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

This paper introduces an adaptive material homotopy continuation framework that guarantees branch identity continuity and enables robust, automated computation of dispersion curves in viscoelastic waveguides of arbitrary cross-section by mapping the non-Hermitian lossy problem to an auxiliary lossless one while effectively handling exceptional points and mode tracking challenges.

The Gist

The Big Picture: Mapping a Foggy Mountain

Imagine you are trying to draw a map of a mountain range. This mountain represents how sound waves travel through a material (like a carbon fiber wing on a plane).

• The "Elastic" Mountain (Clear Day): In a perfect, lossless material (like a stiff spring), the mountain is clear. You can see every peak and valley perfectly. The paths (waves) are distinct and easy to follow.
• The "Viscoelastic" Mountain (Foggy Day): Real-world materials (like the carbon fiber with glue) absorb energy. This is like a thick fog rolling in. The paths get blurry, they twist around each other, and sometimes two paths seem to merge into one before splitting apart again. This is called "mode veering."

The Problem:
Existing methods for drawing this map try to navigate the foggy mountain directly. They start in the fog, guess where a path is, and try to follow it. But because the fog is so thick and the paths twist so wildly, the map-makers often get lost. They might accidentally switch from following Path A to Path B, or they might miss a path entirely. This results in a broken, inaccurate map.

The Solution: The "Homotopy" Elevator

The authors of this paper propose a clever new strategy. Instead of trying to navigate the fog directly, they build an elevator that connects the clear day to the foggy day.

1. Step 1: Map the Clear Day First.
   They start at the bottom of the elevator where the air is perfectly clear (the "elastic" state). Here, the paths are straight and distinct. They draw the entire map perfectly, labeling every single path (Mode 1, Mode 2, etc.) with 100% certainty.

2. Step 2: The Slow Ride Up.
   They then slowly press the button to go up the elevator. As they rise, the fog (material damping/loss) gradually thickens.

   • The Magic Trick: Because they are moving slowly and continuously, they can watch the paths they already labeled. Even as the fog gets thicker, they can see that "Path A" is still "Path A," just slightly distorted. They don't have to guess; they just follow the trail they already made.

3. Step 3: Arriving at the Fog.
   By the time they reach the top (the "viscoelastic" state), they have a complete, accurate map of the foggy mountain. Because they never lost track of the paths during the ride, the labels they gave the paths at the bottom are still correct at the top.

Key Concepts Explained Simply

1. The "Branch Identity" (The Name Tag)
In the foggy world, paths can get very close and look like they are swapping places.

• Old Way: If you look at the foggy mountain, you might think, "Oh, that path looks like it's crossing the other one," and you accidentally swap their names.
• New Way: Because the authors tracked the paths from the clear day, they know for a fact that "Path A" never actually swapped with "Path B." They kept the name tags on the right paths the whole time.

2. The "Exceptional Points" (The Foggy Swirls)
Sometimes, the fog gets so thick that two paths actually merge into a single swirling vortex before splitting again. This is called an "Exceptional Point."

• Type I (Safe Zone): In most common materials, these swirls happen "off to the side" in the mathematical world. The paths on our map just get close, wiggle, and pass each other without getting confused. The new method handles this perfectly.
• Type II (Danger Zone): If the material is extremely lossy (very thick fog), the swirl might move right onto the path. In this rare case, the paths do swap identities. The paper admits that if this happens, the automatic name tags might get mixed up. However, the method is smart enough to sound an alarm: "Hey, the paths are doing something weird here; you might need to swap the labels manually."

3. Why This is Better

• Old Methods: Like trying to walk through a dense forest in the dark, tripping over roots and guessing which way is North. You often end up lost or walking in circles.
• This Method: Like walking through the forest in broad daylight, memorizing the route, and then walking it again in the dark knowing exactly where every tree is.

What the Paper Actually Proved

The authors tested this "elevator" method on several different shapes of materials (flat plates, unsymmetrical stacks, and an L-shaped bar).

