💻 computer science

Bayesian Tendon Breakage Localization under Model Uncertainty Using Distributed Fiber Optic Sensors

This study presents a Bayesian framework that integrates distributed fiber-optic sensor data, finite element modeling with embedded model-form uncertainty, and Gaussian Process surrogates to achieve robust, uncertainty-aware localization of tendon breakage in pre-stressed concrete structures.

Original authors: Daniel Andrés Arcones, Aeneas Paul, Martin Weiser, David Sanio, Peter Mark, Jörg F. Unger

Published 2026-04-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel Andrés Arcones, Aeneas Paul, Martin Weiser, David Sanio, Peter Mark, Jörg F. Unger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, invisible spiderweb made of steel cables (called tendons) running inside a concrete bridge. These cables hold the bridge together under immense pressure. If one of these cables snaps, it's a disaster waiting to happen. The problem is, you can't see inside the concrete, and you can't cut the bridge open to check without destroying it.

This paper presents a clever "digital detective" system to find exactly where a cable has snapped and how deep it is, even when our computer models aren't perfect.

Here is the story of how they did it, broken down into simple steps:

1. The Problem: The "Blind" Detective

Engineers use computers (Finite Element Models) to simulate how a bridge reacts when a cable breaks. But computers are like students who are good at math but bad at reality. They make mistakes because they simplify the world. If you just ask the computer, "Where did the cable break?" it might guess wrong because it doesn't account for all the messy, real-world imperfections.

2. The Solution: The "Smart Sensor" Tape

The researchers used a special technology called Distributed Fiber Optic Sensors (DFOS). Think of this as a long, high-tech measuring tape wrapped around the concrete. Instead of just measuring one spot, this tape has thousands of tiny eyes that can feel the tiniest stretch or squeeze in the concrete surface. When a cable inside snaps, the concrete stretches in a specific pattern, and this tape records the whole story.

3. The "Uncertainty" Trick: Admitting We Don't Know Everything

This is the most important part of the paper. Usually, when engineers calibrate a computer model, they try to make the numbers match perfectly. If they don't match, they force the computer to change its settings until it does, even if those settings make no physical sense.

The authors did something different. They said, "Our model is imperfect, and that's okay."

The Analogy: Imagine you are trying to guess the weight of a watermelon by looking at it. You have a scale (the model), but you know the scale is a bit wobbly.
- Old Way: You force the scale to show the exact weight you think it is, ignoring the wobble.
- This Paper's Way: You tell the scale, "I know you're wobbly. Let's assume the weight isn't a single number, but a range of possible weights." They mathematically "baked" this uncertainty directly into the model's settings.

By doing this, the computer doesn't just give you one answer; it gives you a confidence map. It says, "I'm 90% sure the break is here, but there's a small chance it's over there."

4. The "Speedy Assistant": The Gaussian Process

Running the computer simulation for every possible scenario takes forever (like trying to solve a Rubik's cube blindfolded, one move at a time). To speed this up, they used a "Speedy Assistant" (called a Gaussian Process).

The Analogy: Imagine you are a chef tasting a soup. You don't need to cook the whole pot 1,000 times to know how it tastes. You just taste a spoonful, and your brain (the AI assistant) predicts how the rest of the pot will taste based on that one spoonful. This assistant learned from the expensive simulations and could predict the results instantly, allowing them to test thousands of scenarios in seconds.

5. The "Who Matters?" Check: Influence Analysis

Once they had their model, they asked: "Which parts of our sensor tape are actually helping us solve the mystery?"

The Analogy: Imagine a jury of 100 people trying to solve a crime. Some jurors are paying attention; others are daydreaming. The researchers used a mathematical tool to see which "jurors" (sensor data points) were actually changing the verdict. They found that the sensors closest to the break were the most important, while some far away didn't add much value. This helps engineers know exactly where to put sensors in the future.

