non-Gaussian Random Fields Research Articles

Higher-order spectral representation method: New algorithmic framework for simulating multi-dimensional non-Gaussian random physical fields

This study derives a novel higher-order spectral representation method (HOSRM) to represent and simulate multi-dimensional fourth-order non-Gaussian random physical fields. The method primarily extends the traditional second-order spectral representation method (SRM) for simulating non-Gaussian random physical fields by introducing higher-order cumulant function tensors and trispectrum tensors, thereby accomplishing the modeling of fourth-order non-Gaussian random physical fields (symmetric nonlinear physical fields) from a frequency domain perspective. In an endeavor to enhance the simulation efficiency of this theoretical framework, the Fast Fourier Transform (FFT) algorithm is astutely amalgamated into the simulation. This integration contributes significantly to the augmentation of computational efficiency. Furthermore, exhaustive derivations and proofs are presented for the first-order, second-order, and fourth-order ensemble properties of simulated fourth-order non-Gaussian random physical fields. Subsequently, the reliability and accuracy of the proposed algorithm framework are validated through numerical simulations of two two-dimensional and two three-dimensional non-Gaussian random physical fields. The findings demonstrate that the simulated sample function effectively captures the probability characteristics of the random field, including mean, variance, and kurtosis. Finally, an in-depth analysis of numerical simulation results highlights the difference between the proposed method and the traditional second-order spectral representation method, which further underscores the distinctive attributes and potential superiority of the proposed method.

Probabilistic fatigue life prediction of corroded PC beams considering spatial variability of pitting corrosion

Pitting corrosion intensifies local stress in materials, accelerating the fatigue crack initiation and propagation at the pit root. The randomly distributed corrosion pits can result in fatigue fractures in rebars and wires at non-critical sections, which further increases the fatigue failure probability of structures. However, previous studies have scarcely taken this into account. Here, we propose a novel model for fatigue life prediction of corroded post-tensioned prestressed concrete (PC) beams, considering the spatial variability of pitting corrosion. First, a Weibull distribution model of the pitting factor is developed based on the 3D laser scanning method. Then, the 3D non-Gaussian random field of the pitting factor in PC beams is generated using the inverse Nataf transformation. Next, the 3D pit-induced stress concentration is modeled using pit shape ratios. Following that, the initiation and propagation of fatigue cracks in pitting corroded wires and rebars are considered by the equivalent initial flaw size methodology. Finally, the fatigue analyses of the pitting corroded wires and rebars in all beam elements are conducted step by step until the load-bearing capacity of PC beams becomes lower than the applied load. The proposed prediction model is validated using the experimental data. The results reveal that neglecting the spatial variability of pitting corrosion overestimates the fatigue life of corroded PC beams, especially under high corrosion degrees.

A novel general method for simulating a one dimensional random field based on the active learning Kriging model

Random fields are widely used to represent the uncertainty of some parameters in engineering, and numerous simulation approaches have been developed for Gaussian and non-Gaussian random fields. However, the unified methods among them suffer from low computational accuracy and efficiency or discontinuities in the simulated random fields. Therefore, an easy-to-implement general simulation method based on the active learning Kriging model is proposed for a one dimensional(1D) Gaussian or non-Gaussian random field in this paper. In the proposed method, there are two stages. One stage, called the inner loop, is to construct the Kriging approximation of a random field sample with enough accuracy by some samples of the random variables at some discretized locations, in which an active learning strategy based on the error estimation for the Kriging model is introduced to select adaptively the added locations, and a fast sampling method is presented to determine efficiently the samples at the added locations. In the other stage, called the outer loop, some random field samples are represented accurately by their corresponding Kriging approximations through training iteratively. Furthermore, several numerical examples are presented to show the accuracy, effectiveness and generality of the proposed method for 1D Gaussian and non-Gaussian random fields by comparing with the Karhunen–Loève(KL) expansion method. Meanwhile, the effects of the types of correlation function and the scales of fluctuation on the simulation results are analyzed.

Spatial variability characterization and modelling of 2.5D woven SiO2f/SiO2 composites

2.5D woven SiO2f/SiO2 composites possess geometric variabilities across scales, such as the spatial fluctuations of fibers and yarns or the random distributions of defects and pores formed in the manufacturing process, which induces a complex stress and strain distribution in material testing. In this study, a multivariate cross-correlated non-Gaussian random field based on Vine Copula is proposed to characterize the spatial variability of mechanical properties. A database of material mechanical properties is acquired by stochastic representative volume element (SRVE) simulation results which contains realistic meso-geometry characteristics and spatially random voids. Based on this database, the cross- and auto-correlations of the mechanical properties are quantified by Vine Copula and non-Gaussian random field model. In a comparison of different numerical models and experimental results, the proposed method is verified, which shows great advantages in characterizing the spatial variability and big potential abilities in damage or reliability analysis.

