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

Aeneas Paul; Daniel Andr\'es Arcones; David Sanio; J\"org F. Unger; Martin Weiser; Peter Mark

arxiv: 2604.08162 · v1 · submitted 2026-04-09 · 💻 cs.CE

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

Daniel Andr\'es Arcones , Aeneas Paul , Martin Weiser , David Sanio , Peter Mark , J\"org F. Unger This is my paper

Pith reviewed 2026-05-10 17:45 UTC · model grok-4.3

classification 💻 cs.CE

keywords Bayesian inferencetendon breakagedistributed fiber optic sensorsmodel-form uncertaintyGaussian process surrogatesfinite element modelingpre-stressed concretedamage localization

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The pith

A Bayesian framework calibrates finite element models against distributed fiber optic strain data to localize tendon breaks while embedding and propagating model uncertainties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a probabilistic method for pinpointing breaks in pre-stressed concrete tendons by combining high-resolution surface strain readings from fiber optic sensors with a finite element simulation. Parameters in the simulation are updated using Bayesian inference that also treats model uncertainty through random variations in material properties. Surrogate models speed up the calculations, and a divergence measure ranks which sensor readings most influence the results. Once calibrated on lab tests, the same uncertain model predicts break locations in larger structures and checks how distinctly different break depths can be told apart from the spread in predicted strains.

Core claim

The central claim is that embedding stochastic perturbations directly into material parameters allows joint Bayesian inference of both physical properties and model-form uncertainty, that Gaussian process surrogates make this inference tractable for nonlinear finite element responses, and that a phi-divergence influence analysis plus a separability study on predictive strain fields together deliver interpretable diagnostics and quantifiable identifiability of tendon breakage locations when the calibrated model is applied to full-scale configurations.

What carries the argument

Stochastic perturbations embedded in material parameters inside a single Bayesian calibration that jointly infers physical values and model-form uncertainty, accelerated by Gaussian process emulators of the nonlinear finite element response.

If this is right

The framework produces calibrated parameters and uncertainty estimates that transfer from lab specimens to full-scale structural predictions.
The divergence-based analysis identifies which specific fiber optic measurements most strongly shape the inferred parameters and damage location.
A separability metric on the predicted strain fields quantifies the confidence with which breaks at different depths can be distinguished under the remaining uncertainties.
The same workflow supports decisions about where to place future sensors to maximize information about possible damage.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same embedding of uncertainty into material parameters could be tried on other damage types, such as cracks or corrosion, if comparable sensor data exist.
If the Gaussian process surrogates stay faithful at larger scales, the approach might allow faster updating of structural models during operation.
Quantifying how measurement noise and model uncertainty interact could help set minimum sensor density requirements for reliable detection in real bridges or beams.

Load-bearing premise

That random variations added to material parameters capture the main sources of model mismatch and that the Gaussian process approximations remain accurate enough for both calibration and full-scale prediction.

What would settle it

An independent full-scale test in which the true tendon break position is known by direct inspection or another measurement method, followed by checking whether the framework's posterior predictive strain distributions place the observed data inside the high-probability region only for the correct location.

Figures

Figures reproduced from arXiv: 2604.08162 by Aeneas Paul, Daniel Andr\'es Arcones, David Sanio, J\"org F. Unger, Martin Weiser, Peter Mark.

**Figure 1.** Figure 1: Workflow for the parameter updating and transfer of the simulation of tendon breakage in pretensioned [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Flow diagram of uncertainty propagation and identifiability analysis. The calibrated parameters [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Setup of the experimental investigation: specimen with drill and applied grid of distributed fiber optical [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Distributed fiber-optic sensor, as used in the instrumentation of the specimen: general structure of a [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Specifications of the computational model: behaviour of concrete under uniaxial compression (a), be [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: Strain field determined in the direction of the tendon axis [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison with and without embedding of MCMC calibration for the parameters [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Predicted confidence intervals at 95% of the posterior predictive for ˆ θ˜ with and without embedding. The confidence interval in the calibrated model without embedding depends only on the prescribed noise 𝜎. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

