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A functional integral formulation for stochastic engineering mechanics
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DOI：10.1016/j.probengmech.2026.103987.png)
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Abstract
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A versatile and computationally efficient functional-integral-based methodology is developed for the stochastic response determination of nonlinear engineering mechanics systems, in which uncertainty may arise either as additive excitation or through stochastic operators embedded within the governing equations. The formulation relies on the concept of the probability density functional, representing the infinite-dimensional analogue of the conventional probability density function (PDF), thereby enabling a continuum-level probabilistic description of stochastic fields without a priori discretization. Further, relying on this representation of stochastic fields, a general probabilistic–variational framework is developed, wherein the governing equations are enforced as functional constraints within a functional integral representation of the response PDF. Furthermore, the functional integral is approximated via a variational principle, based on the extremization of an appropriately defined stochastic action functional, yielding the most probable admissible field configuration. The methodology exhibits several significant advantages: a) the complexity of the original stochastic problem is substantially reduced, as it is recast as a standard deterministic boundary value problem for the most probable field configuration; b) the formulation enables the direct determination of the full response PDF, rather than being restricted to low-order statistical moments; c) both additive stochastic excitation and stochastic operators of general form (including nonlinear and nonlocal) can be treated in a unified and direct manner, while the classical Wiener path integral formulation is recovered as a special case when an explicit input–response transformation is available; and d) the framework provides natural localization capabilities, enabling the direct evaluation of marginal PDFs for response quantities of interest at prescribed spatial locations. The accuracy of the methodology is assessed through both a linear benchmark problem, for which exact solutions are recovered, and a geometrically nonlinear, statically indeterminate stochastic beam problem, where comparisons with Monte Carlo simulations demonstrate satisfactory agreement. Overall, the proposed functional integral formulation provides a fundamentally new pathway for addressing stochastic engineering mechanics problems and a foundation for future developments in stochastic finite element approaches.
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