Semi-analytic solutions to edge singularities of three-dimensional axisymmetric bodies

Abstract

Axisymmetric geometries, such as cylindrical elements, are widely used in offshore structures. However, the presence of sharp edges in these geometries introduces challenges in numerical simulations due to singularities. To address this issue, one possible solution is to represent the singularities using analytic eigenfunctions. This approach can provide insights into the essence of the problem and has successfully applied to two-dimensional (2D) corner problems. However, finding appropriate eigenfunctions for the three-dimensional (3D) edges remains an open challenge. This paper proposes a semi-analytic scheme for 3D axisymmetric problems utilizing a scaled boundary finite element method (SBFEM). A dimensional reduction is introduced to the 3D Laplace equation, and a 3D edge is handled on the generatrix plane while governed by a complicated equation. The algorithm for resolving the SBFEM fundamental space is improved, and the singularities are approximated using a fractional-order basis. The effectiveness of the proposed method is demonstrated through its application to solve the radiation problem of a heaving cylinder. The method accurately captures the singular velocity field at the edge tip, ensuring that the boundary condition on the body surface is strictly satisfied in the neighborhood of the singularity. Accuracy of the mean drift force is ensured by performing direct pressure integrations over the body surface using a near-field formulation, which becomes as accurate as the middle-field formulation.