Random Field Discretization Research Articles

Seismic collaborative reliability analysis for a slope considering spatial variability base on optimized subset simulation

Seismic reliability analysis of actual slopes considering the spatial variability of soil materials is crucial. However, for the discretization of large-scale random fields, high-precision finite element analysis and analysis of small failure probability events, the random analysis of slopes under seismic loads is inefficient. To address this situation, this study proposes a collaborative reliability analysis framework based on the modified linear estimation method (MLEM) and optimized subset simulation (OSS). First, the random field of the uncertain parameters of the Jinping-I left bank slope model is efficiently discretized by the MLEM, and a sensitivity analysis is carried out. Then, considering the adoption of different degrees of cross-correlation of the sensitive random parameters, the OSS method is used to perform random finite element analysis on the coarse mesh model. Finally, the fine mesh samples are obtained according to the response conditioning method (RCM). The MLEM is used to ensure the consistency of the two sets of random fields, and the seismic failure probability and reliability index of the slope under different cross-correlation coefficients of uncertain parameters are obtained. The results suggest that the degree of cross-correlation of parameters has a great influence on the seismic reliability of the slope. Considering that the shear strength parameters of geotechnical materials are often negatively correlated, the fine analysis based on a fine model is necessary.

An Interpolation Method for Karhunen–Loève Expansion of Random Field Discretization

An Interpolation Method for Karhunen–Loève Expansion of Random Field Discretization

A decomposed Karhunen–Loève expansion scheme for the discretization of multidimensional random fields in geotechnical variability analysis

A decomposed Karhunen–Loève expansion scheme for the discretization of multidimensional random fields in geotechnical variability analysis

An efficient variable selection-based Kriging model method for the reliability analysis of slopes with spatially variable soils

The random finite element method (RFEM) is an efficient tool to demonstrate the spatial variability of soil properties during reliability analysis of slopes, but it requires remarkable model evaluations and computational efforts. In this paper, an efficient variable selection-based Kriging model method is proposed to approximate the finite element analysis model in reliability analysis of slopes. The variable selection technique successfully remedies the “curse of dimensionality” within Kriging model induced by the numerous random variables in random field discretization. The implementation procedure of this method for the reliability analysis of slopes is introduced in detail. Two typical examples of soil slopes, as well as a real complex slope are subsequently analyzed to illustrate the validity of the proposed method. The results show that the local loss of variability of random field due to the variable selection method has little impact on the safety factor of slopes. The proposed method can significantly reduce the number of finite element analysis and obtain accurate results in reliability analysis of slopes considering the spatial variability of soil properties.

Effect of spatial variability of creep rock on the stability of a deep double-lined drift

This work aims at investigating the effect of aleatoric uncertainty of creep rock properties on the stability of an underground structure. This uncertainty relates to the spatial variability of the mechanical parameters representing the time-dependent behavior of geological rock formation due to the change in its mineralogy. The chosen methodology consists of representing the aleatoric uncertainty of rock properties by random fields, written as correlation functions with respect to the spatial correlation length. The adaptation of the well-known Expansion Optimal Linear Estimationmethod (EOLE) is performed to account for the cross-correlation of the random fields of the viscoplastic parameters of the host rock. Then, the Kriging-based reliability analysis is undertaken with respect to the discretized random fields, which allows elucidating the effect of spatial variability. As an application, the proposed approach is chosen to study the stability in the long-term of a deep double-lined drift within the geological disposal facilities (Cigeo project) conducted by the French National Radioactive Waste Management Agency (Andra). The drift will be excavated in Callovo-Oxfordian (COx) claystone (if the Cigeo project is licensed), considered as a potential host rock for the deep geological nuclear waste disposal in France. The results show that the chosen Kriging metamodel for the reliability analysis can be appropriate for the case of high correlation length represented by a moderate number of random variables (up to about 50) after the discretization of random fields. Further, the consideration of aleatoric uncertainty exhibits a lower probability of exceedance in comparison with the case where spatial variability is ignored. Still, more investigations need to be conducted in the future to conclude this observation.

