Inverse First-order Reliability Method Research Articles

Probabilistic Design Methods for Gust-Based Loads on Wind Turbines

The IEC 61400-1 standard specifies design load cases (DLCs) to be considered in the design of wind turbine structures. Specifically, DLC 2.3 considers the occurrence of a gust while the turbine shuts down due to an electrical fault. Originally, this load case used a deterministic wind event called the extreme operating gust (EOG), but the standard now also includes an approach for calculating the extreme response based on stochastic simulations with turbulent wind. This study presents and compares existing approaches with novel probabilistic design approaches for DLC 2.3 based on simulations with turbulent wind. First, a semiprobabilistic approach is proposed, where the inverse first-order reliability method (iFORM) is used for the extrapolation of the response for electrical faults occurring at a given rate. Next, three probabilistic approaches are formulated for the calculation of the reliability index, which differs in how the aggregation is performed over wind conditions and whether faults are modeled using a Poisson distribution or just by the rate. An example illustrates the methods considering the tower fore-aft bending moment at the tower base and shows that the approach based on iFORM can lead to reductions in material usage compared to the existing methods. For reliability assessment, the probabilistic approach using the Poisson process is needed for high failure rates, and the reliabilities obtained for designs using all semiprobabilistic methods are above the target level, indicating that further reductions may be obtained via the use of probabilistic design methods.

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.

Multivariate probability analysis of wind-wave actions on offshore wind turbine via copula-based analysis

Floating structures in service encounter extreme natural hazards, such as correlated extreme wind-wave loads. This study aims to provide realistic multivariate models of environmental parameters to accurately describe the statistical characteristics of metoceanic conditions, which are essential for the reliability-based design of marine structures. The multiload analysis method includes three major steps: (1) to identify the best-fit univariate marginal distributions of the wind-wave parameters. Mixed models composed of a set of single-mode parametric distributions were found to be more suitable for the measured variables. (2) To construct joint probability distributions of multiple variables using copula-based models, the performance of various models namely, trivariate metaelliptical copulas, Archimedean copulas, and vine copula models. It was observed that joint distributions can be best modelled using vine copula models with considerable flexibility. The final process involves establishing environmental surfaces and contours based on the inverse first-order reliability method. The most unfavourable combination of extreme design conditions can be defined on the environmental surface which is useful for evaluating the response characteristics of offshore wind turbines. These expressions provide a preliminary assessment of wind-wave loads on offshore structures.

Inverse reliability analysis and design for tunnel face stability considering soil spatial variability

The traditional deterministic analysis for tunnel face stability neglects the uncertainties of geotechnical parameters, while the simplified reliability analysis which models the potential uncertainties by means of random variables usually fails to account for soil spatial variability. To overcome these limitations, this study proposes an efficient framework for conducting reliability analysis and reliability-based design (RBD) of tunnel face stability in spatially variable soil strata. The three-dimensional (3D) rotational failure mechanism of the tunnel face is extended to account for the soil spatial variability, and a probabilistic framework is established by coupling the extended mechanism with the improved Hasofer-Lind-Rackwits-Fiessler recursive algorithm (iHLRF) as well as its inverse analysis formulation. The proposed framework allows for rapid and precise reliability analysis and RBD of tunnel face stability. To demonstrate the feasibility and efficacy of the proposed framework, an illustrative case of tunnelling in frictional soils is presented, where the soil's cohesion and friction angle are modelled as two anisotropic cross-correlated lognormal random fields. The results show that the proposed method can accurately estimate the failure probability (or reliability index) regarding the tunnel face stability and can efficiently determine the required supporting pressure for a target reliability index with soil spatial variability being taken into account. Furthermore, this study reveals the impact of various factors on the support pressure, including coefficient of variation, cross-correlation between cohesion and friction angle, as well as autocorrelation distance of spatially variable soil strata. The results also demonstrate the feasibility of using the forward and/or inverse first-order reliability method (FORM) in high-dimensional stochastic problems. It is hoped that this study may provide a practical and reliable framework for determining the stability of tunnels in complex soil strata.