• The Result: Their method produced perfect, continuous maps for almost every test case, even when the material was quite "lossy" (absorbing sound).
• The Comparison: They compared it to the current best software (called "Dispersion Calculator"). The old software often got lost, missed paths, or drew jagged, broken lines in the tricky areas. The new method drew smooth, correct lines every time.
• The Limit: The method works best when the "fog" isn't too thick. If the material is extremely lossy (a very rare case), the automatic labels might get confused, but the math behind the scenes is still accurate; you just need to fix the labels at the end.

Summary

This paper introduces a smart way to calculate how sound waves move through complex, energy-absorbing materials. Instead of struggling to solve the messy problem directly, it solves the clean version first and then slowly transforms the solution into the messy version. This guarantees that the "identity" of the waves is never lost, resulting in a much more reliable and accurate map for engineers.

Technical Summary

Technical Summary: Adaptive Homotopy Continuation for Robust Dispersion Curve Computation

Problem Statement
The accurate computation of dispersion curves in viscoelastic waveguides of arbitrary cross-section presents two intertwined numerical challenges: solving non-Hermitian eigenvalue problems and reliably tracking modal branches across frequency. In lossy materials, such as carbon fiber-reinforced polymers (CFRP), material damping introduces complex arithmetic that destroys the Hermitian structure of system matrices. This leads to a polynomial eigenvalue problem in the complex wavenumber domain where standard iterative solvers struggle with shift-parameter selection, and root-searching methods become cumbersome in densely populated mode regions.

Crucially, the presence of material damping complicates mode tracking. In regions of mode veering, eigenvectors exchange characteristics rapidly over narrow frequency intervals. In non-Hermitian systems, these regions may involve exceptional points (EPs) where eigenvalues and eigenvectors coalesce, causing the Jacobian of the extended system to become singular. Existing methods, which typically fix the material state at the target viscoelastic configuration and attempt direct continuation in the frequency domain, often fail to maintain correct mode identity. This results in "mode jumping," artificial crossings, or the complete omission of modes, particularly in unsymmetric laminates or high-damping scenarios.

Methodology
The paper proposes an adaptive homotopy continuation (HC) framework that decouples the difficult non-Hermitian mode tracking problem from the eigenvalue solution process. The core innovation is a material homotopy parameterized by a scalar attenuation factor $s \in [0, 1]$, which continuously maps the target viscoelastic (lossy, non-Hermitian) problem at $s=1$ to an auxiliary elastic (lossless, Hermitian) problem at $s=0$.

The methodology proceeds in two distinct stages:

1. Stage 1: Hermitian Solution and Mode Tracking ($s=0$):

   • The framework first solves the dispersion problem for the lossless elastic limit. In this regime, the system is Hermitian, ensuring real eigenvalues and mutually orthogonal eigenvectors.
   • Mode tracking is performed using the Modal Assurance Criterion (MAC) combined with an adaptive wavenumber refinement strategy. This strategy inserts sampling points in regions of rapid eigenvector variation (veering regions) to ensure unambiguous modal connectivity.
   • A sparse mapping strategy selects a subset of "key" solution points from the dense Hermitian dataset. These points are chosen to preserve modal continuity (via MAC thresholds) and geometric accuracy (via interpolation error thresholds), significantly reducing the computational load for the subsequent stage.

2. Stage 2: Adaptive Homotopy Path Tracking ($s=0 \to s=1$):

   • The identified key solutions are propagated from the elastic state to the target viscoelastic state using a predictor-corrector homotopy algorithm with arc-length parameterization.
   • The homotopy path is defined in the material parameter space ($s$) at fixed real frequencies, rather than in the frequency domain. This decouples the tracking process from the complex modal interactions (veering) that occur in the frequency domain at the target state.
   • An adaptive step-size control adjusts the homotopy step $\Delta s$ based on the local eigengap and path curvature, ensuring robustness near regions of strong modal interaction.
   • The algorithm employs a backward refinement step to ensure the solution at $s=1$ is obtained with full Newton accuracy.

