Spherical-wave based triangular finite element models for axial symmetric Helmholtz problems

In this paper, six-node hybrid triangular finite element models are devised for axial symmetric Helmholtz problems. In the formulation, boundary and domain approximations to the Helmholtz field are defined for each element. While the boundary approximation is constructed by nodal interpolation, the domain approximation satisfies the Helmholtz equation and is composed of spherical waves with source points located along the axis of symmetry. To formulate rank sufficient six-node elements, a minimal of six wave modes from three source points are required. Two methods of selecting the source points are attempted. In the first method, the directions of the waves passing through the element are essentially parallel to the three lines connecting the parametric center of the element and its three corner (or side) nodes. In the second method, the directions are essentially equally spaced at 2π/3 interval in the r–z-plane. For the attempted examples, the average error ratios of the proposed elements and the conventional element are around 50%.