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

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.

Figures (7)

• Figure 1: Flowchart of the adaptive wavenumber sampling algorithm. The process iteratively solves eigenvalue problems, computes error indicators, and refines the grid where the error exceeds the tolerance.
• Figure 2: Dispersion curves (dimensionless frequency $\omega$ versus dimensionless wavenumber $k$) for the symmetric laminated plate $[0,90,45,-45]_{2s}$: (a) initial coarse sampling ($\Delta k = 0.1$, 70 points) showing mode misidentification in the veering region (black square) due to insufficient resolution; (b) adaptive refinement result (98 points) correctly tracking all modes after automatic refinement guided by the error indicator; (c) SAFEDC with dense uniform sampling ($\Delta k = 0.02$, 350 points) resolving the veering but at high computational cost; (d) Dispersion Calculator (DC) results ($\Delta \omega = 0.0209$) confirming the absence of crossings within each symmetry family and validating the veering structure. Symmetric modes (S) are shown in blue, antisymmetric modes (A) in red. The inset in (d) reveals the avoided crossing between two S-mode branches.
• Figure 3: Error indicator $\varepsilon(k,\Delta k)$ before and after adaptive refinement for all three validation cases: (a) symmetric laminate, (b) unsymmetric laminate, (c) L-shaped bar. In every example, the refinement systematically reduces $\varepsilon$ below the tolerance $\bar{\varepsilon}=0.05$ across the entire wavenumber range, demonstrating that the adaptive algorithm successfully identifies and resolves all regions where the initial step size exceeded the local critical threshold $\Delta k_{\max}(k)$. The post-refinement error indicators provide quantitative confirmation that the correct tracking condition \ref{['eq: theorem_inequality']} holds for all modes and all intervals.
• Figure 4: Dispersion curves ($\omega$ versus $k$) for the unsymmetric laminated plate $[0,90,45,-45]_{4}$: (a) initial coarse sampling ($\Delta k = 0.1$, 70 points) exhibiting multiple mode misidentifications (black squares) due to insufficient resolution in veering regions; (b) adaptive refinement result (185 points) correctly tracking all modes after automatic refinement guided by the error indicator; (c) SAFEDC with dense uniform sampling ($\Delta k = 0.02$, 350 points) reducing but not eliminating tracking errors (one remaining misidentification); (d) DC results ($\Delta \omega = 0.01045$) confirming the absence of true crossings among all coupled modes, but failing to detect all modes (one modes missing).
• Figure 5: Dispersion curves for the L‑shaped aluminum bar: (a) cross‑section geometry; (b) initial coarse sampling ($\Delta k = 0.1$, 50 points) – multiple misidentifications (black squares) occur in veering regions; (c) adaptive refinement (191 points) – all modes correctly tracked after error‑driven refinement; (d) SAFEDC with dense uniform sampling ($\Delta k = 0.02$, 250 points) – reduces but does not eliminate misidentifications (circled residual error). The reference NCM (Fig. 7 in Ref. maruyama_continuation_2025) confirms the absence of true crossings, making veering the sole source of tracking difficulty. Adaptive sampling achieves perfect tracking with fewer points than uniform fine sampling.