New non-intrusive stochastic finite element method for plate structures

Currently, several intrusive stochastic finite element methods (SFEMs) such as perturbation method and spectral SFEM are widely applied for stochastic response analysis of continuous structures. However, the intrusive SFEMs need to modify conventional finite element formulations to establish the stochastic stiffness matrix, and cannot calculate the probability density function of structural response and the reliability straightforwardly. This paper proposes a new non-intrusive SFEM for efficiently computing stochastic responses and reliabilities of plates in a unified way. Firstly, the direct probability integral method (DPIM) is developed to obtain the probability density function of stochastic response by solving probability density integral equation (PDIE). Secondly, the non-intrusive SFEM based on DPIM decouples the deterministic finite element analysis and PDIE to calculate the stochastic responses and reliabilities of uncertain plate structures, and the discretization and quantification of random fields of elastic modulus and thickness are implemented through Karhunen-Loève expansion. Finally, several examples of uncertain Kirchhoff and Mindlin plates demonstrate the efficiency and versatility of the proposed non-intrusive method by comparing with the results from Monte Carlo simulation and literature. The effects of correlation length, mean and variability of random field on the probability distribution of responses and the reliabilities of plates are revealed. For Gaussian random thickness, the linearly elastic plate yields non-Gaussian distributed responses. Increasing the correlation length and variability of random field reduces the reliabilities of plates.

Stochastic isogeometric analysis on arbitrary multipatch domains by spline dimensional decomposition

This paper presents a new stochastic method by integrating spline dimensional decomposition (SDD) of a high-dimensional random function and isogeometric analysis (IGA) on arbitrary multipatch geometries to solve stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-mIGA, involves (1) analysis-suitable T-splines with significant approximating power for geometrical modeling, random field discretization, and stress analysis; (2) Bézier extraction operator for isogeometric mesh refinement; and (3) a novel Fourier-like expansion of a high-dimensional output function in terms of measure-consistent orthonormalized splines. The proposed method can handle arbitrary multipatch domains in IGA and uses standard least-squares regression to efficiently estimate the SDD expansion coefficients for uncertainty quantification applications. Analytical formulae have been derived to calculate the second-moment properties of an SDD-mIGA approximation for a general output random variable of interest. Numerical results, including those obtained for a 54-dimensional, industrial-scale problem, demonstrate that a low-order SDD-mIGA is capable of efficiently delivering accurate probabilistic solutions when compared with the benchmark results from crude Monte Carlo simulation.

The spatial averaging method for non-homogeneous random fields with application to reliability analysis

In probabilistic assessments, inputs with significant spatial variability should be modeled with random fields. Random fields can be non-homogeneous with location-specific marginal distributions, for example, due to site-specific information incorporated through Bayesian analysis or due to spatial trends in the mean or variance of the uncertain quantity. This paper investigates the spatial averaging method for the discretization of non-homogeneous random fields. In this approach, the random field is reduced to a set of random variables representing its local averages over a corresponding set of elemental domains. This is of particular benefit when coupling the random field model with finite elements for structural analysis. We extend the application of the method to non-homogeneous Gaussian and non-Gaussian translation random fields with lognormal, Student’s t- and log-Student’s t-marginal distribution. The latter two distributions are particularly relevant if spatial data is used in a hierarchical Bayesian random field modeling. Two numerical investigations assess the ability of the method to efficiently represent the response variability and probability of failure of structural systems with spatially variable inputs. The investigations include the effect of different element sizes for the spatial averaging on the system response and applicability of the spatial averaging method to assessing local and global failure modes.

An adaptive scheme for random field discretization using KL expansion

With the increase in computational facilities, interest in probabilistic analysis of engineering structures are observed to grow for a realistic assessment of physical systems. While the probabilistic analyses are generally carried out considering random variable models of physical parameters, a spatially varying random field model is more realistic. A discretization method converts the continuous parameter random field to a set of random variables. Truncated Karhunen–Loeve (KL) expansion is one of the popular methods for the discretization of a random field. An accurate discretization can be achieved by considering a fine mesh along with a large number of terms. However, an increase in the number of elements and/or inclusion of more terms leads to an increase in computational cost. It is also found that only an increase in the number of terms in the expansion or number of elements alone does not provide an accurate representation of the random field. An adaptive discretization strategy is presented, which will suitably discretize the concerned domain while ensuring that both global and local level errors are kept within the prescribed limit.

Meta-model based stochastic isogeometric analysis of composite plates

A stochastic isogeometric analysis approach (SIGA) is presented for functionally graded porous plates with graphene platelets reinforcement (FGP-GPLs). Different kinds of random fields and variables are applied to describe the uncertain system inputs which are including material properties of the FGP matrix and graphene platelets, magnitudes and directions of applied loads. A Nyström based Karhunen-Loève expansion is presented for random field discretization within the IGA scheme. The arbitrary polynomial chaos-Kriging (aPCK) method is presented for uncertainty quantification. To sustain the robustness of the aPCK approach for engineering problems involving high-dimension of uncertainty, a new Dagum kernel function is introduced in Kriging. The mean, standard deviation, probability density function (PDF) and cumulative distribution function (CDF) of structural outputs can be effectively estimated. Three illustrative examples are investigated to assess the performance of the proposed method for mathematical and engineering applications.