Artificial Neural Network-Based Prediction of the Extreme Response of Floating Offshore Wind Turbines under Operating Conditions

The development of floating offshore wind turbines (FOWTs) is gradually moving into deeper offshore areas with more harsh environmental loads, and the corresponding structure response should be paid attention to. Safety assessments need to be conducted based on the evaluation of the long-term extreme response under operating conditions. However, the full long-term analysis method (FLTA) recommended by the design code for evaluating extreme response statistics requires significant computational costs. In the present study, a power response prediction method for FOWT based on an artificial neural network algorithm is proposed. FOWT size, structure, and training algorithms from various artificial neural network models to determine optimal network parameters are investigated. A publicly available, high-quality operational dataset is used and processed by the Inverse First Order Reliability Method (IFORM), which significantly reduces simulation time by selecting operating conditions and directly yielding extreme response statistics. Then sensitivity analysis is done regarding the number of neurons and validation check values. Finally, the alternative dataset is used to validate the model. Results show that the proposed neural network model is able to accurately predict the extreme response statistics of FOWT under realistic in situ operating conditions. A proper balance was achieved between prediction accuracy, computational costs, and the robustness of the model.

Efficient adaptive reliability-based design optimization for geotechnical structures with multiple design parameters

Reliability-based design optimization (RBDO) aims to improve the reliability of geotechnical systems while minimizing the cost of design. However, the conventional RBDO method suffers from a large computational cost and intractable convergence problem due to the double-loop structure involving numerical optimization and repeated reliability analysis. To address this issue, this paper proposes an efficient adaptive RBDO framework for geotechnical structure with multiple design parameters. The proposed framework mainly includes two parts: a novel expanded inverse first-order reliability method (E-IFORM) for dealing with the inverse reliability problem with multiple design parameters, and an adaptive surrogate model to decouple the double-loop structure and improve the efficiency. The proposed methodology is applied to four geotechnical systems (i.e., a soil slope, an excavation pit, a strip footing and a retaining wall), and the results demonstrate the effectiveness and efficiency of the proposed framework.

Robust reliability‐based design approach by inverse FORM with adaptive conjugate search algorithm

AbstractThe concept of reliability‐based design (RBD) of geotechnical works has seen great success in recent years, nevertheless, its application in real‐world engineering practice is still limited. This phenomenon could be mainly attributed to two aspects: the engineer's unfamiliarity of reliability algorithms even in their simplest forms, and the lack of direct connection between reliability algorithms and deterministic design tools. To promote the geotechnical RBD application, this study proposes an inverse first‐order reliability method (FORM) integrating adaptive conjugate direction search technique (RBD via conjugate FORM) to solve classical RBD problems where the number of (multiple) limit state functions is usually equal to the number of unknown design parameters. Notably, the RBD via conjugate FORM can readily back‐calculate those unknown design parameters subject to prescribed reliability index or probability of failure, by means of iterative evaluation of the parametric sensitivities. The proposed algorithm is first verified by several well‐documented examples. Comparative studies show that when the reliability problem to solve is highly nonlinear, the RBD results based on steepest descent search direction may fall in oscillation and non‐convergence, while the proposed RBD via conjugate FORM algorithm performs robustly and efficiently. It is then illustrated in combination with standalone numerical codes to conduct risk‐based optimization of unknown design parameters for two soil slope stability models. The first slope RBD model achieves designing the slope angle under specified reliability index, and the second slope RBD model is to optimize multiple design parameters of the stabilization pile, which can meet different slope reliability requirements under static and seismic loading conditions simultaneously.

