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Simulation-based digital twins have emerged as a powerful tool for evaluating the mechanical response of bridges. As virtual representations of physical systems, digital twins can provide a wealth of information that complements traditional inspection and monitoring data. By incorporating virtual sensors and predictive maintenance strategies, they have the potential to improve our understanding of the behavior and performance of bridges over time. However, as bridges age and undergo regular loading and extreme events, their structural characteristics change, often differing from the predictions of their initial design. Digital twins must be continuously adapted to reflect these changes. In this article, we present a Bayesian framework for updating simulation-based digital twins in the context of bridges. Our approach integrates information from measurements to account for inaccuracies in the simulation model and quantify uncertainties. Through its implementation and assessment, this work demonstrates the potential for digital twins to provide a reliable and up-to-date representation of bridge behavior, helping to inform decision-making for maintenance and management.
In recent years, the use of simulation-based digital twins for monitoring and assessment of complex mechanical systems has greatly expanded. Their potential to increase the information obtained from limited data makes them an invaluable tool for a broad range of real-world applications. Nonetheless, there usually exists a discrepancy between the predicted response and the measurements of the system once built. One of the main contributors to this difference in addition to miscalibrated model parameters is the model error. Quantifying this socalled model bias (as well as proper values for the model parameters) is critical for the reliable performance of digital twins. Model bias identification is ultimately an inverse problem where information from measurements is used to update the original model. Bayesian formulations can tackle this task. Including the model bias as a parameter to be inferred enables the use of a Bayesian framework to obtain a probability distribution that represents the uncertainty between the measurements and the model. Simultaneously, this procedure can be combined with a classic parameter updating scheme to account for the trainable parameters in the original model.
This study evaluates the effectiveness of different model bias identification approaches based on Bayesian inference methods. This includes more classical approaches such as direct parameter estimation using MCMC in a Bayesian setup, as well as more recent proposals such as stat-FEM or orthogonal Gaussian Processes. Their potential use in digital twins, generalization capabilities, and computational cost is extensively analyzed.
Simulation-based digital twins must provide accurate, robust and reliable digital representations of their physical counterparts. Quantifying the uncertainty in their predictions plays, therefore, a key role in making better-informed decisions that impact the actual system. The update of the simulation model based on data must be then carefully implemented. When applied to complex standing structures such as bridges, discrepancies between the computational model and the real system appear as model bias, which hinders the trustworthiness of the digital twin and increases its uncertainty. Classical Bayesian updating approaches aiming to infer the model parameters often fail at compensating for such model bias, leading to overconfident and unreliable predictions. In this paper, two alternative model bias identification approaches are evaluated in the context of their applicability to digital twins of bridges. A modularized version of Kennedy and O'Hagan's approach and another one based on Orthogonal Gaussian Processes are compared with the classical Bayesian inference framework in a set of representative benchmarks. Additionally, two novel extensions are proposed for such models: the inclusion of noise-aware kernels and the introduction of additional variables not present in the computational model through the bias term. The integration of such approaches in the digital twin corrects the predictions, quantifies their uncertainty, estimates noise from unknown physical sources of error and provides further insight into the system by including additional pre-existing information without modifying the computational model.
Die Brückeninfrastruktur in Deutschland und Europa steht aufgrund steigender Verkehrslasten und alternder Bauwerke vor erheblichen Herausforderungen. Das DFG-Schwerpunktprogramm 2388 „Hundert plus – Verlängerung der Lebensdauer komplexer Baustrukturen durch intelligente Digitalisierung“ (SPP100+) strebt an, durch digitale Innovationen und prädiktive Instandhaltungsstrategien die Nutzungsdauer bestehender Brückenbauwerke zu verlängern. Der vorliegende Beitrag fokussiert sich auf das SPP100+ zugehörige Cluster „Monitoring und Simulation“, das sieben Teilprojekte umfasst. Die Projekte entwickeln fortschrittliche Methoden zur Überwachung und Zustandsbewertung von Brücken mittels Digitaler Zwillinge, hochauflösender Sensortechnik und numerischer Simulationen. Innovative Ansätze wie nichtlineare Modellanpassungen, stochastische Methoden und künstliche Intelligenz ermöglichen eine präzise und frühzeitige Identifizierung potenzieller Schäden. Die Kombination aus kontinuierlichem Bauwerksmonitoring und effizienter Datenauswertung ist entscheidend für die langfristige Sicherheit und Langlebigkeit bestehender Brücken und trägt darüber hinaus zur Ressourcenschonung bei.
