Adaptive homotopy continuation for robust dispersion curve computation in viscoelastic waveguides: guaranteed branch identity continuity

TL;DR

This paper introduces a homotopy continuation framework for efficiently computing dispersion curves in viscoelastic waveguides, ensuring branch identity continuity and reliable mode tracking across complex material states.

Contribution

It develops a systematic, analytic perturbation-based method that guarantees mode label continuity during viscoelastic dispersion curve computation, even at high loss factors.

Findings

Validated robustness and efficiency across various laminate configurations.

Achieved accurate solutions at high loss factors up to 0.05.

Provided diagnostic signatures for potential mode label mismatches.

Abstract

This paper presents the first systematic application of a material homotopy continuation framework for efficient, automated computation of dispersion curves in viscoelastic waveguides of arbitrary cross-section. A material homotopy continuously maps the original lossy problem to an auxiliary lossless one via an attenuation parameter s in [0,1], addressing the core challenges of the non-Hermitian eigenvalue problem. Grounded in analytic perturbation theory, the method guarantees branch identity continuity--a one-to-one correspondence between solutions at s=0 and s=1--provided the real-parameter path does not cross any exceptional points. Under a Type I exceptional point topology, physical mode labels established at the elastic stage remain valid at the viscoelastic stage without post-processing, yielding the characteristic real-part veering with imaginary-part crossing. The decoupling strategy performs reliable mode tracking in the Hermitian regime via adaptive wavenumber refinement, then propagates a sparse set of key solutions to the target viscoelastic state through predictor-corrector homotopy continuation. Numerical examples across symmetric and unsymmetric laminates validate the framework's robustness and efficiency, with the majority of cases verified at a loss factor of approximately 0.003 and a single symmetric laminate providing additional support at 0.02. For a challenging unsymmetric laminate at a loss factor of 0.05, the method still produces numerically accurate solutions; two complementary diagnostic signatures--an extremely sharp imaginary-part crossing and a discernible discrepancy between spectral group velocity and energy flux velocity--warn of potential label mismatch and guide further analysis.