Integrated Approach for Estimating Extreme Hydrodynamic Loads on Elevated Pile Cap Foundation Using Environmental Contour of Simulated Typhoon Wave, Current, and Surge Conditions

Abstract Typhoon is a disastrous weather system, which usually induces strong waves, currents, and surges along the coastal area, and causes severe hydrodynamic loads on the elevated pile cap foundation, which is widely used to support the sea-crossing bridge. Estimating the hydrodynamic loads under typhoons is essential to ensure the bridge's safety. This paper develops an environmental contour-based framework that can estimate the extreme hydrodynamic loads induced by typhoons while considering the correlation among environmental conditions. The elevated pile cap foundation of the Xihoumen Rail-cum-road Bridge was used to illustrate the framework. The SWAN + ADCIRC model was employed to simulate the environmental conditions under typhoons. The pair-copulas were adopted to construct joint probability distributions, and the environmental contours with a given return period were then established by the inverse first-order reliability method. Given the hydrodynamic model and short-term peak value of the structural response, the AK-LHS method was then used to find the maximum hydrodynamic loads based on the environmental contours. The environmental contour constructing methods and selection methods of short-term peak values were compared and discussed. The main findings include: (1) ignoring correlations of the environmental conditions overestimates the extreme hydrodynamic loads and results in a conservative design; (2) the estimation of extreme hydrodynamic loads is affected by the selection and fitting of short-term peak values significantly; and (3) the extreme hydrodynamic loads estimated by either Rosenblatt transformation or Nataf transformation shows similar results.

Combined wind profile characteristics based on wind parameters joint probability model in a mountainous gorge

Long-span bridges in mountainous areas are greatly disturbed by wind, and the wind field at the mountain gorge bridge site is extremely complex. Therefore, it is of great engineering significance to accurately evaluate the wind field characteristics of this kind of terrain. In this paper, to enhance understanding of this kind of wind field, the wind field in a mountainous gorge is measured for a long time using wind radar, and the mean wind parameters are statistically analyzed. The results show that the mean wind parameters vary greatly under different wind directions, and the wind speed profile does not meet the power-law model. Therefore, a mixed model suitable for the wind speed profile in a mountainous gorge is proposed. Additionally, GEV distribution and Logistic distribution are found to be suitable for describing the distribution characteristics of wind speed and angle of attack, respectively. In addition, considering the correlation between wind parameters, this paper also constructs the joint probability model of wind speed and angle of attack at different heights by the Copula function. Thus, a combined wind parameters profile model is developed under different exceedance probabilities based on the inverse first-order reliability method (IFORM). This study can provide a reference for the construction of the joint probability model of wind parameters.

Estimating Design Loads for Floating Structures Using Environmental Contours

Nonlinear time-domain simulations are often used to predict the structural response at the design stage to ensure the acceptable operation and/or survival of floating structures under extreme conditions. An environmental contour (EC) is commonly employed to identify critical sea states that serve as the input for numerical simulations to assess the safety and performance of marine structures. In many studies, marginal and conditional distributions are defined to construct bivariate joint probability distributions for variables, such as significant wave height and zero-crossing period. Then, ECs can be constructed using the inverse first-order reliability method (IFORM). This study adopts alternative models to describe the generalized dependence structure between environmental variables using copulas and discusses the Nataf transformation as a special case. ECs are constructed using measured wave data from moored buoys. Derived design loads are applied on a semisubmersible platform to assess possible differences. In addition, a linear interpolation scheme is utilized to establish a parametric model using short-term extreme tension distribution parameters and wave data, and the long-term tension response is estimated using Monte Carlo simulation. A 3D IFORM-based approach, in which the short-term extreme response that is ignored in the EC approach is used as the third variable, is proposed to help establish accurate design loads with increased accuracy. Results offer a clear illustration of the extreme responses of floating structures based on different models.

