**6. Reliability analysis based on successive RBF models**

**Figure 1** shows a flowchart of reliability analysis using SRBF-based surrogate modeling technique and MCS. Once the explicit augmented RBF surrogate model is generated in one iteration of the proposed method, MCS is applied to efficiently estimate the failure probability for any sample size. If the convergence criterion is not satisfied in the current iteration, more sample points will be added and another iteration starts. As the sample size increases, the SRBF surrogate models in general become more accurate, a reduction was observed in the failure probability estimation errors. However this results in more function evaluations. Since the number of response simulations is determined by the sample size used to create a surrogate model, the majority of the computational cost is from the response simulations. The detailed procedure is as follows:


**7. Numerical examples**

when surrogate models were used.

SRBF-based MCS, respectively.

**7.2 Example 2: a cantilever beam**

the limit state function is.

*Example 1: numerical results.*

**Table 2.**

**79**

**7.1 Example 1: a nonlinear limit state function**

*Reliability Analysis Based on Surrogate Modeling Methods*

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

Four numerical examples were solved using the proposed reliability analysis method. These include both mathematical and engineering problems found in literature. In this study, the proposed method based on three SRBFs, i.e., SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, is referred to as the SRBF-based MCS. The Direct MCS refers to MCS without using surrogate models. In the Direct MCS, the number of response simulations was determined by the MCS sample size. However, in the SRBF-based MCS, the number of response simulations was based on the surrogate modeling sample size. A total of *N* = 10<sup>6</sup> samples was adopted in MCS

A nonlinear limit state function was studied in literature, as [21, 49, 50]:

SRBF-MQ-LP, SRBF-CS20-, and SRBF-CS30-LP methods to converge,

corresponding to 40, 30, and 20 sample points, respectively. A total of 40, 30, and 20 function evaluations (original limit state function) were required for the three

The reliability analysis of a cantilever beam with a concentrated load is conducted in this example [50]. The beam has a rectangular cross section. The performance requirement is the displacement at tip should be <0.15 in. Therefore,

where *x*<sup>1</sup> and *x*<sup>2</sup> are independent random variables following standard normal distributions (mean = 0; standard deviation = 1). The failure probability *PF* = 0.009372 was obtained based on Direct MCS and used to compare with other solutions. The RBF surrogate models were constructed using the two variables sampled in the range of �3.0 to 3.0. All three surrogate models started with 10 sample points in the first iteration. With 10 sample points, the error of the estimated failure probability was 7.0, 3.0, and 1.8% for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. In each subsequent iteration 10 more sample points were added. At convergence, the accuracy of SRBF models was improved; the error was reduced to 0.9, 0.8, and 1.3% for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. Adequate accuracy of reliability analysis was achieved for all three SRBF surrogate models. The failure probability values obtained based on three surrogate models and the associated errors as compared to the solution obtained using Direct MCS are listed in **Table 2**. It took 4, 3, and 2 iterations for

*g*ð Þ¼ x *exp* ð Þ� 0*:*2*x*<sup>1</sup> þ 6*:*2 *exp* ð Þ 0*:*47*x*<sup>2</sup> þ 5*:*0 (25)

#### **Figure 1.**

*Flowchart of reliability analysis using a SRBF surrogate technique.*