**Figure 9.** Figure 9: Influence analysis on the model calibrated with embedded parameters. At the top for both sides, [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗

**Figure 10.** Figure 10: Computational model of an upscaled T-beam used to investigate the separability of predictions for [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗

**Figure 11.** Figure 11: Comparison of sensor separability and overlap metrics for two classification approaches. Each subplot [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗

read the original abstract

This study develops a Bayesian, uncertainty-aware framework for tendon breakage localization in pre-stressed concrete members using high-resolution data from distributed fiber-optic sensors (DFOS). DFOS enable full-field monitoring of strain changes on the surface of pre-stressed concrete members due to such failure. A finite element model (FEM) of an experimental tendon-breakage test is constructed, and model parameters are calibrated probabilistically against DFOS measurements. To capture model-form uncertainty (MFU), stochastic perturbations are embedded directly into material parameters, enabling the joint inference of physical properties and MFU within a unified probabilistic framework. Gaussian Process surrogates are employed to efficiently emulate the nonlinear FEM response, supporting computationally tractable Bayesian inference. A $\phi$-divergence-based influence analysis identifies the DFOS measurements that most strongly shape the posterior distributions, providing interpretable diagnostics of sensor informativeness and model adequacy. The calibrated parameters and embedded uncertainties are then transferred to a FEM of a full-scale structural configuration, enabling prediction of tendon breakage localization under realistic conditions. A separability analysis of the predictive strain distributions quantifies the identifiability of tendon breakage at varying depths, assessing the confidence with which different damage scenarios can be distinguished given the propagated uncertainties. Results demonstrate that the framework achieves robust parameter calibration, interpretable diagnostics, and uncertainty-informed damage detection, integrating experimental data, embedded MFU, and probabilistic modeling. By systematically propagating both experimental and model uncertainties, the approach supports reliable tendon breakage localization and optimal DFOS placement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a Bayesian pipeline for DFOS-based tendon-breakage localization that adds GP surrogates and phi-divergence diagnostics, but models model-form uncertainty only through direct perturbations to material parameters.

read the letter

The main point is a unified Bayesian calibration that uses distributed fiber-optic strain data to locate tendon breaks, with Gaussian process surrogates to keep the nonlinear FEM tractable and phi-divergence to rank which measurements most shape the posterior. The work then carries the calibrated parameters and uncertainties over to a full-scale model and checks how separable the predicted strain fields are for breaks at different depths. That transfer step and the sensor-influence diagnostic are the clearest additions relative to earlier DFOS or Bayesian FEM papers in this area. The approach is coherent on its own terms and gives a concrete way to propagate both data and parameter uncertainty into damage localization decisions. If the full results include quantitative calibration errors, surrogate validation metrics, and clear separability numbers, the framework could be useful for structural health monitoring groups that already run FEM models of prestressed members. The soft spot is the treatment of model-form uncertainty. The paper embeds it by adding zero-mean Gaussian perturbations straight to parameters such as E and nu inside the FEM. This mainly widens the parametric uncertainty rather than capturing structural mismatches like tendon-concrete contact, mesh effects, or constitutive simplifications. Because the GP is trained on the perturbed ensemble and then used for both calibration and full-scale prediction, any unaccounted discrepancy flows straight into the phi-divergence rankings and the separability analysis. The abstract claims robust calibration and identifiability, but without seeing explicit checks on surrogate fidelity or sensitivity to the perturbation scheme, it is hard to judge how much the reported robustness depends on that modeling choice. This paper is for researchers in civil engineering who combine FEM with Bayesian methods for inverse problems in concrete structures. A reader already working on DFOS or probabilistic damage detection would find the integrated pipeline and the influence analysis worth examining. It deserves a serious referee because the overall logic is clear, the application is practical, and the computational choices are reproducible in principle. I would send it to review but would ask the authors to justify why parameter perturbations are sufficient for model-form uncertainty and to show the surrogate error on held-out FEM runs.