Nearest neighbors weighted composite likelihood based on pairs for (non-)Gaussian massive spatial data with an application to Tukey-hh random fields estimation

A highly scalable method for (non-)Gaussian random fields estimation is proposed. In particular, a novel (a) symmetric weight function based on nearest neighbors for the method of maximum weighted composite likelihood based on pairs (WCLP) is studied.The new weight function allows estimating massive (up to millions) spatial datasets and improves the statistical efficiency of the WCLP method using symmetric weights based on distances, as shown in the numerical examples.As an application of the proposed method, the estimation of a novel non-Gaussian random field named Tukey-hh random field that has flexible marginal distributions (possibly skewed and/or heavy-tailed) is considered. In an extensive simulation study the statistical efficiency of the proposed nearest neighbors WCLP method with respect to the WCLP method using weights based on distances is explored when estimating the parameters of the Tukey-hh random field. In the Gaussian case the proposed method is compared with the Vecchia approximation from computational and statistical viewpoints. Finally, the effectiveness of the proposed methodology is illustrated by estimating a large dataset of mean temperatures in South-America. The proposed methodology has been implemented in an open-source package for the R statistical environment.

An iterative multi-fidelity scheme for simulating multi-dimensional non-Gaussian random fields

This paper presents an efficient multi-fidelity scheme to simulate multi-dimensional non-Gaussian random fields that are specified by covariance functions and marginal probability distribution functions. To develop the multi-fidelity scheme, two numerical algorithms are proposed in turn. The first algorithm is used to simulate random samples of the random field. In this algorithm, initial random samples are first generated to meet the marginal distribution and an iterative procedure is adopted to change the ranking of the random samples to match the target covariance function. By using the above algorithm, random samples satisfying the target covariance function and the target marginal distribution are obtained, but it is computationally intensive and is not suitable to simulate large-scale random fields. By taking advantage of Karhunen–Loève expansion, a multi-fidelity algorithm is then proposed to reduce the computational effort. The random variables in Karhunen–Loève expansion are calculated via performing the first algorithm on the low-fidelity model and the deterministic functions in Karhunen–Loève expansion are solved on the high-fidelity model. In this way, the proposed method has low computational effort and a high fidelity simultaneously. Three numerical examples, including two- and three-dimensional non-Gaussian random fields, are used to verify the good performance of the proposed algorithms.

Tile low-rank approximations of non-Gaussian space and space-time Tukey g-and-h random field likelihoods and predictions on large-scale systems

Large-scale statistical modeling has become necessary with the vast flood of geospace data coming from various sources. In space statistics, the Maximum Likelihood Estimation (MLE) is widely considered for modeling geospace data by estimating a set of statistical parameters related to a predefined covariance function. This covariance function describes the correlation between a set of geospace locations where the main goal is to model given data samples and impute missing data. Climate/weather modeling is a prevalent application for the MLE operation where data interpolation and forecasting are highly required. In the literature, the Gaussian random field is often used to describe geospace data as one of the most popular models for MLE. However, real-life datasets are often skewed and/or have extreme values, and non-Gaussian random field models are more appropriate for capturing such features. In this work, we provide an exact and approximate parallel implementation of the well-known Tukey g-and-h (TGH) non-Gaussian random field in the context of climate/weather applications. The proposed implementation alleviates the computation complexity of the log-likelihood function, which requires O(n2) storage and O(n3) operations, where N is the number of geospace locations, M is the number of time slots, and n=N×M. Based on tile low-rank (TLR) approximations, our implementation of the TGH model can tackle large-scale problems. Furthermore, we rely on task-based programming models and dynamic runtime systems to provide fast execution for the MLE operation in space and space-time cases. We assess the performance and accuracy of the proposed implementations using synthetic space and space-time datasets up to 800K. We also consider a 12-month precipitation dataset in Germany to demonstrate the advantage of using non-Gaussian over Gaussian random field models. We evaluate the prediction accuracy of the TGH model on the precipitation dataset using the Probability Integral Transformation (PIT) tool showing that the TGH model outperforms the Gaussian modeling in the real dataset. Moreover, our performance assessment indicates that TLR computations allow solving larger matrix sizes while preserving the required accuracy for prediction. The TLR-based approximation shows a speedup up to 7.29X and 2.96X over the exact solution.