Theoretical Foundations
The framework relies on analytic perturbation theory, which guarantees that eigenpairs vary analytically with respect to the homotopy parameter $s$ as long as the path does not intersect an exceptional point in the complex $s$-plane. This ensures a branch identity continuity—a one-to-one correspondence between solutions at $s=0$ and $s=1$.

The paper distinguishes between two topological regimes based on the location of EPs in the complex frequency plane relative to the real axis:

• Type I Topology: The two EPs governing a modal interaction lie on opposite sides of the real frequency axis. In this regime, the physical mode labels established at $s=0$ are automatically inherited at $s=1$ without post-processing. The resulting dispersion curves exhibit characteristic "Type I behavior": real-part veering accompanied by imaginary-part crossing.
• Type II Topology: The EPs lie on the same side of the real axis. Here, the automatic inheritance of labels fails, and mode labels must be exchanged at the crossing point to maintain physical continuity.

Key Contributions

1. Theoretical Synthesis: The authors synthesize analytic perturbation theory, EP physics, and non-Hermitian crossing theory to establish that branch identity is preserved along the material homotopy path provided no EP is crossed. They clarify that for Type I topologies, physical mode labels are automatically preserved, eliminating the need for heuristic post-processing.
2. Methodological Framework: A complete computational framework is developed that performs accurate mode tracking in the Hermitian regime and propagates these identities to the viscoelastic state. The method is inherently parallelizable and applies to both frequency-independent and frequency-dependent damping models.
3. Diagnostic Tools: For cases where the Type I assumption is violated (Type II regime), the paper identifies two physically motivated post-hoc diagnostic signatures:
   • An extremely sharp imaginary-part crossing.
   • A discernible discrepancy between the spectral group velocity ($v_g$) and the energy flux velocity ($v_e$).
     These indicators alert users to potential label mismatches, guiding necessary manual label exchanges.

Numerical Results and Validation
The framework was validated against the widely used Dispersion Calculator (DC) toolbox across five numerical examples:

• Symmetric Laminates (Sym1, Sym2): The method correctly resolved symmetry-protected crossings and veering regions where DC misidentified veering as crossings or produced artificial separations in the imaginary wavenumber.
• Unsymmetric Laminates (UnSym1, UnSym2): In these cases, pervasive mode veering caused DC to miss modes entirely or suffer from spurious mode jumps. The HC framework successfully captured the full spectrum with correct modal connectivity, even in the presence of dense veering.
• Arbitrary Cross-Section (L-shaped Bar): The method demonstrated applicability to general 2D geometries, correctly capturing veering behavior and the subtle offset between $v_g$ and $v_e$ caused by non-Hermitian damping.
• High Damping (UnSym3, $\eta=0.05$): In a regime approaching Type II topology, the HC method continued to produce numerically accurate wavenumbers where DC failed. The diagnostic signatures (sharp imaginary crossing and $v_g$-$v_e$ discrepancy) successfully flagged the need for label exchange.

Significance and Claims
The paper claims that the proposed adaptive homotopy continuation framework offers a robust and theoretically grounded alternative to existing dispersion computation methods. By decoupling mode tracking from the non-Hermitian solution process, the method overcomes the fragility of direct frequency-domain continuation in the presence of exceptional points and mode veering.

The authors emphasize that the framework guarantees branch identity continuity along the homotopy path, independent of the target system's EP topology. While physical label inheritance is automatic only for Type I topologies, the method ensures that even in Type II regimes, the numerical solutions remain accurate, with diagnostic tools provided to guide the necessary label corrections. This approach transforms the dispersion solver from a "black box" prone to irreversible errors into a transparent tool capable of handling complex modal interactions in viscoelastic waveguides of arbitrary cross-section. The method is shown to be computationally efficient, particularly for symmetric laminates, and scalable to large finite element models through parallel processing and adaptive sparse mapping.