Model Validation and Uncertainty Quantification, Vol. 3

Structural health monitoring (SHM) involves gathering data to assess the condition of a structure, often to inform maintenance decisions throughout its life cycle. While the information obtained from an SHM system can enhance decision-making, it is essential to evaluate its overall value by considering the expenses associated with designing, developing, installing, maintaining, and operating the system. Quantifying the value of an SHM system requires pre-posterior cost-benefit analysis. To facilitate the evaluation of an SHM system during the design phase, we propose three time-normalized, unitless value of information (VoI) metrics that can be useful in evaluating the performance of an SHM design over an extended period of time. These metrics are (1) average annual expected rate of savings; (2) compounded annual expected rate of savings; (3) exponentially compounded annual expected rate of savings. We investigate the limiting cases of these metrics and demonstrate their use on a miter gate life cycle management application.

In this chapter we consider how to build a structured knowledge graph (KG) for a geometric and behavioral digital twin in the context of modal testing. The concept is based on combining geometric information from computer-aided design (CAD) model(s) and dynamic properties extracted from modal testing data and a finite element analysis (FEA) to create a digital twin. The material properties of the structure are defined in a separate bill of materials that is uploaded to the digital twin. The functionality of the digital twin is to provide a continuous “digital thread” of events during the modal test and gather the information that at a later stage could be used for validation and updating the finite element model of the structure (although that is not discussed in detail here). The modal testing data is taken from a small-scale three-story structure that is used to demonstrate the concept. The structured KG is built using the Neo4j interface, operated by the py2neo Python package. The KG is defined in an entity-event format that can be dynamically updated as new information is received in the digital twin. The CAD information is integrated into the KG using an STL file format. The KG is seeded with the STL file data and the bill of materials. Then as the modal test proceeds, data segments obtained from the sensors are added to a database and simultaneously added to the KG. As the KG evolves, it creates a digital thread of the test that can be interrogated as required to provide information to the user(s) and enable more effective asset management. The KG can also be integrated into a wider digital twin functionality of the structure.

The chapter introduces a software library for addressing the propagation of systematic and random measurement uncertainty in the frequency domain for system identification purposes. The MATLAB-based library aims to overcome the prevailing neglect of systematic uncertainties, e.g., for large data sets, and offers the possibility to evaluate their significance in obtaining accurate and reliable system models. It uses a sophisticated sensor error model in combination with simple propagation laws, initially obtained by Monte Carlo simulations, to propagate the modelled errors into the frequency domain. The modelled errors include four of the most widespread errors for arbitrary sensor signals.

This contribution gives a general background of Bayesian statistics with a real application example. It includes the background and meaning of the essential elements: the prior probability of a hypothetical event, the likelihood of a symptomatic event under the condition that the hypothetical event already happened, the total probability of the symptomatic event, and the posterior probability of the hypothetical event under the condition that the symptomatic event already happened. The Bayesian approach is the framework to estimate the uncertainty of two different analytical model forms used to predict the vibration isolation behavior for an oscillating mass. The first model form describes the oscillating mass as part of a simple one-mass oscillator; the second model form uses the oscillating mass as part of a simple two-mass oscillator. Both model forms contain assumptions about inertia, damping, and stiffness properties representing hypothetical events. The experimental vibrational data measured from a realized test setup featuring the oscillating mass serves as the symptomatic outcome or event. This approach allows the direct and consistent application of the Bayesian framework on a vibration isolation problem in early stage design. This work is part of IMAC’s ongoing round-robin challenge about model form uncertainty quantification on vibration isolation.

Dynamical system state estimation and parameter calibration problems are ubiquitous across science and engineering. Bayesian approaches to the problem are the gold standard as they allow for the quantification of uncertainties and enable the seamless fusion of different experimental modalities. Current techniques such as Kalman, particle, or variational filters apply to discretized dynamical systems. To apply these methods, practitioners must introduce fictitious transition probabilities that might lead to unsatisfactory inference bias. To address these drawbacks, this chapter aims to develop a unifying paradigm that seamlessly combines measurement data and physics to solve dynamical system state and parameter estimation problems. Our method builds upon the information field theory, which is essentially Bayesian statistics for physical fields. Specifically, we construct a physics-informed prior probability measure on the function space of system responses so that functions that satisfy the physics are more likely. This prior allows us to quantify model-form errors. We connect the system’s response to observations through a probabilistic model of the measurement process. Bayes’ rule gives the joint posterior over the system responses and all parameters. We apply and compare two numerical methods to sample from the analytically intractable posterior: (1) sampling using stochastic gradient Langevin dynamics and (2) stochastic variational inference developed in our previous works. Our approach can quantify model-form uncertainties without requiring any numerical solver. The developed methodology offers a powerful framework for Bayesian estimation in dynamical systems.