Seismic responses of suction bucket foundation in liquefiable seabed considering spatial variability

The abundant wind energy happens to be distributed along the seismic regions with potential liquefaction. Offshore wind turbines have to be built in these regions for high power generation efficiency. The earthquake-induced liquefaction affected by the natural variability of the seabed makes the safety assessment of suction bucket foundations a challenge. The seismic responses of the laterally-loaded suction bucket foundation in spatially variable soil modeled by an advanced constitutive model were investigated using an FE- FD method. The SPT-(N1)60cs was considered a random variable and the seabed was modeled as a non-Gaussian random field by the Cholesky decomposition technique. The seismic responses were described by the Monte Carlo simulation and the FE-FD method. Statistics of the tilt and the liquefaction potential index influenced by the coefficient of variation and the scale of fluctuation were investigated. The stochastic model was compared with the deterministic model. The liquefaction mechanism and failure mechanism of the suction bucket foundation in random soil were investigated. The tilts and liquefaction may be underestimated without considering the soil spatial variability. Safety factors were introduced into the calculation of the failure probability of the suction bucket foundations. The simulations provide insight into the nonlinear behavior of suction bucket foundations considering the liquefaction of spatially variable soil.

An innovative method for space-time-dependent reliability analysis

The spatiotemporal variability exists in almost all engineering problems, since the environmental actions and material properties may change both spatially and temporally in a stochastic manner. Therefore, reliability analysis is inherently space–time-dependent. In this study, an innovative space–time-dependent outcrossing rate (STOR) method is proposed for dealing with such reliability problems. The failure probability of a structure is determined based on outcrossing rate which is theoretically derived for the first time, considering the spatiotemporal variability of all basic variables. Then, an efficient numerical algorithm for computing the failure probability is proposed based on Gauss-Legendre quadrature. The application of STOR method is investigated through three examples, including non-Gaussian random processes, non-Gaussian random fields, strong nonlinear limit state functions and small failure probability problems. The proposed STOR method is found to be efficient for space–time-dependent reliability analysis with enough accuracy.

Assembly accuracy analysis of cylindrical parts based on skin model shapes considering form deviations and local surface deformations

Assembly accuracy analysis plays a very important role in verifying whether the design scheme of geometric accuracy in the product design stage can meet the functional requirements of mechanical product. Form deviations, position and orientation deviation are inevitable in manufactured parts, and must be considered during assembly accuracy analysis. With regard to the representation of form deviations, it is verified that skin model shapes are effective. However, the existing simulation approaches of skin model shapes for cylindrical parts are mainly based on gaussian random field, without considering the spatial distribution characteristics of form deviations in dealing with the relationship between geometrical and dimensional deviation for cylindrical parts. Especially, when considering local surface deformations of cylindrical mating surfaces generated during the assembly process, the form deviations with different distribution characteristics leads to different contact deformation, which may have further influence on assembly accuracy analysis of cylindrical parts. To address above problems, this paper proposes a simulation method of skin model shapes of cylindrical parts based on Non-Gaussian random field considering the spatial distribution characteristics of form deviations, and the local surface deformations are analyzed with a boundary element method. Based on above, the relative positioning deviation of cylindrical mating surfaces is obtained considering the influence of form deviations and local surface deformations. Finally, the effectiveness of the proposed method is verified with case studies. Specifically, the results of case study reveal that the influences of form deviations considering spatial distribution characteristics and local surface deformations cannot be ignored for assembly accuracy analysis of cylindrical parts. In addition, for the relative positioning deviation of cylindrical mating surfaces for loose fit, the influence of local surface deformations on translational deviations along the x-axis and y-axis is greater than that on rotational deviations around the x-axis and y-axis.

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.

Tolerance analysis of planar parts with skin modeling considering spatial distribution characteristics of surface morphology and local surface deformations

Purpose This paper aims to develop a method to improve the accuracy of tolerance analysis considering the spatial distribution characteristics of part surface morphology (SDCPSM) and local surface deformations (LSD) of planar mating surfaces during the assembly process. Design/methodology/approach First, this paper proposes a skin modeling method considering SDCPSM based on Non-Gaussian random field. Second, based on the skin model shapes, an improved boundary element method is adopted to solve LSD of nonideal planar mating surfaces, and the progressive contact method is adopted to obtain relative positioning deviation of mating surfaces. Finally, the case study is given to verify the proposed approach. Findings Through the case study, the results show that different SDCPSM have different influences on tolerance analysis, and LSD have nonnegligible and different influence on tolerance analysis considering different SDCPSM. In addition, the LSD have a greater influence on translational deviation along the z-axis than rotational deviation around the x- and y-axes. Originality/value The surface morphology with different spatial distribution characteristics leads to different contact behavior of planar mating surfaces, especially when considering the LSD of mating surfaces during the assembly process, which will have further influence on tolerance analysis. To address the above problem, this paper proposes a tolerance analysis method with skin modeling considering SDCPSM and LSD of mating surfaces, which can help to improve the accuracy of tolerance analysis.

