Random Field Representation Research Articles

A probabilistic identification of the buckling resistance of imperfect cylindrical shells in axial compression

The buckling resistance of imperfect circular cylindrical shells of elastic and isotropic material subject to axial compression is studied focusing on initial geometric imperfections. Considering the imperfections as a random field, the suggested probabilistic representation of the buckling resistance parallels the deterministic dependence of buckling loads on the size and shape of imperfections. It derives from the probability density function of an imperfection size measure and the conditional probability density functions of the resistance confined to the realizations of imperfections given their size levels. The separated representation of the buckling resistance allows: (1) supplementary checking the accuracy of the random field representation, (2) classifying measured and simulated imperfections by their size and shape effects on buckling loads, (3) prospective delimiting the range of working imperfections for fabrication tolerance classes, and (4) defining the characteristic imperfection for the semi-probabilistic design. Introductory calculations illustrate the dependence of the normalized buckling loads of a shell on the shape and size of imperfections.

Bayesian inference for long memory term structure models

In this study, we propose a novel adaptation of the Dynamic Nelson–Siegel term structure model, incorporating long memory properties to enhance its forecasting accuracy. Our approach involves modelling the evolution of latent factors using fractional Gaussian noise processes, approximated by a weighted sum of independent first-order autoregressive components. The resulting formulation allows for a Gaussian Markov Random Field representation, facilitating the application of computationally efficient Bayesian techniques through Integrated Nested Laplace Approximations. Extensive simulation and empirical analysis demonstrate that integrating long memory significantly improves the model's forecasting performance, particularly for longer time horizons. By shedding light on the potential benefits of incorporating long memory concepts into traditional term structure models, our research highlights its utility in capturing intricate temporal dependencies and enhancing prediction precision.

Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning, including Gaussian process regression and statistical finite element analysis. Today’s prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest in the statistics and engineering literature. The SPDE representation is especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on geometrically complex domains. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix do not contain dense matrices and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples.

3-D Adaptive AUV Sampling for Classification of Water Masses

Autonomous underwater vehicles with onboard computing units foster innovative approaches for sampling oceanographic phenomena. Feedback of observations via the onboard model for planning algorithms enable adaptive sampling for such robotic units. In this work, we develop, implement, and test an adaptive sampling algorithm for efficient sampling of water masses in a 3-D frontal system. Focusing on a river plume, salinity variations are used to characterize the water masses. A threshold in salinity is assumed to distinguish the ocean and river waters, so that excursions below the threshold define river waters. The onboard model builds on a Gaussian random field representation of the salinity variations in (north, east, depth) coordinates. This model is initially trained from numerical ocean model data, and then updated with data gathered by the vehicle sensor. The Gaussian random field model further allows closed-form expressions of the expected spatially integrated Bernoulli variance of the salinity excursion set, which is used to reward sampling efforts. Combining these results with forward-looking planning algorithms, we suggest a workflow for 3-D adaptive sampling to map river plume systems. Simulation studies are used to compare the suggested approach with others. Results of field trials in the Nidelva river plume in Norway are presented and discussed.

Wind-induced probabilistic failure analysis of super large cooling tower with random field properties

As the largest civil engineering structure, its failure analysis has become to the focus of researches. However, most of studies ignore the inherent spatial variability of material properties, which exist widely in the actual engineering. Therefore, the influence of the spatial uncertainty is still unknown. In this study, an actual super large cooling tower under strong wind is investigated by the finite element (FE) analysis method first. And a three-stage non-Gaussian random field representation is proposed to characterize the spatial variability of material properties. Then 1200 random FE models with random concrete properties are investigated and analyzed. The results show that the spatial variability of material properties will not influence the wind-induced failure mode, i.e., the failure of the cooling tower is still caused by the loss of material strengths. However, the ultimate capacity will be affected by random material properties. The capacity is generally distributed in a lognormal form, and the mean value is less than the capacity obtained from the deterministic model. In some cases, the deterministic analysis will overestimate the actual capacity of the cooling tower.

