2024 Theses Doctoral

# Nonlinear stochastic dynamics of structural systems: A general and computationally efficient Wiener path integral formalism

This dissertation introduces advances in the Wiener path integral (WPI) technique for determining efficiently and accurately the stochastic response of diverse nonlinear dynamical systems.

First, a novel, general, formalism of the WPI technique is developed to account, in a direct manner, also for systems with non-Markovian response processes. Specifically, the probability of a path and the associated transition probability density function (PDF) corresponding to the Wiener excitation process are considered. Next, a functional change of variables is employed, in conjunction with the governing stochastic differential equation, for deriving the system response joint transition PDF as a functional integral over the space of possible paths connecting the initial and final states of the response vector. In comparison to alternative derivations in the literature, the herein-developed formalism does not require the Markovian assumption for the system response process. Overall, the veracity and mathematical legitimacy of the WPI technique to treat also non-Markovian system response processes are demonstrated. In this regard, nonlinear systems with a history-dependent state, such as hysteretic structures or oscillators endowed with fractional derivative elements, can be accounted for in a direct manner—that is, without resorting to any ad hoc modifications of the WPI technique pertaining, typically, to employing additional auxiliary filter equations and state variables.

Next, a reduced-order WPI formulation is introduced for efficiently determining the stochastic response of diverse nonlinear systems with fractional derivative elements. This formulation can be also construed as a dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the proposed technique can determine, directly, any lower-dimensional joint response PDF corresponding to a subset only of the response vector components. This is accomplished by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in, few only, specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system.

Further, an extrapolation approach within the WPI technique is developed that significantly enhances the computational efficiency of the technique without, practically, affecting the associated degree of accuracy. Overall, the WPI technique treats the system response joint transition PDF as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector.

Next, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited, and it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently and accurately the PDF values at neighboring points without the need for considering additional BVPs.

Last, a joint time-space extrapolation approach within WPI technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of the above space-domain extrapolation scheme to account also for the temporal dimension. Specifically, it is shown that information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs.

In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time-space domain at practically zero additional computational cost. Compared to an alternative brute-force implementation of the WPI technique, and to a standard Monte Carlo simulation (MCS) solution treatment, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude.

Several representative numerical examples are considered to demonstrate the reliability of the developed techniques. Juxtapositions with pertinent MCS data are included as well.

## Subjects

## Files

- Mavromatis_columbia_0054D_18373.pdf application/pdf 3.06 MB Download File

## More About This Work

- Academic Units
- Civil Engineering and Engineering Mechanics
- Thesis Advisors
- Kougioumtzoglou, Ioannis A.
- Degree
- Ph.D., Columbia University
- Published Here
- May 15, 2024