Model Validation and Uncertainty Quantification, Volume 3

Condition-based maintenance (CBM) is a modern maintenance approach that combines data-driven reliability models and information from a condition monitoring process (e.g., inspections and continuous monitoring). Maintenance schedules are predicted based on the results from diagnosis and prognosis. Due to aging, the US Army Corps of Engineers (USACE) has equipped some of its navigation infrastructure with sensors to allow continuous monitoring. Miter gates are one of the most important such structural assets because of their economic impact on navigation corridors. Miter gates prognosis and maintenance schedule capabilities can be improved when a discrete-state deterioration model based on inspection data is used. One of the sources of inspection data available for miter gates is the operational condition assessment (OCA) discrete ratings. However, these discrete ratings are highly abstracted, assigned at variable frequencies, and very prone to human error and to misinterpretations due to inspection protocols. In miter gates, OCA ratings are available for deteriorating components such as quoin blocks. Over time, contact between these quoin blocks deteriorates, ultimately leading to failure, which can be generally avoided with timely maintenance schedules. To overcome these issues, this paper proposes a structural health-monitoring-based CBM framework that accounts for different levels of human observation errors in the inspection data. This proposed framework shows (1) how to use physics-informed (e.g., finite element) simulations to perform damage diagnosis in miter gates and (2) how to account for human observation errors to improve prognosis and maintenance schedule capabilities for deteriorating components (e.g., quoin blocks) in miter gates.

This paper presents a model uncertainty quantification and updating approach for a boundary condition model of a miter gate. A boundary condition model is used as the forward model to predict the boundary load condition of a miter gate for a given gap length. The boundary force prediction is then employed as inputs to a strain analysis model that predicts the strain response of the gate. Due to model simplifications, the boundary condition model may not accurately represent the true physics. By following the Kennedy and O’Hagan (KOH) framework under a Bayesian scheme, this paper corrects the unobservable boundary condition model using the strain measurements by simultaneously estimating the gap length and quantifying the model uncertainty. Results show that the proposed approach can effectively estimate the unknown gap length and improve the prediction of both the boundary condition model and the strain response model.

The National Aeronautics and Space Administration (NASA) is in the midst of bold and exciting next steps in human exploration and spaceflight. The designs of the new Space Launch System (SLS), the Orion spacecraft, and the Exploration Ground Systems (EGS) for vehicle processing and launch are essentially complete, and there has been significant progress in the manufacturing and assembly of specific hardware for the Artemis I and Artemis II missions. Equally as important, the program level and integrated system-level testing and analyses are also well underway to support integrated verification, validation, and Certificate of Flight Readiness (CoFR) for Artemis I. Testing and analysis are key to addressing technical challenges faced by the Artemis missions. Building block approaches are required that provide the right balance between component, element, and/or system-level testing that satisfies verification and validation objectives where uncertainties are quantified and minimized. Artemis I is a system of systems that requires a fusion of test and analysis that adeptly characterizes critical interfaces between major program elements. An example of this fusion involves characterizing the interface between the SLS booster and the Mobile Launcher (ML) Vertical Support Post (VSP) interfaces. Proper characterization of this interface represents a number of challenges, beginning with the fact that it is a mating of ground support structure in the form of a civil structure to flight hardware. Both sides of the interface are built to different construction standards but are governed by interface requirements to ensure compatibility when mated. From past program experience, the flexibility at the booster to the ML interface is critical in developing accurate prelaunch stacking and cryogenic preloads, squat loads, and pad separation release of preloads and squat loads. This same premise holds for Artemis I. To describe the asymmetric characteristics at this interface, careful consideration of static forces due to gravity loading with the commensurate effects resulting from ML VSP leveling, spacing, and shimming during booster stacking and associated geometric nonlinear forces due to booster lateral displacements are necessary for inclusion in pretest assessments. This paper will look at these issues for the upcoming booster pull test, in which two boosters will be installed on the ML and one of these boosters will undergo static lateral loading, followed afterward by dynamic excitation into resonance and free decay. This paper evaluates the booster to the ML interface by characterizing the flexibility between the booster aft skirt and the ML VSP. Furthermore, this paper methodically evaluates the effect of the following on the test outcome: gravitational effects on the booster and ML; the effects of VSP leveling, spacing, and shimming under gravitational loading during booster stacking; the effect of geometric nonlinear follower force due to center of gravity (CG) offset as the booster is laterally displaced; and the system coupling between the booster under test, ML, and the second booster. Simulated results for a static load pull and dynamic excitation provide insight into the differences in prediction vs measured response with and without the inclusion of the abovementioned boundary condition and geometric nonlinear effects.

