Samples Of Random Fields Research Articles

Buckling by disordered growth

Buckling instabilities driven by tissue growth underpin key developmental events such as the folding of the brain. Tissue growth is disordered due to cell-to-cell variability, but the effects of this variability on buckling are unknown. Here, we analyze what is perhaps the simplest setup of this problem: the buckling of an elastic rod with fixed ends driven by spatially varying, yet highly symmetric growth. Combining analytical calculations for simple growth fields and numerical sampling of random growth fields, we show that variability can increase as well as decrease the growth threshold for buckling, even when growth variability does not cause any residual stresses. For random growth, we find numerically that the shift of the buckling threshold correlates with spatial moments of the growth field. Our results imply that biological systems can either trigger or avoid buckling by exploiting the spatial arrangement of growth variability. Published by the American Physical Society 2024

Uncertainty Analysis of Post-Failure Behavior in Landslides Based on SPH Method and Generalized Geotechnical Random Field Theory

In slope stability analysis, the inherent heterogeneity and spatial variability of soil significantly influence the aftermath of landslides, including critical aspects like runout distance, influence distance, and volume. This research integrates Smoothed Particle Hydrodynamics (SPH) with random field theory to precisely model the large deformation events in slopes with anisotropic shear parameters, while leveraging Graphics Processing Unit (GPU) parallel computing to expedite the generation of random field samples and SPH simulations. This approach meticulously examines how spatial heterogeneity of soil affects slope instability, simulating soil parameters across varied geological settings. It considers the fluctuation scale of anisotropy, the cross-correlation of cohesion and the angle of internal friction, along with their coefficient of variation (CV), to elucidate their impact on landslide magnitude and severity. This study furnishes a robust tool for a holistic assessment of slope impacts and landslide volumes. Additionally, by delineating the computational efficiency disparities between GPUs and Central processing units (CPUs), it underscores the pronounced efficiency benefits of GPU computing. The insights garnered offer fresh perspectives and methodologies in slope stability analysis and disaster risk evaluation, contributing to the finer prediction and management of this prevalent and grave geological hazard.

Detection Method for Rice Seedling Planting Conditions Based on Image Processing and an Improved YOLOv8n Model

In response to the need for precision and intelligence in the assessment of transplanting machine operation quality, this study addresses challenges such as low accuracy and efficiency associated with manual observation and random field sampling for the evaluation of rice seedling planting conditions. Therefore, in order to build a seedling insertion condition detection system, this study proposes an approach based on the combination of image processing and deep learning. The image processing stage is primarily applied to seedling absence detection, utilizing the centroid detection method to obtain precise coordinates of missing seedlings with an accuracy of 93.7%. In the target recognition stage, an improved YOLOv8 Nano network model is introduced, leveraging deep learning algorithms to detect qualified and misplaced seedlings. This model incorporates ASPP (atrous spatial pyramid pooling) to enhance the network’s multiscale feature extraction capabilities, integrates SimAM (Simple, Parameter-free Attention Module) to improve the model’s ability to extract detailed seedling features, and introduces AFPN (Asymptotic Feature Pyramid Network) to facilitate direct interaction between non-adjacent hierarchical levels, thereby enhancing feature fusion efficiency. Experimental results demonstrate that the enhanced YOLOv8n model achieves precision (P), recall (R), and mean average precision (mAP) of 95.5%, 92.7%, and 95.2%, respectively. Compared to the original YOLOv8n model, the enhanced model shows improvements of 3.6%, 0.9%, and 1.7% in P, R, and mAP, respectively. This research provides data support for the efficiency and quality of transplanting machine operations, contributing to the further development and application of unmanned field management in subsequent rice seedling cultivation.

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.

Sampling Gaussian stationary random fields: A stochastic realization approach

Generating large-scale samples of stationary random fields is of great importance in the fields such as geomaterial modeling and uncertainty quantification. Traditional methodologies based on covariance matrix decomposition have the difficulty of being computationally expensive, which is even more serious when the dimension of the random field is large. This paper proposes an efficient stochastic realization approach for sampling Gaussian stationary random fields from a systems and control point of view. Specifically, we take the exponential and squared exponential covariance functions as examples and make a decoupling assumption when there are multiple dimensions. Then a rational spectral density is constructed in each dimension using techniques from covariance extension, and the corresponding autoregressive moving-average (ARMA) model is obtained via spectral factorization. As a result, samples of the random field with a specific covariance function can be generated very efficiently in the space domain by implementing the ARMA recursion using a Gaussian white noise input. Such a procedure is computationally cheap due to the fact that the constructed ARMA model has a low order. Furthermore, the same method is integrated to multiscale simulations where interpolations of the generated samples are achieved when one zooms into finer scales. Both theoretical analysis and simulation results show that our approach performs favorably compared with covariance matrix decomposition methods.

