3D Random Field Research Articles

Machine learning analysis of dimensional reduction conjecture for nonequilibrium Berezinskii-Kosterlitz-Thouless transition in three dimensions.

We investigate the recently proposed dimensional reduction conjecture in driven disordered systems using a machine learning technique. The conjecture states that a static snapshot of a disordered system driven at a constant velocity is equal to a space-time trajectory of its lower-dimensional pure counterpart. This suggests that the three-dimensional (3D) random field XY model exhibits the Berezinskii-Kosterlitz-Thouless transition when driven out of equilibrium. To verify the conjecture directly by observing configurations of the system, we utilize the capacity of neural networks to detect subtle features of images. Specifically, we train neural networks to differentiate snapshots of the 3D driven random field XY model from space-time trajectories of the two-dimensional pure XY model. Our results demonstrate that the network cannot distinguish between the two, confirming the dimensional reduction conjecture.

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

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

3D conditional random fields simulation for rockfill compaction quality assessment with sparse EVD measurement

The compaction quality of rockfill materials plays a pivotal role in ensuring the stability and durability of runway structures. However, traditional assessment methods relying on limited random sampling may lack representativeness for the entire construction site and overlook potential areas of insufficient compaction. Addressing this challenge, this paper employs the dynamic deformation modulus (EVD), a rapid and nondestructive testing method, to assess the compaction quality of runway rockfill. Subsequently, this paper propose a 3D Conditional Random Field (CRF) model based on EVD to simulate the uncertain spatial distribution of rockfill compaction quality, allowing for the estimation of the overall construction area's compaction quality with a limited number of sample points. The validity of the model is demonstrated through a case study involving a runway expansion project. Additionally, three critical construction factors--namely, measurement points, compaction passes, and compaction materials--are analyzed throughout the construction process. This analysis provides managers with valuable insights and decision-making tools to promptly evaluate the quality of compaction areas, ensuring the stability and durability of the runway.

Influence of ballast heterogeneity on railway induced vibrations and identification of heterogeneous properties of a ballast layer

In the context of railway induced vibration, the excitation induced by a passing train can be separated into a quasi-static and a dynamic component. The quasi-static component is related to the weight of the train transmitted through the wheels to the track. The dynamic component is related to the dynamic train-track excitation resulting from the irregularities of the running surfaces or from the variation of the track stiffness. At sub-Rayleigh speed on a track with longitudinally invariant properties, quasi-static excitation has no significant influence on the far-field vibration. In this paper, a randomly-fluctuating heterogeneous continuum model of the ballast layer is considered. Mechanical properties are represented as 3D random fields. First, the influence of ballast heterogeneity on the vibration field induced by a moving load is studied. The results show that heterogeneity is responsible for the generation of waves propagating away from the track. Then an identification process of the heterogeneous model of the ballast is presented based on experimental measurements of train passages.

Stochastic free vibration analysis of FG-CNTRC plates based on a new stochastic computational scheme

Stochastic analysis can provide more accurate results compared to deterministic analysis. In this study, the spatial variability of material parameters is introduced into the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates to achieve a higher degree of accuracy, and a new stochastic computational scheme is proposed to handle the low-level uncertainties. First-order shear deformation theory (FSDT) is employed to establish the displacement field of the plates. The Young's moduli of carbon nanotube (CNT) and matrix are regarded as one-dimensional (1D) and two-dimensional (2D) random fields, respectively, which is discretized by Karhunen-Loève (K-L) expansion. The obtained random variables are substituted into modified perturbation stochastic method (MPSM) and radial point interpolation method (RPIM) to calculate the first two order estimates of the stochastic non-dimensional natural frequencies ω¯. The sensitivity of the first to fourth ω¯ to the random fields is analyzed, and the corresponding stochastic bands are plotted. The results indicate that the presented approach can accurately solve the generalized eigenvalue problem in stochastic free vibration; ω¯ is sensitive to random fields E11CNT and Em, and insensitive to random field E22CNT; the CNT distribution mode also affects the sensitivity.

Numerical modelling of the keying process for a suction embedded plate anchor in spatially varying clays

ABSTRACT As the exploration of oil and gas moves into deeper water, suction embedded plate anchors (SEPLAs) are developed to answer the growing demand for anchor-mooring systems. The keying process of SEPLAs has been extensively reported assuming a spatially homogeneous soil which is a departure from reality. This study conducts a comprehensive investigation of the keying process of SEPLA in spatially varying clay through a large deformation random finite element method coupled with the Monte Carlo simulations. The numerical model is validated with the results from the existing centrifuge test and numerical analysis. The spatial randomness of soil strength is assumed to be a three-dimensional (3D) random field to explore the failure mechanism and statistical characteristics of rotational response and pullout behaviour. The normalised factor of the upper 5% fractile of embedment loss is presented as an algebraic expression of the coefficient of variation of soil strength. The relationship between safety factor and failure probability is determined based on the requirement of pullout capacity. Finally, the applicability of the present results to different soil properties is verified.