The primary objective of the structural health monitoring (SHM) system is to continuously oversee and assess the state of the structure and evaluate its integrity at any time based on the appropriate analysis of in situ measured data. Therefore, among other things, an SHM system is an information-gathering mechanism. Gathering the information that is representative of the structural state and correctly analyzing the data help us better understand the state of the structure and mitigate possible losses by taking appropriate actions. However, the design, installation, maintenance, research, and development of an SHM system is an expensive endeavor. Therefore, agreeing to pay for new information is rationally justified if the reduction in the expected losses by new information is more than the intrinsic cost of the information-acquiring mechanism. We investigate the economic advantage of installing an SHM system for damage detection as well as risk and life-cycle management by using the value of information analysis. Among many possible choices of SHM system designs, preposterior decision analysis can be used to pick the most feasible design that can be installed on the structure. We demonstrate the framework on a miter gate application.

Finite element models (FEM) are routinely developed and used during fabrication of high dollar-value hardware. NASA, as part of the pre-flight certification of launch vehicles, routinely conducts vibration and static tests to calibrate models used for flight-risk assessments. During model calibration, certain areas of the model are modified, using engineering judgment and sensitivity analysis, to match the test results. Unfortunately, tools to identify problem areas in the FEM using test data directly are scarce and infrequently applied. Over the years, error localization algorithms have been proposed with very limited success. Recently, the Analytical Dynamics Model Improvement (ADMI) algorithm, which computes closed-form mass and stiffness corrections to match the test data exactly, has been shown to be an effective Error Localization Algorithm (ELA). The paper discusses three examples where ELA is used with simulated test data to locate problem areas. To gain confidence in the approach, the exact answer is shown along with ELA results. Results show that ELA is able to identify general problem areas consistent with known problem areas. In all examples, the ELA identified area is larger than the exact problem area. Nonetheless, with proper optimization tools, calibration results using the ELA identified areas provide excellent results.

In this paper, a best-practice example of a digital twin is presented. For this purpose, the authors choose a test model of a machine frame with a rotating motor to simulate a situation from an industrial context. During the manufacturing process, this type of frame undergoes excitation from the imbalance forces of a single-speed drive. In order to avoid a failure of a structure when operating conditions change, it is important to be able to simulate the impact accurately. This accurate simulation requires an adequate digital twin. In this context, a good digital twin is one that reproduces the modal characteristics of the structure properly.

In this paper, the modal parameters of the machine frame are determined by experimental modal analysis. Afterward, model updating is performed to approximate the simulated modal parameters to the ones obtained from the real structure experiment. The full process is executed using the software WaveImage, which provides an easy-to-use modular kit for experimental modal analysis, finite element analysis, and finite element model updating.

The inaccessibility of wind turbine components after their installation and the stochastic nature of the excitation they are subjected to, render their dynamic behavior challenging for investigation. Wind turbine blade manufacturers are required to carry out extensive test campaigns on each single blade individually prior to proceeding with commercialization. Static and fatigue tests are usually performed on prototypes in order to ensure that each blade can stand extreme loads, even after being subjected to high cyclic loading. The same instrumentation used for static and fatigue tests can be adopted for performing Operational Modal Analysis with the purpose of identifying the blade modal parameters, which can help analyzing failure modes and system stability problems. Measurements performed during tests are usually acquired at accessible locations and with a limited number of sensors due to their high cost. This work focuses on providing alternatives to costly and impractical physical measurements on wind turbine blades by combining information from cost-effective simulated models and more realistic test data through the so-called Virtual Sensing techniques, e.g., Kalman-type filters and Modal Expansion methods.

