Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4317670, 15 pages

https://doi.org/10.1155/2017/4317670

## HLRF-BFGS-Based Algorithm for Inverse Reliability Analysis

School of Computing Engineering and Mathematics, Western Sydney University, Penrith, NSW 2747, Australia

Correspondence should be addressed to Won-Hee Kang

Received 10 March 2017; Accepted 7 June 2017; Published 17 July 2017

Academic Editor: Roman Lewandowski

Copyright © 2017 Rakul Bharatwaj Ramesh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study proposes an algorithm to solve inverse reliability problems with a single unknown parameter. The proposed algorithm is based on an existing algorithm, the inverse first-order reliability method (inverse-FORM), which uses the Hasofer Lind Rackwitz Fiessler (HLRF) algorithm. The initial algorithm analyzed in this study was developed by modifying the HLRF algorithm in inverse-FORM using the Broyden-Fletcher-Goldarb-Shanno (BFGS) update formula completely. Based on numerical experiments, this modification was found to be more efficient than inverse-FORM when applied to most of the limit state functions considered in this study, as it requires comparatively a smaller number of iterations to arrive at the solution. However, to achieve this higher computational efficiency, this modified algorithm sometimes compromised the accuracy of the final solution. To overcome this drawback, a hybrid method by using both the algorithms, original HLRF algorithm and the modified algorithm with BFGS update formula, is proposed. This hybrid algorithm achieves better computational efficiency, compared to inverse-FORM, without compromising the accuracy of the final solution. Comparative numerical examples are provided to demonstrate the improved performance of this hybrid algorithm over that of inverse-FORM in terms of accuracy and efficiency.

#### 1. Introduction

The reliability of structural members is a basic criterion of their design [1–6]. One of the widely used nonsimulation based methods for reliability estimation is the first-order reliability method (FORM) [7]. It uses a linear approximation of the limit state function. Nonnormal variables in their original space are generally mapped to the standard normal space. This is carried out to obtain the most probable point of failure, also called the design point. In the standard normal space,** u** is defined as a column vector of standard normal variables, and is defined as the limit state function. Note that all the vectors mentioned in this paper are column vectors. The design point is defined as the point on the limit state boundary (), which is closest to the origin.

In first-order reliability method (FORM), reliability index is estimated as the magnitude of the design point vector in the standard normal space; that is, when satisfies the following optimization condition: In an inverse reliability problem, a target reliability index is generally given, and an unknown parameter represented by in a limit state function is inversely calculated. In practical cases, can represent any parameter in a limit state function that must be chosen to obtain a specific level of reliability. The value of needs to be determined at the design point that satisfies and . The inverse reliability problem is defined by the following equation set: where is the gradient operator with respect to . Simply, the value of can be found by a* trial and error* method using a forward reliability procedure (e.g., FORM) and then interpolating the value of the unknown parameter for a specific reliability level. However, this method is tedious requiring computational costs [8], and inverse reliability algorithms are far more advantageous than direct reliability measures [9].

The solution for an inverse reliability problem can be obtained using several methods. A reliability contour method was developed by Winterstein et al. [10] and applied to problems in offshore structure problems. As an extension of this method to general limit states, Der Kiureghian et al. [11] developed an algorithm, inverse-FORM, which was based on the Hasofer Lind Rackwitz Fiessler (HLRF) algorithm [12–14]. This inverse reliability method was used by other researchers for various applications including extreme and nominal loads on wind turbines [15, 16] and critical earthquake loads [17]. This HLRF-based method was found to have a fast convergence as demonstrated by the numerical examples considered by Der Kiureghian et al. [11]. However, when the limit state function is highly nonlinear, this method may converge slowly [18].

In solving forward reliability analyses, sequential quadratic programming techniques such as the BFGS (Broyden-Fletcher-Goldfarb-Shanno) formula have been used in the literature [19, 20]. BFGS-based algorithms were found to be a cheaper alternative to HLRF-based algorithms in solving a certain set of problems [21, 22]. These algorithms were proved to be more robust than HLRF-based algorithms in solving forward reliability problems [14]. In addition, they were also found to be more efficient in solving large scale constraint-based reliability problems [23, 24]. Although this method has been used in solving forward reliability problems, this study is the first attempt to extend the applications of the BFGS formula to inverse reliability analysis.

