Reliable computation of roots in analytical waveguide modeling using an interval-newton approach and algorithmic differentiation

For the modeling and simulation of wave propagation in geometrically simple waveguides such as plates or rods, one may employ the analytical global matrix method. That is, a certain (global) matrix depending on the two parameters wavenumber and frequency is built. Subsequently, one must calculate all parameter pairs within the domain of interest where the global matrix becomes singular. For this purpose, one could compute all roots of the determinant of the global matrix when the two parameters vary in the given intervals. This requirement to calculate all roots is actually the method´s most concerning restriction. Previous approaches are based on so-called mode-tracers, which use the physical phenomenon that solutions, i.e., roots of the determinant of the global matrix, appear in a certain pattern, the waveguide modes, to limit the root-finding algorithm´s search space with respect to consecutive solutions. In some cases, these reductions of the search space yield only an incomplete set of solutions, because some roots may be missed as a result of uncertain predictions. Therefore, we propose replacement of the mode-tracer approach with a suitable version of an interval- Newton method. To apply this interval-based method, we extended the interval and derivative computation provided by a numerical computing environment such that corresponding information is also available for Bessel functions used in circular models of acoustic waveguides. We present numerical results for two different scenarios. First, a polymeric cylindrical waveguide is simulated, and second, we show simulation results of a one-sided fluid-loaded plate. For both scenarios, we compare results obtained with the proposed interval-Newton algorithm and commercial software.