Application Of Multipole Expansion Method To Linear Hydrodynamic Computation Of Marine Structures In Deep Oceans

Currently, the deep-sea exploitation has entered an era of the level of a few kilometers depth, which certainly demands higher security requirements on the design and protection of the large maritime floating structures such as ocean platforms. When come to the hydrodynamic analysis and computation of these structures under the action of waves, failure will turn up for the slow convergence of the series of cylindrical functions when using the traditional finite depth calculation method. The multipole expansion method, first proposed by Ursell (1949), is an analytic method for solving hydrodynamic problems in ocean engineering in polar coordinates and spherical coordinates, which will provide an effective and accurate form of the velocity potential in deep water. This thesis studied the multipole expansion method (MEM), mainly focusing on its applications in hydrodynamics, and initially explored the idea of its joint with numerical methods for solving the hydrodynamic problems of arbitrary objects.First, the MEM was adopted to study the analytic solutions of the regular objects. According to Ursell (1950), Evans and Linton (1989), as well as many previous studies, the hydrodynamic model of a two-dimensional submerged horizontal cylinder in infinite water depth was summarized, and also verified by developing a Fortran program. The hydrodynamic model of three-dimensional submerged sphere in infinite depth was independently studied, a new series expression of the multipole expansion coefficient was derived, and the analytic relationship of the hydrodynamic coefficients between the horizontal surge direction and the vertical heave direction when the submergence tends to be infinity was found.Secondly, a joint numerical method combined between MEM and boundary element method (BEM) was developed, to analyze the floating objects in arbitrary shapes. In the two-dimensional (2-D) case, according to the method of Hu (1989), using the simple Green’s function and the multipole expansion expression proposed by Ursell (1950) for floating objects, to couple BEM and MEM; in the mean time, the method of Chau (1989) was extended to the2-D case, by using the BEM based on the2-D free-surface Green’s function to solve the inner domain hydrodynamics, and using the multipole expansion of the2-D free-surface Green’s function, which was proposed by Ursell (1981) and Martin (1981), to express the scattering potential in the outside domain, combining with the outside integral equation. Subsequently, in three-dimensional (3-D) case, by using the method of Chau (1989), the3-D free-surface Green’s function based BEM was adopted to solve the inner hydrodynamics, and the Fourier series expression for the2-D free-surface Green’s function in spherical coordinates, which was proposed by Martin (1981) and Chau (1989), was adopted to express the scattering potential in the outside domain, combining with the outside domain integral equation. These steps laid the foundation for the computation of the free surface integral in the second order problems in the future work.Finally, a numerical algorithm for the2-D free-surface Green’s function in both infinite depth and finite depth within frequency domain was developed. Where, the method suggested by Linton, McIver and Porter, for doing the proper handling of the singularity, was adopted, as well as a new adaptive integration algorithm based on Gauss-Kronrod rule. Subsequently, according to the density of the contour map, the computational domain was divided into several regions, within each of them using the Chebyshev polynomials to approximate the Green’s function’s value, as similar to that done by Newman (1985) and Han Ling (2005), such that the Green’s function could be evaluated most quickly and accurately in the whole range. Numerical examples for both a2-D floating horizontal cylinder and a2-D submerged horizontal cylinder were implemented, which proved the validity and accuracy of the proposed efficient algorithm for Green’s function.This thesis work was part of the research item "Second order wave loads upon marine structures in deep water", which was financed by the National Natural Science Foundation of China (Grant No.11072052).