Stochastic isogeometric analysis for the linear stability assessment of plate structures using a Kriging enhanced Neural Network

The stability of functionally graded porous plates with graphene platelets reinforcement (FGP-GPLs) is investigated in this paper. Combining with a new metamodeling approach, namely the Kriging enhanced Neural Network, a stochastic isogeometric analysis (SIGA) framework is proposed for assessing the structural stability. The uncertainties of material properties of both FGP matrix and graphene platelets are considered in the form of random fields and variables. Karhunen-Loève expansion based Nyström method is applied to random field discretization. The Dagum function is adopted as a new kernel function to further improve the performance of the proposed approach. Statistical information including but not limited to statistical moments, probability density function (PDF), and cumulative distribution function (CDF) of the critical buckling load of the plate structure can be effectively estimated through a non-intrusive fashion. In order to illustrate the effectiveness and applicability of the proposed stochastic computational analysis, both theoretical and real-life engineering examples have been investigated in this study.

Experimental characterization of spatial variability for random field modeling on struts of additively manufactured lattice structures

The accuracy of the numerical predictions of lattice structures relies on the quantification of the uncertainties introduced by additive manufacturing (AM). In this study, an efficient random field discretization framework is proposed to characterize the spatial variability of geometric uncertainties on the strut members of the lattice structures fabricated by a specific AM process called material extrusion. Individual struts are fabricated with various printing angles and diameters using PLA material. Diameter values of the fabricated samples are measured along the printing direction and the radial direction of the cross-section at each layer under an optical microscope. Spatial correlations are characterized based on the measurements and the correlation lengths are evaluated. Two random field discretization methods, namely Karhunen-Loève expansion (KLE) and Expansion Optimal Linear Estimation (EOLE) methods are investigated to reduce the dimensionality of the random field discretization. The KLE method was found to give better accuracy in terms of error means while EOLE can produce better local accuracy. An optimal mesh size determination approach is introduced for the random field discretization with the expansion methods based on the number of expansion terms. The voxel-based strut models are generated with the diameter parameters modeled by the proposed random fields discretization framework. The comparison of the generated models with the fabricated struts shows that the AM-introduced variability in the material extrusion process on the struts is well represented by the random field modeling technique.

Tensor train-Karhunen-Loève expansion: New theoretical and algorithmic frameworks for representing general non-Gaussian random fields

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional physical domain in the tensor train format; secondly, to develop a new algorithmic framework for accurately computing the modes and describing the correlation structure of the latent factors beyond second order within moderate time. Central to the algorithmic framework is the tensor train decomposition of cumulant functions. This decomposition is accurately computed with a novel rank-revealing algorithm. For computing the modes, compared with existing Galerkin-type and collocation-type methods, the proposed computational procedure totally removes the need of selecting the basis functions or collocation points and the quadrature points, which not only greatly enhances adaptivity, but also avoids solving large-scale eigenvalue problems. Moreover, by computing with higher-order cumulant functions, the new theoretical and algorithmic frameworks show great potential for representing general non-Gaussian non-homogeneous random fields. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the efficiency and accuracy of the proposed algorithmic framework.

A stochastic B-spline wavelet on the interval finite element method for beams

The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for beams wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Stochastic beam element formulations based on both Euler-Bernoulli beam theory and Timoshenko beam theory are proposed. BSWI scaling functions are used for the discretization of the random field and the response statistics are obtained using the perturbation approach. Numerical examples are solved and the results from perturbation approach are compared with that obtained from Monte Carlo simulation (MCS). A parametric study is also done to understand the effect of different coefficient of variation (CV) values and correlation length parameters on the response statistics. The study concludes that the proposed BSWI WFEM based perturbation approach for beams produce accurate response statistics for values of CV less than 15%. A comparative study is carried out between the results obtained from the proposed stochastic WFEM with stochastic finite element method (SFEM) wherein the random field discretization is done using Lagrange shape functions. Furthermore, normalized computational times for the execution of perturbation approach and MCS based on WFEM are evaluated and compared with those obtained for SFEM.

Aeroelastic Stability of Conventional and Tow-Steered Composite Plates Under Stochastic Fiber Volume

The numerical investigation of the stochastic aeroelastic characteristics of conventional and tow-steered composite laminates subjected to random uncertainties affecting the laminate fiber volume is addressed in the present paper. A computationally efficient stochastic model is constructed combining the semi-analytical Rayleigh–Ritz approach with the Karhunen–Loève discretization of the two-dimensional random field representing the fiber volume. In addition, polynomial chaos expansions are used as stochastic metamodels of the output random variables characterizing the onset of aeroelastic instability and the mass of the laminate. Such a metamodeling approach enables to alleviate the computational cost involved in the estimation of the statistics of the output variables of interest based on Monte Carlo simulations. In addition, the study encompasses both subsonic and supersonic flow conditions, for which two appropriate aerodynamic models are used. The numerical results provide effective uncertainty quantification in a variety of simulation scenarios, which are found to be useful for the incorporation of uncertainty quantification in the design of aircraft and spacecraft structures when uncertainties induced by the manufacturing process must be dealt with.

