Surface decomposition method for sensitivity analysis of first-passage dynamic reliability of linear systems

Abstract

This work presents a novel surface decomposition method for the sensitivity analysis of first-passage dynamic reliability of linear systems subjected to Gaussian random excitations. The method decomposes the sensitivity of first-passage failure probability into a sum of surface integrals over the constrained component limit-state hypersurfaces. The evaluation of these surface integrals can be accomplished, owing to the availability of closed-form linear expressions of both the component limit-state functions and their sensitivities for linear systems. An importance sampling strategy is introduced to further enhance the efficiency for estimating the sum of these surface integrals. The number of function evaluations required for the reliability sensitivity analysis is typically on the order of 10^2 to 10^3. The approach is particularly advantageous when a large number of design parameters are considered, as the results of function evaluations can be reused across different parameters. Three numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Figures (11)

• Figure 1: Illustration of surface decomposition method using a 3-dimensional problem. Consider a series system limit-state function $G(\boldsymbol{x})=\min_{k=1}^{4} g_k(\boldsymbol{x})$. The four component limit-state functions are $g_1(\boldsymbol{x})=12-(-2x_1+x_2+0x_3)$, $g_2(\boldsymbol{x})=18-(-x_1+3x_2+0x_3)$, $g_3(\boldsymbol{x})=36-(6x_1+7x_2+0x_3)$ and $g_4(\boldsymbol{x})=10-(2x_1-x_2+0x_3)$, respectively. The system failure surface is composed of four constrained component failure planes $g_k(\boldsymbol{x})\mathbb{I}_k(\boldsymbol{x})=0\;(k=1,2,3,4)$, in which the indicator function $\mathbb{I}_k(\boldsymbol{x})$ takes 1 if $g_j(\boldsymbol{x})>0, \forall j\neq k$, and 0 otherwise. The sensitivity of series system failure probability can be decomposed into four surface integrals over these four constrained component failure planes, respectively. These component failure planes will become component failure hyperplanes in higher-dimensional problems.
• Figure 2: Convergence histories of sensitivities of failure probability with respect to $\omega_n$ and $\zeta_n$ for Example 1. (a) Case 1: $c=0.013\mathrm{m}$; (b) Case 2: $c=0.016\mathrm{m}$; (c) Case 3: $c=0.018\mathrm{m}$; (d) Case 4: $c=0.020\mathrm{m}$. The number of function evaluations required in the proposed SDM for Case 1, Case 2, Case 3 and Case 4 are 714, 501, 326 and 252, respectively, while those required in DIS are 1673, 751, 615, 365, respectively. Accordingly, the speedup ratios of the proposed SDM over DIS for these four cases are approximately 2.34, 1.50, 1.89 and 1.45, respectively.
• Figure 3: Comparison among different PMFs. (a) Estimation of component surface integrals with respect to $\omega_n$; (b) Estimation of component surface integrals with respect to $\zeta_n$; (c) Comparison among different PMFs.
• Figure 4: Comparison of computational efficiency of SDM for different input dimensionalities $d$ and numbers of component failure events $n$. (a) Number of function evaluations versus $d$, with $d\in[1000,10000]$ and fixed $n=1000$; (b) Number of function evaluations versus $n$, with $n\in[1000,10000]$ and fixed $d=1000$.
• Figure 5: A 20-degree-of-freedom shear-type structure equipped with viscoelastic dampers.