Pair-Copula-based trivariate joint probability model of wind speed, wind direction and angle of attack

The temporal and spatial distribution of wind fields in the deep-cut gorge bridge site is complex, and the impact of different incoming flows on the structure is also essentially different. Therefore, it is essential to comprehensively consider the correlation between wind direction, wind speed, and angle of attack. The joint probability models between average wind parameters are first established using the Pair-Copula decomposition based on the field-measured data. Then, the concept of the probability-segmentation-based first order inverse reliability method (PSB-IFORM) is proposed for calculating the environmental surface of the multi-peak joint distribution model. The results show that the mixed von Mises distribution is ideal to fit the wind direction with multi-peak characteristics. The trivariate joint probability model constructed in this paper ultimately represents the correlation of average wind parameters, and the most unfavorable combination of design wind parameters can be found on the environmental surface. Furthermore, the proposed PSB-IFORM can effectively avoid the defects of the inverse first-order reliability method (IFORM) and the conservative highest probability contour (HDC) method in solving the multi-peak joint probability distribution model. This study is of particular interest to researchers and engineers engaged in wind resistance of mountain structures.

Bivariate kernel density estimation for environmental contours at two offshore sites

ABSTRACT This paper proposes a novel environmental contour line approach based on measured ocean wave data at two offshore sites. For implementing the novel environmental contour line approach, we propose the use of bivariate kernel density estimation with rigorous bandwidth selection based on the Scott-Tapia-Thompson estimation method. The environmental contours obtained by using the proposed novel approach have been compared with those obtained by using the traditional Inverse-First-Order Reliability Method (IFORM) with Rosenblatt transformation or principal component analysis, and the effectiveness of our proposed novel approach have been clearly substantiated. The research results in this paper demonstrate that our proposed novel approach can be utilised as an effective tool for generating environmental contours if one has sufficient data extending beyond the return period of interest.

Uncertainty quantification of quadratically nonlinear local damage size quantification using nonlinear ultrasonics

Backscattered and forward scattered harmonic waves from local damage modeled as quadratically nonlinear material in a 1D bar are used to solve a deterministic inverse problem to find the local damage size by Ghodake (2020). Due to highly nonlinear (|sin(x)/x|) nature of the proposed forward problem, the constrained nonlinear deterministic optimization problem was solved using non-gradient-based algorithms. The forward model is a function of input frequencies, linear wave velocity, and local damage size. In practice, slight uncertainty of these parameters gives a completely different solution. To address this problem, uncertainty quantification of the modified forward problem is carried out along with solving the inverse problem using reliability-based design optimization (RBDO) and Bayesian Inverse (BI) approaches. In RBDO studies, Quantile Monte Carlo (QMC) approach gives more reliable and accurate results in comparison with approaches such as reliability index approach (RIA), performance measure approach (PMA), sequential optimization and reliability assessment (SORA), and inverse first-order reliability method (FORM) integrated with optimization methods such as interior-point (IP) and sequential programming (SQP). Multiple models are solved using the BI approach. Support vector machines regression (SVR), polynomial chaos expansions (PCE), and Kriging surrogate models are used along with RBDO and BI to reduce computational time.

A new IFORM-Rosenblatt framework for calculation of environmental contours

Environmental contours are useful for designing and assessing risk to offshore structures. The Inverse First-Order Reliability Method (IFORM) and the Rosenblatt transformation are widely used to calculate environmental contours but have several challenges. In this approach, the conditional distribution is often expressed in a parametric form with parameters defined as functions of the independent variable. The distribution should be accurate for both extreme and frequently occurring values, for both the dependent and independent variables, and this is a challenging requirement for complex datasets. A new IFORM-Rosenblatt framework is proposed here to overcome this challenge. The framework includes a method to approximate a distribution with a complex form, including parametric, empirical, and mixed parametric/empirical forms, using a reduced set of parameters that are designed to prioritize accuracy for the quantiles of the distribution that contribute to the contour and a set of metrics to evaluate the goodness-of-fit of the resulting contour. A hindcast dataset is used as an example to calculate the 50-year contour using the proposed framework and a traditional approach. The proposed framework is demonstrated to generate accurate contours, although it requires more parameters and is more complex than conventional approaches.

