Quartic Wiener Path Integral Approximation for Stochastic Response Determination of Nonlinear Systems

A novel Wiener path integral (WPI) technique exhibiting an enhanced degree of accuracy is developed for determining the stochastic response of nonlinear structural systems. Specifically, first, the system response joint transition probability density function (PDF) is represented by an appropriate functional integral series expansion. Ordinarily, this is approximated in the literature by considering only the first term relating to the most probable path. Alternatively, higher-order terms in the expansion can be also accounted for. This yields a localized, state-dependent, coefficient in the approximate calculation of the system response PDF that enhances the accuracy degree of the technique. In this regard, a quadratic WPI approximation was developed recently, where up to second variation terms were considered in the related expansion. Herein, the degree of accuracy exhibited by the WPI technique is increased further by developing a quartic WPI approximation. This is done by considering up to fourth variation terms in the associated functional integral expansion. In fact, following nontrivial analytical calculations involving multidimensional integration in the discretized time domain, the state-dependent coefficient of the quartic approximation is expressed as a correction, toward enhanced accuracy, of the quadratic approximation respective coefficient. Two illustrative examples are considered for demonstrating the enhanced accuracy exhibited by the quartic WPI approximation. These pertain to a Duffing nonlinear oscillator with a bimodal response PDF, and to an oscillator with quadratic and cubic stiffness nonlinearities. Juxtapositions with pertinent Monte Carlo simulation data are included as well.