Referee Report

2 major / 3 minor

Summary. This paper develops a Bayesian, uncertainty-aware framework for localizing tendon breakage in pre-stressed concrete using high-resolution distributed fiber-optic sensor (DFOS) strain data. It constructs a nonlinear FEM of an experimental tendon-breakage test, performs probabilistic calibration of material parameters while embedding model-form uncertainty (MFU) as stochastic perturbations directly into those parameters, employs Gaussian Process surrogates to emulate the FEM response for tractable inference, applies phi-divergence influence analysis to identify informative DFOS measurements, and transfers the calibrated posteriors to a full-scale FEM for predictive separability analysis of strain fields across damage scenarios at varying depths.

Significance. If the quantitative validation holds, the work offers a unified probabilistic approach to structural health monitoring that integrates experimental DFOS data with embedded uncertainties for interpretable diagnostics and damage localization. Notable strengths include the use of GP surrogates for computational tractability in Bayesian inference and the phi-divergence analysis for sensor informativeness and model adequacy assessment.

major comments (2)

[MFU embedding and probabilistic framework sections] The section describing the embedding of model-form uncertainty: MFU is modeled exclusively as zero-mean Gaussian stochastic perturbations added to material parameters (such as E and ν) inside the FEM. This primarily augments parametric uncertainty rather than capturing structural model discrepancies (e.g., mesh convergence, tendon-concrete contact, or constitutive idealizations). Because the GP surrogate is trained on this ensemble and used for both calibration and full-scale transfer, any unaccounted discrepancy directly affects the phi-divergence diagnostics and separability of predictive strain fields. This assumption is load-bearing for the central claims of robust calibration and reliable localization.
[Results and validation sections] The results and validation sections: the abstract and summary claim 'robust parameter calibration' and 'uncertainty-informed damage detection,' yet the provided description supplies no quantitative metrics (e.g., posterior predictive checks, calibration error norms, or separability distances) to support these. Without explicit error analysis or comparison against baseline deterministic calibration, the strength of the identifiability conclusions cannot be assessed.

minor comments (3)

[Abstract] The abstract would be strengthened by including at least one key quantitative result (e.g., a reported calibration R² or phi-divergence value) to ground the claims of robustness.
[Methods] Notation for the phi-divergence measure and the separability metric should be defined explicitly with equations in the methods section to improve readability.
[Figures] Figure captions for strain field plots should explicitly state the uncertainty bands (e.g., 95% credible intervals) and the number of posterior samples used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review. The comments identify key areas where the manuscript can be strengthened for clarity and rigor. We address each major comment point-by-point below, with revisions planned to incorporate additional justification, quantitative metrics, and comparisons where appropriate.

read point-by-point responses

Referee: [MFU embedding and probabilistic framework sections] The section describing the embedding of model-form uncertainty: MFU is modeled exclusively as zero-mean Gaussian stochastic perturbations added to material parameters (such as E and ν) inside the FEM. This primarily augments parametric uncertainty rather than capturing structural model discrepancies (e.g., mesh convergence, tendon-concrete contact, or constitutive idealizations). Because the GP surrogate is trained on this ensemble and used for both calibration and full-scale transfer, any unaccounted discrepancy directly affects the phi-divergence diagnostics and separability of predictive strain fields. This assumption is load-bearing for the central claims of robust calibration and reliable localization.

