(3) Dynamic cable method
In the cases where the influence of mooring line dynamics is appreciable, dynamic cable method should be employed, which simulates the motions of mooring lines based on either the finite element method or finite difference method [82]. In addition to modeling the line weights and tensions with more details, the hydrodynamic rags, added masses, cable inertias and other secondary cable properties such as bending and torsional stiffness are explicitly included in the dynamic cable model. In the model, the mooring lines are no longer simplified into catenary lines staging in the vertical plane. Positions, velocities and accelerations along the mooring line are explicitly solved in the model to show the dynamic responses of the mooring line under the excitation of both foundation movements and surrounding water flows. Since both the inertial and the damping of the mooring line are simulated in the model, it is understandable that the dynamic cable model requires excessive computational powers, which limits its use in a comprehensive computational model concerning the dynamics of the entire floating wind turbine. More details of these three mooring line models can be found in Hall's paper [19].
5.3. Hydrodynamics
The calculation of hydrodynamic loads generally depends on the structures of the foundation. For a semi-submersible foundation, the calculation of hydrodynamic loads is a complex Fluid Structure Interaction (FSI) problem, which need consider the excitations of incident ocean waves, damping resulted from wave radiations, and added mass forces arising from the foundation's motion in the water. In 1950, under the inspiration of simplifying the engineering calculation, Morison et al. [83] presented an empirical Morison's equation to solve for the hydrodynamic loads acting on floating objects. Since the proposition, the equation has been widely used for calculating hydrodynamic loads acting on slender cylindrical bodies emerged into a large body of water [84,85]. In detail, the oscillation amplitude, frequency and random phases are input into a series of superposed linear wave equations [86] from a reasonable wave spectrum to determine the incidentwave kinematics when using Morison's equation. Although it has been demonstrated that it is accurate and reliable to Morison’s equation for the calculation of hydrodynamic forces acting on a bottom-fixed objects, its validity of calculating hydrodynamic loads acting on floating objects are still in doubts. Consequently, alternative methods, such as the first-order hydrodynamic theory, should be employed for the calculation of hydrodynamic loads on floating foundations [78]. It should be noted that the first-order hydrodynamic theory explicitly models the radiation and diffraction effects. The second-order hydrodynamic theory [87] was also proposed for the simulation of a softly-moored foundation floating on the sea surface. Since the semi-submersible foundation, which is moored on seabed by a set of mooring lines, is a typical softlymoored floating object, the second-order hydrodynamic theory may be a good choice for numerically simulating its hydrodynamic responses. In addition, the second-order hydrodynamic theory is