Research PaperAn efficient quasi-Newton approximation-based SORM to estimate the reliability of geotechnical problems
Introduction
Reliability analyses have been proposed as a rational complement to deterministic geotechnical design [1], as they can more directly quantify the influence of the uncertainty about input parameters and their correlation relationships (see e.g., [2], [3]).
Due to its simplicity and efficiency, the first order reliability method (FORM)—that linearly approximates the limit state function (LSF) to estimate the probability of failure—has been widely used in geotechnical reliability analyses (see e.g., [4], [5], [6], [7]). However, the linearization that is inherent to FORM introduces errors in many cases (see e.g., [7], [8]), and the second order reliability method (SORM)—which extends FORM to consider the curvatures of the LSF, hence providing a better approximation—has also been employed as an alternative.
Brza̧kała and Puła [9] and Bauer and Puła [10] analyzed the probability of foundation settlements exceeding an allowable threshold using SORM and a polynomial response surface method (RSM)-based SORM, respectively; Cho [11] combined an artificial neural network (ANN)-based RSM and SORM to compute the reliability of slopes; Lü and co-authors [2], [12], [13] employed various RSMs with SORM to analyze tunnel supports; Chan and Low [14] introduced a practical SORM for foundation reliability analysis using a point-fitted paraboloid method; and Zeng and co-authors [8], [15] applied SORM to evaluate the system reliability of tunnels and slopes, respectively. However, these methods are often more computationally expensive than FORM, due to the need to evaluate the curvatures of the LSF or to construct the response surface function. Therefore, an approach that aims to combine the higher accuracy of SORM-based reliability solutions with the lower computational cost of traditional FORM-based approaches is considered as a useful contribution to the geotechnical field.
This paper proposes an attempt in that direction. In particular, our proposed approach uses the recently proposed Hasofer–Lind–Rackwitz–Fiessler–Broyden–Fletcher–Goldfarb–Shanno (HLRF–BGFS) algorithm [16] to locate the FORM design point efficiently; and it integrates such algorithm with the Symmetric Rank-one (SR1) algorithm [17] to approximate the Hessian matrix (i.e., the second order derivative matrix). The goal is that the identified design point can be used, together with the approximated Hessian matrix, to efficiently estimate the second order probability of failure. Details of the algorithms are discussed first, and then the performance of the proposed method is tested using three typical geotechnical example cases taken from the literature.
Section snippets
Conventional SORM
Conventional SORM estimates the second order probability of failure using (i) the design point and gradient information computed by FORM and (ii) the Hessian matrix computed using finite differences. For completeness, traditional methods to compute FORM solutions and the Hessian matrix, as well as to compute the SORM reliability based on them, are discussed below. Additional details can be found in traditional reliability references such as [18] and [19].
A proposed quasi-Newton approximation-based SORM
In our approach, two types of quasi-Newton approximation—the BFGS and SR1 algorithms—are combined for a more efficient estimation of the second order probability of failure. (Quasi-Newton methods are alternatives to “full” Newton methods, which approximate the Hessian matrices needed at every iteration of gradient-based optimization approaches [38].) In particular, the BFGS algorithm is used, together with the original HLRF algorithm proposed by Hasofer and Lind [22] and Rackwitz and Flessler
Case studies
The reliability of three typical geotechnical problems—settlement of a rectangular foundation, bearing capacity of a shallow footing, and stability of a layered soil slope—are considered in this study as benchmark tests of the proposed method, in which the iHLRF algorithm, the HLRF–BFGS algorithm, and conventional SORM are employed for comparison with our proposed method. The same convergence criteria given in Eq. (15), with , are used for these four methods to make results comparable.
Summary and conclusions
A new method has been proposed in this study for an efficient estimation of the second order probability of failure of geotechnical problems. To reduce the computational cost, the method builds on both the HLRF–BGFS and SR1 algorithms; in particular, instead of using the Hessian matrix computed with a forward difference scheme that is traditionally employed in SORM, it uses good approximations to the Hessian matrix provided by the SR1 algorithm, and it incorporates them to the HLRF–BGFS
Conflict of interest
This work does not have any conflict of interest that needs to be reported.
Acknowledgments
This research was partially supported by the Spanish Ministry of Economy and Competitiveness under Grant BIA2015-69152R and the State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Grant Nos. SKLGP2011Z002 and SKLGP2015Z016). The first author was supported by China Scholarship Council (CSC) and, for insurance coverage, by Fundación José Entrecanales Ibarra. Dr. Solange Regina and Dr. Gislaine Periçaro kindly provided their MATLAB codes for the HLRF–BGFS algorithm. Their
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Department of Ground Engineering, Technical University of Madrid, Madrid, Spain