Tire/road noise has emerged as a significant factor in vehicle noise performance as a result of electrification. In the noise and vibration domain, the industry places great emphasis on capturing the realistic dynamic behavior of substructures. The dynamics of tires are particularly susceptible to variations arising from operational conditions like preload, inflation, and rotational speed. Siemens Digital Industries Software currently provides a lightweight solution combining test and simulation-based approaches for capturing the structure-borne noise of non-rolling tires. The proposed model is a finite element model that expresses the static tire dynamics in the frequency domain and as such fits for efficient integration in full-vehicle models. Experimental modal analysis and frequency response function (FRF) tests are used to validate the proper dynamic behavior in static conditions. In this chapter, the model has been extended with gyroscopic features to include rotational effects in an efficient frequency-domain approach. An advantage with respect to the standard time-domain simulation for rolling tire dynamics is the reduction of the computational effort needed by our method. This approach is validated against experimental results measured via an innovative impact testing technique, allowing the isolation of pure rotational effects from the ones caused by preload deformation.

The Population-Based Structural Health Monitoring (PBSHM) paradigm has recently emerged aiming to enhance data-driven assessment of engineering structures by allowing data to be shared and learning to be transferred between similar structures. In this work, we gear this concept toward automated modal identification of structural systems. Toward modal identification from a PBSHM perspective, we here present a Graph Neural Network (GNN)-based deep learning scheme to identify mode shapes of engineering structures on the basis of monitored (measured) responses. The generation of the training dataset, which includes mode shapes and noise-polluted dynamic responses, relies on availability of an engineering model. Finite element (FE)-based modal and dynamic analyses are first carried out on a population of structures that share certain morphological/typological characteristics but comprise different geometric (size and shape) and material (stiffness) characteristics. The trained model is in a next step fed with dynamic response data from unseen structures and is able to output, in an automated fashion, the corresponding mode shapes. These unseen structures form members of the explored “population” but have not been generated for use within the training set. Moreover, we show that mode shape inference under availability of sparse measurements can be achieved by coupling the GNN with a Feature Propagation operator in what we here term a GNN-OMA approach. A series of numerical experiments are conducted to test the performance of the proposed method. Results show that the proposed model exhibits good accuracy and generalization ability when identifying mode shapes for structures within the same population, rendering its use promising for PBSHM purposes.

Low-fidelity structural dynamics models reveal critical system features, allow for real-time control, and provide ballpark predictions. However, these models are sometimes too simplified to be reliable. In the adaptive vibration isolation round-robin challenge problem, a low-fidelity two-mass oscillator model struggles to capture nonlinear behaviors, resulting in high model-form error and discrepancies between model predictions and experimental data. In this chapter, we explore how to reduce model-form error without increasing the number of states in the model, which would incur greater computational costs. In particular, we embed a stochastic model correction into the low-fidelity model, yielding an enriched model. The correction is informed by physical theory, calibrated with experimental data, and validated using posterior predictive assessments. While the enriched model does not perfectly capture the experimental data, it improves consistency with data over a range of experimental scenarios.

When an unknown input needs to be estimated, one would typically use either a finite element model or a more data-driven modal model. Both types of models have their respective advantages, which we would like to exploit. Choosing only one model, however, leaves the information of the other type unused. An important observation is then that both types of models typically work best in different frequency ranges. In order to exploit both types, we therefore propose a weighted optimization method. For a certain frequency domain, the error with respect to a model is then weighted depending on the reliability of said model in that frequency domain. In order to use the proposed method in the time domain, we first take the Fourier transform of the time data. Then, the input reconstruction is performed in the frequency domain, and finally the inverse Fourier transform is taken of the result. This method was validated using an experimental beam setup. In this validation, the proposed method using two models outperformed the results of using either model separately.

Stress corrosion is a common and complex deterioration type in large steel civil infrastructure such as miter gates. The diagnostics and prognostics of corrosion damage in such structures is a major component of any structural health monitoring (SHM) approach. This chapter performs corrosion damage diagnostics and prognostics on a miter gate structure using multiscale corrosion simulation models developed at micro- and mesoscales. The diagnostic process will integrate measurements from macroscale images and water levels for the damage localization and diagnostics. The prognostic models will use the detected corrosion damage data from a deep learning model to update parameters in corrosion initiation time distribution, propagate uncertainties across the multiscale simulations, and eventually, estimate the remaining useful life.

