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Title:  Methods For Forward And Inverse Problems In Nonlinear And Stochastic Structural Dynamics 
Authors:  Saha, Nilanjan 
Advisors:  Roy, Debasish 
Keywords:  Structural Analysis (Civil Engineering) Stochastic Analysis (Civil engineering) Inverse Problems Nonlinear Oscillations Stochastically Driven Nonlinear Oscillators Nonlinear Oscillators  Linearization Locally Transversal Linearization (LTL) Girsanov Linearization Method Weak VarianceReduced Monte Carlo Simulation Extended Kalman Filter (EKF) Local Linearizations Stochastic Structural Dynamics 
Submitted Date:  Nov2007 
Series/Report no.:  G22181 
Abstract:  A main thrust of this thesis is to develop and explore linearizationbased numericanalytic integration techniques in the context of stochastically driven nonlinear oscillators of relevance in structural dynamics. Unfortunately, unlike the case of deterministic oscillators, available numerical or numericanalytic integration schemes for stochastically driven oscillators, often modelled through stochastic differential equations (SDEs), have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. We propose a few higherorder methods based on the stochastic version of transversal linearization and another method of linearizing the nonlinear drift field based on a Girsanov change of measures. When these schemes are implemented within a Monte Carlo framework for computing the response statistics, one typically needs repeated simulations over a large ensemble. The statistical error due to the finiteness of the ensemble (of size N, say)is of order 1/√N, which implies a rather slow convergence as N→∞. Given the prohibitively large computational cost as N increases, a variance reduction strategy that enables computing accurate response statistics for small N is considered useful. This leads us to propose a weak variance reduction strategy. Finally, we use the explicit derivativefree linearization techniques for state and parameter estimations for structural systems using the extended Kalman filter (EKF). A twostage version of the EKF (2EKF) is also proposed so as to account for errors due to linearization and unmodelled dynamics.
In Chapter 2, we develop higher order locally transversal linearization (LTL) techniques for strong and weak solutions of stochastically driven nonlinear oscillators. For developing the higherorder methods, we expand the nonlinear drift and multiplicative diffusion fields based on backward Euler and Newmark expansions while simultaneously satisfying the original vector field at the forward time instant where we intend to find the discretized solution. Since the nonlinear vector fields are conditioned on the solution we wish to determine, the methods are implicit. We also report explicit versions of such linearization schemes via simple modifications. Local error estimates are provided for weak solutions.
Weak linearized solutions enable faster computation visàvis their strong counterparts. In Chapter 3, we propose another weak linearization method for nonlinear oscillators under stochastic excitations based on Girsanov transformation of measures. Here, the nonlinear drift vector is appropriately linearized such that the resulting SDE is analytically solvable. In order to account for the error in replacing of nonlinear drift terms, the linearized solutions are multiplied by scalar weighting function. The weighting function is the solution of a scalar SDE(i.e.,RadonNikodym derivative). Apart from numerically illustrating the method through applications to nonlinear oscillators, we also use the Girsanov transformation of measures to correct the truncation errors in lower order discretizations.
In order to achieve efficiency in the computation of response statistics via Monte Carlo simulation, we propose in Chapter 4 a weak variance reduction strategy such that the ensemble size is significantly reduced without seriously affecting the accuracy of the predicted expectations of any smooth function of the response vector. The basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the associated stochastic forcing functions through variancereduced functions. In the process, the additional computational cost due to system augmentation is generally far less besides the accrued advantages due to a drastically reduced ensemble size. The variance reduction scheme is illustrated through applications to several nonlinear oscillators, including a 3DOF system.
Finally, in Chapter 5, we exploit the explicit forms of the LTL techniques for state and parameters estimations of nonlinear oscillators of engineering interest using a novel derivativefree EKF and a 2EKF. In the derivativefree EKF, we use oneterm, Euler and Newmark replacements for linearizations of the nonlinear drift terms. In the 2EKF, we use bias terms to account for errors due to lower order linearization and unmodelled dynamics in the mathematical model. Numerical studies establish the relative advantages of EKFDLL as well as 2EKF over the conventional forms of EKF.
The thesis is concluded in Chapter 6 with an overall summary of the contributions made and suggestions for future research. 
URI:  http://hdl.handle.net/2005/608 
Appears in Collections:  Civil Engineering (civil)

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