Stationary Two‐State Variable Problems in Stochastic Mechanics

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Abstract
A numerical procedure is presented for the solution of stationary two‐state Markov‐process problems with a single source. The proposed solution method is an efficient splitting technique that combines finite element and finite difference methods to take advantage of the specific form of the governing differential equation. A standard one‐dimensional finite element technique using linear interpolation and weighting functions is used to determine the solution for each row, while a variable‐weighted finite difference scheme is used to step in the other direction. This method is then illustrated by determining the stationary response of the Duffing oscillator and the statistical moments of the time to reach a critical crack size for the stochastic fatigue crack growth problem. Excellent comparisons are shown between this method, previous analytical studies, and experimental results with a significant reduction in computer processing time and storage.

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