An efficient probabilistic design approach for tunnel face stability by inverse reliability analysis

In order to maintain the safety of underground constructions that significantly involve geo-material uncertainties, this paper delivers a new computation framework for conducting reliability-based design (RBD) of shallow tunnel face stability, utilizing a simplified inverse first-order reliability method (FORM). The limit state functions defining tunnel face stability are established for both collapse and blow-out modes of the tunnel face failure, respectively, and the deterministic results of the tunnel face support pressure are obtained through three-dimensional finite element limit analysis (FELA). Because the inverse reliability method can directly capture the design support pressure according to prescribed target reliability index, the computational cost for probabilistic design of tunnel face stability is greatly reduced. By comparison with Monte Carlo simulation results, the accuracy and feasibility of the proposed method are verified. Further, this study presents a series of reliability-based design charts for vividly understanding the limit support pressure on tunnel face in both cohesionless (sandy) soil and cohesive soil stratums, and their optimal support pressure ranges are highlighted. The results show that in the case of sandy soil stratum, the blowout failure of tunnel face is extremely unlikely, whereas the collapse is the only possible failure mode. The parametric study of various geotechnical uncertainties also reveals that ignoring the potential correlation between soil shear strength parameters will lead to over-designed support pressure, and the coefficient of variation of internal friction angle has a greater influence on the tunnel face failure probability than that of the cohesion.

Environmental wave contours by inverse FORM and Monte Carlo Simulation with variance reduction techniques

Environmental contours of significant wave height and wave period are established for 10, 25 and 100 year-return periods by different approaches. The environmental contours are first derived by Direct sampling and by the Inverse First Order Reliability method (IFORM) using the Rosenblatt transformation. Then, the importance sampling technique is applied at different locations along the environmental contours to reduce the variance of the estimates and, consequently, to increase the accuracy and efficiency of the prediction method. The different environmental contours are derived from a large collection of hindcast wave data from offshore of Canary Islands, which is used as a case study to compare the different approaches.

Development of a Comparison Framework for Evaluating Environmental Contours of Extreme Sea States

Environmental contours of extreme sea states are often utilized for the purposes of reliability-based offshore design. Many methods have been proposed to estimate environmental contours of extreme sea states, including, but not limited to, the traditional inverse first-order reliability method (I-FORM) and subsequent modifications, copula methods, and Monte Carlo methods. These methods differ in terms of both the methodology selected for defining the joint distribution of sea state parameters and in the method used to construct the environmental contour from the joint distribution. It is often difficult to compare the results of proposed methods to determine which method should be used for a particular application or geographical region. The comparison of the predictions from various contour methods at a single site and across many sites is important to making environmental contours of extreme sea states useful in practice. The goal of this paper is to develop a comparison framework for evaluating methods for developing environmental contours of extreme sea states. This paper develops generalized metrics for comparing the performance of contour methods to one another across a collection of study sites, and applies these metrics and methods to develop conclusions about trends in the wave resource across geographic locations, as demonstrated for a pilot dataset. These proposed metrics and methods are intended to judge the environmental contours themselves relative to other contour methods, and are thus agnostic to a specific device, structure, or field of application. The metrics developed and applied in this paper include measures of predictive accuracy, physical validity, and aggregated temporal performance that can be used to both assess contour methods and provide recommendations for the use of certain methods in various geographical regions. The application and aggregation of the metrics proposed in this paper outline a comparison framework for environmental contour methods that can be applied to support design analysis workflows for offshore structures. This comparison framework could be extended in future work to include additional metrics of interest, potentially including those to address issues pertinent to a specific application area or analysis discipline, such as metrics related to structural response across contour methods or additional physics-based metrics based on wave dynamics.

