Random Finite Element Method Research Articles

A novel efficient RFEM for reliability analysis and design of multi-line dynamically installed anchor for floating offshore wind turbines

A novel multi-line dynamically installed anchor was previously proposed by the authors to allow for the mooring of multiple floating offshore wind turbine, resulting in a significant reduction in the total number and cost of anchors required for floating wind farms. Considering the spatial variability of soil properties and the uncertainty of environmental loads, the present study performs reliability analysis and design of the multi-line dynamically installed anchor. Firstly, a strategy to repeatedly use fundamental random variables is proposed and validated for reducing the number of random variables used in Karhunen-Loève expansion in the simulation of random field of soil properties when the ratio of the soil domain dimension to the scale of fluctuation is large. Then, the efficiency, accuracy, and robustness of the RFEM (random finite element method) combined with K-MCS (Kriging model and Monte Carlo simulation) based on the proposed strategy are validated through examples of random capacity of foundations. Next, the random capacities and probabilistic VHMT failure envelopes of the multi-line dynamically installed anchor in spatially variable soil are investigated. Finally, the reliability design of multi-line dynamically installed anchors is conducted and compared with that of multi-line pile anchors, in which both the spatial variability of soil and the uncertainty of loads are condidered. The results show that the costs for multi-line dynamically installed anchors are obviously less than those of multi-line pile anchors.

System reliability analysis of geogrid reinforced retaining wall using random finite element method

Stabilizing earthwalls is extremely important in geotechnical engineering projects and has become an integral part of transportation infrastructures. Meanwhile, geogrid reinforced retaining walls have been the attention of designers due to their advantages. On the other hand, the soil heterogeneity and the requirement of investigating the internal (geogrid rupture and geogrid pullout) and external (global and lateral displacement) stability modes of these structures have necessitated performing system reliability analysis. In this study, finite element in conjunction with random fields is used to evaluate the reliability of these walls. For this purpose, a finite element program is coded in MATLAB to obtain internal and external safety factors considering staged construction. The deterministic program is extended to a stochastic framework to account for spatial variability of soil parameters in retained backfill, foundation soil, and reinforced fill. In the last part of this study, reliability indices of stability modes are calculated to obtain the series system reliability index, utilizing the Sequential Compounding Method (SCM). The results indicated that the effect of soil heterogeneity is more significant on internal stability modes compared to external stability modes especially geogrid pullout. The results of the system reliability analysis demonstrated more critical conditions thus the reliability index of the system was less than the reliability index of individual components.

Calibration of resistance factors for seismic design of masonry infilled frames using the random finite element method

This paper presents a reliability analysis of concrete masonry infill walls with an aim to examine the efficacy of the resistance factor specified in the infill design under seismic loading. A numerical model for masonry infilled reinforced concrete frames was developed and encoded in OpenSees for this study. The model was validated using experimental data. Two features were involved in this reliability study, including 1) that the effect of spatially varying masonry compressive strength f′m on the lateral resistance of masonry infills was considered using the Random Finite Element Method, and 2) that seismic load with different return periods was considered to provide a more complete treatment of a reliability analysis. The study selected six Canadian cities with varying levels of seismic intensity for this analysis. By calculating the probability of failure of infill frames subjected to earthquake loading at different locations, the paper showed that the current resistance factor of 0.6 in the infill design equation suggested by the Canadian masonry design standard is conservative. To achieve a reliability index of 3, this factor could be increased to 0.65 for all cities studied. For cities with low seismicity such as Toronto, the results suggest that an even higher resistance factor is justified.

Reliability and landslide consequence analysis of long heterogeneous soil infrastructure slopes: A parallel computing investigation

Long infrastructures, like flood defense levees, built upon diverse soils, often face complex landslide failures. Risk assessments for such structures involve evaluating slope stability reliability and potential landslide consequences. Historically, embankment slope reliability and failure assessments focused on shorter sections due to computational limitations. This study, however, delves into a longer 3D soil slope section, 100 times the height, using the random finite element method to analyse the size or length effect. It conducts a parametric study varying the soil strength’s horizontal correlation length to understand slope failure initiation in heterogeneous soils. In contrast to prior work, this paper emphasises several facets: the realised factor of safety (FOS) distribution, quantification of discrete unstable zones and failures, and the size impact on the reliability and landslide consequences of long embankment slopes. Findings indicate the Weibull distribution better fits the realised FOS, aiding in predicting long embankment slope system reliability when combined with the weakest link theory. While multiple failures are possible for very long soil embankment slopes, observations suggest a greater likelihood of a single localised failure despite this potential.

