Random Field Parameters Research Articles

CRACKING PATTERN ANALYSIS OF THE TURNING BAND METHOD (TBM) APPLICATION ON A SINGLE-REINFORCEMENT BAR CONCRETE BEAM MODELLING USING THE MAZARS DAMAGE

The development of crack patterns in reinforced concrete structures is a randomly stochastic process influenced by the heterogeneous nature of concrete materials, which impacts the sensitivity to damage and variability in mechanical properties. The numerical simulation of damage pattern appearance on the reinforced concrete for special building structures needs a realistic result, especially in terms of crack distribution. The Turning Band Method (TBM) serves as a tool for assessing the variability of concrete heterogeneity, functioning as an operator to generate a random variable for each specific field. In this study, the investigation of crack patterns in reinforced concrete structures involves using a simple concrete beam sample, measuring 10x10x50 cm3, with a single longitudinal reinforcement cast in the centre. This beam is subjected to an axial tensile loading, while the Mazar Damage Model is employed as the concrete behaviour law. Through the implementation of The Turning Band Method (TBM) and the variation of the random field parameters, distinct crack patterns are observed, not only the number but also the locations of cracks. The use of a smaller correlation length showed 2 cracks while the larger size had around 1 to 3 cracks, with the first crack localizations of each sample generally occurring in the middle surface, in which a noticeable contrast to non-TBM modelling just displays 1 crack. Furthermore, the resulting probability of cracks for ten random draws demonstrates similarity to experimental tests and numerical simulation of the previous study, both at global and local levels.

Analysis of sensitive clay landslide mechanism and impact pipeline bearing characteristics considering spatial variability

Landslides caused by the progressive failure of sensitive clay slopes may cause substantial economic losses and environmental pollution to pipelines. This study utilizes the Karhunen-Loève (K-L) expansion method, the coupled Eulerian-Lagrangian (CEL) method, and the Monte Carlo method to model the large deformation behaviour of slopes with spatial variability. The proposed RCEL-MC framework for joint analysis of extensive deformation-probability statistics of landslide impact on pipelines confirms that the spatial variability clay slope instability failure mechanism, considering the soil softening effect under earthquake action, aligns more closely with real-world scenarios. Furthermore, based on the statistical failure probability of pipelines impacted by landslides, a sensitive clay slope pipeline safety design method is suggested. The research highlights that the random field parameters significantly influence the load-bearing characteristics of pipelines affected by spatially variable slope progressive failure landslides. The uncertainty surrounding the maximum impact force on pipelines notably increases with the rise of the horizontal correlation length and variability coefficient. When accounting for spatial variability, the maximum impact force on landslide pipelines surpasses the deterministic model analysis results by 28-69 %. It is emphasized that the deterministic model analysis, which neglects the spatial variability of soil and the soil softening effect, may seriously underestimate landslide risk. Based on the criteria for pipeline buckling failure and yield failure, it is advised that the safe design parameters for pipelines in practical projects should fall within the intersection of the secure regions defined by the two failure criteria.

Geotechnical correlation field-informed and data-driven prediction of spatially varying geotechnical properties

Geotechnical measurements are often limited, leading to the use of interpolation techniques for interpreting spatial variations in geotechnical properties from sparse geo-data. Traditional geostatistical methods suffer from significant computational complexity. On the other hand, data-driven approaches often lack integration with geotechnical domain knowledge, potentially oversimplifying or complicating predictions related to the spatial variability of geotechnical properties. This study introduces a novel framework that combines geotechnical knowledge with data-driven methods to model inherent soil spatial variability incorporating Geotechnical Correlation Field (GCF) that reflects domain knowledge. The GCF, influenced by Autocorrelation Function (ACF) types and Scale of Fluctuation (SOF), provides a flexible basis for accurately representing spatially varying geotechnical properties. Using a large synthetic database comprising known ACF types and SOFs, we constructed a series of specialized neural networks. These networks identify random field parameters at different sites based on sparse data, and the estimated parameters can be directly used to calculate GCFs for a given site. The performance of the proposed method is validated using a set of synthetic data and a real case history in New Zealand. The results demonstrate the model can accurately predict random field parameters for irregularly spaced geo-data, even with limited information. Significantly, the GCFs offer improved physical interpretations and enhance the performance of subsurface modeling. The computational complexity of this method is independent of the number of soil cells, making it highly efficient and scalable.