Trivariate joint probability model of typhoon-induced wind, wave and their time lag based on the numerical simulation of historical typhoons

Typhoon-induced wind and wave can interrupt the operation and even threaten the safety of moving vehicles and bridges. However, the typhoon-induced maximum wind speed and maximum wave height are not coincident in time. Hence, it is a critical issue to include the time lag in the hazard model of wind and wave conditions for bridges. This paper adopts the concept of the pair-copula decomposed model to develop a trivariate joint probability model of typhoon-induced maximum wind speed, maximum wave height and their time lag. Pingtan Strait, where a sea-crossing bridge is being built, is taken as the example site. Considering the long-term measured wind and wave conditions under typhoons are not available, 58 tropical cyclones from 1990 to 2018 that influenced the example site are selected. The typhoon-induced maximum wind speed, wave height and their time lag at the example site are simulated using the validated SWAN + ADCIRC coupled numerical model. The trivariate joint probability modeling of wind, wave and time lag was carried out based on the simulated data. The trivariate environmental surfaces with the 50-year and 100-year return periods were finally obtained by the inverse first-order reliability method. The results show that more than 50% of typhoons have the maximum wind lagged behind the maximum wave at the example site. The higher typhoon-induced maximum wind speed and wave height tend to occur simultaneously. Two-parameter Weibull distribution is suitable to fit the distribution of the maximum wind speed and wave height, and the GEV distribution is ideal for the distribution of time lag. According to the trivariate environmental surface, neglecting the time lag might slightly overestimate the demand of the wind and wave loads. This study is of particular interest to the researchers and engineers in developing the metocean conditions.

Marginal and total exceedance probabilities of environmental contours

Various methods have been proposed for defining an environmental contour, based on different concepts of exceedance probability. In the inverse first-order reliability method (IFORM) and the direct sampling (DS) method, contours are defined in terms of exceedances within a region bounded by a hyperplane in either standard normal space or the original parameter space, corresponding to marginal exceedance probabilities under rotations of the coordinate system. In contrast, the more recent inverse second-order reliability method (ISORM) and highest density (HD) contours are defined in terms of an isodensity contour of the joint density function in either standard normal space or the original parameter space, where an exceedance is defined to be anywhere outside the contour. Contours defined in terms of the total probability outside the contour are significantly more conservative than contours defined in terms of marginal exceedance probabilities. In this work we study the relationship between the marginal exceedance probability of the maximum value of each variable along an environmental contour and the total probability outside the contour. The marginal exceedance probability of the contour maximum can be orders of magnitude lower than the total exceedance probability of the contour, with the differences increasing with the number of variables. For example, a 50-year ISORM contour for two variables at 3-h time steps, passes through points with marginal return periods of 635 years, and the marginal return periods increase to 10,950 years for contours of four variables. It is shown that the ratios of marginal to total exceedance probabilities for DS contours are similar to those for IFORM contours. However, the marginal exceedance probabilities of the maximum values of each variable along an HD contour are not in fixed relation to the contour exceedance probability, but depend on the shape of the joint density function. Examples are presented to illustrate the impact of the choice of contour on simple structural reliability problems for cases where the use of contours defined in terms of either marginal or total exceedance probabilities may be appropriate. The examples highlight that to choose an appropriate contour method, some understanding about the shape of a structure's failure surface is required.

A robust methodology for predicting extreme structural responses of offshore wind turbines

ABSTRACT This paper proposes a robust methodology for predicting extreme structural responses of offshore wind turbines. This robust methodology starts with forecasting an improved IFORM (Inverse-First-Order Reliability Method) environmental contour line through the use of principal component analysis. It then performs stochastic time domain simulations of an offshore wind turbine based on the information of the highest sea state identified along the aforementioned environmental contour line. Next, an enhanced Peaks-Over-Threshold (POT) method is utilised for fitting an extreme value distribution based on which a long-term extreme structural response can finally be obtained. A specific example of predicting the 50-year extreme structural responses of a monopile-supported 5MW offshore wind turbine will be provided in this paper for demonstrating the robustness of our proposed methodology.