Study on HOBEM Based on Analytical Panel Integrals Related to Translating-Pulsating Source for Hydrodynamic Responses of Vessels Sailing in Waves

A higher-order boundary element method(HOBEM)incorporated with analytical panel integrals related to translat-ing-pulsating source Green’s function is proposed for the hydrodynamic response prediction of ships advancing in waves.In this method,the 9-node bi-quadratic curvilinear elements employed to discretize the mixed-source/dipole boundary integral equation are mapped into the parametric plane through a coordinate transformation.Then in order to ease the numerical instability problem,a novel analytical quadrature is derived to calculate the influence coefficients by changing the integral order and using integration by parts.The singularity caused by infinite discontinuity is analyzed and eliminated by adopting some mathematical techniques.Through the calculations of panel integrals of Green’s function and its x-derivative,the analytical integral method is proved to be always accurate even for field points approaching the free surface,where numerical quadrature is impossible to give reasonable results.Based on this,a higher-order seakeeping program is developed and applied in the motion response prediction of two different types of ships(i.e.,a wall-sided ship Wigley III and a non-wall-sided ship S175).By comparing the computed results with the corresponding experimental data and numerical solutions of the translating-pulsating and higher-order Green’s function methods based on traditional Gauss quadrature,it is found that the HOBEM based on analytical quadrature is of better accuracy and stability.For the non-wall-sided ship,only the present method can produce reasonable pre-diction of motion responses,while obvious oscillatory phenomenon is observed in the results of the other two numerical methods based on Gauss quadrature.