Probabilistic investigations on the elastic stiffness coefficients for suction caisson considering spatially varying soils

The importance of accurate estimation of elastic stiffness coefficients of suction caisson has been already highlighted in previous works. Nonetheless, studies related to the spatial non-uniformity of the clayey seabed are somewhat limited, and current theoretical solutions are usually established using deterministic soil properties, overlooking the inherent uncertainty of the soil. This study focuses on the elastic performance of suction caisson in spatially random soils using three-dimensional (3D) small strain random finite element analyses within a Monte-Carlo framework. The non-uniformity of soil is mapped by modeling the shear modulus as a random field. The influences of soil profiles, embedment ratios and random field parameters on the elastic stiffness coefficients are comprehensively quantified. Results demonstrate that the spatially varying shear modulus notably alters the failure mechanism, which in turn significantly affects the elastic stiffness coefficients, ignoring soil variation will provide non-conservative predictions. To deal with the uncertainty of soil shear modulus, a series of expressions are developed by adding reliability-related parameters to the existing theoretical solutions, which can be adopted to calculate quantiles of stiffness coefficients corresponding to several given probabilities. The findings of this research may potentially facilitate the elastic performance assessment of suction caisson.

Near surface sediments introduce low frequency noise into gravity models

3D geologic modeling and mapping often relies on gravity modeling to identify key geologic structures, such as basin depth, fault offset, or fault dip. Such gravity models generally assume either homogeneous or spatially uncorrelated densities within modeled rock bodies and overlying sediments, with average densities typically derived from surface and drill-hole sampling. The noise contributed to the gravity anomaly by these density assumptions is zero in the homogeneous case and typically <200 μGal in the uncorrelated case. Rock bodies and sediments, however, show both a range of densities and spatial correlation of these densities, in both surface and drill-hole samples, and this correlation causes an increase in power in the low frequency content of the resulting gravity anomaly. Spatial correlation of densities can be modeled as a Gaussian random field (GRF), with the random field parameters derived from drill-hole and geologic map data. Data from alluvial fan sediments in southern Nevada indicate correlation lengths of up to 300 m in the vertical dimension and kilometers in the horizontal dimension. The resulting GRF density models show that the noise contributed to the measured gravity anomaly is of low frequency and can be several mGal in amplitude, contradicting the common attribution of lower frequencies to deeper sources. This low-frequency noise increases in power with an increase in sediment thickness. Its presence increases the ambiguity of interpretations of subsurface geologic body shape, such as basin analyses that attempt to quantify concealed basement fault depths, offsets, and dip angles. In the southwestern United States, where basin analyses are important for natural resource applications, such ambiguity increases the uncertainty of subsequent process modeling.

Additively manufactured spinodoid sound absorbers

Spinodoid structures, also called spinodoid metamaterials, are non-periodic cellular structures that mimic spinodal topologies that are observed during diffusion-driven phase separation processes. Computationally efficient to model, spinodoid structures can be fabricated using additive techniques and offer an attractive route to the design of multifunctional structures. In this paper, we investigate the normal incidence sound absorption behavior of four distinct spinodoid topologies: isotropic, cubic, columnar, and lamellar. We fabricate the test samples using the fused filament fabrication process and systematically study the effect of the Gaussian Random Field (GRF) parameters on their underlying open pore network and sound absorption behavior. We employ a watershed segmentation-based image processing approach to correlate their pore and throat radii distributions to the GRF parameters. The normal incidence sound absorption properties are experimentally measured using the two-microphone impedance tube test method. Finally, we use a particle-swarm-based inverse characterization approach to extract the bulk properties necessary to model their acoustical behavior by using the Johnson-Champoux-Allard formulation. Our results show that the open pore network and acoustical properties of spinodoid structures are primarily a function of their relative density and wavenumber. Further, while the absorption behavior of the isotropic, cubic, and columnar spinodoids is similar, the lamellar spinodoids display unique low-frequency sound absorption behavior. Overall, all four spinodoid topologies provide favorable sound absorption characteristics, which may be tuned by varying the GRF parameters. The presented work advances the state-of-the art by establishing the feasibility of using additive manufacturing to enable non-periodic porous structures for that can be tuned to simultaneously provide mechanical stiffness and noise dampening capabilities.

