Browsing by Author "P. A. Gremaud, Committee Member"

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Numerical Methods for the Wigner-Poisson Equations
(2005-10-06) Lasater, Matthew; M. Shearer, Committee Member; R. H. Martin, Committee Member; C. T. Kelley, Committee Chair; D. L. Woolard, Committee Member; P. A. Gremaud, Committee Member
This thesis applies modern numerical methods to solve the Wigner-Poisson equations for simulating quantum mechanical electron transport in nanoscale semiconductor devices, in particular, a resonant tunneling diode (RTD). The goal of this dissertation is to provide engineers with a simulation tool that will verify earlier numerical results as well as improve upon the computational efficiency and resolution of older simulations. Iterative methods are applied to the linear and nonlinear problems in these simulations to reduce the computational memory and time required to calculate solutions. Initially the focus of the research involved updating time-integration techniques, but this switched to implementing continuation methods for finding steady-state solutions to the equations as the applied voltage drop across the device varied. This method requires the solution to eigenvalue problems to produce information on the RTD's time-dependent behavior such as the development of current oscillation at a particular applied voltage drop. The continuation algorithms/eigensolving capabilities were provided by Sandia National Laboratories' software library LOCA (Library of Continuation Algorithms). The RTD simulator was parallelized, and a preconditioner was developed to speed-up the iterative linear solver. This allowed us to use finer computational meshes to fully resolve the physics. We also theoretically analyze the steady-state solutions of the Wigner-Poisson equations by noting that the solutions to the steady-state problems are also solutions to a fixed point problem. By analyzing the fixed point map, we are able to prove some regularity of the steady-state solutions as well provide a theoretical explanation for the mesh-independence of the preconditioned linear solver.
Optimal Control and Shape Design: Theory and Applications
(2003-10-05) Lewis, Brian M.; P. A. Gremaud, Committee Member; Z. Li, Committee Member; H. T. Banks, Committee Member; Hien T. Tran, Committee Chair
This work focuses on the spectrum of problems connected with the analysis and the development of computational tools and models for engineering and scientific applications. This includes: (i) reduced order modeling techniques; (ii) linear and nonlinear feedback control design methodologies and real-time implementation; and (iii) shape optimization techniques. Excluding shape optimization techniques, most of the research herein can be seen as extensions of linear quadratic regulation (LQR) techniques. First, we consider the synthesis of control methodologies for the attenuation of beam vibrations caused by a narrow-band exogenous force. By a narrow-band exogenous force we mean periodic force over a narrow frequency band or an exact harmonic. The control methods under consideration are based on the minimization of two specific quadratic cost functionals. One of these cost functionals is a typical time domain cost functional constrained by an affine plant. The other is a cost functional that is frequency dependent. These control methods have been used successfully in various applications but this investigation differs in that it emphasizes the development of real-time control methodologies based on reduced order models derived from physical first principles. Specifically, an integral component of this research is the proper orthogonal decomposition (POD) reduction technique and its application to real-time control of beam vibrations. The second LQR extension involves a particular nonlinear control methodology that mimics standard LQR formulation. State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and state estimators for a broad class of nonlinear regulator problems. The technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Then LQR techniques are used on the state-dependent coefficients to formulate a suboptimal control law. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This work addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. A previous numerical method, which is based on the Taylor series, works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method, introduced here, can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and feasibility of the SDRE technique for the design of nonlinear compensator based feedback controllers. Finally, this work also includes an optimization technique in which the objective is attained via alterations to the physical geometry of the system. This optimization framework, to be considered in the context of electron guns, is known as optimal shape design. Optimal shape design has been used in a number of applications including wing design, magnetic tape design, and nozzle design, among others. In this investigation, we use the methods of shape optimization to design the shape of the cathode of an electron gun. The dynamical equations modeling the electron particle path as well as the generalized shape optimization problem will be presented. Illustrative examples of the technique on gun designs that were previously limited to spherical cathodes will be given.
Temporal and Pseudo-Temporal Numerical Integration Methods
(2002-10-28) Coffey, Todd Stirling; C. T. Kelley, Committee Member; C. S. Woodward, Committee Member; D. S. McRae, Committee Member; M. Shearer, Committee Member; P. A. Gremaud, Committee Member
Numerical methods for integrating partial differential equations are used in nearly every scientific field. In this dissertation we study two types of numerical integration methods, transient methods and pseudo-transient methods. Transient methods for partial differential equations look for time-accurate solutions that explain the evolution of the equation (although a steady state solution may evolve). Pseudo-transient methods look for steady-state solutions of partial differential equations while paying attention to the transient behavior to aid in stability. In contrast, root-finding methods, e.g. line-search methods, look only for a steady-state solution often not paying attention at all to the transient behavior of the problem. Pseudo-transient continuation is a method for solving steady state solutions of partial differential equations, and is used when traditional line-search methods fail to converge or converge to non-physical solutions. The method is a hybrid between implicit Euler and Newton's method where variable step-sizes are used to transfer from one method to the other. We demonstrate the performance of pseudo-transient continuation both numerically and theoretically on a variety of problems. We extend the global convergence theory, which currently covers a class of ordinary differential equations, to include a class of semi-explicit index-1 differential-algebraic equations. We also studied CVode, a transient code for solving nonlinear partial differential equations. In this work, we explain how CVode was extended to allow for a two-grid nonlinear solver. The two-grid solver coarsens a given mesh and solves the nonlinear problem on the coarse mesh, which is then moved back to the fine mesh for refining. This approach can be less expensive than computing the full nonlinear solution on the fine mesh. We explore some of the theoretical and computational issues involved in implementing this method for a radiative transfer problem as might be seen in stellar fusion.