Non-parametric simulation of random field samples from incomplete measurements using generative adversarial networks

ABSTRACT Random field theory is an effective tool for modelling spatial or temporal variability and uncertainty in natural phenomena, and it has been widely applied in many areas such as structural dynamics, geology, hydrology, and meteorology. Conventional methods of random field simulations are often parametric, in which explicit function forms for trend function, auto-correlation function, and marginal probability density function (PDF) should be selected, together with their corresponding parameters estimated from measurements. Complete random field measurements with many data points are indispensable for selection of the appropriate function forms and accurate estimates of their corresponding parameters. However, the measurements in practice are often incomplete. Without sufficient measurement data, the random field samples (RFSs) generated by conventional methods might contain unexpected uncertainty and might be misleading. To tackle this problem, a purely non-parametric random field simulation method is developed in this study that generates RFSs directly from incomplete measurement data using generative adversarial networks (GAN). Statistical analysis is performed to estimate statistical properties from the generated RFSs, including mean, standard deviation (SD), autocovariance function (AF), and cumulative density function (CDF). The results show that the proposed method properly simulates RFSs from incomplete measurement data in a purely data-driven manner.

Data-Driven Development of Three-Dimensional Subsurface Models from Sparse Measurements Using Bayesian Compressive Sampling: A Benchmarking Study

With the rapid development of computing and digital technologies recently, three-dimensional (3D) subsurface models for accurate site characterization have received increasing attention, for example, with various data-driven methods developed for 3D subsurface modeling. This leads to a need for validating the 3D modeling results obtained from each method and comparing the performance of different methods in a fair and consistent manner. To address this need, a benchmarking study, which is often used in machine learning (ML), is presented in this study to compare the performance of different 3D subsurface modeling methods in four aspects, including accuracy, uncertainty, robustness, and computational efficiency. A suite of performance metrics is proposed for the four aspects above. Multiple sets of real cone penetration test (CPT) data are compiled in the benchmarking study for quantifying performance of 3D modeling methods using sparse measurements as input, a typical scenario in geotechnical practice. The benchmarking study is illustrated using an in-house software package called Analytics of Sparse Spatial Data based on Bayesian compressive sampling/sensing (ASSD-BCS), which can directly generate high-resolution 3D random field samples (RFSs) from sparse measurements. The evaluation results show that ASSD-BCS provides accurate estimates with quantified uncertainty from sparse measurements. In addition, ASSD-BCS exhibits remarkably high computational efficiency and performs robustly under different benchmarking cases.

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.

Data augmentation for CNN-based probabilistic slope stability analysis in spatially variable soils

A novel methodology that involves the coupling of Convolutional Neural Networks (CNNs) and a data augmentation technique is proposed for slope reliability calculations. The methodology starts from generating a small set of random field samples, which are then calculated using the shear strength reduction method in the finite-difference scheme to obtain the associated factors of safety. Based on the theoretical relationship between the factor of safety and soil property values derived from the underlying mechanism of the shear strength reduction method, an innovative data augmentation technique is developed to enhance both the quantity and comprehensiveness of the dataset. A CNN model is then trained using the augmented dataset to learn the relationship between the factors of safety and random fields and predict the probability of slope failure. The effectiveness of the methodology is illustrated and validated using a c-φ soil slope and a multi-layered Su slope. The results show that CNNs are effective in interpreting high-dimensional random fields. In addition, through using the data augmentation technique, not only has the predictive capability of the CNN model been effectively boosted, particularly for cases involving low probability levels of failure, but the computational efficiency of the slope reliability analysis has also been improved.

Stochastic analysis of excavation-induced wall deflection and box culvert settlement considering spatial variability of soil stiffness

In this study, a three-dimensional (3D) finite element modelling (FEM) analysis is carried out to investigate the effects of soil spatial variability on the response of retaining walls and an adjacent box culvert due to a braced excavation. The spatial variability of soil stiffness is modelled using a variogram and calibrated by high-quality experimental data. Multiple random field samples (RFSs) of soil stiffness are generated using geostatistical analysis and mapped onto a finite element mesh for stochastic analysis of excavation-induced structural responses by Monte Carlo simulation. It is found that the spatial variability of soil stiffness can be described by an exponential variogram, and the associated vertical correlation length is varied from 1.3 m to 1.6 m. It also reveals that the spatial variability of soil stiffness has a significant effect on the variations of retaining wall deflections and box culvert settlements. The ignorance of spatial variability in 3D FEM can result in an underestimation of lateral wall deflections and culvert settlements. Thus, the stochastic structural responses obtained from the 3D analysis could serve as an effective aid for probabilistic design and analysis of excavations.