In the previous work, the authors proposed an uncertainty-aware metric called the information measure of calibration flexibility (ICaF). ICaF addresses the trade-off between goodness of fit and model generalizability, and its efficacy has been proved in calibration parameter selection and model selection. This study further investigates the impact of four influential factors on the performance of ICaF through a regression example. These factors include the model form, selection of calibration parameters, prior knowledge of the system being studied, observation characteristics (i.e., the quantity and distributions of observations), and experimental uncertainty. Models with different model simplicity affect the goodness of fit of a calibrated model and the subsequent model predictions. Prior knowledge reflects initial beliefs about the underlying system. The characteristics of observations comprehensively consider the influence of quantity and distributions of observations on the calibration process. Lastly, experimental uncertainty associated with various sources is included. This study provides a comprehensive analysis by systematically varying influential factors and assesses how these factors individually influence the effectiveness of ICaF in model calibration and forecasting. These findings contribute to a collective understanding and insights into the behavior and the robustness of ICaF, which informs the design and implementation of ICaF in scientific and engineering domains.

Bridges form a salient part of critical infrastructure networks that are faced with the adverse effects of aging. At the same time, the growing need for mobility has created new demands for higher traveling speeds and increased imposed loads, which introduce additional requirements on such aged structures. This motivates monitoring of the condition of bridges utilizing Structural Health Monitoring (SHM) schemes, which aim at identifying changes in the characteristics of the response of the respective structures. Such changes may signify the presence of damage or deterioration; thus, SHM is imperative for deciding on the remaining life of bridges and accordingly scheduling maintenance procedures. Focusing on railway bridges, SHM typically relies on stationary sensors mounted on the bridge system, with direct assessment of the collected data. Although reliable, such an approach hinders the comprehensive inspection of multiple bridges of a railway network, while the short life span of sensors poses limitations to the continuous supply of data from the structure. As an alternative, vibration-based mobile sensing that relies on traversing trains has the potential to provide data from multiple railway bridges based solely on a few sensor networks installed on the trains. At the same time, when running the network at frequent intervals (e.g., in the case of in-service trains), the sensor-equipped trains can also provide continuous data that give insight into the deterioration of bridges over time. To this end, this work proposes a model-based methodology to extract modal parameters of bridges based on acceleration data collected by traversing trains. The proposed approach relies on Kalman filtering for the estimation of the train’s state and input and a subspace identification method for the identification of the frequencies and modes of the bridge. Long-term monitoring of bridge frequencies and modes can contribute to the timely restoration in case of damage and, thus, ensure the safety and reliability of rail transportation.

The sensitivity analysis in the framework of the finite element method requires the calculation of the derivatives of the system matrices with respect to the parameters of interest. When the sensitivity with respect to geometric modifications is investigated, the problem becomes very complicated due to two factors. First, the geometry is represented by a very large number of parameters, the position of the nodes of the FE mesh, which cannot be considered explicitly in a sensitivity analysis. Second, the system matrices do not depend explicitly on the nodes’ position; thus, the calculation of the derivatives needs a proper formulation. This chapter addresses these two problems using a modal parametrization of the geometric variations and exploiting the use of the directional derivatives to represent the sensitivity of the system matrices. The proposed method is illustrated through a simple two-dimensional example and validated using the finite differences method.

Recently, increasing attention has been devoted to analyzing nonlinear phenomena in engineering design due to rapid advancements in complex lightweight structures and novel materials. These structures are generally modeled using Finite Element Methods, resulting in large systems of ordinary differential equations. For large degrees of freedom, parametric studies are often infeasible, and thus, engineers look for techniques to reduce the computational cost and ultimately expedite the design process. Linear projection-based Reduced-Order Model (ROM) solves this problem by identifying a linear modal basis (e.g., using full-scale training data) that aims to embed the relevant nonlinear normal modes utilizing a subset of these modal vectors. Since this modal basis is usually identified using training data from the full-scale nonlinear system, the identified projection basis is expected to be robust for small variations of the forcing energy levels used in the training data. The Proper Orthogonal Decomposition (POD) has been the standard approach to identify this reduced projection basis. Since POD identifies the modes based on their energy contribution within the training data, arguments are based on including enough modes in the reduced projection basis to achieve some energy-preserving criteria. Unfortunately, this approach to model order reduction is highly specific to the training data used and, more importantly, lacks a physical meaning when considering the underlying dynamics of the system. This work uses a Generalized Characteristic Value Decomposition (GCVD) to construct physics-informed reduced-order models by considering the identified modes’ decay rates and oscillation frequencies when selecting the reduced projection basis. Thus, our reduced-order models can be built based on a slow, intermediate, and fast time dynamics hierarchy. A significant advantage of this approach is its versatility for free and forced vibrations over a wide forcing frequency range. The proposed model order reduction strategy is demonstrated on several geometrically nonlinear finite element models. A numerical continuation is performed on both the reduced order model and full-scale systems to verify the accuracy of the GCVD-ROMs for predicting the primary frequency response curves, stability of solutions, and sub- and superharmonic resonances.