Seismic stability and failure mechanisms of gentle and steep slopes considering soil rotated anisotropy and spatial variability

The present study focuses on investigating the effects of soil rotated anisotropy and spatial variability on slope failure in seismic conditions. The random finite element method aided by subset simulation is implemented, which ensures an efficient quantification of both the probability of failure and its associated consequence of failure. Several slope angles of gentle and steep slopes and soil properties that lead to a low probability of failure were selected for parametric studies. The comprehensive parametric studies of seismic slope stability analysis consider various factors such as slope inclinations, seismic coefficients, scales of fluctuation, and orientations of the major principal scale of fluctuation (i.e., the rotation angle of random field). The results underscore the importance of considering the combined effects of soil anisotropy and the orientation of the major principal scale of fluctuation for designing both gentle and steep slopes under seismic conditions and emphasize that care must be given to ensure the worst-case scenario is considered. Visual observations of failure mechanisms of gentle and steep slopes under seismic conditions were also shown to be very helpful in interpreting the variation of probability of failure due to soil spatial variation.

Unsaturated soil nailing wall system reliability analysis using random finite element

This study focuses on the evaluation of stability for a typical unsaturated Soil Nailing Wall System (SNWS), considering variations in soil shear strength, suction, and elasticity modulus. The Random Finite Element Method (RFEM) is employed to derive the reliability indices for stability modes, including Global Stability (GS), Lateral Displacement Stability (LDS), Tensile Strength Stability (TSS), and Pullout Resistance Stability (PRS), which serve as essential components of the SNWS. Finally, the Sequential Compounding Method (SCM) is utilized to combine these modes of stability and their direct influence on the system reliability index. The results show although the maximum Coefficient of Variation (COV) of the soil parameters was 11%, the COV of lateral displacements at the top of the wall was obtained as 19%, which shows the significant influence of considering the soil heterogeneity. Furthermore, uncertainties of elasticity modulus and matric suction lead to an increase in GS, LDS, and PRS. The calculated system reliability index was found to be lower than the lowest reliability index of the individual components. The findings indicate that while the inclusion of matric suction and elasticity modulus may enhance the mean values of stability modes, there is a possibility that the system reliability index could decrease.

Investigating the Influence of Non-Uniform Characteristics of Layered Foundation on Ground Vibration Using an Efficient 2.5D Random Finite Element Method

High-speed train operation may cause vibration near track facilities and propagate far through the ground, affecting people’s lives, work, and normal use of precision instruments in an urban environment. An efficient numerical method is proposed to calculate the non-uniform ground vibration under a moving high-speed railway load. The theory of stochastic variables is used to describe the soil spatial variability of the non-uniform layered elastic ground, and the coupled 2.5D random finite element method (FEM) is proposed to reduce the computational cost without losing accuracy. Vibration propagation and attenuation of the non-uniform layered ground are investigated and the effect of train speed and soil non-homogeneity are analyzed. Results show that (1) at cross speed and high speed, the homogeneity coefficient of the layered ground has the most important influence on the ground vibration amplitude; (2) the upward acceleration is much larger than the downward acceleration at most speeds, and at cross speed and high speed, the acceleration amplitude decreases with the increase in the homogeneity coefficient; (3) as train speed increases from 60 m/s to 130 m/s, the influencing range of the homogeneity coefficient increases to 10 m from 2 m; and (4) the phenomenon of an in increase in local rebound can be observed in the velocity and acceleration attenuation curve at cross speed when the ground soil has a weaker homogeneity.

Reliability-based evaluation of ultimate embedment depths of torpedo anchors in spatially variable clay

In this study, a large deformation random finite element (LDRFE) method was used to investigate the dynamic penetration of anchor piles in spatially variable clay. Validation of the rationality of the proposed methodology using data from published studies. The randomness of the undrained shear strength of the soil in the system is considered by three-dimensional random fields to investigate the anchor penetration mechanism and quantitatively assess the ultimate embedment depths (Hran ) of the anchor in spatially variable clay. The strain softening and strain rate effects of the soft clayey soil were considered. The findings indicate that the variability in soil strength significantly affects the flow pattern and failure mechanism of the soil, consequently influencing the Hran of the anchor. Furthermore, according to the computed Hran results, a coefficient of variation of soil shear strength φ can be used to connect the deterministic (Hdet ) and random analysis results. This correlation may facilitate further development of reliability-based designs.