Data-driven simulation of two-dimensional cross-correlated random fields from limited measurements using joint sparse representation

Cross-correlated random fields are an essential tool for simultaneously modeling both auto- and cross-correlation structures of spatial or temporal quantities in stochastic analysis of structures or systems. Existing cross-correlated random field simulation methods often require explicit information about random field parameters as inputs. However, in engineering practice, site-specific measurements of different quantities are often limited, non-co-located and irregularly distributed within a given site because of time, budget, or space constraints as well as missing data. It is notoriously difficult to properly estimate reliable random field parameters from limited non-co-located measurements with an irregular spatial pattern, particularly the auto-correlation and cross-correlation structures of a two-dimensional (2D) cross-correlated random field. To deal with this issue, this study proposes a novel 2D cross-correlated random field generator for simulating 2D cross-correlated random field samples (RFSs) directly from sparsely measured non-co-located data points with unequal measurement intervals. Using a joint sparse representation, auto- and cross-correlation structures of different spatial/temporal quantities are exploited simultaneously from sparse measurements, followed by the generation of cross-correlated RFSs using Bayesian compressive sampling (BCS) and Markov chain Monte Carlo (MCMC) simulation in a data-driven manner. The proposed generator is demonstrated using 2D data of two correlated geotechnical properties. The results indicate that the RFSs generated using the proposed method from sparse measurements can properly characterize the spatial auto- and cross-correlation structures of different geotechnical properties.

Improving the estimation of soil spatial variability by considering transformation uncertainty based on LDRFE analysis

Transformation uncertainty is an imperative consideration in the process of transforming measurement data to desired design parameters in geotechnical engineering. In the reliability-based design framework, an improved understanding of transformation uncertainty is expected to produce more appropriate design of geotechnical structures. This paper quantifies the site-specific and aleatory transformation uncertainty and investigates how to consider it in estimation of random field parameters through numerically simulated cone penetration tests in spatially variable clay by means of large deformation random finite element analysis. The transformation uncertainty of a transformation model is modelled as a random field. Random field parameters of transformation uncertainty, including the standard deviation, scale of fluctuation and sample path smoothness, are estimated using maximum likelihood estimation along with Whittle-Matérn autocorrelation function model. Random field parameters of predicted shear strength with and without considering transformation uncertainty are thereafter compared with those random field parameters of actual shear strength. Results show that the consideration of transformation uncertainty in the transformation model can considerably improve the prediction accuracy of random field parameters. It is important to consider the spatial correlation structure of transformation uncertainty. This study offers a simple-to-use framework to estimate random field parameters considering the site-specific and aleatory transformation uncertainty.

Uncertain Dynamic Characteristic Analysis for Structures with Spatially Dependent Random System Parameters.

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represent the random uncertainty of parameters, and the stochastic dynamic characteristics of different structural systems are then analyzed by incorporating the presented parameter setting model with finite element method. First, Gauss random field theory is used to describe the uncertainty of structural material parameters, the random parameters are then characterized as the standard deviation and correlation length of the random field, and the random field parameters are then discretized with the Karhunen-Loeve expansion method. Moreover, based on the discretized random parameters and finite element method, structural dynamic characteristics analysis is addressed, and the probability distribution density function of the random natural frequency is estimated based on multi-dimensional kernel density estimation method. Finally, the random field parameters of the structures are quantified by using the maximum likelihood estimation method to verify the effectiveness of the proposed method and the applicability of the constructed model. The results indicate that (1) for the perspective of maximum likelihood estimation, the parameter setting at the maximum value point is highly similar to the input parameters; (2) the random field considering more parameters reflects a more realistic structure.

Application of the Coupled Markov Chain in Soil Liquefaction Potential Evaluation

The evaluation of localized soil-liquefaction potential is based primarily on the individual evaluation of the liquefaction potential in each borehole, followed by calculating the liquefaction-potential index between boreholes through Kriging interpolation, and then plotting the liquefaction-potential map. However, misjudgments in design, construction, and operation may occur due to the complexity and uncertainty of actual geologic structures. In this study, the coupled Markov chain (CMC) method was used to create and analyze stratigraphic profiles and to grid the stratum between each borehole so that the stratum consisted of several virtual boreholes. The soil-layer parameters were established using homogenous and random field models, and the subsequent liquefaction-potential-evaluation results were compared with those derived using the Kriging method. The findings revealed that within the drilling data range in this study, the accuracy of the CMC model in generating stratigraphic profiles was greater than that of the Kriging method. Additionally, if the CMC method incorporated with random field parameters were to be used in engineering practice, we recommend that after calculating the curve of the mean, the COV should be set to 0.25 as a conservative estimation of the liquefaction-potential interval that considers the evaluation results of the Kriging method.

Modeling Point Referenced Spatial Count Data: A Poisson Process Approach

Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution considering a sequence of independent copies of a random field with an exponential marginal distribution as “inter-arrival times” in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson counting process. Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters. Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. Supplementary materials for this article, including technical proofs and R code for reproducing the work, are available as an online supplement.

