Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 569016, 14 pages

http://dx.doi.org/10.1155/2015/569016

## Improved Reliability-Based Optimization with Support Vector Machines and Its Application in Aircraft Wing Design

^{1}College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA

Received 25 September 2014; Revised 20 January 2015; Accepted 13 April 2015

Academic Editor: Marc Dahan

Copyright © 2015 Yu Wang 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

A new reliability-based design optimization (RBDO) method based on support vector machines (SVM) and the Most Probable Point (MPP) is proposed in this work. SVM is used to create a surrogate model of the limit-state function at the MPP with the gradient information in the reliability analysis. 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 RBDO, the Sequential Optimization and Reliability Assessment (SORA) is employed as well, which decouples deterministic optimization from the reliability analysis. The improved SVM-based reliability analysis is used to amend the error from linear approximation for limit-state function in SORA. A mathematical example and a simplified aircraft wing design demonstrate that the improved SVM-based reliability analysis is more accurate than FORM and needs less training points than the Monte Carlo simulation and that the proposed optimization strategy is efficient.

#### 1. Introduction

There are many uncertainties encountered in both of the aircraft manufacturing process and its subsequent flight operation. The physical properties of materials are uncertain. Manufacturing errors produce the aerodynamic shape and structural dimensions different from the original design [1]. Furthermore, the load on the aircraft is not constant during operation. Fuel is consumed continually during cruise; thus, fight parameters keep changing. After an aircraft has been produced, some parameters sometimes need to be adjusted for a new type. If these uncertainties are considered in the conceptual design, the aircraft performance will be more reliable than deterministic design [2], and both of the risk and cost in the design will be reduced.

Reliability analysis is the key part of reliability-based design optimization (RBDO). Reliability is the probability of success. In physics-based reliability, the status of success is specified by limit-state functions, which are derived from physics principles [3]. Let the limit-state function be , where is a vector of random variables with length . If the event of success is specified by , then the failure event is . Consequently, the probability of failure is computed bywhere is the joint probability density function (PDF) of .

Since it is difficult to analytically evaluate the probability integral, many approximation methods have been developed. Among them, the First Order Reliability Method (FORM) [4–6] is commonly used. The FORM linearizes the limit-state function at the Most Probable Point (MPP), which in the standard normal space has the highest probability of producing the value of limit-state function. Then, (1) becomes a linear combination of normal variables. Since only the first order derivatives of and some basic information are needed, the FORM is efficient. Its accuracy, however, may not be satisfactory for highly nonlinear limit-state functions.

Several methods are available to improve the accuracy of the FORM. One of the methods is the Second Order Reliability Method (SORM) [7]. The SORM provides a second order approximation to at the MPP. As a result, it is generally more accurate than the FORM. The MPP-based importance sampling [8, 9] is another alternative method. Random samples are drawn from distributions whose center is shifted to the MPP. A relatively small sample size can then produce a good estimation of the probability of failure. The other strategy is to approximate the safety-failure boundary at the MPP with higher accuracy. In the point-fitting method [10], a piecewise paraboloid surface is built with the fitting points selected from each side of MPP along both forward and backward directions of each random variable. Similarly, response surface modeling has also been used to create a surrogate model for the limit-state function at the MPP [11, 12]. A surrogate model can be created using artificial neural network [13] as well.

Recently, another strategy, the support vector machine (SVM) [14] method, has been introduced in reliability analysis. SVM is a statistical classification method. As indicated in [15], reliability analysis can be viewed as a classification problem where SVM is applicable. Examples of using SVM include the fast Monte Carlo simulation (MCS) [16], the limit-state function identification for discontinuous responses and disjoint failure domains [17], SVM-based MCS [18], and virtual SVM for high-dimensional problems [19]. It is worthwhile to further study SVM for reliability analysis and RBDO.

The conventional approach for solving a reliability-based design optimization problem is to employ a double-loop strategy in which the analysis and the synthesis are nested in such a way that the synthesis loop performs the reliability analysis iteratively for meeting the probabilistic constraints. As the double-loop strategy may be computationally infeasible, various single loop strategies have been studied to improve its efficiency. The method “approximately equivalent deterministic constraints” creates a link between a probabilistic design and a safety-factor based design [20]. The reliability constraints are formulated as deterministic constraints that approximate the condition of the MPP for reliability analysis [21]. A single loop method, Sequential Optimization and Reliability Assessment (SORA), is a very efficient method for RBDO [22]. In this method, optimization and reliability analysis are decoupled from each other; no reliability analysis is required within optimization and the reliability analysis is only conducted after the optimization. Hence, the design is quickly improved from cycle to cycle and the computational efficiency is improved significantly. However, because of FORM employed in SORA based on the limit-state function linearization, its precision may not be high enough for the highly nonlinear problem.

In this work, the accuracy of SVM-based reliability analysis was improved firstly. In addition to the training points around the MPP, the gradient of a limit-state function at the MPP was included in approximating the limit-state function, to guarantee that the surrogate model not only passes through the MPP but also is tangent to the limit-state function at the MPP. And importance sampling is used to estimate the probability of failure based on the surrogate model. Then, the improved SVM-based reliability analysis was integrated into SORA for wing optimization. Results of the two examples showed that this strategy is more accurate than before with a moderately increased computational cost.

#### 2. FORM, IS, and SVM

In this section, the three methods used in this work are reviewed. The methods include the First Order Reliability Method (FORM), importance sampling (IS), and support vector machines (SVM).

##### 2.1. First Order Reliability Method (FORM)

In this work, we assume all random variables in are independent. FORM involves the following three steps.

*(1) Transformation of Random Variables into Standard Normal Variables*. The original random variables (in the -space) are transformed into random variables (in the -space) whose elements follow a standard normal distribution. The transformation is given by [23]where is the cumulative distribution function (CDF) of and is the inverse CDF of a standard normal variable.

*(2) Search the Most Probable Point (MPP)*. The MPP is the point at the limit-state , and, at the MPP, the PDF of is at its maximum. Maximizing the joint PDF of and noting that a contour of the PDF of is a concentric hypersphere, we obtain the MPP by solvingwhere stands for the norm (length) of a vector.

Geometrically, the MPP is the shortest distance point from surface to the origin of the -space. The minimum distance is called the* reliability index*. Then, is approximated by the first Taylor expansion series at aswhere is the gradient of at the MPP and is given by

*(c) Calculate the Probability of Failure*. As shown in (4), is now a linear combination of normal variables. As a result, can be easily computed bywhere is the CDF of a standard normal variable.

##### 2.2. Importance Sampling (IS)

The FORM is commonly used due to its good balance between accuracy and efficiency. For highly nonlinear limit-state functions, however, the accuracy may not be good enough. IS can be used to improve accuracy.

The probability of failure in (1) can be rewritten aswhere is an indicator function and is defined as

is therefore the expectation of ; namely,where denotes an expectation. With the direct Monte Carlo simulation (MCS), can be estimated by averaging : where are the samples drawn from the joint probability density and is the number of failures.

A large sample size is required when is small because the chance of getting samples in failure region is small. IS draws samples from a new set of distributions such that more samples will be in the failure region. One strategy is to shift the mean values of the random variables to the MPP. As shown in Figure 1, all the samples (the lower cloud) generated from the original distributions of and are in the safe region. They do not contribute to the probability estimation. After the mean values are shifted to the MPP, sufficient samples are now in the failure region.