Physics-guided impact localisation and force estimation in composite plates with uncertainty quantification

This paper presents a hybrid framework that combines a physics-guided First-Order Shear Deformation Theory model with machine learning and uncertainty quantification to achieve accurate, robust, and data-efficient impact localisation and force estimation in composite plates.

The Gist

Imagine you have a large, delicate composite plate (like the wing of a plane or a drone) made of layers of carbon fiber. This plate is like a giant, invisible drum. If something bumps into it—a dropped tool, a hailstone, or a bird strike—it makes a sound. But unlike a drum you can hear, this sound is a complex vibration that travels through the material.

The goal of this research is to answer two questions just by listening to the "rumble" with a few tiny sensors glued to the surface:

1. Where did the bump happen?
2. How hard did it hit?

The problem is that in the real world, we rarely have enough sensors or enough data to know the answer perfectly. It's like trying to guess where a stone was thrown into a dark, foggy lake just by looking at a few ripples on the edge.

Here is how the authors solved this puzzle using a mix of physics and AI, explained through simple analogies.

1. The Problem: The "Black Box" vs. The "Empty Room"

• Old AI Methods (The Black Box): Traditional machine learning is like a student who memorizes answers to a specific test. If you ask a question slightly different from the test, the student panics and guesses wrong. In engineering, this means if you train an AI on a specific plate, it fails if you change the plate slightly or if the impact happens in a spot it hasn't seen before.
• Old Physics Methods (The Empty Room): Traditional physics models are like trying to solve a math problem without knowing the numbers. You know the laws of motion, but if you don't know the exact thickness of the plate, the material density, or how tightly the edges are clamped, your calculations are useless.

2. The Solution: The "Smart Detective" Framework

The authors created a hybrid system that acts like a smart detective. Instead of just memorizing data or guessing with pure math, it builds a mental model of the plate using very little information, then uses that model to teach the AI.

Step A: Building the Mental Model (The "Ghost Plate")

The researchers didn't need to know the plate's blueprint. Instead, they hit the plate a few times in known spots and listened to the echoes.

• Finding the Material: They analyzed how fast the "ripples" (waves) traveled. Just like how sound travels faster in water than in air, different materials carry waves at different speeds. By measuring these speeds, they figured out the plate's "personality" (its stiffness and density).
• Finding the Edges: They listened to how the plate "sang" (its natural vibrations). A plate clamped tight sounds different from one that is loose. By matching the "song" to a physics model, they figured out how the edges were held.

Analogy: Imagine you are blindfolded and someone taps a drum. You don't know the drum's size or what it's made of, but by listening to the pitch and how long the sound rings, you can guess, "Ah, this is a small, tight drum made of wood." That's what they did with the composite plate.

Step B: Teaching the AI with "Fake" Data (Data Augmentation)

Once they built this "Ghost Plate" (a computer model that mimics the real plate's physics), they used it to generate thousands of fake impact scenarios.

• They told the AI: "Here are 4 real impacts we measured. Now, here are 10,000 simulated impacts based on our physics model."
• The AI learned the rules of how waves travel on this specific plate, not just the specific answers.
• The Result: When a real impact happened in a spot the AI had never seen, it didn't guess blindly. It used the physics rules it learned to make an educated guess. It's like a student who understands the concept of gravity, so they can predict where a ball will land even if they've never thrown a ball from that specific height before.

Step C: Guessing the Force (The "Inverse Problem")

Figuring out how hard the plate was hit is even harder. It's like trying to guess how hard someone kicked a soccer ball just by watching the ball fly through the air.

• The system uses a technique called Regularization. Think of this as a "noise filter."
• Usually, when you try to reverse-engineer a force, the math gets messy and amplifies tiny errors (like static on a radio).
• The authors created a smart filter that knows the physics of the plate. It says, "Okay, the math suggests a huge force, but the physics of this plate says that's impossible. Let's dial it back." This filter adapts depending on the frequency, ensuring the answer is smooth and realistic.

Step D: The "Confidence Score" (Uncertainty Quantification)

This is the most crucial part for safety. The system doesn't just say, "The hit was at (5, 5)." It says, "The hit was likely at (5, 5), but there is a 95% chance it was within this circle."

