A Wiener Path Integral Formalism for Treating Nonlinear Systems with Non-Markovian Response Processes

Nonlinear stochastic dynamics of an array of coupled micromechanical oscillators

“…In this regard, the L N 2 boundary value problems required to be solved by the standard formulation of the technique decrease to L p q + problems only, where L 30 < < 50 is a reasonable range of values for various diverse engineering applications. [16][17][18][19] Note that for small values of p q + relative to N 2 , it has been shown 31 that the technique becomes orders of magnitude more efficient than an alternative MCS solution treatment.…”

“…This capability is particularly advantageous for problems where the interest lies in few $$(p+q)$$ specific DOF whose stochastic response is critical for the design and optimization of the $$2N$$‐DOF system. In this regard, the $${L}^{2N}$$ boundary value problems required to be solved by the standard formulation of the technique decrease to $${L}^{p+q}$$ problems only, where $$30\lt L\lt 50$$ is a reasonable range of values for various diverse engineering applications 16–19 . Note that for small values of $$p+q$$ relative to $$2N$$, it has been shown 31 that the technique becomes orders of magnitude more efficient than an alternative MCS solution treatment.…”

“…In this section, the WPI technique developed by Kougioumtzoglou and coworkers 12,13,16–22 for determining the stochastic response of diverse dynamical systems is employed for computing the joint response PDF of the array of coupled microbeams presented in Section 2.1. Specifically, the reduced‐order WPI formulation developed by Petromichelakis and Kougioumtzoglou 31 is utilized next for determining any $$p$$‐dimensional joint response PDF of the system of Equation (), where $$p\le 2N$$.…”

“…In this regard, the most probable path, which is used for determining the system response joint transition PDF approximately, is computed by solving a functional minimization problem that takes the form of a deterministic boundary value problem 17 . It is remarked that the WPI technique is capable of treating systems exhibiting diverse nonlinear/hysteretic behaviors 18,19 and subjected to non‐white and non‐Gaussian stochastic excitations 20 . Further, it was shown by Psaros et al 21 and by Katsidoniotaki et al 22 that the associated computational cost can be reduced drastically by using sparse representations for the system response PDF in conjunction with compressive sampling concepts and tools 23 …”