Numerical Analysis of Second-Order Mean Wave Forces by a Stabilized Higher-Order Boundary Element Method

A stabilized Higher-Order Boundary Element Method (HOBEM) based on cubic shape functions is presented to solve the linear wave-structure interaction with the presence of steady or slowly varying velocities. The m-terms which involve second derivatives of local steady flow are difficult to calculate accurately on structure surfaces with large curvatures. They are also not integrable at the sharp corners. A formulation of the Boundary Value Problem (BVP) in a body-fixed coordinate system is thus adopted, which avoids the calculation of the m-terms. The use of body-fixed coordinate system also avoids the inconsistency in the traditional perturbation method when the second order slowly varying motions are larger than the first order motions. A stabilized numerical method based on streamline integration and biased differencing scheme along the streamlines will be presented in this paper. The presence of convective terms in the kinematic and dynamic free surface conditions will lead to unstable solutions if the explicit method is used. Thus, an implicit scheme is used in this paper for the time integration of kinematic and dynamic free surface conditions. In an implicit scheme, solution of an additional matrix equation is normally required because the convective terms are discretized by using the variables at current time step rather than that from the previous time steps. A novel method that avoids solving such matrix equation is presented in this paper, which reduces the computational efforts significantly in the implicit method. The methodology is applicable on both structured and unstructured meshes. It can also be used in general second order wave-structure interaction analysis with the presence of steady or slowly varying velocities.