6. The Final Test: Can We Tell the Difference?

Finally, they asked the ultimate question: "If we have uncertainty, can we still tell the difference between a cable breaking 10cm deep vs. 20cm deep?"

The Analogy: Imagine two twins wearing slightly different hats. If you are standing right next to them, you can easily tell them apart. If you are standing a mile away, they look identical.
- The researchers mapped out the bridge to find the "sweet spots." They discovered that near the break, the computer can clearly tell the difference between depths. But further away, the "uncertainty fog" gets so thick that the computer can't tell the twins apart.

Why This Matters

This paper isn't just about math; it's about safety and trust.

For Bridges: It gives engineers a reliable way to find hidden damage without destroying the structure.
For the Future: It teaches us how to build computer models that admit their own mistakes. Instead of giving a false sense of certainty, these models tell us, "Here is where we are confident, and here is where we need more data."

In short, they built a smart, self-aware digital twin of a bridge that knows how to find a broken wire, even when the world is messy and unpredictable.

1. Problem Statement

The localization of tendon breakage in pre-stressed concrete structures is critical for structural health monitoring (SHM) and safety assessment. However, direct calibration of mechanical models against real-world tendon failures is impossible because destructive testing cannot be performed on operational infrastructure (e.g., bridges). Consequently, models must be calibrated using controlled laboratory experiments and then transferred to full-scale structural simulations.

The core challenges identified are:

Model-Form Uncertainty (MFU): Finite Element Models (FEM) inevitably contain errors due to idealized constitutive laws, boundary conditions, and geometric simplifications. Traditional calibration often ignores this, leading to overconfident predictions that fail to capture observed variability.
Transferability: Standard discrepancy correction methods (like the Kennedy-O'Hagan formulation) are tied to the specific calibration setup and do not naturally transfer to new structural configurations.
Identifiability: It is unclear whether the strain patterns caused by tendon breakage at different depths are distinguishable given the propagated uncertainties from the calibration process.

2. Methodology

The authors propose a unified, uncertainty-aware workflow integrating experimental data, high-fidelity modeling, surrogate modeling, and Bayesian inference.

A. Experimental Setup & Data

Specimen: A pre-stressed concrete beam instrumented with a 2D grid of Distributed Fiber Optic Sensors (DFOS) on the surface.
Event: A controlled tendon breakage was induced, and DFOS measured the resulting high-resolution strain redistribution ( $\Delta \varepsilon$ ).
Data: 55 measurement points capturing strain changes along the beam surface.

B. Numerical Modeling & Surrogates

FEM: A 3D nonlinear FEM (using ABAQUS) modeled the concrete using the Concrete Damage Plasticity (CDP) law and the tendon-concrete interface using a Hoyer effect contact formulation.
Surrogate Modeling: To make Bayesian inference computationally feasible, Gaussian Process (GP) surrogates were trained to emulate the FEM response.
- One GP formulation was used for parameter calibration (mapping parameters to strain).
- A second GP formulation was used for uncertainty propagation (mapping parameters and tendon depth to strain).

C. Bayesian Calibration with Embedded MFU

Instead of using a separate discrepancy term (which is non-transferable), the authors adopted an embedded uncertainty approach:

Stochastic Parameters: The Young's modulus of the concrete ( $E_{cm}$ ) was treated not as a fixed value but as a stochastic variable ( $\tilde{E}_{cm}$ ) following a log-normal distribution.
Mechanism: This allows the model to absorb structural/model-form discrepancies directly into the variability of the material stiffness.
Inference: A Markov Chain Monte Carlo (MCMC) sampler (emcee) was used to infer the posterior distributions of the mean stiffness, its variance, and other contact parameters (friction, contact pressure).
Uncertainty Propagation: Polynomial Chaos Expansion (PCE) was used to propagate the stochastic parameters through the GP surrogate to compute predictive means and variances efficiently.

