Reliability-based optimal design of linear dynamical systems under stochastic stationary excitation and model uncertainty

The reliability-based design under stochastic stationary excitation of linear dynamical systems with higher-dimensional output is discussed in this paper. An analytical approximation is initially presented for the calculation of this reliability for applications for which the model parameters are treated as known. This approach is based on a computationally efficient approximation to the conditional out-crossing rate for higher-dimensional vectors. The extension to cases where the model parameters are treated as unknown and are characterized by probability models is then addressed. This requires the evaluation of a multidimensional probability integral over the uncertain model parameter space and a methodology based on a Taylor series expansion around the local maxima of the integrand, called design points, is considered for this purpose. A novel approach is developed for addressing cases with multiple design points. The estimation of this probability integral by Monte Carlo simulation with importance sampling is also considered. Implementation details for applications to reliability-based design problems are extensively discussed. In particular, the effect of the errors introduced by the various, numerical and asymptotic, approximations is addressed and methods for reducing their relative or absolute importance are presented. Also practical guidelines are provided for improvement of the computational efficiency for using the analytical reliability approximation within the algorithm that searches for the optimal system design configuration.