Physics-Informed Single Atom Matching Pursuit: Guided-Waves Wavenumbers and Propagation Distance Estimation for Damage Localization in Structural Health Monitoring

Technical Summary: Physics-Informed Single Atom Matching Pursuit for Guided-Wave Damage Localization

Problem Statement
Structural Health Monitoring (SHM) of thin-walled components often relies on Ultrasonic Guided Waves (GWs) generated by piezoelectric transducers (PZTs). A primary challenge in this domain is the simultaneous excitation of multiple dispersive wave modes (specifically the $A_0$ and $S_0$ modes) by a single actuator. As these wave packets propagate, they interact with boundaries and potential damages, creating complex signals with overlapping and dispersed contributions. Conventional signal processing techniques face significant limitations:

• Dictionary-based methods (e.g., standard Matching Pursuit) require pre-computed dictionaries of all possible modal contributions, which is impractical for continuous structures and limits adaptability.
• Non-dictionary-based methods (e.g., EMD, VMD, or data-driven AutoEncoders) often lack a strong connection to the underlying physics of wave propagation, resulting in extracted features that are mathematically sound but physically uninterpretable.

The core difficulty addressed in this work is the need for a decomposition framework that is both computationally efficient and grounded in the physical principles of wave propagation, enabling the direct extraction of modal wavenumber functions and propagation distances without prior knowledge of the structure's specific dispersion curves.

Methodology: Physics-Informed Single Atom Matching Pursuit (PISAMP)
The authors propose the Physics-Informed Single Atom Matching Pursuit (PISAMP) method, a signal decomposition technique that models measured signals as delayed and dispersed impulse responses of an external excitation. The methodology is built upon the following principles:

1. Physical Formulation: The method models the signal in the frequency domain. A measured signal $\hat{s}(\omega)$ is approximated as a linear combination of $N$ atoms, where each atom represents a wave packet traveling a specific distance $d_i$ with a specific wavenumber function $k_i(\omega)$:
   $$\hat{s}(\omega) \approx \sum_{i=1}^{N} \alpha_i(d_i) \hat{x}(\omega) e^{-j k_i(\omega) d_i}$$
   Here, $\hat{x}(\omega)$ is the excitation, $\alpha_i$ is the amplitude (modeling attenuation), and the exponential term encodes both the propagation distance and the dispersive effects via the wavenumber function $k_i(\omega)$.

2. Iterative Optimization: The problem is solved using a greedy, incremental approach. For each term $i$, the algorithm jointly optimizes the amplitude $\alpha_i$, the propagation distance $d_i$, and the wavenumber function $k_i(\omega)$ to minimize the reconstruction error in the frequency domain.

   • Initialization: The wavenumber is initialized using a non-dispersive estimate (e.g., the $S_0$ mode).
   • Amplitude and Distance: These are determined by minimizing the error with respect to the distance, treating the wavenumber as fixed from the previous iteration.
   • Wavenumber Update: The wavenumber function is updated using a first-order Taylor linearization. The correction $\Delta k_i(\omega)$ is approximated using Chebyshev polynomials of the second kind to ensure smoothness and numerical efficiency.

3. Scale Calibration and Damage Localization: A critical challenge identified is the scale indeterminacy of the exponential term, where the product $k_i(\omega)d_i$ remains constant even if $k_i$ and $d_i$ are scaled inversely. To resolve this:

   • The method uses the direct actuator-to-sensor path in an undamaged structure as a reference to define a global scaling factor, converting numerical distances into physical units.
   • A modified error criterion is introduced to specifically identify "first arrival" wave packets, which often have negligible amplitude compared to the total signal but are crucial for localization.
   • Once physical distances are established, the damage location is determined using Elliptical Localization. The total path length from the actuator to the damage and then to the sensor is modeled as the sum of two Euclidean distances. A non-linear least squares problem (solved via Gauss-Newton or Levenberg-Marquardt algorithms) is used to find the damage coordinates that best fit the estimated path lengths from multiple sensors.

Key Results
The paper validates the PISAMP method through numerical investigations:

• Signal Approximation: In an academic case involving an isotropic plate, PISAMP achieved a 4% reconstruction error using 6 submodes, successfully distinguishing between non-dispersive ($S_0$) and dispersive ($A_0$) modes.
• Complex Structures: Applied to experimental data from an Airbus A380 fan cowl structure (a composite plate with stiffeners), the method achieved a 20% reconstruction error using 40 submodes, effectively handling reflections and material damping.
• Damage Localization: In a 2D square plate simulation with 30 different damage configurations, the proposed approach (combining PISAMP distance estimation with Elliptical Localization) achieved an average relative error of 2.35% for the x-coordinate and 2.26% for the y-coordinate. The method successfully identified damage locations using only the scattered wave components.

Significance and Claims
The authors claim that the principal novelty of this study lies in integrating wave propagation physics directly into a compact signal decomposition framework. Unlike purely data-driven or numerically intensive techniques, PISAMP embeds strong physical constraints to produce a low-dimensional, computationally efficient representation.

The significance of the work is twofold:

1. Interpretability: It enables the direct identification of physically meaningful features, specifically modal wavenumber functions and propagation distances, without requiring a pre-computed dictionary of all possible modes.
2. Localization Capability: It provides a robust, interpretable methodology for damage localization in thin plate structures by linking the decomposed signal features directly to geometric path lengths.

The paper modestly notes that the current implementation is restricted to planar surfaces where Euclidean distance applies. It identifies the extension to curved geometries (requiring geodesic distance calculations) and validation with real-world experimental data as necessary steps for future research.