The effect of random field parameter uncertainty on the response variability of composite structures

The accurate quantification of the random spatial variation of material properties at different scales is crucial for the systematic propagation of uncertainties through engineering models. In a previous work, the spatial variability of the apparent material properties of two-phase composites has been quantified in a Bayesian framework. This framework enables a consistent modeling of the statistical uncertainty in the parameters of the respective mesoscale random fields and also allows selecting the most plausible correlation model among different models belonging to the Matérn family. In this work, the most plausible random field model is employed in the context of uncertainty propagation of composite structures. Sample functions of the mesoscale random fields are generated using a covariance decomposition approach and the response variability of various composite structures is computed through Monte Carlo simulation. Parametric investigations are conducted to highlight the effect of the identified parameter uncertainty on structural response variability.

Bayesian inference for random field parameters with a goal-oriented quality control of the PGD forward model’s accuracy

Numerical models built as virtual-twins of a real structure (digital-twins) are considered the future of monitoring systems. Their setup requires the estimation of unknown parameters, which are not directly measurable. Stochastic model identification is then essential, which can be computationally costly and even unfeasible when it comes to real applications. Efficient surrogate models, such as reduced-order method, can be used to overcome this limitation and provide real time model identification. Since their numerical accuracy influences the identification process, the optimal surrogate not only has to be computationally efficient, but also accurate with respect to the identified parameters. This work aims at automatically controlling the Proper Generalized Decomposition (PGD) surrogate’s numerical accuracy for parameter identification. For this purpose, a sequence of Bayesian model identification problems, in which the surrogate’s accuracy is iteratively increased, is solved with a variational Bayesian inference procedure. The effect of the numerical accuracy on the resulting posteriors probability density functions is analyzed through two metrics, the Bayes Factor (BF) and a criterion based on the Kullback-Leibler (KL) divergence. The approach is demonstrated by a simple test example and by two structural problems. The latter aims to identify spatially distributed damage, modeled with a PGD surrogate extended for log-normal random fields, in two different structures: a truss with synthetic data and a small, reinforced bridge with real measurement data. For all examples, the evolution of the KL-based and BF criteria for increased accuracy is shown and their convergence indicates when model refinement no longer affects the identification results.

Failure probability analysis of corroded RC structures considering the effect of spatial variability

A reasonable assessment of the probability of structural failure is essential to ensure the safe performance of corroded reinforced concrete (RC) structures. In this paper, a novel cross-sectional area loss model of corroded steel bars is proposed and the scale of fluctuation of some key parameters is estimated using the semi-variogram function method. A stochastic resistance deterioration model considering the effect of unbalanced corrosion and spatial variability and a time-dependent reliability assessment method of corroded concrete structures (CCSs) are then presented. A simple case analysis of a CCS is used to illustrate the application of the proposed method. The case analysis results show that the cumulative failure probability (CFP) is overestimated using the popular pit area model, while it is underestimated without considering the spatial variability of calculation parameters. Random field parameter analysis shows that an appropriate choice of the element size is very important for the safety evaluation of RC structures. Additionally, compared with a marine atmospheric environment, the CFP of a case study structure in tidal and splash environments was found to be increased by 20.15% and 70.40%, respectively, due to the gradual increase of corrosion current density. Of the three service environments, the splash environment thus has the greatest impact on structural failure probability.

A Random‐XFEM technique modeling of hydraulic fracture interaction with natural void

In this paper, a Random-XFEM technology was employed to simulate hydraulic fracture interaction with the natural void in a random field of Young's modulus. The random distribution of Young's modulus is characterized by random field theory, while the stress and pressure fields fracturing were solved by the extended finite element method. And a Random-XFEM iteration algorithm was proposed to solve the hydraulic fracture propagation problem. Then the proposed model was verified against the KGD model and numerical study. A series of numerical examples were presented to analyze the interaction mechanism of hydraulic fracture and void in a random field. The numerical results show that the random field of Young's modulus has a great impact on the hydraulic fracture propagation path. Under the random distribution of Young's modulus, the hydraulic fracture would deviate from the direction of maximum horizontal principal stress. And the random field parameters (Young's modulus mean and point variance) have different effects on the hydraulic fracture propagation path. Also, the interaction patterns of hydraulic fracture and void are greatly affected by random fields.