D. Influence Analysis

A $\phi$ -divergence (specifically reverse Kullback-Leibler divergence) was used to quantify the influence of specific subsets of DFOS data on the posterior parameter distributions. This identified which sensor locations provided the most information and which regions drove model discrepancies.

E. Identifiability & Separability Analysis

The calibrated stochastic parameters were transferred to a new FEM of a realistic T-beam (bridge scale).

Goal: Determine if the model can distinguish between tendon breakages at different depths ( $a$ ).
Metric: A "Min- $\Delta$ " metric and a PDF-overlap metric were calculated. These measure the smallest detectable change in tendon depth and the degree of overlap between predictive distributions for different depths.
Optimization: A maximin optimization problem was solved to find the "worst-case" tendon depth where distinguishability is lowest, generating spatial maps of sensor separability.

3. Key Results

Calibration Performance

Embedded vs. Non-Embedded: The approach with embedded uncertainty significantly outperformed the standard approach.
- Coverage: The 95% predictive intervals covered 87% of experimental data points with embedded uncertainty, compared to only 60% without.
- Outliers: The percentage of data points with normalized residuals ( $|Z|$ ) exceeding 2 dropped from 40% (non-embedded) to 13% (embedded).
- Physical Meaning: The non-embedded calibration drove parameters to unrealistic boundaries (e.g., friction coefficient $\mu$ at its maximum). The embedded approach yielded physically realistic values (e.g., $E_{cm} \approx 31,000$ MPa with a standard deviation of $\approx 3,500$ MPa).

Influence Analysis

Dominant Data: The subset of DFOS measurements closest to the breakage point exerted the strongest influence on the posterior distributions.
Discrepancy Source: The analysis revealed that local discrepancies near the breakage point were partially absorbed by the increased variability of the embedded stiffness parameter ( $\sigma_{E_{cm}}$ ), suggesting a localized model deficiency in that region.

Identifiability & Separability

Spatial Regimes: The separability analysis on the T-beam revealed four distinct regions:
1. High Separability ( $x < 450$ mm): Near the breakage, strain patterns for different depths are distinct despite uncertainty.
2. Poor Separability ( $450 < x < 900$ mm): Predictive distributions overlap significantly due to the interaction of bending moments and embedded uncertainty.
3. Recovered Separability ( $900 < x < 1900$ mm): Strain patterns become distinct again as bending effects stabilize.
4. Low Signal ( $x > 1900$ mm): Strain changes are too small to be distinguished from noise/uncertainty.
Implication: Sensor placement strategies should prioritize regions of high separability (near the breakage) to reliably identify tendon depth.

4. Key Contributions

Embedded MFU Framework: Demonstrated a robust method for handling model-form uncertainty by embedding it directly into material parameters, ensuring the uncertainty is physically interpretable and transferable to new structural configurations.
Uncertainty-Aware Transfer: Successfully transferred calibrated parameters and their associated uncertainties from a laboratory-scale experiment to a full-scale bridge simulation without relying on non-transferable discrepancy terms.
Diagnostic Influence Analysis: Applied $\phi$ -divergence to identify which specific sensor data drives parameter inference and model discrepancy, offering a tool for targeted model improvement.
Quantitative Identifiability Maps: Generated spatial maps quantifying the "detectability" of tendon breakage depths under propagated uncertainty, providing a rigorous basis for optimizing sensor placement in SHM systems.

5. Significance

This work addresses a fundamental bottleneck in Structural Health Monitoring: the inability to validate models on real structures. By rigorously quantifying and propagating uncertainty from lab to field, the framework provides a reliable, probabilistic basis for decision-making. It moves beyond simple "best-fit" modeling to provide confidence intervals for damage localization. The results demonstrate that while uncertainty reduces the "perfect" distinguishability of damage scenarios, the framework can still identify optimal regions for sensor placement to maximize diagnostic capability, thereby extending the service life and safety of critical infrastructure like bridges.