Fig. 1

Schematics of three envelope barriers: (a) circular domain; (b) elliptical domain; (c) rectangular domain (color online)"

Fig. 2

The first-passage time probability FS(τ, x0) and its density function p(τ, x0) of the Duffing oscillator defined in the elliptical safe domain starting from different initial conditions (color online)"

Fig. 3

The first-passage time probability FS(τ, x0) and its density function p(τ, x0) of the Duffing oscillator defined in the circular safe domain starting from different initial conditions (color online)"

Fig. 4

The reliability functions of the Duffing oscillator defined in the three safe domains at time t = 5T. The first column represents results computed with the MCS. The second column displays results calculated with the proposed method. The third column shows contour plots of the errors between the MCS results and the results obtained with the proposed method. Each row corresponds to results calculated in (a)–(c) the circular safe domain, (d)–(f) the elliptical safe domain, and (g)–(i) the rectangular safe domain (color online)"

Table 1

The computational time of the proposed method and the MCS for the reliability functions of the Duffing oscillator starting from all the grid points in the safe domain. ${\bar R}$S denotes the results when the RBF-NN is not incorporated into the boundaries of the safe domain. The CPU time is in seconds"

Fig. 5

The first-passage time probability FS(τ, x0) and its density function p(τ, x0) of the Van der Pol oscillator defined in the elliptical safe domain starting from different initial conditions (color online)"

Fig. 6

The first-passage time probability FS(τ, x0) and its density function p(τ, x0) of the Van der Pol oscillator defined in the circular safe domain starting from different initial conditions (color online)"

Fig. 7

The reliability functions of the Van der Pol oscillator defined in the three safe domains at time t = 2T. The layout and interpretation of the columns and rows are the same as those described in Fig. 4 (color online)"

Table 2

The computational time of the proposed method and the MCS for the reliability functions of the Van der Pol oscillator starting from all the grid points in the safe domain"

Fig. 8

Schematics of the boundary of the sinusoidal shallow arch for different λ1. The filled regions represent the domain of safe operation of the sinusoidal shallow arch (color online)"

Fig. 9

The first-passage time probability FS(τ, x0) and its density function p(τ, x0) of the sinusoidal shallow arch defined in the safe domain for different λ1 starting from the initial condition x0 = (0, 0) (color online)"

Fig. 10

The reliability functions of the sinusoidal shallow arch defined in the safe domains at time t = 2T for different λ1. The layout and interpretation of the columns are the same as those described in Fig. 4. Each row corresponds to the results calculated for (a)–(c) λ1 = 2.1, (d)–(f) λ1 = 2.3, and (g)–(i) λ1 = 3.1 (color online)"

Table 3

The computational time of the proposed method and the MCS for the reliability functions of the sinusoidal shallow arch starting from all the grid points in the safe domain"