Assembled systems typically contain mechanical joints that are in physical contact and heavily influenced by friction and vibration. Friction is affected by contact stress, temperature, material, and roughness of contacting parts, from geometrical features at the macro- to nanoscale. Understanding and predicting the friction of contact helps to create designs that reduce wear, crack propagation, damage, and energy consumption. Recently, digital twins have been used in different mechanical engineering mechanisms and systems to predict crack, damage, and frequency response functions. Digital twins, with their system-level thinking, have promoted the idea of cross-industry development and ideology. The aim of the current study is to develop the digital twin-enabling technology for a simple dry contact under reciprocating motion. This enabling technology (digital twins) is the development of a grey-box model using conventional tribometer experimental data under cyclic loading and advanced multi-scale (contact mechanics to macro-scale dynamics) finite element analysis to provide an accurate estimation in a realistic time scale for digital twins. To demonstrate this, a ball and a flat plate made of steel (304) were used to create a physical twin. The test was run using a Universal Mechanical Tester (Broker UMT-3 tribometer) under speed and load sweep conditions to determine the coefficient of friction at different operating conditions. The experimental data for friction were collected and used for machine learning along with an FEA model using Abaqus which makes the digital twin. The machine learning part of the digital twin was used to predict the coefficient of kinetic friction under different operating conditions and can interoperate with other models to greatly expand the digital twin functionality. The predicted coefficient of friction was fed to FEA model to predict the mechanical behaviour of the system such as Frequency Response Function.

Even though Bayesian inference has been widely applied in the field of structural health monitoring (SHM), it often encounters noteworthy challenges if the likelihood function lacks a closed-form expression or is numerically intractable. This is especially true when the computational simulation models involve hierarchically connected sub-models. While likelihood-free approaches have been developed using neural networks to deal with this issue, currently available methods are not suitable for estimating variables that dynamically change with time. This study proposes an innovative likelihood-free inference method for efficient dynamic state estimation using continuously collected data. The proposed framework leverages two complementary neural networks, namely a posterior network and a likelihood network, within the Bayesian framework. The posterior network approximates the posterior distributions for any given observation, while the likelihood network emulates the likelihood of the underlying probabilistic model. These two networks are jointly implemented using conditional invertible neural networks (cINN) based on the normalizing flow. To facilitate continuous model updating over an extended monitoring period, we extended a recursive model updating strategy that was proposed in our previous research to the cINN-based posterior neural approximator and likelihood neural approximator. The proposed framework allows for dynamic parameter estimation over time while quantifying the uncertainty in the estimation. To validate the efficacy of the proposed likelihood-free inference framework, we apply it to a four-story shear frame. The results demonstrate the effectiveness of the proposed framework in estimating time-varying uncertain model parameters.

Engineers are often faced with the decision to select the most appropriate model for simulating the behavior of engineered systems, among a candidate set of models. Experimental monitoring data can generate significant value by supporting engineers toward such decisions. Such data can be leveraged within a Bayesian model updating process, enabling the uncertainty-aware calibration of any candidate model. The model selection task can subsequently be cast into a problem of decision-making under uncertainty, where one seeks to select the model that yields an optimal balance between the reward associated with model precision, in terms of recovering target Quantities of Interest (QoI), and the cost of each model, in terms of complexity and compute time. In this work, we examine the model selection task by means of Bayesian decision theory, under the prism of availability of models of various refinements, and thus varying levels of fidelity. In doing so, we offer an exemplary application of this framework on the IMAC-MVUQ Round-Robin Challenge. Numerical investigations show various outcomes of model selection depending on the target QoI.