Probabilistic estimation of thermal crack propagation in clays with Gaussian processes and random fields

Prediction of thermal crack propagation in desiccated soils is imperfect due to the obscure field measurements, modeling approximations, and the underlying soil uncertainties. To address these issues, a probabilistic framework is developed to quantify the uncertainties and enhance the crack estimation reliability. Hence, the uncertainties associated with the soil properties, including the tensile strength, matric suction, Poisson’s ratio, elastic modulus and suction ratio are associated in a probabilistic model. Then, the probabilistic model is utilized with the improved Monte Carlo simulation to calculate the failure probability due to the exceedance of the crack propagation. Furthermore, failure probability optimization is exemplified by evaluating the spatial correlation between variables. The Bayesian optimization is employed to optimize the covariance kernels via the Gaussian process regression. Then, the covariance matrices are decomposed to generate random fields to further enhance the calculated failure probabilities. The results indicate that the cracking probability in near-surface layers is imminent. However, with the increase of the tensile strength and decrease in the matric suction, a sharp drop in crack propagation is highly expected. The findings also suggest that the importance of variable uncertainties is in the order of Matric suction> Elastic modulus and Suction ratio > Tensile strength> Poisson’s ratio. Also, the operation of the probabilistic model with different random field sampling outperforms the improved and directed Monte Carlo simulations because of the comprehension of the spatial correlation. However, the Gaussian fields are restricted to the second-order statistics, while lognormal fields show relatively lesser constraints. Hence, the random field sampling with a larger length scale as well the Karhunen–Loève expansion followed by the approximated eigenvalue decomposition offer viable options for the probabilistic estimation of the crack depth.

Improving species delimitation for effective conservation: a case study in the endemic maple-leaf oak (Quercus acerifolia).

Species delimitation is challenging in lineages that exhibit both high plasticity and introgression. This challenge can be compounded by collection biases, which may downweight specimens morphologically intermediate between traditional species. Additionally, mismatch between named species and observable phenotypes can compromise species conservation. We studied the species boundaries of Quercus acerifolia, a tree endemic to Arkansas, U.S. We performed morphometric analyses of leaves and acorns from 527 field and 138 herbarium samples of Q. acerifolia and its close relatives, Q. shumardii and Q. rubra. We employed two novel approaches: sampling ex situ collections to detect phenotypic plasticity caused by environmental variation and comparing random field samples with historical herbarium samples to identify collection biases that might undermine species delimitation. To provide genetic evidence, we also performed molecular analyses on genome-wide SNPs. Quercus acerifolia shows distinctive morphological, ecological, and genomic characteristics, rejecting the hypothesis that Q. acerifolia is a phenotypic variant of Q. shumardii. We found mismatches between traditional taxonomy and phenotypic clusters. We detected underrepresentation of morphological intermediates in herbarium collections, which may bias species discovery and recognition. Rare species conservation requires considering and addressing taxonomic problems related to phenotypic plasticity, mismatch between taxonomy and morphological clusters, and collection biases.

A quasi-brittle fracture investigation of concrete structures integrating random fields with the CSFEM-PFCZM

Design and maintenance of concrete structures require efficient methods for predicting quasi-brittle fracture while incorporating mesoscale-induced structural uncertainties. Toward this aim, a phase field cohesive zone model combined with the cell-based smoothed finite element method (CSFEM-PFCZM) is developed in the study, with mesoscale heterogeneity of concrete characterised by Weibull random fields. An automatic quadtree decomposition technique is presented to adaptively refine meshes around mesoscale areas, in which the hanging-node problem is tackled by treating hierarchical quadtree elements as CSFEM polygons. The crack driving forces are evaluated as history variables at the integration points of CSFEM subcells, and only the boundary geometry and Wachspress shape functions are needed to calculate strain matrices and stiffness matrices without the Jacobian inversion problems. The developed method is validated by extensive Monte Carlo simulations of typical nonlinear fracture benchmarks of concrete structures, demonstrating that the well-captured variability of crack paths and load–displacement curves can supplement the limited statistical results obtained from a small number of experiments. It is also found that larger correlation length and higher variance underline higher heterogeneity that leads to more dispersed responses. The developed method holds promise for the efficient evaluation of structural reliability taking into account stochastic mesoscale effects, due to the fast generation of random field samples in large quantities and flexible simulation of complicated fracture through the CSFEM-PFCZM.

