**5. Estimation of failure probability**

Eqs. (11) and (17) are the RBF and augmented RBF surrogate model functions of *g*ð Þ x . The surrogate models have explicit expressions; therefore their function values where *<sup>λ</sup>* <sup>¼</sup> ½ � *<sup>λ</sup>*<sup>1</sup> <sup>⋯</sup> *<sup>λ</sup><sup>n</sup>* T, and *<sup>A</sup>* is given as:

2 6 4

<sup>e</sup>*g*ð Þ¼ <sup>x</sup> <sup>∑</sup>*<sup>n</sup>*

∑*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*λifj*

*F* ¼

Solve the linear system of Eq. (19) to get *λ* and *c*, as:

RBFs. The following RBF models were studied:

**5. Estimation of failure probability**

*λ c* � �

2 6 4

solve for all unknowns, as:

where *c* ¼ *c*<sup>1</sup> ⋯ *cp*

polynomials.

polynomials.

**76**

� �<sup>T</sup>

*ϕ*ð Þ k k x1 � x1 ⋯ *ϕ*ð Þ k k x1 � x*<sup>n</sup>* ⋮⋱⋮ *ϕ*ð Þ k k x*<sup>n</sup>* � x1 ⋯ *ϕ*ð Þ k k x*<sup>n</sup>* � x*<sup>n</sup>*

*<sup>λ</sup>* <sup>¼</sup> *<sup>A</sup>*�<sup>1</sup>

Since highly nonlinear basis functions are used, the RBF surrogate models in Eq. (11) can approximate nonlinear responses very well. However, they were found to have more errors for linear responses [58]. In order to overcome this drawback,

*<sup>i</sup>*¼<sup>1</sup>*λiϕ*ð Þþ k k <sup>x</sup> � <sup>x</sup>*<sup>i</sup>* <sup>∑</sup>*<sup>p</sup>*

where the second part represents *p* terms of polynomial functions, and *cj* (*j* = 1*,… p*)

Eqs. (17) and (18) consist of (*n* þ *p*) equations in total, and they can be rewritten, as:

<sup>¼</sup> *<sup>g</sup>* 0 � �

*g* 0 � �

*c* � �

*f* <sup>1</sup>ð Þ x1 ⋯ *f <sup>p</sup>*ð Þ x1 ⋮⋱⋮ *f* <sup>1</sup>ð Þ x*<sup>n</sup>* ⋯ *f <sup>p</sup>*ð Þ x*<sup>n</sup>*

<sup>¼</sup> *A F F<sup>T</sup>* 0 � ��<sup>1</sup>

**SRBF-MQ-LP**: sequential multiquadric function with linear polynomials. **SRBF-CS20-LP**: sequential compactly supported function *ϕ*2*,*<sup>0</sup> with linear

**SRBF-CS30-LP**: sequential compactly supported function *ϕ*3*,*<sup>0</sup> with linear

Eqs. (11) and (17) are the RBF and augmented RBF surrogate model functions of *g*ð Þ x . The surrogate models have explicit expressions; therefore their function values

For augmented RBFs, either linear or quadratic polynomial functions can be used. In this study, only linear polynomial functions were added to Eq. (17). For the rest of the paper, a suffix "-LP" is used to represent linear polynomials added to

Solve the linear system of Eq. (14) to calculate coefficients *λ*, as:

the RBF model in Eq. (11) can be augmented by polynomial functions, as:

are the unknown coefficients to be determined. There are more unknowns than available equations; therefore the following orthogonality condition is required to

> *A F F<sup>T</sup>* 0 � � *λ*

, and *F* is given as:

*g* (16)

ð Þ x (17)

*<sup>j</sup>*¼<sup>1</sup>*cjfj*

ð Þ¼ x*<sup>i</sup>* 0*,* for *j* ¼ 1*,* …*p* (18)

(15)

(19)

(20)

(21)

*A* ¼

*Reliability and Maintenance - An Overview of Cases*

can be efficiently calculated in each iteration of the SRBF approach. Based on the surrogate model <sup>e</sup>*g*ð Þ <sup>x</sup> , the failure probability *PF* can be computed using a sampling method, such as MCS, as:

$$P\_F \equiv P(\mathbf{g}(\mathbf{x}) \le \mathbf{0}) = \frac{1}{N} \sum\_{i=1}^{N} \Gamma\left[\widetilde{\mathbf{g}}\left(\mathbf{x}^i\right) \le \mathbf{0}\right] \tag{22}$$

where *N* is the total number of MCS samples, x*<sup>i</sup>* is the *i*th realization of x, and *Γ* is a deciding function, as:

$$\Gamma = \begin{cases} 1 \, \text{if } \tilde{\text{g}}\,(\mathbf{x}^i) \le \mathbf{0} \\ 0 \, \text{if } \tilde{\text{g}}\,(\mathbf{x}^i) > \mathbf{0} \end{cases} \tag{23}$$

The reliability index *β* can be further determined, as [49]:

$$
\beta = -\Phi^{-1}(P\_F) \tag{24}
$$

where *Φ* is the standard normal cumulative distribution function.
