**1. Introduction**

The probabilistic analysis of practical engineering problems has been a traditional research field [1–3]. The first category of engineering reliability analysis methods are the most probable point (MPP) methods [4–7]. In this category of methods, a design point, or the so-called most probable point in the design space is sought. The limit state function is often transformed into a standard Gaussian space and approximated using Taylor series expansions. Depending on the order of approximation used, FORM/SORM are available [4–7]. These methods require the derivatives of system responses, i.e., sensitivity analysis. For complex engineering systems that require expensive response simulations such as nonlinear explicit finite element (FE) analysis, the integration of the MPP-based methods and a commercial FE code is not straightforward. An alternative category of methods are the direct sampling-based methods, including MCS and some other simulation methods [8–12]. These methods can be integrated fairly easily with an existing simulation program because they do not require the derivation or calculation of gradient

information. However the direct application of MCS can be computationally prohibitive in complex engineering problems that require expensive response simulations.

Since the most appropriate sample size is not known before the creation of the surrogate models, it remains a challenge to determine the appropriate sample size to use. One viable approach is to create and test a few different sample sizes, and the best sample size for the problem can be determined. To improve this process, the concept of SRBF surrogate models is developed and it is intended to automate this process and find the proper sample size iteratively and automatically for the augmented RBF surrogate models that can be used for reliability analysis of practical

*Reliability Analysis Based on Surrogate Modeling Methods*

*DOI: http://dx.doi.org/10.5772/intechopen.84640*

This chapter presents an engineering reliability analysis method based on a SRBF surrogate modeling technique. In each iteration of the new method, augmented RBFs can be used to generate surrogate models of a limit state function. Three accurate augmented RBFs surrogate models, which were identified from a previous study, are adopted. The failure probability can be calculated using the SRBF surrogate models combined with MCS. Section 3 describes the general concept of engineering reliability analysis. Section 4 briefly reviews some surrogate modeling methods, and explains the augmented SRBF surrogate modeling technique. Sections 5 and 6 presents the MCS method and the overall reliability analysis procedures. In Section 7, the proposed approach is applied to the probability analysis of several mathematical and practical engineering problems. The failure probabilities are compared with those computed based on the direct implementation of MCS without surrogate models. The numerical accuracy and efficiency of the proposed

A time-invariant reliability analysis of an engineering problem is to compute the

where x is an *s*-dimensional real-valued vector of random variables, *g*ð Þ x is the limit state function, and *pX*ð Þ x is the joint probability density function. Eq. (1) is difficult to obtain for practical engineering applications, since *pX*ð Þ x is unknown and *g*ð Þ x is usually an implicit and nonlinearity function. A detailed response analysis model, such as the FE analysis of the engineering system is often required to

An implicit function *<sup>g</sup>*ð Þ <sup>x</sup> is considered, where x = *<sup>x</sup>*<sup>1</sup> <sup>⋯</sup> *xs* ½ �<sup>T</sup> is an input variable vector and *s* is the number of input variables. Before a surrogate model of function *g*ð Þ x can be created, some sample points shall be generated using design of experiments (DOE). Some routinely used DOE approaches include factorial design, Latin hypercube sampling (LHS) [68], central composite design, and Taguchi orthogonal array design [69]. Assume x*<sup>i</sup>* is the input variable vector at the *i*th (*i* = 1,…*n*) sample point, the limit state function *g*ð Þ x needs to be evaluated at all the sample points to

� �<sup>T</sup> <sup>=</sup> ½ � *<sup>g</sup>*ð Þ x1 <sup>⋯</sup> *<sup>g</sup>*ð Þ <sup>x</sup>*<sup>n</sup>* <sup>T</sup>

.

ð

*pX*ð Þ x *d*x (1)

*g*ð Þ x ≤0

approach using MCS and SRBF surrogate models is studied.

failure probability, *PF*, using the following integral [1–3]:

*PF* � *P g* ð Þ¼ ð Þ x ≤0

**3. Engineering reliability analysis**

evaluate function values of *g*ð Þ x .

**4.1 Design of experiments**

**73**

**4. Surrogate modeling methods**

obtain the function values, i.e., *g* ¼ *g*<sup>1</sup> ⋯ *gn*

engineering systems.

To reduce the complexity of implementation and improve the computational efficiency, various approximate modeling techniques have been applied to the reliability analysis of practical engineering systems [13, 14]. These approximate models are referred to as surrogate models. There are abundant literature that presented surrogate models and their applications to numerical optimization and reliability-based design optimization. However, the focus of this chapter and the review of literature here is primarily on the applications of surrogate models to engineering reliability analysis. In surrogate modeling methods, the analysis software is replaced by approximate surrogate models, which have explicit functions and are very efficient to evaluate. FORM/SORM or a sampling method can then be applied using the explicit surrogate model instead of the original implicit numerical model. In all the surrogate models developed, the most basic and popular surrogate model is the conventional polynomial-based response surface method (RSM). The RSM has been shown to be useful for different engineering reliability analyses and applications [15–25]. The entire response space is approximated using a single quadratic polynomial function in a global RSM model. To improve model accuracy for reliability analysis using a global RSM model, different techniques were proposed such as efficient sampling methods [26, 27] and inclusion of higher order effects [28, 29]. When combined with gradient-based search methods, it is more efficient to use RSM in an iterative manner or a local window of the response space [30]. Local RSM methods such as the moving least square technique were developed to handle highly nonlinear limit state functions [31]. Other commonly used surrogate modeling methods have also been developed over the years, such as artificial neural networks (ANN) [32–37], Kriging [38–46], high-dimensional or factorized high-dimensional model representation [47–51], support vector machine [52–57], radial basis functions (RBFs) [58], and even ensemble of surrogates [59–62].

An RBF surrogate model is a multidimensional interpolation approach using available scattered data. Due to their characteristics in global approximation, RBFs could create accurate surrogate models of various responses [63, 64]. An RBF model provides exact fit at the sample points. In the studies by Fang and coauthors [65, 66], various basis functions were investigated including Gaussian, multiquadric, inverse multiquadric, and spline functions. Some compactly supported (CS) basis functions developed by Wu [67] were also studied. Mathematical functions and practical engineering responses were tested and their surrogate models were created using different basis functions. Augmented compactly supported functions worked well and were found to create more accurate surrogate models than non-augmented models.