A framework for conditional simulation of nonstationary non-Gaussian random field and multivariate processes

We propose a conditional simulation framework for the nonstationary/nonhomogeneous non-Gaussian/Gaussian field and multivariate processes. There are three distinct stages within the proposed framework. The first one is to sample directly the unconditional nonstationary non-Gaussian field or multivariate processes without resorting to the underlying Gaussian characteristics. The second stage maps the unconditional samples and the conditions (i.e., observations) to the Gaussian space and uses them to find the conditional mean and covariance. In the final stage, the conditional samples by considering the effect of the conditioning are generated in Gaussian space and mapped back to the original non-Gaussian field. The efficiency of the framework relies on the efficient unconditional simulation algorithms (i.e., the iterative rank-dependent shuffling algorithm, and the iterative power and amplitude correction algorithm), and the reuse of the unconditional samples for the conditional simulation. The conditional samples match the conditional mean and conditional variance–covariance. The framework can directly apply to the random field and multivariate processes defined by the marginal probability distribution function and the covariance-variance information, or the time–frequency dependent PSD function. The framework is illustrated using several examples.

A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems

Constructing surrogate models for uncertainty quantification (UQ) on complex partial differential equations (PDEs) having inherently high-dimensional O(10n), n≥2, stochastic inputs (e.g., forcing terms, boundary conditions, initial conditions) poses tremendous challenges. The “curse of dimensionality” can be addressed with suitable unsupervised learning techniques used as a pre-processing tool to encode inputs onto lower-dimensional subspaces while retaining its structural information and meaningful properties. In this work, we review and investigate thirteen dimension reduction methods including linear and nonlinear, spectral, blind source separation, convex and non-convex methods and utilize the resulting embeddings to construct a mapping to quantities of interest via polynomial chaos expansions (PCE). We refer to the general proposed approach as manifold PCE (m-PCE), where manifold corresponds to the latent space resulting from any of the studied dimension reduction methods. To investigate the capabilities and limitations of these methods we conduct numerical tests for three physics-based systems (treated as black-boxes) having high-dimensional stochastic inputs of varying complexity modeled as both Gaussian and non-Gaussian random fields to investigate the effect of the intrinsic dimensionality of input data. We demonstrate both the advantages and limitations of the unsupervised learning methods and we conclude that a suitable m-PCE model provides a cost-effective approach compared to alternative algorithms proposed in the literature, including recently proposed expensive deep neural network-based surrogates and can be readily applied for high-dimensional UQ in stochastic PDEs.

Ultraefficient reconstruction of effectively hyperuniform disordered biphase materials via non-Gaussian random fields.

Disordered hyperuniform systems are statistically isotropic and possess no Bragg peaks like liquids and glasses, yet they suppress large-scale density fluctuations in a similar manner as in perfect crystals. The unique hyperuniform long-range order in these systems endow them with nearly optimal transport, electronic, and mechanical properties. The concept of hyperuniformity was originally introduced for many-particle systems and has subsequently been generalized to biphase heterogeneous materials such as porous media, composites, polymers, and biological tissues for unconventional property discovery. Existing methods for rendering realizations of disordered hyperuniform biphase materials reconstruction typically employ stochastic optimization such as the simulated annealing approach, which requires many iterations. Here, we propose an explicit ultraefficient method for reconstructing effectively hyperuniform biphase materials, based on the second-order non-Gaussian random fields where no additional tuning step or iteration is needed. Both the effectively hyperuniform microstructure and the latent material property field can be simultaneously generated in a single reconstruction. Moreover, our method can also incorporate hierarchical uncertainties in the heterogeneous materials, including both uncertainties in the disordered material microstructure and material property variation within each phase. The efficiency and feasibility of the proposed reconstruction method are demonstrated via a wide spectrum of examples spanning from isotropic to anisotropic, effectively hyperuniform to nonhyperuniform, and antihyperuniform systems. Our ultraefficient reconstruction method can be readily incorporated into material design, probabilistic analysis, optimization, and discovery of novel disordered hyperuniform heterogeneous materials.

