State variance-based uncertainty quantification in structural response via Malliavin formulation

Committee Mandate Concern for the quasi-static response of ship and offshore structures, as required for safety and serviceability assessments. Attention shall be given to uncertainty quantification of quasi-static load and response analysis approaches, and their limitations, including exact and approximate methods for derivation of different acceptance criteria. Introduction In the design of ship and offshore structures, Naval Architects and Structural Engineers require access to a wide range of analysis methods to successfully progress from concept brief through to a production ready design, that will safely operate for the duration of its service life. Significant development in computational analysis techniques have occurred over the preceding decades, coupled with increased availability of high-performance computing; however, computationally intensive methods regularly do not fit the requirements of a design team, particularly in the early design phases. Whilst some quasi-static methods may have arisen at times of lesser computational capabilities, quasi-static analysis methods remain relevant, providing an appropriate balance between accuracy and speed, often having an ability to provide a quick result based on minimal input data, facilitating rapid design iteration. In ship and offshore structures, the loading, whether local or global, is predominantly caused by a dynamic motion that is cyclic or oscillating, for example the wave loading of a ship hull girder in a seaway, the sloshing loading due to the motions of fluid in a tank, the loading on a deck or equipment foundation, etc. True dynamic analysis of such scenarios is complex and time consuming to undertake, and often can’t be successfully completed until the structural design details are in a progressed state. Therefore, quasi-static methods implementing a simplified approach that resembles the scenario, whether through a defined loading or to induce a seemingly equivalent structural response, have been developed. Not all loading scenarios can be suitably represented by quasi-static methods, particularly where loading is complex or structural response of the individual parts of the system may interact. However, where a quasi-static method can be implemented to develop a structure with sufficient reserve to facilitate safe operation, the benefits to the design process can be significant. In structural response analysis, a method may be considered to be quasi-static where the effects of structural dynamics (structural inertia and damping) may be neglected. In this regards the time component, or time derivatives, may be neglected. To adopt a quasi-static method, the true time dependant loading must be sufficiently slow in relation to the structural response not to coincide with resonant response frequencies. Due to this ‘slow’ progression, during analysis the system may be considered to be in static equilibrium at all time instances. These points are true for many quasi-static analyses, where loading may be through incremental application of force or displacement to a structure, and static equilibrium of the system is achieved before the next increment is applied. Therefore, time associated with the loading is only implied and not explicitly included in the assessment. In other words, the structural responses at any time instant will be only determined by the loads at that time instant, and the structural responses have no memory effect. Whilst the applied loading may be incremental, it need not be entirely linear, and in the same regard the structural response also need not be linear. For example, the loading and response could be coupled, such that as the structure deforms the load is iterated to reflect the new state of the system. However, in the application of quasi-static methods, the relative accuracy and therefore suitability of the method should always be considered. Quasi-static analysis covers a broad spectrum of methods from hand calculations to finite element analysis (FEA) and may even combine methods such as computational fluid dynamics (CFD). The methods may be used directly for structural assessment, or as part of broader method, such as the input to a reliability analysis or optimisation routine, or to derive a peak stress or stress sequence for fatigue assessment. For this reason, there may be perceived overlap between this and other ISSC committees. However, this committee has specifically focussed the presented report around methods that are quasi-static in nature, including where the topics, such as fatigue, ultimate and accidental limit states, that are covered in depth from a different perspective by other ISSC Technical Committees.

We present here an uncertainty quantification methodology assisting fracture/damage detection that accounts for different sources of uncertainty involving in real-time damage identification. The relationship between structural responses (i.e., the natural frequencies) and the uncertain crack locations and lengths in mechanical systems is quantitatively assessed using variance-based sensitivity analysis. Sparse variational Gaussian process is employed to replace the computationally expensive finite element model. Dynamic behaviors of two mechanical systems which are (1) aluminum frame with adjustable masses and (2) cracked car suspension arm are analyzed through experimental and numerical data. The results demonstrate the effectiveness and robustness of combining these methods for structural assessments, providing precise and reliable insights applicable to complex mechanical models. We believe that this study enhances the understanding of vibration-induced fracture/damage and highlights the potential of this framework for advancing structural reliability analysis and dynamic system optimization.