When utilizing numerical models for decision-making in the absence of experimental data, analysts often face the challenge of extrapolating simulation predictions to untested application domains. Although this extrapolation is recognized as potentially hazardous, limited methods exist to establish the credibility of such predictions. In this chapter, we present a novel diagnostic tool for quantifying the robustness of forecasting errors in the presence of uncertainties arising from the calibration process, encompassing compensating effects and measurement errors. Our approach leverages Bayesian inference to calibrate a stochastic numerical model, which enables the incorporation of probabilistic assessments. To establish a robustness indicator, we rely on numerically generated data from both the validation and forecasting domains. Through proxy simulations that emulate the application domain, we account for the absence of experimental data and consider the uncertainties inherent to the measurement process. By combining these elements, we formulate a robustness criterion that quantifies the permissible level of calibration errors while maintaining a critical threshold for forecasting accuracy. The resulting indicator guides decision-makers on whether extrapolation should be avoided, particularly when only small errors can be tolerated. To illustrate the effectiveness of our methodology, we apply it to an aluminum frame structure, showcasing its applicability in a practical context. By utilizing validated physics-based simulations and incorporating data uncertainties, our approach provides a comprehensive framework for assessing the robustness of numerical models in forecasting scenarios. This research contributes to bridging the gap between simulation-based predictions and real-world decision-making, aiming to mitigate the inherent challenges associated with extrapolation and offering valuable insights for informed decision-making.

The present work focuses on applying model predictive methods to control nonlinear vibrations. Various approaches to derive the model used within the controller, including linearization (modal and Carleman/Koopman) and nonlinear model reduction techniques, are reviewed and compared. The structure used for the numerical and experimental investigations is a straight cantilever beam with geometric nonlinearity and a 1:3 internal resonance between its first two modes, resulting in the usual saddle-node and Neimark–Sacker bifurcations and the presence of an isola. Approximate analytical and finite element models of the beam are first derived and then used to construct the linearized and nonlinear reduced-order models. The models are subsequently exploited by the model predictive controller to reach a range of stable and unstable periodic responses of the structure under harmonic excitation.

Detecting fatigue cracks in underwater structures presents unique challenges due to the complexities of the underwater environment. With the advancements in deep learning and computer vision techniques, inspection methods are transitioning towards automated machine learning–based approaches. However, the inherent uncertainties of the underwater realm pose difficulties in effectively training and utilizing these models. To address this, the research introduces a framework employing a graphics-based digital twin, derived from a finite element model, to generate synthetic images under diverse conditions. With a deep learning–based crack detection, this approach facilitates a quantitative evaluation of the impact of specific environmental factors on crack detection probabilities. By comprehending these impacts, more effective inspection strategies for large underwater structures can be developed. The utility of this framework is demonstrated through a case study on a miter gate, demonstrating its potential to improve crack detection by considering the dynamic and uncertain environmental conditions in real-world contexts.

Bayesian filtering techniques involve recursively combining a mathematical model and a system’s response measurements to enhance the model’s estimation and prediction capabilities. These techniques rely on reduced-order surrogate models for estimation, which limits the type of models used. It is important to choose a model class and its parameters with sufficient complexity to accurately represent the physical system. However, increasing the number of model parameters reduces computational efficiency and poses challenges for modeling and filtering. To tackle this issue, this chapter applies Bayesian filtering for estimating and quantifying uncertainty in a large-scale suspension strut system called the Modular Active Spring-Damper System (MAFDS). Specifically, the performance of the Kalman filter using a 2-degree-of-freedom model is investigated in terms of estimation accuracy and statistical error behavior when estimating the response of the MAFDS.

Simplifications and theoretical assumptions are often incorporated into numerical modeling of structures; however, these assumptions may reduce the accuracy of simulation results. Model updating techniques have been developed to minimize the error between experimental response and modeled structures by updating their parameters based on observed data. Structural numerical models are typically constructed using a deterministic approach, obtaining a single best-estimated value for each structural parameter. However, structural models are often complex and involve many uncertain variables, making it impossible to find a unique solution that captures all the variability. Updating techniques using Bayesian inference (BI) have been developed to quantify parametric uncertainty in analytical models. This chapter presents the implementation of BI in the parametric updating of a five-story building model and the quantification of associated uncertainty. The Bayesian framework is implemented to update the model parameters based on experimental information provided by modal frequencies and mode shapes. The main advantage of this approach is considering the uncertainty in the experimental data, leading to a better representation of the actual building behavior. Additionally, the implications of Bayesian modeling are discussed, highlighting the importance and implications of using a multivariate normal likelihood function in the analysis. The results show that this Bayesian model updating approach effectively allows for a statistically rigorous update of model parameters, characterizing the uncertainty and increasing confidence in the model’s predictions. This is particularly useful in engineering applications where model accuracy is critical.