Data augmentation guided breast tumor segmentation based on generative adversarial neural networks

Breast tumor segmentation built on Dynamic Contrast-Enhanced Magnetic Resonance Imaging is a noteworthy phase for the quantifiable radiomics study of breast cancer. Manual tumor explanation is an inefficient method and encompasses medical associates, influenced, persuaded to error, and inter-user divergence. The quantities of contemporary preparations have exposed the competence of deep learning depictions in image segmentation. At this point, we designate a 3D Connected-AUNets for tumor segmentation from 3D MRIs built on encoder–decoder design. Owing to a constrained training dataset size, a generative adversarial networks channel is complementary to modernizing the input image itself to recognize the shared decoder and implement added controls on its layers. Based on the preliminary segmentation of Connected-AUNets, a fully connected 3D condition random field is used to improve segmentation results by determining 2D neighbor areas and 3D volume statistics. Furthermore, 3D connected components assessment is used to sustain around large components and reduce segmentation noise. The proposed system has been estimated on two regularly available datasets, apparently INbreast and the Curated Breast Imaging Subset of Digital Database for Screening Mammography. The proposed system has also been evaluated using a private dataset. The experimental significances show that the proposed model outperforms the state-of-the-art processes for breast tumor segmentation.

Estimation of hydraulic conductivity in a watershed using sparse multi-source data via Gaussian process regression and Bayesian experimental design

Enhanced water management systems depend on accurate estimation of subsurface hydraulic properties. However, geologic formations can vary significantly, so information from a single source (e.g., widely spaced boreholes) is insufficient in characterizing subsurface aquifer properties. Therefore, multiple sources of information are needed to complement the hydrogeology understanding of a region. This study presents a numerical framework in which information from different measurement sources is combined to characterize the 3D random field in a multi-fidelity prediction model. Coupled with the model, a Bayesian experimental design was used to determine the best future sampling locations. The Upper Sangamon watershed in east-central Illinois was selected as the case study site, where the multi-fidelity Gaussian process model was used to estimate the hydraulic conductivity in the region of interest. Multi-source observation data were obtained from electrical resistivity and borehole pumping tests. The accuracy of the model prediction is dependent on the locations and the distribution of both high- and low-fidelity data. Furthermore, the multi-fidelity model was compared with the single-fidelity model. The uncertainties and confidence in the measurements and parameter estimates were quantified and used to design future cycles of data collection to further improve the confidence intervals.

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.

Hierarchical reconstruction of 3D well-connected porous media from 2D exemplars using statistics-informed neural network

The relationships between porous microstructures and transport properties are of fundamental importance in various scientific and engineering applications. Due to the intricacy, stochasticity and heterogeneity of porous media, reliable characterization and modeling of transport properties often require a complete dataset of internal microstructure samples. However, it is often an unbearable cost to acquire sufficient 3D digital microstructures by purely using microscopic imaging systems. This paper presents a machine learning-based technique to hierarchically reconstruct 3D well-connected porous microstructures from one isotropic or several anisotropic low-cost 2D exemplar(s). To compactly characterize the large-scale microstructural features, a Gaussian image pyramid is built for each 2D exemplar. Local morphology patterns are collected from the Gaussian image pyramids, and then they serve as the training data to embed the 2D morphological statistics into feed-forward neural networks at multiple length levels. By using a specially-developed morphology integration scheme, the 3D morphological statistics at different levels can be inferred from the statistics-informed neural networks. Gibbs sampling is adopted to hierarchically reconstruct 3D microstructures by using multi-level 3D morphological statistics, where the large-scale, regional and local morphological patterns are statistically generated and successively added to the same 3D random field. The proposed method is tested on a series of porous media with distinct morphologies, and the statistical equivalence between the reconstructed and the real microstructures is systematically evaluated by comparing morphological descriptors and transport properties. The results demonstrate that the proposed 2D-to-3D microstructure reconstruction method is a universal and efficient approach to generating morphologically and physically realistic samples of porous media.

Differential Equations for the KPZ and Periodic KPZ Fixed Points

The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for the height function of models in the KPZ universality class. Similarly, the periodic KPZ fixed point is a conjectured universal field for spatially periodic models. For both fields, their multi-point distributions in the space-time domain have been computed recently. We show that for the case of the narrow-wedge initial condition, these multi-point distributions can be expressed in terms of so-called integrable operators. We then consider a class of operators that include the ones arising from the KPZ and the periodic KPZ fixed points, and find that they are related to various matrix integrable differential equations such as coupled matrix mKdV equations, coupled matrix NLS equations with complex time, and matrix KP-II equations. When applied to the KPZ fixed points, our results extend previously known differential equations for one-point distributions and equal-time, multi-position distributions to multi-time, multi-position setup.

