Wiener Path Integral Formulation in Modal Space for Efficient Stochastic Response Determination of Structural Systems

Abstract A Wiener path integral (WPI) formulation in modal space is developed for the efficient stochastic response determination of nonlinear dynamical systems. Specifically, the governing equations of motion are transformed into the classical modal basis of the associated linear system, while the original nonlinear contributions are retained in full through their projection onto the modal coordinates. Next, a mixed fixed/free boundary-condition formulation is applied directly in modal space, whereby only a selected subset of modal coordinates is prescribed, and the remaining modal variables are treated as free optimization parameters. In this manner, the computational complexity of the WPI analysis depends only on the dimension of the retained modal subspace rather than on the dimension of the complete physical system. Notably, the developed methodology constructs a low-dimensional probabilistic representation in modal space rather than directly in physical coordinates. Owing to the explicit transformation between modal and physical variables, the resulting modal probability density function (PDF) constitutes a reusable low-dimensional stochastic representation from which arbitrary physical-coordinate marginal and joint PDFs can be recovered without performing additional WPI-based analyses or constructing separate inverse mappings. Further, an approximate probabilistic correction based on Gaussian convolution and low-order covariance information is introduced to account for the contribution of neglected-modal coordinates while preserving the dominant non-Gaussian response characteristics captured by the retained modal variables. Two representative examples pertaining to a chain-like nonlinear system subjected to both white-noise and nonwhite Gaussian excitation are considered for validation. Comparisons with pertinent Monte Carlo simulation data demonstrate that the developed methodology accurately recovers multiple physical-coordinate marginal and joint PDFs from a single modal-coordinate WPI-based analysis. Consequently, different physical quantities of interest can be evaluated without performing additional WPI-based computations, while preserving the substantial computational savings afforded by the reduced probabilistic representation.