A finite element-scaled boundary finite element coupled method for corner singularities in 2D second-order radiation problems

Abstract

In potential flow analyses, singularities at the corners of bodies lead to erroneous pressure integrations and unsolved high-order velocity potentials. While much research has been devoted to addressing the former difficulty, fewer solutions have been proposed for the latter. In non-linear problems, higher derivatives are required on the boundaries, but usually, they are inaccurate and even singular at the corners. This paper introduces a coupling method that combines the scaled boundary finite element method (SBFEM) and the finite element method (FEM) to overcome this challenge. The SBFEM is semi-analytic and can accurately describe the singularities; the FEM is enhanced with Adini elements to achieve higher continuity. By integrating them, the proposed method allows for evaluating second derivatives on the body surface. The singular boundary conditions at the corners are automatically satisfied by the SBFEM, enabling the well-posed solution of boundary value problems (BVPs). The implementation of the proposed method is demonstrated through the solution of the second-order radiation problem of a rectangular box, with fine convergence observed for the first- and second-order velocity potentials and their gradients. The method also provides accurate and efficient direct pressure integrations for second-order wave forces, with results in good agreement with established indirect methods.