Crossover from mechanical stabilization of martensite to spanning avalanche-type reverse transformation of deformed Cu-Al-Ni shape memory crystals

A new interpretation of factors controlling the kinetics of reverse transformation of martensites stabilized by prestrain is suggested. We investigate the effect of amount of prestrain in the β1´martensite and in the β-phase on temperature and kinetics of the reverse martensitic transformation in Cu-Al-Ni shape memory alloy single crystals. Calorimetry and video recording of the jumping samples, provoked by the burst strain recovery during temperature-induced reverse transformation, are used in experiments. For crystals deformed both in the β´1 martensite and in the β-phase, we report the existence of a sharp transition (crossover) from the mechanical stabilization of martensite to the burst reverse martensitic transformation which proceeds as a single avalanche. The crossover occurs for prestrain values close to the maximum transformation strain. The specific energy of the jump during burst strain recovery depends on the maximum stress applied during prestraining the sample. The crossover from a broad transformation range to an infinite or spanning avalanche is predicted by 3D random field Ising model for disordered systems undergoing the first order phase transition. Based on this model, we explain the crossover in deformed Cu-Al-Ni crystals by a rapid decrease of structural disorder towards the critical one at the final stage of the formation of detwinned γ´1martensite. A simple solution is obtained for the critical transformation rate which produces jumping of the sample during burst reverse transformation after prestrain.

Stochastic numerical analysis for static performance of composite laminates based on resampling method

Due to the unstable properties of composite materials, it is of great significance to obtain the strength and its mechanical dispersion of composite laminates. The dispersion can be simply simulated using a recently developed Stochastic Numerical Analysis Architecture based on resampling method. This paper attempts to quantify the inconsistency of mechanical uncertainty between the composite mechanical test panels and the components by resampling method in two scales. Researchers have given different scale distribution prediction methods based on manufacturing and experiments for this phenomenon. By introducing resampling method and 3D random field into the Monte-Carlo simulation framework, a fast prediction method for structural uncertainty is established. Three sets of laminate strength experiment were done to verify the analysis architecture, and the results showed that the predicted strength and distribution are in good agreement with the experimental ones. Some intelligent algorithms regarding applications of the 3D random field are also addressed.

Optimized active learning Kriging reliability based assessment of laterally loaded pile groups modeled using random finite element analysis

This paper presents an optimization of the Active learning Kriging (AK) reliability assessment of laterally loaded pile groups modeled using random nonlinear finite element (FE) where the spatial variability in the soil response is discretized using three dimensional (3D) random fields. The input functions of the AK including the regression, correlation, and the learning functions (called AK configuration herein) are optimized for four AK based reliability estimation techniques: AK Monte Carlo simulation (MCS), AK first order reliability method (FORM), AK importance sampling (IS), and AK self-normalizing importance sampling (LS). The optimal AK configurations are obtained using an extensive parametric analysis (4320 reliability assessment analyses). Two rational-based methods are adopted to propose the optimal AK configurations: accuracy-based priority ranking and consistency-based priority ranking. Summary tables of the optimal AK configurations are included in the paper. The proposed optimal AK configurations can be used by engineers to assess the reliability of pile group-soil interaction problems using AK and random FE analysis.

Optimization of Site-exploration Programs in Slope Designs Using 3D Conditional Random Fields

Background: In situ soil properties exhibit inherent spatial variability, which is often described by a 3D random field. Soil properties at particular portions are available by site investigation. Wider site investigation scope provides a more accurate description of the geologic profile. However, limited by budget, choosing an effective site exploration scope is of significance. Objective: This study introduces a framework to determine the optimal site investigation strategy in the 3D domain, which yields the lowest mean risk of slope designs. Method: The mean risk of slope designs is considered to be a function of the costs of site investigation, under-design, and over-design. The unconditional random fields are generated by the spectral representation method initially. Subsequently, the sampled data are incorporated into the random fields via the Kriging algorithm, and the conditional random fields are simulated. A 3D undrained slope is evaluated for illustration. Results: The effects of sampling locations and spacing on the risk of slope designs are examined. The results indicate that the optimal sampling location is close to the zone where slope failure may occur. Moreover, there exists an optimal sampling spacing that minimizes the mean risk of slope designs. Conclusion: This investigation can provide guidance for determining the optimal site exploration programs on the 3D domain with knowledge of the associated risks.