Dimension-Reduction Spectral Representation of Soil Spatial Variability and Its Application in the Efficient Reliability Analysis of Seismic Response in Tunnels

Reliability analysis of tunnels considering soil spatial variability was traditionally carried out using Monte Carlo simulations, which are computationally expensive. This study proposes a dimension-reduction spectral representation method (SRM) for simulating parameter stochastic fields to capture uncertainties in soil properties, which can be coupled with the probability density evolution method (PDEM). Additionally, the PDEM is employed to conduct seismic reliability assessments of tunnels in a more efficient manner. By integrating a non-intrusive dynamic random finite element method (FEM), this study explores the horizontal drift angle and vertical relative settlement of tunnel dynamic response under different coefficient of variations (COVs) and horizontal autocorrelation lengths of elastic modulus (E). Results show the changes in COVs have more pronounced effects on the mean and standard deviation of horizontal drift angle and vertical relative settlement of tunnel than alterations in horizontal autocorrelation lengths. A limit state pertaining to the horizontal drift angle and vertical relative settlement is established through the use of a probability density function (PDF) and cumulative distribution function (CDF). The results suggest that the combination of soil spatial variability and PDEM can serve as an effective approach for practical tunnel design.

Probabilistic analysis of a sustainable landfill cover considering stress-dependent water retention model and copula-based random fields

Material properties, crucial design parameters in landfill cover systems, can exhibit spatial variability due to non-uniform particle size and uneven compaction, affecting cover performance. Additionally, cover thickness, another key design parameter, can induce stress variation and influence soil-water retention capability. Previous designs always ignored these uncertainties and stress effects, potentially leading to unreasonable design schemes. This study aims to provide design recommendations for a three-layer landfill cover system under heavy rainfall, considering these factors simultaneously by the random finite element method (RFEM). The stress effects are captured using a stress-dependent soil-water retention curve (SDSWRC) and the hysteresis in the SDSWRC is considered in probabilistic analyses. For practical applications, uncertainties in two easily controllable particle size parameters, D10 and D60, are depicted by copula-based cross-correlated random fields. Based on empirical equations, the spatial variability of SDSWRC and water permeability function (WPF) can be further characterised by D10 and D60 random fields. It is found that considering SDSWRC and its spatial variability (e.g., shape) can overestimate percolation by at least 1.7 times compared to deterministic analysis using SWRC. The spatial variability of SDSWRC shape has a more significant impact than spatially variable saturated water permeability on cover system performance. Among candidate copulas, the CClayton copula (survival copula of Clayton copula) yields the most conservative results due to its upper tail dependence. The particle size and thickness of the bottom layer influence the cover system performance more than the upper two layers. A bottom layer with D10≤ 0.009 mm and D60/D10≥ 7.54, and thickness ≥ 0.8 m is recommended to achieve a satisfactory performance level with a 100-year design life.

Reliability analysis of offshore monopile foundations considering multidirectional loading and soil spatial variability

Monopiles are the primary type of foundations for offshore wind turbines. Current monopile design methods do not adequately account for the effects of multidirectional loading and soil spatial variability, which may underestimate the failure probability of the foundations. This study aims to investigate the reliability of offshore monopile foundations considering the effects of multidirectional loading and soil spatial variability. A non-intrusive random finite element method combined with Monte Carlo simulation is employed. The behaviors of monopiles in different spatially variable marine soils and loading paths are simulated. Through the reliability analysis, safety factors (FSdesign) are recommended for different scenarios. The results demonstrate that the combined effects of multidirectional loading and soil spatial variability significantly affect the monopile foundation performance and that ignoring these factors will underestimate the failure probability. If the effect of multidirectional loading is more prominent or the marine soil exhibits stronger spatial variability, a larger FSdesign is required to assure an acceptable failure probability.

Machine learning based prediction model for the pile bearing capacity of saline soils in cold regions

The difficulty in determining the bearing capacity of pile foundations in saline soil environments in cold regions can pose a challenge when developing a bearing capacity prediction model. To address this, the study uses data from the construction of the De Xiang Expressway project in Qinghai Province, China, and considers pile length, pile diameter, corrosion depth, and spalling thickness as influential parameters. We combine Python with ABAQUS to build a finite element model of a single pile in a salt marsh environment with random parameters and develop a database for machine learning. We use the kernel ridge regression (KRR) and multilayer perceptron (MLP) algorithms to establish load-bearing capacity prediction models for vertical and horizontal loads, respectively. The SHAP (SHapley Additive exPlanations) method is employed to perform explanatory analysis on the prediction models, providing insight into influential parameter effects on prediction accuracy. The results demonstrate that the database established through the random parameter finite element method is suitable for machine learning. While the KRR model exhibits good performance in predicting vertical load-bearing capacity, the MLP model performs well in predicting horizontal load-bearing capacity. Moreover, the SHAP method effectively explains the importance of influential parameters on the prediction results and enhances the reliability of the prediction models. The vertical bearing capacity of a saline soil pile foundation is more affected by the depth of salt erosion than the thickness of spalling, while the horizontal bearing capacity is more affected by the thickness of spalling than the depth of salt erosion.