This study focuses on the development of efficient surrogate models by polynomial chaos expansion (PCE) for the prediction of the long-term extreme surge motion of a moored floating offshore structure. The structure is subjected to first-order and second-order (difference-frequency) wave loading. Uncertainty in the long-term response arises from contrasting sea state conditions, characterized by significant wave height, Hs, and spectral peak period, Tp, and their relative likelihood of occurrence; these two variables are explicitly included in the PCE-based uncertainty quantification (UQ). In a given sea state, however, response simulations must be run for the associated Hs and Tp; in such simulations, typically, a set of random amplitudes and phases define an irregular wave train consistent with that sea state. These random amplitudes and phases for all the frequency components in the wave train introduce additional uncertainty in the simulated waves and in the response. The UQ framework treats these two sources of uncertainty—from Hs and Tp on the one hand, and the amplitude and phase vectors on the other—in a manner that efficiently yields long-term surge motion extreme predictions consistent with more expensive Monte Carlo simulations (MCS) that serve as the “truth” system. To reduce uncertainty in response extremes that result from sea states with a low likelihood of occurrence, importance sampling is employed with both MCS- and PCE-based extreme response predictions. Satisfactory performance with such efficient surrogate models can help in assessing the long-term response of various offshore structures.

Bayesian based nonlinear model updating using instantaneous characteristics of structural dynamic responses

The manufacturing and fabrication of complex polymer sandwich composite plates involve various processes and parameters, and the lack of control over them causes uncertain system parameters. It is essential to consider randomness in varying parameters to analyse polymer sandwich composite plates. The present study portrays uncertainty quantification in structural responses (such as natural frequencies) of polymer sandwich composite plates using the surrogate model. The comparative study of artificial neural network (ANN) and polynomial neural network (PNN) for uncertain structural responses of the sandwich plate is presented. The proposed ANN as well as PNN algorithm is found to be convergent with intensive Monte Carlo simulation (MCS) for uncertain vibration responses. The predictability of PNN is observed to be more efficient than that of ANN. Typical material properties, skew angle, fibre orientation angle, number of laminate and core thickness are randomly varied to quantify the uncertainties. The use of both the surrogate models (PNN and ANN) results in a significant saving of computational time and cost compared to that of full-scale intensive finite element-based MCS approach.

Stochastic characterization of multiscale material uncertainties on the fibre-matrix interface stress state of composite variable stiffness plates

A hierarchy model for the uncertainty quantification of spatial variability in the constitutive parameters of concrete in structures

Bridge Moving Load Identification aims to reconstruct the spatiotemporal evolution of vehicle loads from measured structural dynamic responses, supporting load rating, fatigue assessment, and overload management in structural health monitoring. In bridge engineering practice, MFI is widely recognized as a severely ill-posed inverse problem. The system matrix derived from the vehicle–bridge dynamic model is typically ill-conditioned, leading to instability and non-uniqueness, so that small perturbations in measurements or modeling assumptions can produce large deviations in the identified loads. These difficulties are further aggravated by vehicle–bridge interaction effects, uncertain vehicle kinematics such as speed and axle configuration, limited sensor coverage, and operational and environmental variability. This review synthesizes recent progress along three complementary lines. First, classical time-domain and state-space identification frameworks are revisited, emphasizing regularization strategies and Krylov-subspace iterative solvers for stabilizing large ill-conditioned systems. Second, modern signal-processing approaches are summarized that exploit sparse sensing and sparse representations, including redundant dictionary modeling, non-convex and reweighted regularization to improve identification accuracy under noise and undersampling, and adaptive dictionary learning to mitigate representation mismatch. Third, deep-learning-based technologies are reviewed, covering data-driven inverse operator learning for near-real-time inference and physics-guided learning that embeds structural operators and governing equations, with particular attention to physics-informed neural networks for vehicle–bridge coupling and limited-label scenarios. Remaining challenges are discussed in real-time deployability, robustness to domain shift, uncertainty quantification, and data efficiency. Future directions point to physics–data dual-driven frameworks and digital-twin-enabled calibration for reliable load sensing in complex service environments.