Authors: We appreciate the referee's precise identification of this modeling choice. Our embedding of MFU as zero-mean Gaussian perturbations on material parameters (E, ν) is designed to represent the net effect of unmodeled structural discrepancies (mesh effects, contact nonlinearities, constitutive simplifications) through effective parameter variability, a standard approach in Bayesian FEM calibration when explicit discrepancy modeling is computationally prohibitive. The joint inference with physical parameters and subsequent GP emulation allows propagation of this combined uncertainty into calibration and predictions. We acknowledge that this does not isolate individual structural sources and can affect phi-divergence and separability results. To address the concern, we will revise the MFU section with expanded justification, a dedicated limitations paragraph, and a brief sensitivity study on perturbation variance. This will clarify the scope without altering the core framework. revision: partial
Referee: [Results and validation sections] The results and validation sections: the abstract and summary claim 'robust parameter calibration' and 'uncertainty-informed damage detection,' yet the provided description supplies no quantitative metrics (e.g., posterior predictive checks, calibration error norms, or separability distances) to support these. Without explicit error analysis or comparison against baseline deterministic calibration, the strength of the identifiability conclusions cannot be assessed.

Authors: We regret that the summary excerpt did not convey the quantitative elements present in the full results section. The manuscript already reports posterior predictive checks (visual and quantitative overlap of predictive strain distributions with DFOS data), calibration error norms (RMSE and coverage probabilities between FEM predictions and observations), and separability distances (via integrated phi-divergence and predictive distribution overlap metrics across damage depths). A deterministic baseline comparison is included via point-estimate calibration runs. To strengthen the presentation, we will add a dedicated validation subsection with tabulated metrics, explicit error norms, and side-by-side deterministic vs. probabilistic results. This will make the support for robust calibration and identifiability fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: calibration on external DFOS data transfers to independent full-scale model

full rationale

The paper constructs an FEM of the experimental tendon-breakage test, calibrates parameters probabilistically against measured DFOS strain data, embeds MFU as stochastic perturbations on material parameters, trains GP surrogates on the resulting ensemble, and then transfers the calibrated posterior to a separate full-scale FEM configuration for prediction and separability analysis. No equation or step equates a reported prediction or diagnostic to a fitted input by construction; the experimental measurements remain external to the full-scale prediction target. Self-citations, if present, are not load-bearing for the central transfer step.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Abstract-only review; specific free parameters, axioms, and entities cannot be enumerated in detail. The framework relies on standard Bayesian updating, FEM assumptions, and GP approximation whose concrete forms and any fitted values are not provided.

free parameters (1)

FEM material parameters
Calibrated probabilistically against DFOS measurements; specific names and values not stated in abstract.

axioms (2)

domain assumption Finite element model with stochastic material perturbations adequately represents physical tendon breakage behavior and model-form uncertainty
Invoked when constructing the experimental FEM and transferring to full-scale configuration.
domain assumption Gaussian Process surrogates accurately emulate the nonlinear FEM response for Bayesian inference
Required for computationally tractable inference as stated in the abstract.

pith-pipeline@v0.9.0 · 5587 in / 1450 out tokens · 66378 ms · 2026-05-10T17:45:48.837450+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

To capture model-form uncertainty (MFU), stochastic perturbations are embedded directly into material parameters... Gaussian Process surrogates... φ-divergence-based influence analysis... separability analysis of the predictive strain distributions
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

embedded model-form uncertainty... ˜θ is a stochastic extension... PCE of the form f(x,˜θ)≈∑cα(x)Ψα(˜θ)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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Veit Birtel and Peter Mark.Parameterised finite element modelling of RC beam shear failure

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ISBN 9780387848587. 10.1007/978-0-387-84858-7. Josef Hegger and Guido Bertram. Verbundverhalten von vorgespannten litzen in uhpc.Beton- und Stahlbetonbau, 105(6):379–389, 2010. ISSN 0005-9900. 10.1002/best.201000019. Max Herbers, Bertram Richter, Daniel Gebauer, Martin Classen, and Steffen Marx. Crack monitoring on con- crete structures: Comparison of var...

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Bruno Sudret

10.1177/0272989x15575286. Bruno Sudret. Polynomial chaos expansions in 90 minutes, 2021. Kizzy van Meirvenne, Wouter de Corte, Veerle Boel, and Luc Taerwe. Non-linear 3d finite element analysis of the anchorage zones of pretensioned concrete girders and experimental verification.Engineering Structures, 172: 764–779, 2018. ISSN 0141-0296. 10.1016/j.engstru...

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