Closed-cell polymer foams are commonly employed as support structures to absorb shock and vibration in mechanical systems. Engineering analysts responsible for system designs that incorporate these closed-cell foams have a need to understand the effect that extrinsic environmental conditions have on the dynamic response of the mass supported by the foams. Environmental conditions such as preload, forcing energy, and forcing frequency, along with inherent model form uncertainty, have the potential to drive such a system into nonlinear, and sometimes chaotic, behavior. A suite of studies utilizing finite element (FE) analysis and numerical simulations of a material meta-model was used to perform a parametric study on the extrinsic forcing conditions to understand such effects on the nonlinear system dynamics. A simple two-dimensional (2D) plane strain FE model of a mass supported on both sides by foam was used to perform two tasks—to implicitly determine stress states from precompression in the foam and to explicitly solve for the system’s response when subject to a dynamic input. Using prior knowledge of uncertain quantities in the model and ranges of possible environmental conditions, parameter sets of extrinsic conditions were used as inputs to the models to obtain the time-domain responses of the suspended mass. The corresponding frequency domain characteristics of the suspended mass were used in conjunction with the input parameter distributions to form conclusions about the influence of variation in each parameter on the natural frequency of the system. Considering the results from the extrinsic property sensitivity analysis, the models were perturbed from linear, harmonic oscillation to showing signs of nonlinear motion. Small changes in extrinsic conditions while oscillating near the system’s nonlinear stiffness regime are hypothesized to cause sudden changes in a holistic response. This work aims to discuss the effects of model form uncertainty and the effects of changing the extrinsic conditions in an inherently nonlinear dynamic system.

Researchers have studied cantilevered pipelines conveying fluid for many decades for their practical applications and interesting dynamic behaviors. These applications can include oil pipelines and risers, mechanical pumps, and micro-/nanofluidic nanotubes, which can be used in drug delivery. Because the stability of these systems is so important to companies and industries, many researchers have investigated methods to improve their overall system stability. Some of the implemented methods include adding a mass to the tip of the pipe and motion-limiting constraints. The constraints introduce an impacting force that can produce interesting nonlinear chaotic behavior. The implementation of ideas and strategies that improve system stability leads to increasingly more complex systems with a greater number of input parameters. The cantilevered pipe conveying fluid itself already has many parameters to consider, including the flow speed, smoothness of the internal structure, size of the pipe, and position of motion-limiting constraints, to name a few. Additionally, each input parameter possesses its own uncertainty. The propagation of uncertainty in these parameters can significantly alter the dynamic response and stability of the proposed system. It is important for engineers to design the system with a firm understanding of how the system is going to behave and respond in the presence of these uncertainties. Therefore, it is necessary to employ uncertainty quantification methods. In this study, a sensitivity analysis is performed on a cantilevered pipeline conveying fluid with motion-limiting constraints to determine which parameters in the system are most dominant. By finding these dominant parameters, more uncertainty quantification methods can be employed to better understand the response of the system.

Mechanical stresses due to strings are imposed on an instrument when it is played. Such stress can lead to long-term strains or damages. In the cultural heritage domain, this can prevent an instrument from being played if the risk of damage is too high. Most of the properties of such instrument are uncertain, such as mechanical parameters, relative humidity or already existing cracks. Model-based approaches dealing with deep uncertainties can be a very efficient approach for decision support. In this paper, an example is given with an antique cello, which exhibits damages, especially boreholes or galleries created by wood-eating insects. A model is created for static analysis to compute the stress field that will be used as a basis for the info-gap robustness analysis of the uncertainties and their impact on the sustainability of the instrument, considering defects, probabilistic distributions of elastic constants and Knightian uncertainties of yield stresses of wood.

Manufacturing process errors and inaccurate readings of material properties in beam-based systems can result in uncertainties in the static and dynamic responses of the system. For this reason, this study focuses on the critical buckling loads, buckling mode shapes, and natural frequencies of the axially loaded beam by introducing uncertainties in the length, width, height, density, and Young’s modulus. Ideal fixed boundary conditions are first considered, and the output uncertainties in the natural frequencies and critical buckling load are determined. Then flexibility in the beam’s boundary conditions is introduced by modeling them with torsional springs, allowing for uncertainty in the boundary conditions to be studied. The results show that a 5% uncertainty in the input parameters may lead to 15% uncertainties in the system’s outputs. The results also show the importance of accurately determining the input parameter properties and boundary conditions of the system in order to avoid any wrong estimation of the critical buckling load and natural frequencies of the system.