Three‐stage non‐Gaussian homogeneous random field representation over manifolds

AbstractRandom fields are widely used to characterize the inherent spatial uncertainty in engineering applications. However, for widely existing manifolds, few studies focus on the simulation of Gaussian/non‐Gaussian fields over manifolds due to the geometry complexity. Recently, we proposed a two‐stage method to simulate Gaussian fields over manifolds. In this study, we extend it to a three‐stage method for non‐Gaussian fields. In the first stage, the original manifold is dimensional reduced to a Euclidean domain by the isometric feature mapping. By this step, the geodesic distance is preserved to guarantee the correlation structure unchanged. In the second stage, conventional representations can be used to simulate the underlying Gaussian field over the Euclidean domain. In the third stage, an enhanced translation procedure is proposed to translate the underlying Gaussian field to the target non‐Gaussian field over the manifold. To explain the applicability and accuracy of the three‐stage method, two kinds of non‐Gaussian fields over three different manifolds are simulated. Then, a global error is proposed to evaluate the performance of the method, and the error analysis demonstrates the method is accurate enough. Finally, two engineering structures, that is, a compressed half‐cylinder and a Canopy shell with complex geometry, are investigated to illustrate the potential applications of the method.

Multilevel representations of isotropic Gaussian random fields on the sphere

Abstract Series expansions of isotropic Gaussian random fields on $\mathbb {S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localized structure provide an alternative to the standard Karhunen–Loève expansions of isotropic random fields in terms of spherical harmonics. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analysed.

Continuous random field representation of stochastic moving loads

While the problem of random loads moving across a beam has been studied extensively, the existing methods are limited in terms of their applicability. The most common approach uses modal superposition, which requires that the load be represented in terms of the structure’s mode shapes, making the representation of the load intrinsically a function of the structure to which it is applied. To address this problem, this paper approximates the discrete stochastic moving load as a white noise passed through filters constructed with Padé approximants. The resulting model of the load is independent of the structure and enables efficient random vibration methods to be applied for solution of the problem. By representing the structure and the loading together in an augmented state space system, the variance of responses can be found directly and accurately, enabling the traffic-induced responses of a broader class of bridge structures to be analyzed. The effectiveness and usefulness of the proposed approach is demonstrated through two examples of bridge structures.

A non-Gaussian stochastic model from limited observations using polynomial chaos and fractional moments

The reasonable representation of input random fields is the key element in the reliability analysis of practical engineering systems. In most engineering applications, the characterization of a random field often relies on limited measurements. Although the simulation of random fields with complete probabilistic information has been quite well-established, reconstructing a random field from limited observations is still a challenging task. In this paper, we develop a methodology for constructing non-Gaussian random model from limited observations based on polynomial chaos (PC) and fractional moments for real-life problems. Our method begins with the reduce-order representation of measurements by Karhunen-Loève (KL) expansion, followed by the PC representation of KL coefficients. The PC coefficients are further modeled as random variables, whose distributions are determined by a modified maximum entropy principle with fractional moments (ME-FM) procedure and a ME-FM-based bootstrapping. In this way, the developed non-Gaussian model enables to quantify the inherent randomness and the statistical uncertainty of the observed non-Gaussian field simultaneously. Since the developed non-Gaussian model is embedded into the well-established PC framework, our method facilitates the implementation of PC-based stochastic analysis in practical engineering applications, in which only limited probabilistic measures are available. Two numerical examples demonstrate the application of the developed method.

Spatial 3D Matérn Priors for Fast Whole-Brain fMRI Analysis

Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the spatial priors used have several less appealing properties, such as being improper and having infinite spatial range. We propose a statistical inference framework for whole-brain fMRI analysis based on the class of Matérn covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of possibly anisotropic spatial Matérn fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditionally on the inferred hyperparameters, we make a fully Bayesian treatment of the brain activity. The Matérn prior is applied to both simulated and experimental task-fMRI data and clearly demonstrates that it is a more reasonable choice than the previously used priors, using comparisons of activity maps, prior simulation and cross-validation.

Failure assessment of corrosion affected pipeline networks with limited failure data availability

Many pipeline networks around the world do not have a failure history. Because of financial reasons or lack of technology, this is a common issue for asset managers in developing and under-developed countries. In the case of data scarcity, it is not straightforward to assess the risk and reliability of such systems. This research proposes a probabilistic framework for analysing the reliability of corroded pipeline networks with limited failure history availability. To achieve this aim through a comprehensive reliability assessment methodology, state-of-the-art corrosion modeling, component and system failure analysis techniques are utilized. A variety of spatio-temporal reliability assessment methods are employed along with the random-field representation of spatially variable parameters to estimate the failure in a large-scale corrosion affected pipeline network. The applicability of the methodology is evaluated through a case-study of buried pipeline network, and the results are verified with failure history data reported by the industry. The results obtained from the proposed methodology, combined with the cost data provided by asset managers make accurate risk assessment of process/water pipeline possible.