<strong class="journal-contentHeaderColor">Abstract.</strong> Uncertainty quantification is necessary in wind turbine design due to the random nature of the environmental inputs, through which the uncertainty of structural loads and response under specific situations can be quantified. Specifically, wind turbulence (described by the standard deviation of the longitudinal wind speed over a 10 min time duration) has a significant impact on the extreme and fatigue design envelope of the wind turbine. The wind parameters (mean and standard deviation of longitudinal wind speed over 10 min time duration) are not independent stochastic variables, and structural reliability analysis or uncertainty quantification therefore requires these wind parameters to be correlated stochastic parameters. An accurate probabilistic model should be established to model the correlation among wind parameters. Compared to univariate distributions, theoretical multivariate distributions are limited and not flexible enough to model the wind parameters from different sites or direction sectors. Copula-based models are often used for correlation description, but existing parametric copulas may not model the correlation among wind parameters well, due to limitations of the copula structures. The Gaussian mixture model is widely applied for density estimation and clustering in many domains, but limited studies have been conducted in wind energy and few have used it for density estimation of wind parameters. In this paper, the Gaussian mixture model is used to model the joint distribution of mean and standard deviation of longitudinal wind speed over 10 min time duration, which is calculated from 15 years of wind measurement time series data. As a comparison, the Nataf transformation (Gaussian copula) and Gumbel copula are compared with the Gaussian mixture model in terms of the estimated marginal distributions and conditional distributions. The Gaussian mixture model is then adopted to estimate the extreme wind turbulence (wind parameters for extreme load), which could be taken as an input to design loads used in the ultimate design limit state of turbine structures. The wind parameter contour associated with a 50-year return period computed from the Gaussian mixture model is compared with what is used in the design of wind turbines as given in IEC 61400-1. The Gaussian mixture model is able to model the joint distribution of wind parameters well, where the estimated tail distributions of both the marginal distributions and conditional distribution have good accuracy, and it is a good candidate for extreme turbulence estimation.

Uncertainty quantification is a necessary step in wind turbine design due to the random nature of the environmental loads, through which the uncertainty of structural loads and responses under specific situations can be quantified. Specifically, wind turbulence has a significant impact on the extreme and fatigue design envelope of the wind turbine. The wind parameters (mean and standard deviation of 10-minute wind speed) are usually not independent, and it will lead to biased results for structural reliability or uncertainty quantification assuming the wind parameters are independent. A proper probabilistic model should be established to model the correlation among wind parameters. Compared to univariate distributions, theoretical multivariate distributions are limited and not flexible enough to model the wind parameters from different sites or direction sectors. Copula-based models are used often for correlation description, but existing parametric copulas may not model the correlation among wind parameters well due to limitations of the copula structures. The Gaussian mixture model is widely applied for density estimation and clustering in many domains, but limited studies were conducted in wind energy and few used it for density estimation of wind parameters. In this paper, the Gaussian mixture model is used to model the joint distribution of mean and standard deviation of 10-minute wind speed, which is calculated from 15 years of wind measurement time series data. As a comparison, the Nataf transformation (Gaussian copula) and Gumbel copula are compared with the Gaussian mixture model in terms of the estimated marginal distributions and conditional distributions. The Gaussian mixture model is then adopted to estimate the extreme wind turbulence, which could be taken as an input to design loads used in the ultimate design limit state of turbine structures. The wind turbulence associated with a 50-year return period computed from the Gaussian mixture model is compared with what is utilized in the design of wind turbines as given in the IEC 61400-1.