Material and geometric input parameter uncertainties and their effects on the static and dynamic responses of a cylindrical structure subject to thermal loading with clamped-clamped boundary conditions are studied. Different thermal loads acting as an induced axial force are considered, including uniform, linear, and nonlinear thermal distributions. Following the derivation of the governing equations of motion and boundary conditions, several methods for sensitivity analysis and uncertainty quantification are introduced. Then static and dynamic analyses are performed for different variations of the material and geometric parameters and thermal loads. Specifically, the critical buckling temperatures and prebuckling natural frequencies are discussed via sensitivity analysis methods, including the Morris method, correlation coefficients, parametric studies, and output distributions. Utilizing these methods allows for the determination of the most influential input parameters for the proposed system as well as the output uncertainties.

This work introduces the beginnings of an error-based mathematical framework for digital twins, with the intention of providing an effective platform from which digital twins of engineering applications can be built. The framework assumes a digital twin to be some optimal combination of a physics- and data-based model and operates by weighting the contribution of each model depending on its relative mean square error compared to data measured from the physical system being twinned (the so-called physical twin). These weightings then provide a quantifiable measure of the ratio of physics- to data-based components in the resulting digital twin, and this offers a means of consistently comparing different digital twin models. The framework aims to improve the initial physics-based model of the system over time by updating it to the optimal model combination.

The initial framework is applied to a simulated Duffing oscillator, where the equivalent linear system is assumed as the physics-based model. The data-based model is learnt by identifying the system parameters from the measured system response with polynomial regression. The digital twin framework aims to detect the type of nonlinearity from the measured data (cubic in this case) and is successful in improving the physics-based model.

The framework is then extended to acceleration data recorded from the vibration response of a physical 3 degree-of-freedom structure, in order to analyse its performance in a real-world application. In this case, the assumed physics-based model uses estimated system parameters, and the data-based model is trained with a genetic algorithm to improve the accuracy of results. The digital twin framework improves the parameter estimations of the physics-based model.

It is anticipated that by developing this error-based framework and incorporating other aspects such as uncertainty analysis and optimisation, a unified method of implementing digital twins will be enabled for future research efforts.

This paper presents a novel hierarchical Bayesian modeling (HBM) framework for the model updating and response predictions of dynamic systems with material nonlinearity using multiple data sets consisting of measured response time histories. The proposed framework is capable of capturing the uncertainties originating from both structural and prediction error parameters. To this end, a multilevel probabilistic model is proposed aiming to characterize the variability of both model and noise parameters. Moreover, a new Laplace approximation is formulated within the HBM framework to reduce the computational burden up to a great extent. Finally, a multidegree of freedom (MDOF) nonlinear system modeled by Bouc-Wen hysteresis elements is employed to demonstrate the effectiveness of the method.

Uncertainty in structural loading during launch is a significant concern in the development of spacecraft and launch vehicles. Small variations in launch vehicle and payload mode shapes and their interaction can result in significant variation in system loads. In many cases involving large aerospace systems, it is difficult, not economical, or impossible to perform a system modal test. However, it is still vital to obtain test results that can be compared with analytical predictions to validate models. Instead, the “Building Block Approach” is used in which system components are tested individually. Component models are correlated and updated to agree as best they can with test results. The Space Launch System consists of a number of components that are assembled into a launch vehicle. Finite element models of the components are developed, reduced to Hurty/Craig-Bampton models, and assembled to represent different phases of flight. The only opportunity to obtain modal test data from an assembled Space Launch System will be during the Integrated Modal Test. There is always uncertainty in every model, which flows into uncertainty in predicted system results. Uncertainty quantification is used to determine statistical bounds on prediction accuracy based on model uncertainty. For the Space Launch System, model uncertainty is at the Hurty/Craig-Bampton component level. Uncertainty in the Hurty/Craig-Bampton components is quantified using the hybrid parametric variation approach, which combines parametric and nonparametric uncertainties. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. This type of uncertainty cannot be represented by variations in finite element model input parameters and thus cannot be included in a parametric approach. However, model-form uncertainty can be modeled using a nonparametric approach based on the random matrix theory. The hybrid parametric variation method requires the selection of dispersion values for the Hurty/Craig-Bampton fixed-interface eigenvalues and the Hurty/Craig-Bampton stiffness matrices. Component test/analysis frequency error is used to identify fixed-interface eigenvalue dispersions, while test/analysis cross-orthogonality is used to identify stiffness dispersion values. The hybrid parametric variation uncertainty quantification approach is applied to the Space Launch System Integrated Modal Test configuration. Monte Carlo analysis is performed, and statistics are determined for modal correlation metrics, frequency response from Integrated Modal Test shakers to selected accelerometers, as well as other metrics for determining how well target modes are excited and identified. If the predicted uncertainty envelopes future Integrated Modal Test results, then there will be increased confidence in the utility of the component-based hybrid parametric variation uncertainty quantification approach.