Isotropic non-Lipschitz regularization for sparse representations of random fields on the sphere

In this paper, we consider an infinite-dimensional isotropic non-Lipschitz optimization problem with ℓ 2 , p \ell _{2,p} ( 0 > p > 1 0>p>1 ) regularizer for random fields on the unit sphere with spherical harmonic representations. The regularizer not only gives a group sparse solution, but also preserves the isotropy of the regularized random field represented by the solution. We present first order and second order necessary optimality conditions for local minimizers of the optimization problem. We also derive two lower bounds for the nonzero groups of stationary points, which are used to prove that the infinite-dimensional optimization problem can be reduced to a finite-dimensional problem. Moreover, we propose an iteratively reweighted algorithm for the finite-dimensional problem and prove its convergence. Finally, numerical experiments on Cosmic Microwave Background data are presented to show the efficiency of the non-Lipschitz regularization.

Stochastic modeling of geometrical uncertainties on complex domains, with application to additive manufacturing and brain interface geometries

We present a stochastic modeling framework to represent and simulate spatially-dependent geometrical uncertainties on complex geometries. While the consideration of random geometrical perturbations has long been a subject of interest in computational engineering, most studies proposed so far have addressed the case of regular geometries such as cylinders and plates. Here, standard random field representations, such as Karhunen–Loève expansions, can readily be used owing, in particular, to the relative simplicity to construct covariance operators on regular shapes. On the contrary, applying such techniques on arbitrary, non-convex domains remains difficult in general. In this work, we formulate a new representation for spatially-correlated geometrical uncertainties that allows complex domains to be efficiently handled. Building on previous contributions by the authors, the approach relies on the combination of a stochastic partial differential equation approach, introduced to capture salient features of the underlying geometry such as local curvature and singularities on the fly, and an information-theoretic model, aimed to enforce non-Gaussianity. More specifically, we propose a methodology where the interface of interest is immersed into a fictitious domain, and define algorithmic procedures to directly sample random perturbations on the manifold. A simple strategy based on statistical conditioning is also presented to update realizations and prevent self-intersections in the perturbed finite element mesh. We finally provide challenging examples to demonstrate the robustness of the framework, including the case of a gyroid structure produced by additive manufacturing and brain interfaces in patient-specific geometries. In both applications, we discuss suitable parameterization for the filtering operator and quantify the impact of the uncertainties through forward propagation.

Efficient Simulation of Random Fields by Trigonometric Polynomial and Low‐rank Tensor

AbstractWe propose a fast and economic representation of stationary random fields in trigonometric polynomial, utilizing the prowess of fast Fourier transform (FFT) and low‐rank tensor approximation. With the method we are able to generate large random fields with discretization size up to 220 which are otherwise well beyond the capacity of PCs. We also illustrate the approach to the specified property of random field by increasing rank in the tensor approximation.

Cyclic performance of a monopile in spatially variable clay using an advanced constitutive model

The performance of monopiles in cohesive soils is of great interest for future offshore wind farm developments, particularly under the cyclic loads experienced in the ocean environment. Clay behaviour during undrained cyclic loading is complex and involves the accumulation of plastic strains, generation of excess pore-water pressures and degradation of initial stiffness. In this paper, the cyclic performance of a laterally-loaded monopile in spatially variable clay is investigated for the first time. A kinematic hardening constitutive model is used in a 3D finite element analysis to capture the hysteretic stress-strain behaviour of the clay. The monopile is installed in overconsolidated London Clay, which is present at several offshore wind farms in the Thames Estuary. The finite element model is coupled with random field representations of initial stiffness and clay structure. The statistical characterisation of the random fields was undertaken considering parameter ranges observed in laboratory tests. Under one-way cyclic loading, the monopile showed ratcheting behaviour, where pile rotation accumulates with increasing numbers of load cycles. The cyclic secant stiffness also increased due to the generation of negative excess pore-pressures in the clay. This behaviour occurred in both homogeneous and spatially variable clay. The monopile was also subjected to an extreme dynamic event and the soil response around the monopile showed increasing variability in stress-strain response and generation of excess pore-water pressure over time as plastic strain accumulated. However, the overall behaviour of the foundation was governed by a spatial average of the mobilised clay. The range in monopile response demonstrates how the natural spatial variability of clay can have a strong influence on monopile performance.