This paper presents a hierarchical Bayesian modeling framework for the uncertainty quantification in modal identification of linear dynamical systems using multiple vibration data sets. This novel framework integrates the state-of-the-art Bayesian formulations into a hierarchical setting aiming to capture both the identification precision and the variability prompted due to modeling errors. Such developments have been absent from the modal identification literature, sustained as a long-standing problem at the research spotlight. Central to this framework is a Gaussian hyper probability model, whose mean and covariance matrix are unknown, encapsulating the uncertainty of the modal parameters. Detailed computation of this hierarchical model is addressed under two major algorithms using Markov chain Monte Carlo (MCMC) sampling and Laplace asymptotic approximation methods. Since for a small number of data sets the hyper covariance matrix is often unidentifiable, a practical remedy is suggested through the eigenbasis transformation of the covariance matrix, which effectively reduces the number of unknown hyper-parameters. It is also proved that under some conditions the maximum a posteriori (MAP) estimation of the hyper mean and covariance coincide with the ensemble mean and covariance computed using the optimal estimations corresponding to multiple data sets. This interesting finding addresses relevant concerns related to the outcome of the mainstream Bayesian methods in capturing the stochastic variability from dissimilar data sets. Finally, the dynamical response of a prototype structure tested on a shaking table subjected to Gaussian white noise base excitation and the ambient vibration measurement of a cable footbridge are employed to demonstrate the proposed framework.

Adaptive sub-interval perturbation-based computational strategy for epistemic uncertainty in structural dynamics with evidence theory

In this paper, we propose a numerical procedure for the quantification of uncertainties in wave–structure interaction. We utilize the smoothed particle hydrodynamics (SPH) scheme for modeling the wave mechanics, coupled one-way with a finite element method (FEM) for the structural response. Physical wave flumes are extensively used in the study of hydrodynamics especially in wave–structure interaction (WSI) and prediction of forces to near-shore structures in disaster mitigation and offshore structures in the oil and gas, and more recently renewable energy sector. Over the years, numerical wave flumes have been developed extensively to enable the modeling of complex wave–structure interaction. However, most of these studies are deterministic and limited to using either simple flexible beams or rigid monolithic structures to model the structural part in the WSI. Additionally, uncertainties are commonly observed in both wave and structural parameters and need to be accounted for. This work presents a numerical framework to enable uncertainty quantification for wave–structure interaction problems in terms of the forces experienced by the structure. A one-way coupling between SPH with the FEM and uncertainty quantification is proposed. We employ the so-called Tokyo wave flume geometry, which has potential for future surrogate modeling in WSI. The developed model is validated using numerical and experimental results from the literature and is used to demonstrate the prediction of probabilistic responses of structures under breaking and non-breaking wave scenarios.

Modern deep learning models achieve state-of-the-art results for many tasks in computer vision, such as image classification and segmentation. However, its adoption into high-risk applications, e.g. automated medical diagnosis systems, happens at a slow pace. One of the main reasons for this is that regular neural networks do not capture uncertainty. To assess uncertainty in classification, several techniques have been proposed casting neural network approaches in a Bayesian setting. Amongst these techniques, Monte Carlo dropout is by far the most popular. This particular technique estimates the moments of the output distribution through sampling with different dropout masks. The output uncertainty of a neural network is then approximated as the sample variance. In this paper, we highlight the limitations of such a variance-based uncertainty metric and propose an novel approach. Our approach is based on the overlap between output distributions of different classes. We show that our technique leads to a better approximation of the inter-class output confusion. We illustrate the advantages of our method using benchmark datasets. In addition, we apply our metric to skin lesion classification—a real-world use case—and show that this yields promising results.

The probabilistic evaluation of the capacity and response of structural systems is often done by probabilistic simulation. This approach relies on a considerable number of deterministic analyses to obtain an accurate estimation of the desired statistical characteristics. However, the need of performing a large number of analyses hinders the application of probabilistic simulation in practice, especially when the deterministic calculation of the structural response is computationally expensive in itself (e.g. nonlinear finite element analysis). To increase the appeal of probabilistic simulation, the number of analyses should be very small (i.e. ten or less). This study shows how different sampling techniques can be adopted to select a very small sample subset of analyses to be run, and how they can yield accurate results in estimating the shear capacity of prestressed reinforced concrete beams. Based on the results of the study, the Fractile Based Sampling (FBS) method emerges as a more promising sampling strategy than other sampling techniques, like Latin Hypercube Sampling (LHS). For the same small number of samples, the results show that FBS is preferable over LHS and other sampling techniques for capturing both the general distribution of the response and the tails of the distribution, which shows its usefulness in assessing probability of failure and reliability.