• If the sensors are far apart, the "circle" gets bigger.
• If the AI is unsure, it tells you. This is vital for engineers: knowing how unsure you are is often more important than a precise but wrong guess.

The Real-World Test

They tested this on a real carbon fiber plate.

• The Setup: They only used 4 sensors and 4 reference hits to train the system.
• The Result: The system could accurately locate impacts in areas where it had never been trained, and it could estimate the force of the hit with high accuracy.
• The Benefit: In the past, you might need hundreds of sensors and thousands of test hits to get this level of accuracy. This method does it with a fraction of the effort, making it cheap and fast to deploy on real airplanes or wind turbines.

Summary

Think of this research as teaching a computer to be a structural doctor.

1. Instead of needing a full MRI (expensive, detailed data), it listens to a few heartbeats (sparse sensor data).
2. It builds a mental map of the patient's body using basic laws of physics.
3. It uses that map to simulate millions of scenarios to learn how the body reacts.
4. When a new injury happens, it diagnoses the location and severity, and even tells you how confident it is in its diagnosis.

This approach makes it possible to monitor the health of giant, complex structures (like planes) without needing to know every single detail about them beforehand, saving time, money, and potentially lives.

Technical Summary

1. Problem Statement

Composite aerostructures are susceptible to low-velocity impacts (e.g., tool drops, hail) that cause Barely Visible Impact Damage (BVID), threatening structural integrity. Structural Health Monitoring (SHM) systems aim to identify impact location and force history from sparse sensor data. However, existing methods face significant challenges:

• Model-based approaches (e.g., Finite Element Analysis) require precise knowledge of geometry, material properties, and boundary conditions, which are often unavailable or inaccurate in real-world scenarios.
• Data-driven approaches (e.g., Deep Learning) rely heavily on large, high-fidelity experimental datasets. They struggle with generalization to untrained impact areas, lack interpretability, and are difficult to scale to new structures without retraining.
• Inverse Problem Nature: Impact identification is an ill-posed inverse problem where noise and limited sensor coverage lead to unstable solutions, particularly for force reconstruction.

The paper addresses the need for a robust, scalable, and data-efficient framework that can operate with sparse training data, handle structural uncertainties, and provide probabilistic outputs for safety-critical applications.

2. Methodology

The authors propose a hybrid physics-augmented framework that integrates a data-driven First-Order Shear Deformation Theory (FSDT) plate model with machine learning and uncertainty quantification. The workflow consists of three main phases:

A. Sparse Data-Driven FSDT Modelling

Instead of relying on prior structural knowledge, the FSDT model is constructed entirely from observed structural responses (sparse reference impacts and sensor data).

1. Elastic Property Identification: The group velocity profile (GVP) of flexural waves is estimated by fitting dispersion relations to Time Difference of Arrival (TDOA) data from reference impacts. A Maximum Likelihood Estimation (MLE) approach is used to identify effective material properties (stiffness, density) by optimizing the fit between observed TDOAs and theoretical dispersion curves.
2. Boundary Condition Identification: Using the identified material properties, boundary conditions are calibrated by matching modal characteristics (natural frequencies and excitation transmissibility) extracted from the data. Artificial spring stiffnesses at the plate edges are optimized using a Rayleigh-Ritz variational method to minimize discrepancies between predicted and observed modal data.

B. Physics-Augmented Impact Localisation

The constructed FSDT model serves as a "low-fidelity" (LF) physics engine to augment sparse "high-fidelity" (HF) experimental data.

• Data Augmentation: The LF FSDT model generates synthetic TDOA data across the entire plate surface, capturing wave propagation physics.
• Multi-Fidelity Gaussian Process Regression (GPR): A GPR model is trained using a multi-fidelity kernel. It combines the sparse HF experimental data with the dense LF synthetic data. The model treats the HF output as a scaled version of the LF prediction plus a correction term. This allows the model to generalize to untrained impact regions (extrapolation) while maintaining accuracy near reference points.
• Uncertainty Propagation: The GPR provides not only a predicted impact location but also a variance (uncertainty), which is propagated to the subsequent force estimation stage.

C. Physics-Guided Impact Force Reconstruction

Force reconstruction is performed via frequency-domain deconvolution using transfer functions.

