Keywords: first passage reliability, stationarity, model uncertainty, asymptotic approximation, robust stochastic design, multiple design points.

Calculation of system reliability is one of the most difficult problems in stochastic analysis of dynamical systems. This task is often referred to as solving the first-passage problem and it is defined as the determination of the probability that within some given time duration, the output trajectory of a system out-crosses the boundary of a safe region that defines acceptable performance. The system model parameters, based on the available knowledge, may either be certain or include some level of uncertainty. This paper focuses on an analytical approximation for the stationary reliability of linear dynamic systems with higher dimensional output and on its application to the robust design of engineering systems.

Initially a recently published [1] analytical approximation for the first-passage problem (assuming known system parameters) is reviewed, that calculates the system reliability using the out-crossing rate over the boundary of the safe region. This rate is approximated by a summation of the individual out-crossing rates over each linear surface of this boundary. Computational issues are discussed for taking into account the spatial (at different parts of the boundary) and temporal (at different times) correlation of the failure events. Numerical approximations are established for evaluating both these effects. Then, the extension to systems with uncertain model parameters is presented. This requires evaluation of multidimensional probability-integrals and an existing asymptotic expansion is suggested for this task. This expansion involves solving (numerically) for the local maxima (design points) of the integrand and accurately calculating the hessian matrix at these locations. Problems arise in both these tasks due to the numerical approximations (as discussed before) that were established for calculating the out-crossing rate. Efficient approaches are discussed for addressing these problems. A novel methodology is also developed for accounting for the contributions of multiple design points. An alternative calculation, based on stochastic simulation with importance sampling is also considered for the reliability integral.

The implications to robust stochastic design are discussed, when the above methods are used for evaluation of system reliability. In particular, the effect of the errors introduced by the numerical and asymptotic approximations is addressed and methodologies for reducing their relative or absolute importance are presented.

An illustrative example is presented that addresses the concepts discussed in this paper. It considers the robust-optimization of a mass damper for improvement of the reliability of a three-story structure under stochastic earthquake loading. The approach for addressing the existence of multiple design points is shown to be highly efficient. Also, the importance of explicitly considering the uncertainties in the system description is demonstrated.

1

A.A. Taflanidis, J.L. Beck, "Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher dimensional output", Earthquake Engineering and Structural Dynamics, 35, 1247-1267, 2006. doi:10.1002/eqe.581

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