Structural Health Monitoring (SHM) involves determining the health state of an engineered structure based upon measured, damage-sensitive features such as natural frequencies, modeshapes and time-domain model coefficients. One of the key challenges in SHM is the difficulty associated with gathering experimental data from a structure in its damaged state. This challenge is particularly acute for purely data-based supervised learning methods. Numerical modelling offers the potential to overcome the lack-of-data problem by making physically informed predictions of how the structure will behave once damaged. However, numerical modelling raises challenges of its own, with a major question being how one incorporates uncertainties and errors arising from the model prediction process within SHM decision-making. In addition, variability inevitably arises in the observed experimental responses and this, too, should be incorporated in the decision process. Finally, it is desirable that the cost of misclassification be incorporated within the decision process, with risk-based approaches being an attractive option for moving from classification to decision-making. This paper introduces a practical application of a Forward Model Driven (FMD) paradigm for SHM. A key tenet of the approach is that numerical model predictions may be used to inform a statistical classifier. The method is demonstrated for the case of damage detection on an experimental truss bridge structure for which an associated finite element (FE) model has been developed. A framework based upon a sequence of binary classifiers is introduced, with attention drawn to the importance both of the choice of individual classifier and the strategy for their combination.

The low-pressure fuel pump inducer of the new Space Launch System RS25 core stage engine operates in a highly complex environment that substantially affects its modal characteristics. Some of the more important effects are fluid-added mass (FAM) resulting from operation within a light liquid (hydrogen) and the magnification of this effect due to tight tip clearance (TC). Since higher-order cavitation has been identified as a significant harmonic driver, knowledge of the natural frequency of potentially excitable modes is critical for safe operation, but this frequency cannot be measured during the severe operational environment. A comprehensive testing and analysis program has therefore been performed over the last 4 years to identify the nominal value and uncertainty of the frequency by modeling and testing two simpler structures in several configurations that share some of the characteristics of the operational inducer. This testing was used to assess and adjust modeling techniques, and an excellent correlation was achieved. Identification of the uncertainty in the inducer frequency itself was still problematic, however. This difficulty led to an investigation of Bayesian uncertainty quantification techniques and to the application of the relatively simple technique of multivariate normal conditional distributions to calculate the inducer natural frequency uncertainty. Assumptions on the prior distributions of the uncertainty of the fluid-added mass and tip clearance effect are initially applied to the models of each of the simple structures and the inducer itself, and these uncertainties are propagated to generate natural frequencies using the design of experiments. Simple response surfaces are then created from this data in order to calculate a covariance matrix relating all of these natural frequencies. Finally, the results from the modal test of the simple structures are considered to be observations and used to calculate the conditional variance of the desired inducer frequencies. As this method is less rigorous than more complicated Bayesian methods reported in the literature, a conservative factor is applied to the result, but the resulting uncertainty is still significantly less than originally estimated and will greatly assist the certification of the inducer for use in the engine.

