First order reliability method for time-variant problems using series expansions

Abstract

Time-variant reliability is often evaluated by Rice’s formula combined with the First Order Reliability Method (FORM). To improve the accuracy and efficiency of the Rice/FORM method, this work develops a new simulation method with the first order approximation and series expansions. The approximation maps the general stochastic process of the response into a Gaussian process, whose samples are then generated by the Expansion Optimal Linear Estimation if the response is stationary or by the Orthogonal Series Expansion if the response is non-stationary. As the computational cost largely comes from estimating the covariance of the response at expansion points, a cheaper surrogate model of the covariance is built and allows for significant reduction in computational cost. In addition to its superior accuracy and efficiency over the Rice/FORM method, the proposed method can also produce the failure rate and probability of failure with respect to time for a given period of time within only one reliability analysis.

Appendices

Appendix A: Expansion optimal linear estimation (EOLE)

EOLE (Sudret and Der Kiureghian 2002) is used to generate samples for a Gaussian stochastic process Y(t) on [0, T], which is divided into sintervals with a step size ${\Delta } t=\frac {T}{s}$. The s time instants are t _i =(i−1)Δt, where i=1, 2, ⋯, s. The covariance matrix Σ is given by

$$ {\mathbf{\Sigma }}=\left( {{\begin{array}{*{20}c} {c_{Y} \left( {t_{1} ,\;t_{1} } \right)} \hfill & {c_{Y} \left( {t_{1} ,\;t_{2} } \right)} \hfill & {\cdots} \hfill & {c_{Y} \left( {t_{1} ,\;t_{s} } \right)} \hfill \\ {c_{Y} \left( {t_{2} ,\;t_{1} } \right)} \hfill & {c_{Y} \left( {t_{2} ,\;t_{2} } \right)} \hfill & {\cdots} \hfill & {c_{Y} \left( {t_{2} ,\;t_{s} } \right)} \hfill \\ {\vdots} \hfill & {\vdots} \hfill & {\ddots} \hfill & {\vdots} \hfill \\ {c_{Y} \left( {t_{s} ,\;t_{1} } \right)} \hfill & {c_{Y} \left( {t_{s} ,\;t_{2} } \right)} \hfill & {\cdots} \hfill & {c_{Y} \left( {t_{s} ,\;t_{s} } \right)} \hfill \\ \end{array} }} \right)_{s\times s} $$

(A1)

where c _Y (t _i , t _j ) is the covariance of Y(t) at t _i and t _j .

Let the eigenvalues and eigenvectors of Σ be η _i and φ _i , i=1, 2, ⋯, s, respectively. Then Y(t) is approximated by the following series expansion (Sudret and Der Kiureghian 2002):

$$ Y\left( t \right)\approx \mu_{Y} (t)+\sigma_{Y} (t)\sum\limits_{i=1}^{p} {\frac{Z_{i} }{\sqrt {\eta_{i} } }{\mathbf{\varphi }}_{i}^{T} } {\boldsymbol\rho }_{Y} (t) $$

(A2)

in which Z _i (i=1, 2,⋯,p≤s) are independent standard Gaussian random variables, ρ _Y (t)=[ρ _Y (t, t ₁),ρ _Y (t, t ₂),…,ρ _Y (t, t _p )]^T, and μ _Y (t) and σ _Y (t) are the mean and standard deviation of Y(t), respectively; p is the number of terms, and p≤s. When p=s, no truncation is made, and the error is minimum.

The accuracy of EOLE is affected by the size of the finite element mesh Δt, and the selection of the size depends on the correlation length of the stochastic process (Sudret and Der Kiureghian 2002). The shorter is the mesh length, the more accurate are the results.

Appendix B: Orthogonal series expansion (OSE)

Different from EOLE, OSE does not need finite element meshes. Its accuracy is therefore not affected by the mesh size. OSE approximates a Gaussian process Y(t) as Zhang and Ellingwood (1994)

$$ Y(t)=\mu (t)+\sum\limits_{i=1}^{M} {\gamma_{i} } h_{i} (t) $$

(B1)

in which Γ=[γ ₁,γ ₂,…,γ _M ] is a vector of correlated zero-mean Gaussian random variables, and h _i (t), i=1, 2, ⋯, M, are orthogonal functions.

The correlated random variables γ _i are then transformed into independent standard Gaussian random variables with the following steps.

Construct an M×M square matrix Γ by

$$ ({\boldsymbol{\Gamma }})_{ij} ={{\int}_{0}^{T}} {{{\int}_{0}^{T}} {C_{YY} (t,\;\tau )h_{i} (t)h_{j} (\tau )dtd\tau } } $$

(B2)

where C _{Y Y} (t, τ) is the ato-covariance of Y(t).

Then obtain the eigenvectors of Γ. Use

$$ {\boldsymbol{\Gamma }}={\mathbf{P\uplambda P}}^{-1}={\mathbf{\Lambda {\Lambda} }}^{T} $$

(B3)

where λ is a diagonal matrix with its diagonal elements being eigenvalues of Γ, P is a M×M square matrix whose i-th column is the i-th eigenvector of Γ, and Λ is a lower triangular matrix.

Then Γ=[γ ₁,γ ₂,…,γ _M ] is transformed into independent standard Gaussian random variables Z=[Z ₁,Z ₂,…,Z _M ] through

$$ {\boldsymbol{\Gamma }}={\mathbf{\Lambda Z}} $$

(B4)

For the orthogonal functions,

$$ \int h_{i} (t)h_{j} (t)dt=\left\{ {\begin{array}{l} 1,\;\text{if}\;i=j \\ 0,\;\text{otherwise} \\ \end{array}} \right. $$

(B5)

The Legendre polynomials may be chosen as the orthogonal functions. When the eigenfunctions and covariance functions are approximated using the same orthogonal functions, OSE can be regarded as an approximation to the K−L expansion (Zhang and Ellingwood 1994).