Quantifying Uncertainty of Damage in Composites Using a Quasi Monte Carlo Technique

Abstract Property variations in a structure strongly impact the macroscopic mechanical performance as regions with lower strength will be prone to damage initiation or acceleration. Consideration of the variability in material property is critical for high-resolution simulations of damage initiation and propagation. While the recent progressive damage analyses consider randomness in property fields, accurately quantifying the uncertainty in damage measures remains computationally expensive. Stochastic damage analyses require extensive sampling of random property fields and numerous replications of the underlying nonlinear deterministic simulations. This paper demonstrates that a Quasi-Monte Carlo (QMC) method, which uses a multidimensional low discrepancy sobol sequence, is a computationally economical way to obtain the mean and standard deviations in cracks evolving in composites. An extended finite element method (XFEM) method with spatially random strength fields simulates the damage initiation and evolution in a model composite. We compared the number of simulations required for Monte Carlo (MC) and QMC techniques to measure the influence of input variability on the mean crack-length in an open-hole angle-ply tensile test. We conclude that the low discrepancy sampling and QMC technique converges substantially faster than traditional MC methods.

Assessment of reclamation-induced consolidation settlement considering stratigraphic uncertainty and spatial variability of soil properties

Consolidation analysis is a key task for reclamation design. Although consolidation is a long-term process, acceleration of consolidation is often preferred for speeding up the reclamations. Before proposing measures to accelerate the consolidation and reclamation process, it is imperative to have an accurate prediction of consolidation settlement for fine-grained materials, which is greatly affected by spatial distribution of subsurface zones with different soil types (i.e., stratigraphic heterogeneities and uncertainty) and spatial variability of soil properties. In current practice, calculation of consolidation settlement often uses simplified stratigraphic boundaries and deterministic consolidation parameters without considering stratigraphic uncertainty or soil property spatial variability. The oversimplified practice might result in unconservative estimations of consolidation settlement and pose threats to safety and serviceability of constructed facilities on reclaimed lands. In this study, a stochastic framework is proposed for consolidation settlement assessment with explicit modeling of stratigraphic uncertainty and spatial variability of soil properties by machine learning and random field simulation from limited site investigation data. Performance of the proposed framework is demonstrated using an illustrative example. Results indicate that the proposed method effectively generates multiple realizations of geological cross section and random field samples of geotechnical properties from limited measurements and offers valuable insights into spatial distribution of the estimated total primary consolidation settlement curves and angular distortion.

Deep learning-based evaluation of factor of safety with confidence interval for tunnel deformation in spatially variable soil

The random finite difference method (RFDM) is a popular approach to quantitatively evaluate the influence of inherent spatial variability of soil on the deformation of embedded tunnels. However, the high computational cost is an ongoing challenge for its application in complex scenarios. To address this limitation, a deep learning-based method for efficient prediction of tunnel deformation in spatially variable soil is proposed. The proposed method uses one-dimensional convolutional neural network (CNN) to identify the pattern between random field input and factor of safety of tunnel deformation output. The mean squared error and correlation coefficient of the CNN model applied to the newly untrained dataset was less than 0.02 and larger than 0.96, respectively. It means that the trained CNN model can replace RFDM analysis for Monte Carlo simulations with a small but sufficient number of random field samples (about 40 samples for each case in this study). It is well known that the machine learning or deep learning model has a common limitation that the confidence of predicted result is unknown and only a deterministic outcome is given. This calls for an approach to gauge the model's confidence interval. It is achieved by applying dropout to all layers of the original model to retrain the model and using the dropout technique when performing inference. The excellent agreement between the CNN model prediction and the RFDM calculated results demonstrated that the proposed deep learning-based method has potential for tunnel performance analysis in spatially variable soils.

CPT-based probabilistic liquefaction assessment considering soil spatial variability, interpolation uncertainty and model uncertainty

