ASS-GPR: Adaptive Sequential Sampling Method Based on Gaussian Process Regression for Reliability Analysis of Complex Geotechnical Engineering

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

For obtaining correct reliability-based optimum design, an input model needs to be accurately estimated in identification of marginal and joint distribution types and quantification of their parameters. However, in most industrial applications, only limited data on input variables is available due to expensive experimental testing costs. The input model generated from the insufficient data might be inaccurate, which will lead to incorrect optimum design. In this paper, reliability-based design optimization (RBDO) with the confidence level is proposed to offset the inaccurate estimation of the input model due to limited data by using an upper bound of confidence interval of the standard deviation. Using the upper bound of the confidence interval of the standard deviation, the confidence level of the input model can be assessed to obtain the confidence level of the output performance, i.e. a desired probability of failure, through the simulation-based design. For RBDO, the estimated input model with the associated confidence level is integrated with the most probable point (MPP)-based dimension reduction method (DRM), which improves accuracy over the first order reliability method (FORM). A mathematical example and a fatigue problem are used to illustrate how the input model with confidence level yields a reliable optimum design by comparing it with the input model obtained using the estimated parameters.

A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis

An articulated tower is one of the compliant offshore structures connected to the sea bed through a universal joint. In the random sea environment, this joint is subjected to reversal of shear stresses, which makes it susceptible to fatigue damage. In this study, fatigue damage accumulation in articulated joint under random loading is studied. The dynamic analysis of the tower has been carried out for twelve simulated sea states under random wave alone and random wave with wind. Nonlinearities due variable submergence, instantaneous position of the tower and hydrodynamic loading have been taken into account in the derivation of equation of motion. Fatigue life of the joint has been determined by S-N curve and fracture mechanics (F-M) approaches. Advanced First Order Reliability Method (FORM) and Monte Carlo Simulation method have been used for the reliability estimation. The results of the analyses are presented in terms of probability of failure and reliability indices. Sensitivity analysis is carried out to study the effect and participation of various random variables on the joint reliability. Most probable point (MPP) or design points have been located on the failure surface. Important parametric studies have been carried out which yield important information for reliability based design. The results of the study indicate that Miner’s rule, which is generally used in the design against fatigue in steel structures, yields a conservative estimate of probability of failure as compared to the fracture mechanics approach.

Reducing the surrogate model-based method computation without loss of prediction accuracy remains a significant challenge in structural reliability analysis. The unbalanced probability density, important information in critical region and information redundancy of added sample points are ignored in most of traditional surrogate-based methods, resulting in heavy computational burden. In this work, an active learning combining adaptive Kriging method and weighted penalty (AK-WP) is proposed to analyze the reliability of engineering structures. Firstly, an active learning and weighted penalty function (WPLF) is the result of integrating active learning method, weighted function and penalty function, which is proposed to find the most probable point (MPP). Meanwhile, to avoid redundant information, the best suitable MPP is determined by a proposed distance law established between the found MPP and the existing design of experiment (DoE). Secondly, the Kriging model is refined according to best suitable MPP in each iteration. Thirdly, the failure probability is estimated by the Monte Carlo sample points from the n-ball domain until the convergence condition is satisfied. The accuracy and efficiency of the proposed method are demonstrated by some numerical examples including the highly nonlinear, the small probability problems and implicit function, as well as a real engineering application.

Neural Networks Combined with Importance Sampling Techniques for Reliability Evaluation of Explosive Initiating Device

This paper proposes a novel second-order reliability method (SORM) using non-central or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point (MPP) in the standard normal U-space, (2) error due to approximating the quadratic function in U-space by a hyperbolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two errors by describing the quadratic failure surface with the linear combination of non-central chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function (PDF) over the linear failure surface and the second approach uses the cumulative distribution function (CDF) of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method (FORM), conventional SORM, and Monte Carlo simulation (MCS) results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.

An adaptive sequential sampling method based on ANN_MCD and RF: Application in geotechnical problems

In this paper, an efficient method for the reliability analysis of systems with nonlinear limit states is described. It combines optimization-based and simulation-based approaches and is particularly applicable for problems with highly nonlinear and implicit limit state functions, which are difficult to solve by conventional reliability methods. The proposed method consists of two major parts. In the first part, an optimization-based method is used to search for the most probable point (MPP) on the limit state. This is achieved by using adaptive response surface approximations. In the second part, a multi-modal adaptive importance sampling method is proposed using the MPP information from the first part as the starting point. The proposed method is applied to the reliability estimation of a vehicle body-door subsystem with respect to one of the important quality issues -- the door closing effort. The superiority of the proposed method, in terms of efficiency and accuracy, is demonstrated with a numerical example of highly nonlinear limit state problem, as well as an automotive door closing effort application. A generalized framework for reliability estimation is also proposed for problems with large numbers of random variables and complicated limit states.