Certified and fast computations with shallow covariance kernels

Many techniques for data science and uncertainty quantification demand efficient tools to handle Gaussian random fields, which are defined in terms of their mean functions and covariance operators. Recently, parameterized Gaussian random fields have gained increased attention, due to their higher degree of flexibility. However, especially if the random field is parameterized through its covariance operator, classical random field discretization techniques fail or become inefficient. In this work we introduce and analyze a new and certified algorithm for the low-rank approximation of a parameterized family of covariance operators which represents an extension of the adaptive cross approximation method for symmetric positive definite matrices. The algorithm relies on an affine linear expansion of the covariance operator with respect to the parameters, which needs to be computed in a preprocessing step using, e.g., the empirical interpolation method. We discuss and test our new approach for isotropic covariance kernels, such as Mat\'ern kernels. The numerical results demonstrate the advantages of our approach in terms of computational time and confirm that the proposed algorithm provides the basis of a fast sampling procedure for parameter dependent Gaussian random fields.

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.

Imprecise random field analysis with parametrized kernel functions

Abstract The application of isotropic random fields in engineering analysis requires the definition of their first two central moments, as well as their covariance function. In general, insufficient data are available to make a fully objective crisp estimate on these quantities, and hence, subjectivity enters implicitly into the analysis. The framework of imprecise probabilities is gaining popularity in this context, as it allows to explicitly separate the epistemic uncertainty, present due to data insufficiency, from the aleatory nature of the random parameters. However, an approach that is capable of handling imprecision in the complete definition of the imprecise random field is lacking to date. This paper proposes a framework for imprecise random field analysis with parametrized covariance functions. As such, the functional form of the covariance function is assumed to be known deterministically, whereas the governing parameters are subjected to imprecision. First, a comprehensive analysis of the effect of imprecise random fields, given imprecision on both the mean and auto-covariance structure, is presented. It is shown that the discretization of an imprecise random field, given interval-valued correlation lengths cannot be performed using interval arithmetical procedures as the resulting basis functions are in that case no longer a complete basis on L 2 . Therefore, an iterative procedure is proposed where the bounds on the imprecise random field basis are determined via an optimization procedure. This procedure provides exact bounds on the response of a system in case the system is monotonic with respect to the random field realizations. Two illustrative case studies on a mass-damper-spring system and a dynamic state-space model of a car suspension are included to illustrate the methodology. These case studies illustrate that indeed a separation between aleatory and epistemic uncertainty is possible in random field analysis, and hence, more objective results are obtained at only slightly increased computational cost.

A stochastic B-spline wavelet on the interval finite element method for problems in elasto-statics

The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for one dimensional (1D) and two dimensional (2D) problems in elasto-statics wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Random field discretization is done using BSWI scaling functions. Perturbation approach is used to calculate the response statistics. The results are compared with that obtained from Monte Carlo simulation (MCS) and a parametric study is carried out by considering different coefficient of variation values by varying the standard deviation. For 1D problem in particular, results from proposed stochastic WFEM method are compared with those found using stochastic finite element method (FEM) wherein random field discretization is done using Lagrange shape functions. A comparative study of the computational time needed for the execution of perturbation approach and MCS based on WFEM and FEM is also performed.

Spectral stochastic isogeometric analysis for linear stability analysis of plate

A spectral stochastic isogeometric analysis (SSIGA) scheme is proposed for the stochastic linear stability analysis of plate with uncertain material properties. Within the proposed SSIGA scheme, the first-order shear deformation theory of plate is adopted for modelling the kinematic relationship. Both homogeneous and functionally graded material (FGM) models can be incorporated. The considered spatially dependent uncertainties (i.e., Young’s modulus and Poisson’s ratio) are modelled as random fields with Gaussian and lognormal distributions, and the spatially independent uncertainty (i.e., gradient index of FGM) is modelled as random variable. The generalized isogeometric basis function is adopted for both the random field geometry representation and random field discretization through the Karhunen–Loève (K–L) expansion. An extended support vector regression (X-SVR) with a new generalized Gegenbauer polynomial kernel is developed to model the nonlinear relationship between the structural uncertainties and the buckling load. By further implementing various nonparametric statistical inference methods, the mean, standard deviation, probability density function (PDF), and cumulative distribution function (CDF) of the buckling load can be effectively established to determine the strength limit of the plate. The accuracy, efficiency, and applicability of the proposed approach are illustrated through two numerical examples.