Two‐phase flow behavior in CO2 geological storage considering spatial parameter heterogeneity

AbstractSaline aquifer rocks exhibit significant spatial randomness due to geological sedimentation processes. To address the issue of the heterogeneity of rock formations in numerical simulations, it is common practice to homogenize rock layers with similar lithologies. However, the acceptability of the errors generated during homogenized computations is a major concern and should be investigated. Therefore, to study the influence of heterogeneity at the storage site on the CO2 migration behavior, the Monte Carlo simulation–random finite element method (MCS‐RFEM) was combined with a CO2 two‐phase flow model to compare the effects of the coefficient of variation (Cv) and correlation length (λx) of random reservoir permeability fields on the migration distance and extent of CO2 storage under the same mean conditions. The results showed that higher Cv and λx values significantly reduced the CO2 migration distance while increasing the spread extent. Compared to the homogeneous model, at a λx value of 100 m, the CO2 migration distance decreased by 5.05%, while the profile sweep area increased by 6.20%. Concurrently, with increasing Cv, the area with a CO2 volume fraction higher than 0.75 decreased by 20.22%, while an increase in λx resulted in a 42.35% increase in the area with a CO2 volume fraction higher than 0.75. Therefore, reservoirs with high Cv and low λx values are more suitable for safely storing CO2. © 2023 Society of Chemical Industry and John Wiley & Sons, Ltd.

Hazard Curves for Reliability Assessment of Strip Footings on Spatially Varying Cohesive Soils

The present study aimed to create a series of hazard curves against maximum total settlement and angular rotation of strip footings for probabilistic shallow foundation design on clays. Random field finite element method (RFEM) was adopted with elasto-plastic clay-like soil behavior, deformation modulus (Ed) and shear strength parameters (c and ) were employed as random field inputs. Parameters were defined and assigned to the analysis models with varying correlation lengths (h, v). Models have been iteratively solved one thousand times, and output distributions of maximum settlement and angular rotations were recorded. Probability density functions (PDF) were fitted to the outputs, and probability of failure (Pf) for footing deformation limits was subsequently estimated. Proposed hazard curves for two anisotropy and three variability categories were developed employing the estimated Pfs. The method proposed has been validated using an independent database of in-situ results, and a worked example was provided to illustrate the implementation of the process. The key contribution of the research is to form hazard curves for shallow foundations considering elasto-plastic soil behavior with the impact of all influencing parameters, respecting the limit values for foundation deformation in the design codes. The proposed technique offers a probabilistic evaluation of strip footings with spatial variation of clayey soils and a valid method for the reliability-based design of foundations in the serviceability limit state.

Horizontal bearing capacity of monopile in three-dimensional spatially varying soils with linearly increasing mean strength

This paper aims to study the effects of the non-stationary soil property on the horizontal bearing capacity of three-dimensional monopile in spatially variable soils. The soil undrained shear strength is assumed to obey lognormal distribution and is simulated as non-stationary random fields. The mean value of the undrained shear strength linearly increases with depth, while the standard deviation keeps constant. The random finite-element method is applied to analyze the reliability of the bearing capacity. The influence of the correlations and non-stationary property on the mean and coefficient of variation of the bearing capacity are discussed. It is found that the correlation distance has no obvious effect on the bearing capacity and the bearing capacity increases with the increase of non-stationary coefficient. The results can guide the reliability-based design of horizontally loaded piles embedded in spatially variable soil.

Effect of resonance compaction method on the bearing capacity considering geotechnical spatial variability characteristics

Using the resonance compaction method (RCM) (Gong, Zhao, C. Hsein Juang, et al. 2020), energy is transferred to non-cohesive soil by the vibratory probe. Consequently, soil spatial changes occur and liquefaction is reduced. It should be noted, however, that the spatial variability of soil is determined by the deposition process. Therefore, it is worthwhile to investigate whether and what changes in the spatial variability of the soil will occur if the soil is treated with the RCM. And how the change in spatial variability affects the bearing capacity of the soil. The random field theory was used in this study to characterize the spatial variability of soils. A Bayesian framework was used for the solution of a random field model for the non-cohesive soils before and after RCM. This involves determining the random field correlation function and determining the model parameters. Furthermore, the random finite element method (RFEM) incorporating the random field theory (RFT) and the Monte Carlo simulation (MCS) were used to assess the effects of changes in the spatial variability of the soil on the bearing capacity before and after treatment. According to the results, the RCM could not only improve the soil's bearing capacity, but also reduce its spatial variability.

