**3. Engineering reliability analysis**

A time-invariant reliability analysis of an engineering problem is to compute the failure probability, *PF*, using the following integral [1–3]:

$$P\_F \equiv P(g(\mathbf{x}) \le \mathbf{0}) = \int\_{g(\mathbf{x}) \le \mathbf{0}} p\_X(\mathbf{x}) d\mathbf{x} \tag{1}$$

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 evaluate function values of *g*ð Þ x .