• Transfer Function Estimation: Radial Basis Function (RBF) interpolation is used to estimate transfer functions at target locations. The FSDT model provides a physical prior to guide this interpolation.
• Adaptive Regularisation: To solve the ill-posed inverse problem, a frequency-dependent adaptive regularisation scheme is introduced. Unlike traditional constant $\ell_2$ regularisation (which often underestimates force magnitude), this method calculates a regularisation parameter $\lambda_{reg}(\omega)$ based on the discrepancy between the RBF-interpolated transfer function and the theoretical FSDT prediction. High discrepancies (indicating high uncertainty) trigger stronger regularisation.
• Probabilistic Force Estimation: The localisation uncertainty from the GPR is propagated through the force deconvolution process, yielding probabilistic force estimates with confidence bounds.

3. Key Contributions

1. Data-Driven FSDT Modelling: A novel method to construct a physically interpretable FSDT plate model using only sparse experimental data, inferring material properties via dispersion analysis and boundary conditions via modal matching.
2. Physics-Augmented Machine Learning: A multi-fidelity GPR framework that leverages physics-generated synthetic data to enable accurate impact localisation with minimal experimental training data (demonstrated with as few as 4 reference impacts).
3. Adaptive Regularisation: A physics-informed regularisation strategy for force deconvolution that adapts to frequency-dependent interpolation errors, significantly improving force magnitude accuracy compared to fixed-parameter methods.
4. Uncertainty Quantification: A systematic approach to propagate localisation errors into force estimates, providing confidence intervals essential for safety-critical decision-making.

4. Experimental Results

The framework was validated on a M21/T800 CFRP composite plate (290 × 200 × 4 mm) subjected to low-velocity hammer impacts, monitored by four PZT sensors.

• Material & Boundary Identification: The method successfully identified effective material properties and boundary conditions using only four reference impacts. The identified Group Velocity Profiles (GVPs) closely matched reference data, particularly in the low-frequency range (1–10 kHz) critical for impact dynamics.
• Impact Localisation:
  • Multi-fidelity vs. Single-fidelity: The multi-fidelity GPR model significantly outperformed models trained solely on sparse HF data (which failed to extrapolate) or solely on LF data (which lacked accuracy).
  • Accuracy: With only 4 reference impacts, the mean localisation error was 5.0 mm. Increasing reference impacts to 9 reduced the error to 5.9 mm for HF-only and 6.0 mm for MF, demonstrating that the physics-augmented approach achieves high accuracy with minimal data.
  • Frequency Sensitivity: Lower frequency ranges (1–2 kHz) yielded more robust localisation results than higher frequencies due to reduced noise sensitivity in TDOA measurements.
• Force Reconstruction:
  • Adaptive Regularisation: The adaptive scheme reduced peak force underestimation common in constant regularisation. For impacts within the reference coverage, peak force errors were kept within 7%.
  • Extrapolation: While force reconstruction accuracy decreased for impacts far from reference points (due to transfer function extrapolation errors), the adaptive regularisation and uncertainty quantification correctly identified these regions as having higher uncertainty.
  • Uncertainty Propagation: The framework successfully quantified how localisation errors (e.g., 5–7 mm) translated into force estimation variance, providing a realistic confidence bound for the reconstructed force history.

5. Significance and Conclusion

This paper presents a scalable solution for Structural Health Monitoring (SHM) in composite aerostructures where prior model information is incomplete or unavailable.

• Reduced Experimental Cost: The ability to achieve high accuracy with only a few reference impacts (4–9) drastically reduces the cost and time associated with extensive impact testing campaigns.
• Generalisability: By integrating physical laws (FSDT) with data-driven learning, the method overcomes the "black box" limitations of pure deep learning and the data-hunger of standard ML, enabling transferability to new structures.
• Safety-Critical Reliability: The inclusion of uncertainty quantification ensures that predictions are accompanied by confidence bounds, making the system suitable for real-time operational monitoring where false negatives or positives could have severe consequences.

The proposed framework offers a robust pathway for deploying impact monitoring systems in complex, real-world composite structures, balancing physical interpretability with the flexibility of data-driven adaptation.