Mathematical derivation of simplified calculation of symmetrical Green function

Abstract

[Objectives] This paper studies the way to reduce the elements used for calculating the hydrodynamic coefficient for movement of a large floating body with boundary element method and to quicken the calculation.[Methods] Based on the characteristic of the symmetrical floating body that the coefficient matrix can be considered as a circulant matrix during solvement of source intensity of singular point,a simplified calculation method of symmetrical Green function is proposed. Besides,detailed mathematical derivation is carried out and the physical meanings are explained.[Results] It is found that, through this method, the process of solving the hydrodynamic coefficient for wave movement of symmetrical floating body can be quickened significantly. For the floating body with one symmetrical plane,it only needs to solve half of the elements of the coefficient matrix of source intensity;meanwhile the time of solving linear equations is reduced by one quarter. And for the floating body with two symmetrical planes,it only needs to solve a quarter of elements of the coefficient matrix of source intensity;meanwhile the time of solving linear equations is reduced by one sixteenth.[Conclusions] The numerical results show that the proposed symmetrical simplification method is effective,and can greatly improve the calculation efficiency.

Keywords

• symmetrical floating body
• Green function
• circulant matrix