Homotopy continuation of viscoelastic waveguide dispersion curves: from intra-manifold tracking to inter-manifold transport

Abstract

Conventional mode tracking operates in the dark: it traces dispersion branches on the non-Hermitian eigenvalue manifold using only local continuity, unaware of the global Riemann-sheet topology. When exceptional points (EPs) lie close to the real frequency axis, the eigenvector similarity that local trackers rely on degrades, and mode tracking becomes unreliable, failing silently. This paper replaces blind intra-manifold tracking with inter-manifold transport. A material attenuation parameter s in [0,1] continuously maps the target lossy problem to an auxiliary lossless one whose Hermitian eigenvalue problem yields a well-posed anchor manifold on which each dispersion branch possesses a globally unique and continuous identity. These identities are defined once on the elastic anchor and then transported to the viscoelastic target via predictor-corrector homotopy continuation; as long as the path avoids all EPs, branch identity is preserved throughout the transport. For any mode pair whose EPs have not crossed the real frequency axis (Type I), the transported identities are inherited automatically. In contrast, when an EP crosses the real axis and becomes Type II, the topology differs from the elastic anchor and a label swap is required. The framework is validated on symmetric and unsymmetric laminates, with most cases at loss factors of 0.003 to 0.02; for all Type I pairs in these cases the identities are inherited without alteration. For a challenging unsymmetric laminate at 0.05, several EP pairs have become Type II, yet the homotopy transport still produces numerically accurate solutions. Two diagnostic signatures--an extremely sharp imaginary-part crossing and a marked discrepancy between spectral group velocity and energy flux velocity--identify where the underlying EP topology demands a label swap.