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.

Reliability Analyses of Soil Slopes with Multiple Spatially Varying Parameters Using Multi-Input Convolutional Neural Networks

Constructing meta-models and selecting a suitable deterministic analysis method are important to improve the computational efficiency and accuracy of the nonintrusive reliability analysis of a spatially varying soil slope. However, existing meta-models are not applicable to the slopes considering multiple parameters with high spatial variability. Moreover, it is difficult to identify the failure modes when the spatial variability is high by using deterministic analysis methods based on slip surface search. Therefore, a nonintrusive stochastic strength reduction finite-element method (SRFEM) is developed based on the multi-input convolution neural networks (CNNs) and ABAQUS 2016. The SRFEM developed based on ABAQUS is adopted as the deterministic analysis method to avoid the uncertain search for the critical slip surfaces of slopes with high spatial variability. A multi-input CNN is proposed to construct the meta-model to avoid the “curse of dimensionality” and replace the overmuch times of time-consuming finite-element simulations. It can fit the relationships between multiple spatially varying parameters and the factor of safety by processing different parameters with different streams of CNNs. Two illustrative examples show that the proposed method can accurately identify the failure modes of slopes with different degrees of spatial variability. The agreement of the reliability results based on the proposed method and the general random finite-element method (RFEM) shows the high accuracy of the proposed method. The time cost of the proposed method can be reduced to 6.0 × 10−3 times that of the general RFEM, verifying the high computational efficiency of the proposed method. The multi-input CNN also shows higher fitting accuracy and better interpretability than the single-stream CNN and the support vector machines (SVMs). The generalization ability, accuracy, and efficiency of the proposed method show its potential to carry out the reliability analyses of slopes with multiple spatially varying parameters.

Probabilistic Failure Envelopes of Monopiles in Scoured Seabed Based on a New Nonstationary Random Field Model

Monopiles, which have been most extensively used in the offshore wind industries, are usually subject to threats from local scour. The effects of local scour on the bearing capacity of monopiles have been studied based on deterministic soil properties and local scour dimensions. However, the properties of the seabed soil and the dimensions of the local scour both have uncertainty. In this study, a new nonstationary random field model, which can better simulate the nonstationary characteristics of the seabed clayey soil and result in a safer probabilistic design of foundations, is proposed to characterize the spatial variability of the seabed clayey soil. Then, the effects of the spatial variability of the seabed clayey soil and the uncertainty of the local scour depth on the bearing capacity of monopiles are investigated under both uniaxial loading and vertical-horizontal-moment (VHM) combined loading using the random finite-element method combined with kriging metamodeling technique and Monte Carlo simulation. The undrained shear strength of the seabed clayey soil was simulated with a lognormally distributed nonstationary random field, and the local scour depth was treated as a uniform random variable. The negative correlation between them was also considered. The results show that the factor of safety required for the local scoured seabed decreased from 3.3 to 2.6 as the negative correlation was enhanced, greater than or equal to that [factor of safety (FS)=2.6] for the intact seabed. The procedure of analyzing the design failure envelop of a monopile in spatially variable clayey soil with a local scour proposed in this study can provide a reference for geotechnical engineers.

Prediction of Undrained Bearing Capacity of Skirted Foundation in Spatially Variable Soils Based on Convolutional Neural Network

Skirted foundations are widely used in offshore and subsea engineering. Previous studies have shown that soil undrained shear strength variability has a notable impact on probabilistic analyses of skirted foundation bearing capacity. This study proposes an efficient machine-learning method to predict the uniaxial bearing capacity factors of skirted foundations under pure horizontal and moment loads, without relying on traditional time-consuming random finite element methods. A two-dimensional convolutional neural network is adopted to capture the potential correlation between soil random fields and bearing capacity factors. The proposed CNN-based model exhibits satisfactory prediction performance with regard to coefficients of variation and scale of fluctuations in two directions. Specifically, coefficient of determination (R2) values exceed 0.97, while root mean square error (RMSE) values remain below 0.13 for the surrogate model. In addition, more than 96% of the predictions are associated with a relative error of 5% or less, providing evidence of the proposed 2D-CNN model’s satisfactory prediction performance.