A Sylvester equation approach for the computation of zero-group-velocity points in waveguides

TL;DR

This paper introduces a Sylvester equation-based method to efficiently compute zero-group-velocity points in waveguides, enabling analysis of larger and more complex structures than previous approaches.

Contribution

It develops a novel Sylvester equation approach that improves the efficiency of computing ZGV points, extending applicability to larger and more complex waveguide problems.

Findings

Enables computation of ZGV points for large matrices

Applicable to multi-layered plates and 3D structures

Improves efficiency over existing methods

Abstract

Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest. A particular example is found in the dispersion curves of elastic waveguides, where such points are called zero-group-velocity (ZGV) points. Recently, it was revealed that the problem of computing ZGV points can be modeled as a multiparameter eigenvalue problem (MEP), and several numerical methods were devised. Due to their complexity, these methods are feasible only for problems involving small matrices. In this paper, we improve the efficiency of these methods by exploiting the link to the Sylvester equation. This approach enables the computation of ZGV points for problems with much larger matrices, such as multi-layered plates and three-dimensional structures of complex cross-sections.