Accurate and efficient hydrodynamic analysis of structures with sharp edges by the Extended Finite Element Method (XFEM): 2D studies

Achieving accurate numerical results of hydrodynamic loads based on the potential-flow theory is very challenging for structures with sharp edges, due to the singular behavior of the local-flow velocities. In this paper, we introduce, perhaps the first time in the literature on marine hydrodynamics, the Extended Finite Element Method (XFEM) to solve fluid-structure interaction problems involving sharp edges on structures. Compared with the conventional FEMs, the singular basis functions are introduced in XFEM through the local construction of shape functions of the finite elements. Four different FEM solvers, including conventional linear and quadratic FEMs as well as their corresponding XFEM versions with local enrichment by singular basis functions at sharp edges, are implemented and compared. To demonstrate the accuracy and efficiency of the XFEMs, a thin flat plate in an infinite fluid domain and a forced heaving rectangle at the free surface, both in two dimensions, will be studied. For the flat plate, the mesh convergence studies are carried out for both the velocity potential in the fluid domain and the added mass, and the XFEMs show apparent advantages thanks to their local enhancement at the sharp edges. Three different enrichment strategies are also compared, and suggestions will be made for the practical implementation of the XFEM. For the forced heaving rectangle, the linear and 2nd order mean wave loads are studied. Our results confirm the previous conclusion in the literature that it is not difficult for a conventional numerical model to obtain convergent results for added mass and damping coefficients. However, when the 2nd order mean wave loads requiring the computation of velocity components are calculated via direct pressure integration, the influence of singularity is significant, and it takes a tremendously large number of elements for the conventional FEMs to get convergent results. On the contrary, the numerical results of XFEMs converge rapidly even with very coarse meshes, especially for the quadratic XFEM. Unlike other methods based on domain decomposition when dealing with singularities, the FEM framework is more flexible to include the singular functions in local approximations.

Automated summary

This paper introduces the Extended Finite Element Method (XFEM) for solving fluid-structure interaction problems involving sharp edges on structures, which introduces singular basis functions through the local construction of shape functions of the finite elements. Studies on a thin flat plate and a forced heaving rectangle show that XFEM has apparent advantages at sharp edges, achieving higher accuracy and efficiency compared to conventional finite element methods.