A meso-scale size effect study of concrete tensile strength considering parameters of random fields

This study analyses size effects of concrete under uniaxial tension by Monte Carlo simulations, where heterogeneous strength at meso-scale is modelled by Weibull random fields with statistical parameters including correlation length and variance. For a given sample size and different random field parameters, a sufficient number of random field realisations are simulated to obtain statistical information from macroscopic stress-strain curves, while the complex meso-crack initiation and propagation is captured by the phase-field regularized cohesive zone model (PF-CZM). The effects of sample size and material heterogeneity on macroscopic tensile strength are analysed, and the quasi-brittle transition between plasticity and linear elastic fracture mechanics (LEFM) is well simulated using the nonlocal PF-CZM. It is also found that both the correlation length and the variance affect the trend of size effect in varying degrees: larger correlation length and higher variance with higher heterogeneity lead to more dispersed responses that approach the LEFM descending line. A modified law in three-dimensional parametric space is proposed by data regression for effective assessment of size effect and structural reliability.

Robust topology optimization of structures under uncertain propagation of imprecise stochastic-based uncertain field

This study introduces a novel computational framework for Robust Topology Optimization (RTO) considering imprecise random field parameters. Unlike the worst-case approach, the present method provides upper and lower bounds for the mean and standard deviation of compliance as well as the optimized topological layouts of a structure for various scenarios. In the proposed approach, the imprecise random field variables are determined utilizing parameterized p-boxes with different confidence intervals. The Karhunen–Loève (K–L) expansion is extended to provide a spectral description of the imprecise random field. The linear superposition method in conjunction with a linear combination of orthogonal functions is employed to obtain explicit mathematical expressions for the first and second order statistical moments of the structural compliance. Then, an interval sensitivity analysis is carried out, applying the Orthogonal Similarity Transformation (OST) method with the boundaries of each of the intermediate variable searched efficiently at every iteration using a Combinatorial Approach (CA). Finally, the validity, accuracy, and applicability of the work are rigorously checked by comparing the outputs of the proposed approach with those obtained using the particle swarm optimization (PSO) and Quasi-Monte-Carlo Simulation (QMCS) methods. Three different numerical examples with imprecise random field loads are presented to show the effectiveness and feasibility of the study.

Uncertainty quantification for initial geometric imperfections of cylindrical shells: A novel bi-stage random field parameter estimation method

Thin-walled cylindrical shells are widely used as main load-carrying structures in the aerospace industry. Considering the high sensitivity of the buckling load capacity to initial geometric imperfections (GIs), uncertainty quantification (UQ) of GIs is crucial to both structural safety as well as lightweight design. However, the inherent strong random characteristics of GIs and limited experimental data exacerbate the difficulty. Therefore, a novel bi-stage random field parameter estimation method with limited samples is proposed for the UQ of GIs. In stage I, a new optimization formula based on average coefficient of determination (AR) is established to overcome the complexity caused by both parameter coupling and limited samples and then, Kriging-based method via chunking and selecting mechanism (KM-CMS) is proposed to find the optimal correlation length and number of truncated terms. In stage II, a maximum likelihood (ML) based method for estimating standard deviation (SD) of random field is established, benefitted from the SD-independence of AR. Numerical results show that the proposed random field parameter estimation method achieves the best performance in terms of accuracy, efficiency and stability. Finally, a high-fidelity UQ model of measured GIs of cylindrical shells is obtained by proposed method in a rational manner.

Tweedie hidden Markov random field and the expectation-method of moments and maximisation algorithm for brain MR image segmentation

ABSTRACT In this paper, a segmentation algorithm of brain magnetic resonance imaging is elaborated. The proposed method rests on Tweedie hidden Markov random field processing and the Expectation Method of moments and Maximisation algorithm. This new type of stochastic process stands for a Markov Random Field whose state sequence cannot be observed directly and whose spatial information is encoded through the mutual influences of neighbouring sites. Tweedie models are a special case of exponential dispersion models with power mean variance. In image processing problems as medical image segmentation no work has been recorded in terms of demonstrating the applicability of the Tweedie models with hidden Markov random field and different Tweedie parameters values p. Hence, our research stands for the pioneering work aiming at proving the efficiency of the Tweedie models in brain image segmentation. Expectation- Method of moments and Maximisation algorithm results from the combination of both EM (Expectation-Maximisation) algorithm and the method of moments. The algorithm was validated on synthetic data and tested on real images. The basic merit of our proposed algorithm of segmentation lies in its ability to identify the brain tumour. Experiments using Tweedie hidden Markov random field showed that the proposed method produces more intuitive results with a recognition rate of 93.72. In fact, the experimental results highlighted that our approach outperforms other approaches that are reported in literature.

Approximate reference priors for Gaussian random fields

AbstractReference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.