Application of quintic order parabolic arcs in the analysis of waveguides with arbitrary cross-section

Abstract

A simple and efficient higher order finite element scheme is presented for obtaining highly accurate numerical solution for the two-dimensional Helmholtz equation in waveguides of arbitrary cross-section subjected to dirichlet boundary conditions. The above approach makes use of the Quintic order (5th order) parabolic arcs for accurately mapping the irregular cross section of the waveguide and then transforming the entire waveguide geometry to a standard isosceles triangle. In case of waveguides with regular geometry the transformation is done by straight sided quintic order finite elements. A unique and accurate point transformation technique is developed that ensures high accuracy of mapping by this quintic order curved triangular elements. This point transformation procedure gives a simple interpolating polynomial that defines the transformation from the global coordinate system to the local coordinate system. The above higher order finite element method is found to be highly optimal and accurate considering the various computational parameters like the number of triangular elements, degrees of freedom, nodal point distribution on the entire geometry, etc.