Evolution of probability densities in the phase space for reliability analysis of non-linear structures

In this paper a numerical solution procedure is presented to compute the evolution of the probability density in the state space for a single degree of freedom system excited by white noise. Furthermore, crossing rates and first excursion rates are evaluated for required thresholds. This work aims at an accurate reference solution, providing a basis for the comparison with Monte Carlo based procedures for the first passage problem of non-linear systems.The densities are represented as a continuous surface in logarithmic scale, specified in terms of discrete points in the state space. For interpolation, the state space is discretized by isoparametric elements, which allows one to represent piecewise quadratic functions and hence to represent Gaussian densities exactly. The random loading is represented by impulses, which allows for a separation of the dynamics into a loading and free motion part within small time steps. The impulsive loading is applied independently at discrete instants by a convolution in the direction of the velocity. The excitation can be additive and multiplicative white noise. It is shown here that a linear transformation has the capability to represent the free motion of the densities within the short time step, i.e. π(t+Δt)=Aπ(t)π(t+Δt)=Aπ(t), where ππ is the vector of the probability densities and the matrix AA captures the non-linear dynamics of the system in the state space.For a verification of the algorithm, the analytically known crossing rates of a Duffing oscillator under stationary excitation are compared. In a next step, absorbing boundaries are introduced to study first excursion rates. The results are then compared with direct Monte Carlo simulation.