Computing zero-group-velocity points in anisotropic elastic waveguides: Globally and locally convergent methods

TL;DR

This paper introduces three numerical methods for accurately computing zero-group-velocity points in anisotropic elastic waveguides, which are crucial for nondestructive testing and structural characterization.

Contribution

It develops a two-parameter eigenvalue model for ZGV points and presents globally convergent, locally convergent, and hybrid algorithms for their computation.

Findings

The globally convergent method finds all ZGV points but is limited to small problems.

The Newton-type method is fast and applicable to larger problems with initial guesses.

The hybrid method combines approaches to handle large problems without initial guesses.

Abstract

Dispersion curves of elastic waveguides exhibit points where the group velocity vanishes while the wavenumber remains finite. These are the so-called zero-group-velocity (ZGV) points. As the elastodynamic energy at these points remains confined close to the source, they are of practical interest for nondestructive testing and quantitative characterization of structures. These applications rely on the correct prediction of the ZGV points. In this contribution, we first model the ZGV resonances in anisotropic plates based on the appearance of an additional modal solution. The resulting governing equation is interpreted as a two-parameter eigenvalue problem. Subsequently, we present three complementary numerical procedures capable of computing ZGV points in arbitrary nondissipative elastic waveguides in the conventional sense that their axial power flux vanishes. The first method is globally convergent and guarantees to find all ZGV points but can only be used for small problems. The second procedure is a very fast, generally-applicable, Newton-type iteration that is locally convergent and requires initial guesses. The third method combines both kinds of approaches and yields a procedure that is applicable to large problems, does not require initial guesses and is likely to find all ZGV points. The algorithms are implemented in "GEW ZGV computation" (doi: 10.5281/zenodo.7537442).