Stochastic reduced order modelling and analysis of rotating bladed discs

A computational finite-element (FE)-based technique is proposed for developing a stochastic reduced order model for rotating bladed disc with spatial random inhomogeneities. The spatial inhomogeneities imply the system to be randomly mistuned. The formulation assumes the availability of a high fidelity FE model for the tuned system. The corresponding FE matrices are antisymmetric on account of the Coriolis forces due to rotation. The spatial inhomogeneities, available from limited point measurements on the blades, are modelled as non-Gaussian random fields with arbitrary distributions. A low order stochastic computational model is developed by projecting the FE model onto a reduced dimensional state space defined in terms of specified observable nodal points and expressing the stochasticity through an arbitrary polynomial chaos (aPC) basis. This model enables probabilistic quantification of the variabilities in the system response and estimating failure probabilities. The methodology enables drastic reduction in the state space and stochastic dimensions, addresses the practical difficulties with having limited measurable data points, antisymmetric FE matrices, aPC representation in complex irregular geometries and carrying out probabilistic analyses on industrial systems, at significantly reduced computational costs. The methodology is illustrated through an academic rotor and an industrial rotor blade.

Stochastic analysis of geometrically imperfect thin cylindrical shells using topology-aware uncertainty models

Buckling of thin-shell structures is one of the most canonical problems in mechanics. In practice, the buckling load and its deviation from theoretical prediction is often handled through the development of knock-down factors in shell structure design. Uncertainty-informed analysis based design is an alternative path that has been aggressively pursued in recent years. In this study, we use a novel representation of the stochastically imperfect shell geometry to investigate the buckling behavior of imperfect cylindrical shells. The representations rest on a non-Gaussian random field model, obtained by translating a latent Gaussian field defined as the solution to a stochastic partial differential equation and allows for the construction of topology-aware spatially correlated imperfections on nonconvex domains. We perform finite element analysis of buckling for imperfect shells and gain new insights into how the interplay between random imperfections and topological features influences buckling behavior.

Construction a Non-parametric Prediction Model for Spatial Random Field Using Projection Theorem

Identifying the best prediction of unobserved observation is one of the most critical issues in spatial statistics. In this line, various methods have been proposed, that each one has advantages and limitations in application. Although the best linear predictor is obtained according to the Kriging method, this model is applied for the Gaussian random field. The uncertainty in the distribution of random fields makes researchers use a method that makes the nongaussian prediction possible. In this paper, using the Projection theorem, a non-parametric method is presented to predict a random field. Then some models are proposed for predicting the nongaussian random field using the nearest neighbors. Then, the accuracy and precision of the predictor will be examined using a simulation study. Finally, the application of the introduced models is examined in the prediction of rainfall data in Khuzestan province.

A new perspective on the simulation of cross-correlated random fields

Cross-correlated random fields are widely used to model multiple uncertain parameters and/or phenomena with inherent spatial/temporal variability in numerous engineering systems. The effective representation of such fields is therefore the key element in the stochastic simulation, reliability analysis and safety assessment of engineering problems with mutual correlations. However, the simulation of such fields is generally not straightforward given the complexity of correlation structure. In this paper, we develop a unified framework for simulating non-Gaussian and non-stationary cross-correlated random fields that have been specified by their correlation structure and marginal cumulative distribution functions. Our method firstly represents the cross-correlated random fields by means of a new general stochastic expansion, in which the fields are expanded in terms of a set of deterministic functions with corresponding random variables. A finite element discretization scheme is then developed to further approximate the fields, so that the sets of deterministic functions reflecting the cross-covariance structure can be straightforwardly determined from the spectral decomposition of the resulting discretized fields. For non-Gaussian random fields, an iterative mapping procedure is developed to generate random variables to fit non-Gaussian marginal distribution of the fields. By virtue of the remarkable property of the presented stochastic expansion, i.e., various random fields share an identical set of random variables, the framework we develop is conceptually simple for simulating non-Gaussian cross-correlated fields with arbitrary covariance functions, which need not be stationary. In particular, the developed method is further generalized to a consistent framework for the simulation of multi-dimensional random fields. Five illustrative examples, including a spatially varying non-Gaussian and nonstationary seismic ground motions, are used to demonstrate the application of the developed method.

The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matérn covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space–time fields on arbitrary manifolds, and practical computational considerations.