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The nonlinear dynamic analysis of aeroelastic systems is a topic that has been covered extensively in the literature. The two main sources of nonlinearities in such systems, structural and aerodynamic nonlinearities, have analyzed numerically, analytically and experimentally. In this research project, the aerodynamic nonlinearity arising from the stall behavior of an airfoil is analyzed. Experimental data was used to fit a piecewise linear curve to describe the lift versus angle of attack behavior for a NACA 0012 2 DOF airfoil. The piecewise linear system equilibrium points are found and their stability analyzed. Bifurcations of the equilibrium points are analyzed and applying continuation software the bifurcation diagrams of the system are shown. Border collision and rapid/Hopf bifurcations are the two main bifurcations of the system equilibrium points. Chaotic behavior represented in the intermittent route to chaos was also observed and shown as part of the system dynamic analysis. Finally, sets of initial conditions associated with the system behavior are defined. Numerical simulations are used to show those sets, their subsets and their behavior with respect to the system dynamics. Poincaré sections are produced for both the periodic and the chaotic solutions of the system. The proposed piecewise linear model introduced some interesting dynamics for such systems. The introduction of the border collision bifurcation and the existence of periodic and chaotic solutions for the system are some examples. The model also enables the understanding of the mapping of initial conditions as it defines clear boundaries with different dynamics that can be used as Poincaré sections to understand further the global system dynamics. One of the constraints of the system is its validity as it is dependent on the range of the experimental data used to generate the model. This can be addressed by adding more linear pieces to the system to cover a wider range of the dynamics.

In this thesis, we discuss the development, implementation and testing of a piecewise linear (PWL) continuous Galerkin finite element method applied to the threedimensional diffusion equation. This discretization is particularly interesting because it discretizes the diffusion equation on an arbitrary polyhedral mesh. We implemented our method in the KULL software package being developed at Lawrence Livermore National Laboratory. This code previously utilized Palmer's method as its diffusion solver, which is a finite volume method that can produce an asymmetric coefficient matrix. We show that the PWL method produces a symmetric positive definite coefficient matrix that can be solved more efficiently, while retaining the accuracy and robustness of Palmer's method. Furthermore, we show that in most cases Palmer's method is actually a non-Galerkin PWL finite element method. Because the PWL method is a Galerkin finite element method, it has a firm theoretical background to draw from. We have shown that the PWL method is a well-posed discrete problem with a second-order convergence rate. We have also performed a simple mode analysis on the PWL method and Palmer's method to compare the accuracy of each method for a certain class of problems. Finally, we have run a series of numerical tests to uncover more properties of both the PWL method and Palmer's method. These numerical results indicate that the PWL method, partially due to its symmetric matrix, is able to solve large-scale diffusion problems very efficiently.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 106-108). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Oregon State University, 1996. / Typescript (photocopy). Includes bibliographical references (leaves 68-69). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1973. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

This dissertation applies policy improvement and successive approximation or value iteration to a general class of Markov decision processes with discounted costs. In particular, a class of Markov decision processes, called piecewise-linear, is studied. Piecewise-linear processes are characterized by the property that the value function of a process observed for one period and then terminated is piecewise-linear if the terminal reward function is piecewise-linear. Partially observable Markov decision processes have this property. It is shown that there are e-optimal piecewise-linear value functions and piecewise-constant policies which are simple. Simple means that there are only finitely many pieces, each of which is defined on a convex polyhedral set. Algorithms based on policy improvement and successive approximation are developed to compute simple approximations to an optimal policy and the optimal value function. / Business, Sauder School of / Graduate

The piecewise linear group is the set of all piecewise linear orientation preserving homeomorphisms from the interval to itself under the operation of composition. We present here a complete set of invariants to classify the conjugacy classes of this group. Our approach to this problem relies on the factorization of elements into elements having only a single breakpoint.

The (nonlinear) response behavior of a piecewise-linear system is investigated under both deterministic and stochastic excitations in this study. A semi-analytical procedure is developed to predict the system behavior by evaluating the joint probability density functions of the Fokker-Planck equations of the stochastic piecewise-linear system under random excitations via a path-integral solution procedure. From an analysis of the deterministic system, nonlinear system behavior is derived for further analysis of the corresponding stochastic system. The influence of variations in other parameters on the response behavior are also examined. In the stochastic system analysis, which represents a more realistic approach to understanding the system behavior, a Markov process approach is used. By introducing random perturbations in the harmonic excitation, the stochastic responses are examined and compared to those of the corresponding deterministic system. Stability of the various responses including regular and chaotic motions, is investigated by varying the noise intensity. Routes to chaos in period-doubling processes with and without external random perturbations are studied and compared. Stationarity and ergodicity of the chaotic responses of the system are examined via time domain and probability domain simulations. Potential coexisting responses of the piecewise-linear system are examined via the path-integral solutions. It is found that the path-integral solutions can provide global information of the system behavior in the probability domain, and can also verify the potential coexisting responses and discern the relative strength of the coexisting response attractors. / Graduation date: 1996

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