A survey on approaches for reliability-based optimization

Abstract

Reliability-based Optimization is a most appropriate and advantageous methodology for structural design. Its main feature is that it allows determining the best design solution (with respect to prescribed criteria) while explicitly considering the unavoidable effects of uncertainty. In general, the application of this methodology is numerically involved, as it implies the simultaneous solution of an optimization problem and also the use of specialized algorithms for quantifying the effects of uncertainties. In view of this fact, several approaches have been developed in the literature for applying this methodology in problems of practical interest. This contribution provides a survey on approaches for performing Reliability-based Optimization, with emphasis on the theoretical foundations and the main assumptions involved. Early approaches as well as the most recently developed methods are covered. In addition, a qualitative comparison is performed in order to provide some general guidelines on the range of applicability on the different approaches discussed in this contribution.

Appendix A: Methods for reliability analysis

This appendix presents a brief overview on methods that have been developed in order to compute the integral associated with probability. These methods can be broadly classified into two categories (Schuëller et al. 2004): approximate methods and simulation methods.

1.1 A. 1 Approximate reliability methods

The key concept of approximate reliability methods is introducing an asymptotic approximation of the limit state function (LSF), i.e. $g(\boldsymbol{y},\boldsymbol{\theta}) = 0$, using a Taylor series. Although this approximation of the limit state function can be regarded as a meta-model, its scope is different from the meta-models discussed in Section 6.1. This is due to the fact that in approximate reliability methods, the objective of generating a Taylor series is replacing an unknown probability integral by a known one. The approximation using a Taylor series is constructed around the so-called design point. For defining the design point, assume that the vector $\boldsymbol{\theta}$ is composed by independent, Gaussian standard distributed random variables. Thus, the design point (which is denoted as $\boldsymbol{\theta}^\ast$) can be defined using two equivalent criteria (Freudenthal 1956). According to the geometrical criterion, the design point is the realization in the standard normal space which lies on the LSF ($g(\boldsymbol{y},\boldsymbol{\theta})=0$) with the minimum Euclidean norm (β) with respect to the origin; this is shown schematically in Fig. 6. According to the probabilistic interpretation, the design point is the failure point with highest probability density. This means, it is the point that maximizes $f(\boldsymbol{\theta})$ subject to $g(\boldsymbol{y},\boldsymbol{\theta})\leq0$, where f(·) is the standard normal probability density function in $\mathcal{R}^{n_\theta}$ (see Fig. 6). It should be noted that the norm of the design point ($\beta={\left|\left|{\boldsymbol{\theta}^{\ast}}\right|\right|}$) has been denoted in the literature as reliability index.

Fig. 6

Schematic representation of the design point and the FORM/SORM approximations in the standard normal space

From the discussion above, it is clear that the identification of the design point is also an optimization problem, as it is necessary either to minimize the Euclidean norm or maximize the probability density function. For details on how to determine the design point, it is referred to, e.g. Liu and Der Kiureghian (1991), Au (2006), Koo et al. (2005) and Wu et al. (1990).

Once the design point has been determined, the integral associated with the probability of failure can be approximated using the First or Second Order Reliability Method (FORM and SORM, respectively). In the case of FORM, the LSF is replaced by a first order Taylor expansion centered around the design point. In the case of SORM, the LSF is replaced with an incomplete second order Taylor expansion (also centered around the design point). A more detailed explanation of FORM and SORM is outside the scope of this paper; for more details on these reliability techniques, it is referred to, e.g. Rackwitz (2001). However, it is important to note that no estimator of the error introduced when approximating the probability integral using FORM and/or SORM is available. Moreover, these methods may not always be applicable, e.g. in cases where the performance function is highly non linear and/or the dimensionality of $\boldsymbol{\theta}$ is high (Katafygiotis and Zuev 2008; Valdebenito et al. 2010).

Besides FORM and SORM, another technique that can be classified as an approximate reliability method is the so-called Dimension Reduction Method (DRM), which was introduced in the field of structural reliability analysis in (Rahman and Xu 2004; Xu and Rahman 2004). The key idea of this approach is approximating the original performance function—with an associated n _θ -dimensional domain—as a summation of a number of simpler functions, where each of the domains of the latter functions has lower dimensionality. This approximate representation of the performance function can then be used to perform reliability analysis at reduced numerical costs using, e.g. uni-dimensional numerical integration (Rahman and Wei 2008), an appropriate response surface (Rahman and Wei 2006), etc.

1.2 A.2 Simulation methods

Simulation methods estimate the value of the probability integral by generating samples of the uncertain parameters according to some prescribed rule. The most widely known method of this class is Monte Carlo Simulation (MCS) (Metropolis and Ulam 1949). This method is based on generating N _S samples of $\boldsymbol{\theta}$ which are distributed according to $f(\boldsymbol{\theta})$. Then, the failure probability can be estimated as:

$$p\approx\hat{p}=\frac{1}{N_S}\sum\limits_{i=1}^{N_S}{I\left( \boldsymbol{y},\boldsymbol{\theta}^{(s)}\right) },~~~\boldsymbol{\theta}^{(s)}\sim f(\boldsymbol{\theta}) $$

(27)

where I(·) is an indicator function which is equal to one in case $g\left( \boldsymbol{y},\boldsymbol{\theta}^{(s)}\right) \leq0$ and zero, otherwise. The error in the estimator of the failure probability can be estimated by means of the coefficient of variation δ _MC , i.e. $\delta_{MC}=\sqrt{(1-\hat{p})/(N_S\hat{p}})$.

The MCS method is a general simulation technique, i.e. it is applicable to linear and non linear problems indifferently. Moreover, its efficiency is independent of the number of random variables involved in the problem under analysis. However, its major drawback is that for calculating low failure probabilities, a large number of samples (proportional to 1/p) is required for generating a reliable estimator, i.e. with sufficient accuracy (or, equivalently, a low coefficient of variation). Hence, the numerical costs involved in estimating probabilities of rare occurrence of failure events may be extremely high and even prohibitive, especially when a structural system is modeled using large FE models. In view of this shortcoming, the so-called advanced simulation methods have been developed, which allow estimating low failure probabilities with increased efficiency if compared with MCS.

Advanced simulation methods are also based on generating samples of the uncertain parameters. However, specific sampling procedures are followed in order to increase the efficiency. An important characteristic of several advanced simulation methods is that they are specially designed for addressing reliability problems involving a large number of uncertain parameters (Katafygiotis and Zuev 2008; Valdebenito et al. 2010). Some examples of these advanced simulation methods are Importance Sampling (Schuëller and Stix 1987), Line Sampling (Schuëller et al. 2003), Subset Simulation (Au and Beck 2001), Domain Decomposition Method (Katafygiotis and Cheung 2006), Auxiliary Domain Method (Katafygiotis et al. 2007), Linked Importance Sampling (Katafygiotis and Zuev 2007; Neal 2005), Horseracing Simulation method (Katafygiotis and Zuev 2009; Zuev 2009), etc.