Rigorous foundations of adaptive mode tracking in single-parametric Hermitian eigenvalue problems: existence theorems, error indicators, and application to SAFE dispersion analysis

TL;DR

This paper develops a rigorous theoretical framework for mode tracking in Hermitian eigenvalue problems from SAFE formulations, introducing adaptive sampling and error indicators to improve accuracy and efficiency in dispersion analysis.

Contribution

It provides explicit eigenvector derivative expressions, guarantees for mode identification, and an adaptive sampling algorithm with a novel error indicator for reliable dispersion curve computation.

Findings

Robust mode tracking in veering regions with fewer sampling points.

Validation on various structures confirms accuracy and efficiency.

Theoretical guarantees ensure unambiguous mode identification.

Abstract

The Semi-Analytical Finite Element (SAFE) method is widely used for computing guided wave dispersion curves in waveguides of arbitrary cross-section. Accurate mode tracking across consecutive wavenumber steps remains challenging, particularly in mode veering regions where eigenvalues become nearly degenerate and eigenvectors vary rapidly. This work establishes a rigorous theoretical framework for mode tracking in single-parameter Hermitian eigenvalue problems arising from SAFE formulations. We derive an explicit expression for the eigenvector derivative, revealing its inverse dependence on the eigenvalue gap, and prove that for any wavenumber and mode there exists a sufficiently small step ensuring unambiguous identification via the Modal Assurance Criterion. For symmetry-protected crossings, the Wigner-von Neumann non-crossing rule guarantees bounded eigenvector derivatives and reliable tracking even with coarse sampling. For continuous symmetries leading to degenerate subspaces, we introduce a rotation-invariant subspace MAC that treats each degenerate pair as a single entity. Based on these insights, we propose an adaptive wavenumber sampling algorithm that automatically refines the discretization where the MAC separation falls below a tolerance, using a novel error indicator to quantify tracking confidence. Validation on symmetric and unsymmetric laminates, an L-shaped bar, and a steel pipe demonstrates robust tracking in veering regions with substantially fewer points than uniform sampling or continuation-based approaches, while comparisons with open-source codes SAFEDC and Dispersion Calculator confirm accuracy and efficiency. The framework provides both theoretical guarantees and practical tools for reliable dispersion curve computation.