Bias Identification Approaches for Model Updating of Simulation-based Digital Twins of Bridges
(2024)
Simulation-based digital twins of bridges have the potential not only to serve as monitoring devices of the current state of the structure but also to generate new knowledge through physical predictions that allow for better-informed decisionmaking. For an accurate representation of the bridge, the underlying models must be tuned to reproduce the real system. Nevertheless, the necessary assumptions and simplifications in these models irremediably introduce discrepancies between measurements and model response. We will show that quantifying the extent of the uncertainties introduced through the models that lead to such discrepancies provides a better understanding of the real system, enhances the model updating process, and creates more robust and trustworthy digital twins. The inclusion of an explicit bias term will be applied to a representative demonstrator case based on the thermal response of the Nibelungenbrücke of Worms. The findings from this work are englobed in the initiative SPP 100+, whose main aim is the extension of the service life of structures, especially through the implementation of digital twins.
Embedded Model Form Uncertainty Quantification with Measurement Noise for Bayesian Model Calibration
(2026)
A key factor in ensuring the accuracy of computer simulations that model physical systems is the proper calibration of their parameters based on real-world observations or experimental data. Inevitably, uncertainties arise, and Bayesian methods provide a robust framework for quantifying and propagating these uncertainties to model predictions. Nevertheless, Bayesian methods paired with inexact models usually produce predictions unable to represent the observed datapoints. Additionally, the quantified uncertainties of these overconfident models cannot be propagated to other Quantities of Interest (QoIs) reliably. A promising solution involves embedding a model inadequacy term in the inference parameters, allowing the quantified model form uncertainty to influence non-observed QoIs.
This paper introduces a more interpretable framework for embedding the model inadequacy compared to existing methods. To overcome the limitations of current approaches, we adapt the existing likelihood models to properly account for noise in the measurements and propose two new formulations designed to address their shortcomings.
Moreover, we evaluate the performance of this inadequacy-embedding approach in the presence of discrepancies between measurements and model predictions, including noise and outliers. Particular attention is given to how the uncertainty associated with the model inadequacy term propagates to the QoIs, enabling a more comprehensive statistical analysis of prediction’s reliability. Finally, the proposed approach is applied to estimate the uncertainty
in the predicted heat flux from a transient thermal simulation using temperature bservations.
Modeling complex physical systems such as they arise in civil engineering applications requires finding a trade-off between physical fidelity and practicality. Consequently, deviations of simulation from measurements are ubiquitous even after model calibration due to the model discrepancy, which may result from deliberate modeling decisions, ignorance, or lack of knowledge.
If the mismatch between simulation and measurements are deemed unacceptable, the model has to be improved. Targeted model improvement is challenging due to a non-local impact of model discrepancies on measurements and the dependence on sensor configurations. Many approaches to model improvement, such as Bayesian calibration with additive mismatch terms, gray-box models, symbolic regression, or stochastic model updating, often lack interpretability, generalizability, physical consistency, or practical applicability.
This paper introduces a non-intrusive approach to model discrepancy analysis using mixture models. Instead of directly modifying the model structure, the method maps sensor readings to clusters of physically meaningful parameters, automatically assigning sensor readings to parameter vector clusters. This mapping can reveal systematic discrepancies and model biases, guiding targeted, physics-based refinements by the modeler. The approach is formulated within a Bayesian framework, enabling the identification of parameter clusters and their assignments via the Expectation-Maximization (EM) algorithm.
The methodology is demonstrated through numerical experiments, including an illustrative example and a real-world case study of heat transfer in a concrete bridge.
Um die Lebensdauer von Bauwerken unter Wahrung derer Standsicherheit und Funktionsfähigkeit zu verlängern, bedarf es effektiver Monitorings- sowie Instandhaltungskonzepte. Im Rahmen des von der Deutschen Forschungsgemeinschaft (DFG) geförderten Schwerpunktprogramms 2388 „Hundert plus – Verlängerung der Lebensdauer komplexer Baustrukturen durch intelligente Digitalisierung“ (kurz: SPP 100+) werden hierfür innovative, interdisziplinäre Methoden entwickelt und an der Nibelungenbrücke in Worms (NBW) validiert. Der vorliegende Beitrag stellt einige dieser neuentwickelten digitalen Methoden vor. Unter anderem umfasst dies zwei Systeme des Structural Health Monitoring (SHM) und deren zielorientierte Verknüpfung von mehreren Beschleunigungsmessdaten zur umfassenden Zustandsbewertung. Ergänzend werden innovative datenbasierte Simulationsmethoden zur Bestimmung des Temperaturfelds des Brückenüberbaus vorgestellt sowie mehrere Finite-Elemente-Modelle unterschiedlicher Detailtiefe präsentiert und miteinander verglichen. Abschließend werden innovative Methoden zum Verwalten des Bestandswissens von Brückenbauwerken diskutiert. Die Methoden wurden überwiegend unabhängig voneinander entwickelt und an der NBW validiert. Im nächsten Schritt werden die Methoden integriert, um die Instandhaltung der NBW zu unterstützen.
Bayesian Tendon Breakage Localization under Model Uncertainty Using Distributed Fiber Optic Sensors
(2026)
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 ϕ-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.
