effects must be analyzed, particularly the second-order sum- and difference-frequency loads which occur at relatively high and low frequencies compared to the first-order wave spectrum. High-frequency loads are important in cases where resonant structural response is encountered. Examples include hull deflections of long slender ships, bending of vertical monotowers, and vertical motions of tension-leg platforms. Conversely, low-frequency loads are important for rigid-body motions where the restoring forces are relatively weak. Examples include the horizontal oscillations of moored and towed vessels, and the vertical response of vessels with small waterplane areas. The analysis of these second-order loads can be performed using a low-order panel method, but the computational burden is quite large and much care is necessary to ensure robust results. The extension of HIPAN to include second-order loads is currently in progress, and we are not able to show complete computations, but preliminary results which have been obtained demonstrate the improvements that may be achieved using the B-spline methodology in the second-order analysis.
It should be noted that we refer to ‘order' in two completely different contexts here, one specifying the numerical approximation of the geometry and velocity potential on the body surface, and the other referring to the order of the perturbation expansion in terms of the amplitude of the waves and body motions. Low-order and higher-order panel methods are distinguished by the type of numerical approximations used, whereas first-and higher-order wave loads are defined with respect to the corresponding powers of the wave amplitude.
Recent attention in the offshore community has been directed toward higher-order nonlinear loads which are thought to cause ‘ringing', a hydrodynamic/structural resonance which has been observed for large platforms in extreme wave conditions. The relevant frequencies suggest that ringing may be caused by third-harmonic wave loads, and this has motivated two recent studies which are restricted to the simplest possible geometrical configuration, a vertical circular cylinder. In the work of Faltinsen et al.  a perturbation scheme is employed appropriate to the regime where the cylinder radius and wave amplitude are of comparable magnitude, and both are small compared to the wavelength. In the complementary work of Malenica & Molin , a conventional Stokes expansion is used with the wave amplitude assumed small compared to the cylinder radius and wavelength and without restricting the radius/wavelength ratio. Partly to clarify the differences between these two works, and also to develop a more general computational tool for analyzing practical bodies, we have extended HIPAN to include third-order loads. Here too our present results are incomplete, but they do indicate that the Stokes expansion may be more appropriate for contemporary platforms.
In Sections 2–3 we review the boundary-value problems to be addressed, and the formulation of the panel method for solving these problems based on B-splines. Solutions of several illus-