In this paper, a hybrid inverse reliability algorithm based on the BFGS update formula is proposed. In inverse-FORM, the HLRF-based algorithm, the Hessian of the Lagrangian is approximated by the identity matrix for every iteration as shown by Periçaro et al. [18]. However, the proposed hybrid method uses the BFGS update formula in some iterations to calculate the Hessian of the Lagrangian.

Initially, when developing this hybrid algorithm, the idea of using the BFGS update formula for every iteration was tried as an intermediate algorithm. Although this modification improved the efficiency of solving the numerical examples, it affected the accuracy of the final solution. Hence, further modifications were made to arrive at the hybrid algorithm that uses the identity matrix in few iterations and the BFGS update formula in other iterations for approximating the Hessian of the Lagrangian. This hybrid algorithm achieves better efficiency without compromising the accuracy of the final solution. In the following sections, the mathematical formulations of inverse-FORM and the proposed hybrid algorithm are provided for inverse reliability analysis. The method proposed in this paper can be used to analyze the reliability of a component in a system.

#### 2. Inverse-FORM Algorithm

The inverse-FORM algorithm was proposed by Der Kiureghian et al. [11] to solve (2a), (2b), and (2c). They proposed using a merit function to induce convergence to the solution and a step direction vector , which represents a direction vector towards the next searching point in the subsequent iteration. The merit function is composed of two subfunctions, and . ensures the convergence of the reliability index at the search point () to the target reliability index ; ensures that the search point converges to a point on the limit state boundary. The step direction vector () also has two parts: and . represents a direction vector towards the next values of the design variables in the subsequent iteration (). represents a direction constant towards the next value of the unknown parameter in the subsequent iteration ().

In the inverse-FORM algorithm, the first starting values for and are selected as and , respectively, and they are updated using the following equation until converged: where is the step size for the th iteration; is the step direction vector; and are the values of in the th and th iterations, respectively; and are the values of in the th and th iterations, respectively.

The step size in (3) is determined satisfying the following condition:where is the merit function given by the following equation:where and are given by the following equations:where is given by the following equation:The step direction vector () in (3) is calculated as follows:where , , , and are calculated as follows:where , , , and are calculated as follows:This inverse-FORM algorithm based on the HLRF algorithm successfully provided inverse reliability solutions as demonstrated by the two examples in the work done by Der Kiureghian et al. [11]. However, the algorithm converges slowly for highly nonlinear examples [18] and can be further improved by using the BFGS-based-hybrid algorithm that is proposed in the next section.

#### 3. BFGS-Based Algorithm

##### 3.1. Intermediate Algorithm

In this section, an intermediate algorithm that was developed by modifying the HLRF algorithm in inverse-FORM using the Broyden-Fletcher-Goldarb-Shanno (BFGS) update formula is explained. This intermediate algorithm uses calculation steps similar to those in inverse-FORM to solve an inverse reliability problem described in (2a), (2b), and (2c) incorporating the BFGS formula. The starting points for and are adopted as and , respectively, and the values of** u** and are updated using the step direction vector, using (3). The step size in (3) is found such that the condition in (4) is satisfied.

The difference of this intermediate algorithm is as follows: in the calculation of the merit function in (5), is calculated using the following equation instead of (6), while is calculated using (7) unchangingly:where is calculated as follows:Note that, in (18) and (19), the Hessian of the Lagrangian for the th iteration is given by the BFGS matrix . The BFGS matrix in each iteration is updated using the BFGS update formula shown in (20). In the first step of the iterations however, the inverse of Hessian of the Lagrangian is taken as the identity matrix () instead of using the following formula:where and are calculated using the following formula:where is calculated to beThe same as the inverse-FORM, the step direction vector () in (3) is calculated using (9). In the calculation of , the values of , , , and are calculated using (10), (11), (12), and (13), respectively.

In the calculation of using (12), is calculated using the following formula, instead of (14), while is calculated unchangingly using (15):In the calculation of in (13), the values of , and are calculated using the following formulae instead of (16) and (17):A pseudocode for this intermediate algorithm is provided in Algorithm 1, in which it is noted that the BFGS update formula is used to update the inverse of Hessian in every iteration. The proposed use of the BFGS update formula increases the computational efficiency of this intermediate algorithm when compared to that of the inverse-FORM because it provides more optimized step changes.