In engineering practice, simplified procedure based on cone penetration test (CPT) results is widely used for evaluating soil liquefaction potential. Since the CPT-based simplified procedure was developed from observations during past earthquakes, it is semi-empirical and involves significant uncertainty (e.g., model uncertainty). In addition, due to time, budget and access constraints, CPTs are often sparsely conducted at a specific site, leading to a significant uncertainty associated with interpolation of the limited CPT soundings, particularly along horizontal direction. Furthermore, it is well-recognized that spatial variability of soil properties has a remarkable effect on soil liquefaction. All these variability and uncertainties greatly affect the seismic liquefaction assessment results, particularly spatial distribution of liquefiable soils in a site, and liquefaction-induced damage. This underscores a question of how to properly incorporate these variability and uncertainties in liquefaction assessment, e.g., how to characterize spatial distribution of soil liquefaction potential in a site with quantitative consideration of the abovementioned variability and uncertainty. To address this issue, this paper develops a novel probabilistic method for characterizing spatial distribution of soil liquefaction potential through factor of safety, FS against liquefaction in a vertical cross-section using Bayesian compressive sampling and Monte Carlo simulation. Using the proposed method, many random field samples of FS cross-section are obtained directly from limited CPT measurements. The proposed method is illustrated using both a simulated data example and a set of real CPT data from Christchurch, New Zealand. It is shown that the proposed method performs well and provides reasonable liquefaction assessment results.

Quantification of multiple-variate random field by synthesizing the spatial correlation function of prime variable and copula function

The probabilistic dependence in a multi-variate random field consists of two parts: the spatial dependence of a random quantity at different positions and the probabilistic dependence between different random variables at the same position. The classical model cannot capture the possible non-Gaussian dependence or nonlinear dependence between different random variables. To this end, in this paper, an approach by synthesizing the spatial correlation function and the multi-variate copula function (SCFVCF) is proposed. In this model, the correlation function model is adopted to quantify the spatial dependence involved in the random field of the prime variable, and the copula function model is adopted to quantify the dependence configuration between subordinate variables. The properties of such multi-variate random fields are then studied. To generate samples of such multi-variate random fields, the spectral representation method is incorporated with the conditional sampling method. As an example, to illustrate the application of SCFVCF, the random field of the constitutive parameters of concrete is adopted. The results demonstrate that the proposed method can capture the spatial dependence of compressive strength, and at the same time the dependence configuration between different parameters is consistent with the test complete compressive stress-strain curves.

Correlated disorder in rainbow metamaterials for vibration attenuation

Metastructures are typically composed of periodic unit cells designed to present enhanced dynamic properties in which either single or multiple resonators are periodically distributed. Even though the periodic metamaterials can obtain bandgaps with outstanding vibration attenuation, the widths of bandgaps can still be narrow for some practical applications. Rainbow metamaterials have been proposed based on gradient or random profiles to provide further improved attenuation. Nonetheless, the effects of correlated random disorder on their attenuation performance remains an open challenge. This work presents an investigation on the effects of correlated disorder on the vibration attenuation of rainbow metamaterials. An analytical model using the transfer matrix approach is used to calculate the receptance functions in a finite length metastructure composed of evenly spaced non-symmetric resonators attached to a beam with Π-shaped cross-section, thus a multi-frequency metastructure. The correlated disorder is modelled using random fields and an analytical expression of the Karhunen-Loève expansion is used such that spatial correlation on the resonator properties is modified by various correlation lengths, i.e., the level of spatial smoothness. Individual samples of random fields are used to investigate the effects of the correlated disorder in the vibration attenuation of a multi-frequency metastructure. It is shown that the bandgap can be further widened when compared to uncorrelated disorder. The obtained results indicates that a combination of the gradient profile with some level of disorder, typically resulting from random fields with larger correlation lengths, tends to give improved vibration attenuation when compared to a optimized gradient rainbow metamaterial. It opens new and innovative ways for the design of broadband rainbow metastructures for vibration attenuation.

Novel approach to efficient slope reliability analysis in spatially variable soils

The random field finite element method (RF-FEM) provides a robust tool for carrying out slope reliability analysis that incorporates the spatial variability of soil properties. However, it has a major drawback of being computationally very time-consuming. To address this common criticism, the current study proposes a novel metamodel-based method for efficient slope reliability analysis in spatially variable soils. The proposed method involves the use of Convolutional Neural Networks (CNNs) as metamodels of the random field finite element model. With proper training using a small but sufficient number of random field samples, the CNN can potentially replace the computationally demanding random field finite element analyses for Monte-Carlo simulations. This paper examines the capability of CNNs to learn high-level features that contain information about the random variabilities in both spatial distribution and intensity, and the accuracy of the subsequent predictions of the RF-FEM results. Application of the proposed method to slope reliability analysis in spatially variable soils is illustrated and compared against other metamodel-based approaches, using a case study involving a multi-layered soil system with randomly varying cohesion c and the friction angle ϕ. The results show that (i) the proposed CNN approach predicts a probability of slope failure that is within 5% of the corresponding value obtained using direct RF-FEM Monte-Carlo simulations, but at a small fraction of the computational cost, and (ii) the proposed method also compares favourably against other metamodel-based methods in terms of computational efficiency and accuracy.