Time-dependent finite element reliability assessment of cast-iron water pipes subjected to spatio-temporal correlated corrosion process

Reliable prediction of the service-life of water pipes is of great importance for asset managers and decision makers. This paper introduces a framework for evaluating the reliability of corroded pipelines. Incorporating the random field representation of corrosion into a finite-element analysis have always been a daunting task especially when the time-dependent reliability analysis is intended. This research addresses the issue by representing the cross-section reduction of a buried pipe due to corrosion through a combination of the Gamma process concept and copula. Moreover, spatial and temporal evolutions of the correlation structure that exists among the corrosion pits over the pipe surface are considered using a time-dependent correlation length model recently introduced by the authors. A three-dimensional non-linear finite element analysis is used to model the residual strength of pipes in terms of time. The method is applied to a case study for estimating the failure probability of a corrosion-affected cast iron water pipe. Furthermore, the impact of the correlation structure of the corrosion depths on the estimated probability of failure is investigated. The research concludes that the proposed method is able to predict the service life of corroding buried pipeline efficiently.

Comparative Investigation of Two Random Medium Models for Concrete Mesostructure

Concrete is intrinsically endowed with randomness on meso-scale due to the random distribution of aggregates, mortar, etc. In this paper, two random medium models of concrete mesostructure are developed and comparative studies are provided based on random field representation approach. In the first place, concrete is considered as a kind of one-phase random field, where stochastic harmonic function is adopted as the approach to simulate the random field. Secondly, in order to represent the stochastic distribution of the multi-phase of concrete such as aggregates and mortar, two-phase random field based on the Nataf transformation and the Hermite polynomials are introduced. Then, the proposed two random medium models are testified by the multi-scale simulation results, specifically, the mean value of the homogenized stress-strain relationship and the damage evolution curve. Meanwhile, the generalized density evolution equation is utilized to measure the consistency of these two random medium models by the probability distribution of damage.

Buckling assessment of thin-walled plates with uncertain geometrical imperfections based on non-probabilistic field model

Abstract It is well-known that the buckling response of thin-walled structures is greatly affected by initial geometrical imperfections. In this study, the uncertainty in such geometrical imperfections is modelled by a non-probabilistic field model that is a reasonable and convenient alternative to the probabilistic random field representation of uncertainties with limited samples. After representing the bounded field uncertainty as a function of a reduced set of uncorrelated uncertain coefficients using the series expansion method, the buckling assessment problem of thin-walled plates is then constructed to minimize the critical buckling load under the non-probabilistic field description of imperfections and the volume constraint of plates. The optimization problem is solved effectively using a standard gradient-based algorithm with adjoint-variable sensitivity analysis. Numerical examples are given to demonstrate the validity and applicability of the proposed model for assessing both the critical buckling load and the worst imperfection pattern in thin-walled plates.

Bayesian inference of random fields represented with the Karhunen–Loève expansion

The integration of data into engineering models involving uncertain and spatially varying parameters is oftentimes key to obtaining accurate predictions. Bayesian inference is effective in achieving such an integration. Uncertainties related to spatially varying parameters are typically represented through random fields discretized into a finite number of random variables. The prior correlation length and variance of the field, as well as the number of terms in the random field discretization, have a considerable impact on the outcome of the Bayesian inference, which has received little attention in the literature. Here, we investigate the implications of different choices in the prior random field model on the solution of Bayesian inverse problems. We employ the Karhunen–Loève expansion for the representation of random fields. We show that a higher-order Karhunen–Loève discretization is required in Bayesian inverse problems as compared to standard prior random field representations, since the updated fields are non-homogeneous. Furthermore, the smoothing effect of the forward operator has a large influence on the posterior solution, particularly if we are analyzing quantities of interest that are sensitive to local random fluctuations of the inverse quantity. This is also reflected in posterior predictions, such as the estimation of rare event probabilities. We illustrate these effects analytically through a 1D cantilever beam with spatially varying flexibility, and numerically using a 2D linear elasticity example where the Young’s modulus is spatially variable.

Series representations of isotropic vector random fields on balls

This paper deals with a class of second-order vector random fields in the unit ball of Rd, whose direct/cross covariances are invariant or isotropic with respect to a distance defined on the ball, and gives a series representation of such an isotropic vector random field. A necessary format of covariance matrix functions is also derived for isotropic and mean square continuous vector random fields on the ball.