A finite element method with cover functions for underwater acoustic propagation problems

Abstract

To improve the performance of the classical finite element method (FEM) in acoustic analyses, in this work the interpolation cover functions are incorporated into the standard linear nodal shape functions to give the finite element with cover functions (FEwC). Since the original linear approximation space of the standard FEM is significantly enriched, more accurate solutions can be obtained. However, this FEwC encounters the intractable linear dependence (LD) problem. In this paper the LD problem is studied systematically and then an effective method is given to completely avoid this problem. The imposition of Dirichlet boundary conditions in FEwC which is different from that in FEM is also illustrated. Besides, to handle the acoustic scattering problems in the unbounded domains, the Dirichlet-to-Neumann (DtN) map technique is employed as the non-reflecting boundary condition. The dispersion error analyses and several typical numerical experiments associated with the joint frequency-domain and time-domain analysis are performed to exhibit the high performance of the FEwC compared with the classic FEM and the edge-based smoothed FEM in solving underwater acoustic propagation problems.