The domain overlap between speech and structural vibration presents opportunities to leverage advances in speaker recognition for structural health monitoring. Classification of x-vectors, which are the outputs of a pre-final layer from a time-delay neural network (TDNN) acoustic model, has been used to recent success in speaker discrimination. x-Vectors present a flexible speaker representation for increased classification robustness, as they contain intermediate speaker parameterizations rather than distinct class predictions for a specific identification task. In investigation of the parallels between speech and structural acoustics, this paper explores the viability of the x-vector speaker recognition system for structural damage detection. A TDNN following the x-vector structure is trained to classify damage scenarios from the Z24 Bridge Benchmark, using cepstral and pitch features from accelerometer measurements. x-Vectors are calculated for each measurement, which are used to train a probabilistic linear discriminant analysis (PLDA) model for Z24 damage scenario categorization. This approach yields strong performance in damage detection and classification, and we attempt a transfer learning approach to use this developed TDNN for training a modified x-vector system for local damage. We also apply the developed x-vector system to the LANL SHM Alamosa Canyon Bridge and UC-Irvine Bridge Column studies to explore generalization of this method, obtaining strong results in damage detection. We find that the x-vector system demonstrates the feasibility of speaker recognition techniques for structural health monitoring and shows significant potential for output-only structural health assessment.

Model updating algorithms are used to minimise the differences between the experimental results of a structure and the analytical solutions of its finite element model (FEM). In simple model updating procedures, iterative optimisation techniques can be easily used to update models and reduce the errors between experimental and analytical results. Unfortunately, experimental results as well as analytical models may have some degree of uncertainty that comes from different sources. As a result, iterative optimisation techniques may not be enough to quantify the uncertainty associated with structures. Uncertainty quantification approaches, such as the Bayesian approach, have the ability to incorporate the uncertainties associated with experiments as well as the modelling process into the updating procedure. In Bayesian finite element model updating, the uncertainty associated with the structural system is described by a posterior distribution function, while numerical tools are essential to approximate the solution of the complex posterior distribution function. In this paper, an improved evolution Markov chains Monte Carlo algorithm is used to solve the Bayesian model updating problem. In the proposed approach, the Markov chain Monte Carlo (MCMC) method is combined with the differential evolution optimising algorithm, while the final updating procedure is modified and extended with a snooker updater. The proposed approach is tested by updating a structural example, and the results are compared with the results obtained by the Metropolis-Hastings and the standard Differential Evolution Markov Chain (DE-MC) methods.

To enhance the maintenance process of the Indian River Inlet Bridge, the Delaware Department of Transportation worked with the Center for Innovative Bridge Engineering (CIBrE) of the University of Delaware (UD) to install a structural health monitoring (SHM) system on the bridge during construction. The SHM system collects data in real time 24/7. For this research, data collected are transformed into 10-minute average values. These average values represent the response of the bridge to slowly changing thermal loads or to constant loads such as dead loads.

This paper presents a method for identifying damage from the structures’ strain vs. temperature response. The methodology is evaluated based on both actual response and response simulated using a calibrated finite element model (FEM).

Using data collected over 8 years since the bridge was opened to traffic, a finite element model (FEM) was used to evaluate the ability of the SHM to identify different types and severity of damage. To do this, different levels of severity were simulated, and their effect on the structural response was compared with the observed response, including the variability of that response. Using this approach, the ability to assess various levels of damage has been determined.

The main objective of this work is to present a vibration-based damage estimation framework for structural systems by integrating vibration experimental measurements in a high-fidelity, large-scale, finite element (FE) model. Using the measured responses of a healthy structural system under operational vibrations, a parameterized FE model could be tuned using state-of-the-art FE model updating techniques in order to develop a high-fidelity model of the structural system, representing the healthy reference model. These methods provide much more comprehensive information about the condition of the monitored system than the analysis of raw data. The diagnosed degradation state, along with its identified uncertainties, can be incorporated into robust reliability tools for updating predictions on the residual useful lifetime of structural components and safety against various failure modes, taking into account stochastic models of future loading characteristics. A fault or damage would cause a sudden change in the operational responses of the structure. Incorporating the unhealthy response under measured operational excitations, a series of FE model updating runs of incrementally reparameterized FE models could be automated. The sensitivity of the unhealthy response to the parameter change pertains to the sensitive parts of the FE model, where damage or fault is located. A recursive reparameterization of those sensitive parts, followed by an FE model updating, leads toward both the detection localization and the type and magnitude of the fault or damage. The proposed framework is applied on a small-scale laboratory steel truss bridge.