T. Wichtmann, T. Triantafyllidis: Behaviour of granular soils under environmentally induced cyclic loads. - D. Muir Wood: Constitutive modelling. - C. di Prisco: Creep versus transient loading effects in geotechnical problems. - M. Pastor et al.: Mathematical models for transient, dynamic and cyclic problems in geotechnical engineering. - M. Pastor: Discretization techniques for transient, dynamics and cyclic problems in geotechnical engineering: first order hyperbolic partial diffential equations. - M. Pastor et l.: Discretization techniques for transient, dynamic and cyclic problems in geotechnical engineering: second order equation. - C. di Prisco: Cyclic mechanical response of rigid bodies interacting with sand strata. - D. Muir Wood: Macroelement modelling. - M. F. Randolph: Offshore design approaches and model tests for sub-failure cyclic loading of foundations. - M.F. Randolph: Cyclic interface shearing in sand and cemented solis and application to axial response of piles. - M. F. Randolph: Evaluation of the remoulded shear strength of offshore clays and application to pipline-soil and riser-soil interaction.

Introduction:Reliability analysis is a good tool to deal with the uncertainty and has been widely used in the engineering system. The first order reliability method (FORM) is generally used to calculate the reliability index but FORM is time-consuming and requires derivative computing.Methods:Artificial Bee Colony (ABC) algorithm is a very simple, robust and population-based stochastic optimization algorithm. In this study, an ABC-based reliability analysis was proposed to calculate the reliability index of engineering system through combining Artificial Bee Colony (ABC) algorithm with FORM. FORM was adopted to calculate the reliability index and design point. ABC is used to solve the constrained optimization about FORM. The procedure of ABC-based reliability analysis was presented in detail.Results and Conclusion:The proposed method was verified by two classic examples and then applied to geotechnical engineering. The results show that the ABC algorithm can effectively solve the global optimization problem in FORM. Results demonstrate that ABC-based reliability analysis is a good approach to obtain the reliability index and design point with a good accuracy so that it can be applied to analyze the reliability of a complex engineering system.

In this study, an effective method for reliability-based design optimization (RBDO) is proposed enhancing sequential optimization and reliability assessment (SORA) method by convex approximations. In SORA, reliability estimation and deterministic optimization are performed sequentially. The sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the reliability analysis loop. In this study, the convex approximations for probabilistic constraint are constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. Hence, the proposed method requires much less function evaluations of probabilistic constraints in the deterministic optimization than the original SORA method. The efficiency and accuracy of the proposed method were verified through numerical examples.

This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.

Profust reliability analysis is investigated in this article. The main aim is to address problems with very small failure probabilities. First, a single-layer multimodal importance sampling (MMIS) method based on the most probable points (MPPs) in the fuzzy failure region is proposed. Secondly, to reduce function calls, an adaptive method fusing kriging model and MMIS is developed. In this method, a surrogate fuzzy failure region (SFFR) based on the kriging prediction information is defined. The MPPs in the SFFR are explored, the important samples are generated, the optimal training point is chosen, and the kriging model and the SFFR are updated. The learning process is executed iteratively until the SFFR converges to the true one. To find all MPPs and maintain unbiasedness, an intelligent multimodal optimization method based on multi-objective optimization is introduced. Four case studies are used to verify the advantages of the proposed method.

Traditional reliability-based design optimization(RBDO)is either computational intensive or not accurate enough.In this work,a new RBDO method based on Support Vector Machines(SVM)is proposed.For reliability analysis,SVM is used to create a surrogate model of the limit-state function at the Most Probable Point(MPP).The uniqueness of the new method is the use of the gradient of the limit-state function at the MPP.This guarantees that the surrogate model not only passes through the MPP but also is tangent to the limit-state function at the MPP.Then Importance Sampling(IS)is used to calculate the probability of failure based on the surrogate model.This treatment significantly improves the accuracy of reliability analysis.For optimization,the Sequential Optimization and Reliability Assessment(SORA)is employed,which decouples deterministic optimization from the SVM reliability analysis.The decoupling makes RBDO more efficient.The two examples show that the new method is more accurate with a moderately increased computational cost.