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Uncertainty Quantification of Dynamical Systems and Stochastic Symplectic Schemes Open Access


Other title
stochastic symplectic integrator
Uncertainty Quantification
Stochastic differential equations
Type of item
Degree grantor
University of Alberta
Author or creator
Deng, Jian
Supervisor and department
Christina Adela Anton (Department of Mathematical and Statistical Sciences)
Yau Shu Wong (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Henry Van Roessel (Department of Mathematical and Statistical Sciences)
JianHong Wu (Department of Mathematical and Statistical Sciences)
Bin Han (Department of Mathematical and Statistical Sciences)
Yau Shu Wong (Department of Mathematical and Statistical Sciences)
Chong-Qing Ru (Department of Mechanical Engineering)
Christina Adela Anton (Department of Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
It has been known that for some physical problems, a small change in the system parameters or in the initial/boundary conditions could leas to a significant change in the system response. Hence, it is of importance to investigate the impact of uncertainty on dynamical system in order to fully understand the system behavior. In this thesis, numerical methods used to simulate the effect of random/stochastic perturbation on dynamical systems are studied. In the first part of this thesis, an aeroelastic system model representing an oscillating airfoil in pitch and plunge with random variations in the flow speed, the structural stiffness terms and initial conditions are concerned. Two approaches, stochastic normal form and stochastic collocation method, are proposed to investigate the Hopf bifurcation and the secondary bifurcation behavior, respectively. Stochastic normal form allows us to study analytically the Hopf bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation; while numerical simulations demonstrate the effectiveness of stochastic collocation method for long term computation and discontinuous problems. In the second part of this work, we focus the construction of efficient and robust computational schemes for stochastic system, and the stochastic symplectic schemes for stochastic Hamiltonian system are developed. A systematic procedure to construct symplectic numerical schemes for stochastic Hamiltonian systems is presented. The approach is an extension to the stochastic case of the methods based on generating functions. The idea is also extended to the symplectic weak scheme construction. Theoretical analysis of the convergence is reported for strong/weak symplectic integrators. The numerical simulations are carried out to confirm that the symplectic methods are efficient computational tools for long-term behaviors. Moreover, the coefficients of the generating function are invariant under permutations for the stochastic Hamiltonian system preserving Hamiltonian functions. As a consequence the high-order symplectic weak and strong methods have simpler forms than the Taylor expansion schemes with the same order.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
C. Anton, J. Deng, and Y.S. Wong. Hopf bifurcation analysis of an aeroelastic model using stochastic normal form. Journal of Sound and Vibration, 331:3866 – 3886.J. Deng, C. A. Popescu, and Y. S. Wong. Application of the stochastic normal form for a nonlinear aeroelastic model. In Proceedings of the 52nd AIAA/ ASME/ ASCE/ AHS/ ASC/ Structures, Structural Dynamics, and Materials Conference, Denver, U.S., 2011. 42-58.J. Deng, C. A. Popescu, and Y. S. Wong. Stochastic collocation method for secondary bifurcation of a nonlinear aeroelastic system. Journal of Sound and Vibration, 330:3006–3023, 2011.

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