Effect of spatial variability on stability and failure mechanisms of 3D slope using random limit equilibrium method

It is well known that soils are prone to spatial non-uniformity, which affects evaluations of slope stability and failure mechanisms. This paper presents a probabilistic slope stability evaluation, considering the 3D spatial variation in the soil properties, by the random limit equilibrium method (RLEM). Specifically, 3D random fields of cohesion c, friction angle ϕ, and soil unit weight γ are generated using a fast Fourier transform. The RLEM is applied to evaluate the effects of the 3D spatial variability of the soil properties on slope stability and failure mechanisms. A Monte Carlo simulation is used to interpret the slope reliability and variation in slope failure dimension. Based on the critical slip surface passing different portions of a slope (slope base, inclined face, and crest), four main failure mechanisms (two base failures and two face failures), and one additional failure mechanism (toe failure), are identified for spatially variable slopes, and the corresponding distributions of the stability number (Ns) and sliding volume (V) are investigated in detail. The results show that the large variation in the soil properties induces changes in the failure mechanisms, and a threshold of c-ϕ values is found for a shift from base failure to toe failure. Lastly, associated sensitivity studies are performed to explore the effects of the uncertainties of the input parameters on the uncertainty of the output. The results estimated by partial Spearman correlation coefficients show that cohesion has the greatest influence on the stability number, and that a positive influence of the unit weight, contributing to slope stability, is found for a base failure mechanism.

A novel peridynamic approach for fracture analysis of quasi-brittle materials

In the present paper, a novel peridynamic approach is proposed to simulate the fracture behaviour of quasi-brittle materials. It consists in: (i) an improved bond force-stretch relationship to implement into peridynamic models, in order to simulate the fracture behaviour of quasi-brittle materials, and (ii) the proposal to consider, into the above models, the material energy release rate as a 3D random field, to take into account the stochastic nature of quasi-brittle material properties. Such a novel peridynamic approach is verified by analysing four case studies related to quasi-brittle materials, that is: rock, expanded polystyrene, and concrete.

Simulation of Gaussian random field in a ball

Abstract We address the problem of statistical simulation of a scalar real Gaussian random field inside the unit 3D ball. Two different methods are studied: (i) the method based on the known homogeneous isotropic power spectrum developed by Meschede and Romanowicz [M. Meschede and B. Romanowicz, Non-stationary spherical random media and their effect on long-period mantle waves, Geophys. J. Int. 203 2015, 1605–1625] and (ii) the method based on known radial and angular covariance functions suggested in this work. The first approach allows the extension of the simulation technique to the inhomogeneous or anisotropic case. However, the disadvantage of this approach is the lack of accurate statistical characterization of the results. The accuracy of considered methods is illustrated by numerical tests, including a comparison of the estimated and analytical covariance functions. These methods can be used in many applications in geophysics, geodynamics, or planetary science where the objective is to construct spatial realizations of 3D random fields based on a statistical analysis of observations collected on the sphere or within a spherical region.

Diffusion in a Frozen Random Velocity Field

Particle diffusion in a frozen isotropic 2D random velocity field is studied by simulation, and the results are compared with the prediction of a simple model. The model accounts for the effects of particle trapping and infinite correlation time.

Influence of horizontally variable soil properties on nonlinear seismic site response and ground motion coherency

AbstractThe objective of this study is to quantify the effect of the spatial variability of soil properties on site response and ground motion coherency. A 2D random field theory is used to simulate soil parameters, including shear wave velocity (Vs) and nonlinear stress–strain curves. Different levels of spatial variability are controlled by coefficient of variation (COV) and correlation length (CL). The wave prorogation is simulated through the 2D finite difference software FLAC2D. Different types of analysis conditions, such as 1D or 2D model, linear or nonlinear analysis, and input motion, are investigated. The increase of Vsvariability leads to a decrease in the mean amplification ratio (surface response spectrum to bedrock response spectrum) but an increase in its variation. The effect of the variability of soil nonlinearity on the mean amplification ratio is marginal compared with that induced by Vsvariability. However, as the ground motion intensities increase (or more nonlinearity induced), the standard deviation of the simulation normalized by the baseline response increases. Compared with the 2D result, the 1D site response analysis that is commonly used in practice may potentially underestimate the mean amplification ratio but overestimate its variability due to the ignorance of the horizontal soil property variability. In the evaluation of the spatially correlated site response, the ground motion coherency decays significantly, as the COV of Vsand the ground motion intensity increase. The effect of CL on ground motion coherency is minor and limited to a separation distance within two‐times of CL. Therefore, a coherency model, which eliminates the CL term but includes the ground motion intensity (i.e., nonlinear effect) parameter, is proposed in this study. It is revealed to yield a better agreement with the simulation results than other coherency models.