A new method for transient simulation of integrated circuits has been developed and investigated. The method utilizes expansion of circuit variables into Chebyshev series. A prototype computer simulation program based on this technique has been implemented and applied in the transient simulation of several MOS circuits. The results have been compared with those generated by SPICE. The method has been also combined with the waveform relaxation technique. Several algorithms were developed using the Gauss-Seidel and Gauss-Jacobi iterative procedures. The algorithms based on the Gauss-Seidel iterative procedure were implemented in the prototype software. They offer substantial CPU time savings in comparison with SPICE without compromising the accuracy of solutions. A description of the prototype computer simulation program and a summary of the results of simulation experiments are included.

Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter method combining recursive subdivision and edge detection techniques. For interpolation by rational functions with free poles, a novel method is presented. When the interpolation nodes are roots of unity or Chebyshev points the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Computed rational interpolants are compared with the behaviour expected from the theory of convergence of these approximants, and the difficulties due to truncated arithmetic are explained. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is characterised by the behaviour of the singular values of the linear system associated with the rational interpolation problem. Finally, new Remez algorithms for the computation of best polynomial and rational approximations are presented. These algorithms rely on interpolation, for the computation of trial functions, and on Chebfun, for the location of trial references. For polynomials, the algorithm is particularly robust and efficient, and we report experiments with degrees in the thousands. For rational functions, we clarify the numerical issues that affect its application.

In this thesis, a 2D CHebyshev spectral domain decomposition method is developed for simulating the generation and propagation of internal waves over a topography. While the problem of stratified flow over topography is by no means a new one, many aspects of internal wave generation and breaking are still poorly understood. This thesis aims to reproduce certain observed features of internal waves by using a Chebyshev collation method in both spatial directions. The numerical model solves the inviscid, incomprehensible, fully non-linear, non-hydrostatic Boussinesq equations in the vorticity-streamfunction formulation. A number of important features of internal waves over topography are captured with the present model, including the onset of wave-breaking at sub-critical Froude numbers, up to the point of overturning of the pycnoclines. Density contours and wave spectra are presented for different combinations of Froude numbers, stratifications and topographic slope. / by Arjun Jagannathan. / Thesis (M.S.C.S.)--Florida Atlantic University, 2012. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

The solution of initial value problems (IVPs) provides the evolution of dynamic system state history for given initial conditions. Solving boundary value problems (BVPs) requires finding the system behavior where elements of the states are defined at different times. This dissertation presents a unified framework that applies modified Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs. Existing methods for solving IVPs and BVPs have not been very successful in exploiting parallel computation architectures. One important reason is that most of the integration methods implemented on parallel machines are only modified versions of forward integration approaches, which are typically poorly suited for parallel computation. The proposed MCPI methods are inherently parallel algorithms. Using Chebyshev polynomials, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. Combining Chebyshev polynomials with Picard iteration, MCPI methods iteratively refine estimates of the solutions until the iteration converges. The developed vector-matrix form makes MCPI methods computationally efficient. The power of MCPI methods for solving IVPs is illustrated through a small perturbation from the sinusoid motion problem and satellite motion propagation problems. Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, prior to parallel implementation. Modifying the algorithm to do double integration for second order systems, and using orthogonal polynomials to approximate position states lead to additional speedups. Finally, introducing perturbation motions relative to a reference motion results in further speedups. The advantages of using MCPI methods to solve BVPs are demonstrated by addressing the classical Lambert’s problem and an optimal trajectory design problem. MCPI methods generate solutions that satisfy both dynamic equation constraints and boundary conditions with high accuracy. Although the convergence of MCPI methods in solving BVPs is not guaranteed, using the proposed nonlinear transformations, linearization approach, or correction control methods enlarge the convergence domain. Parallel realization of MCPI methods is implemented using a graphics card that provides a parallel computation architecture. The benefit from the parallel implementation is demonstrated using several example problems. Larger speedups are achieved when either force functions become more complicated or higher order polynomials are used to approximate the solutions.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 8 uppsatser.

Thesis (Ph. D.)--West Virginia University, 1998. / Title from document title page. Document formatted into pages; contains xi, 116 p. : ill. (some col.) Includes abstract. Includes bibliographical references (p. 94-98).

Orientadores: Jose Manoel Balthazar / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-04T04:06:35Z (GMT). No. of bitstreams: 1 Peruzzi_NelsonJose_D.pdf: 9438459 bytes, checksum: 1e95acc28fd5e0b87f7b964ca5a2f34e (MD5) Previous issue date: 2005 / Resumo: Neste trabalho, apresenta-se um novo método numérico para aproximar matriz de transição de estados (STM) para sistemas com coeficientes periódicos no tempo. Este método, é baseado na expansão polinomial de Chebyshev, no método iterativo de Picard e na transformação de Lyapunov-Floquet (L-F) e aplica-se na análise da dinâmica e o controle de sistemas lineares e periódicos. Para o controle, aplicam-se dois projetos para eliminar o comportamento caótico de sistemas periódicos no tempo. O primeiro, usa o projeto de controle realimentado baseado na aplicação da transformação L-F, e o objetivo do controlador é conduzir a órbita do sistema para um ponto fixo ou para uma órbita periódica. No segundo, utiliza-se o controle não-linear para bifurcação, e o objetivo, neste caso, é modificar (atrasar ou eliminar) as características de uma bifurcação ao longo de sua rota para o caos. Como exemplo, aplicou-se, com sucesso, a técnica para análise e o controle da dinâmica: num pêndulo com excitação paramétrica, no oscilador de Duffing, no sistema de Rõssler e sistema pêndulo duplo invertido. O método, também, mostrou-se satisfatório na análise e controle de um sistema monotrilho não ideal / Abstract: In thiswork, a new numericalmethodto approximatestatetransitionmatrix(STM) for systems with time-periodic coefficients is presented. This method is based on the expansion Chebyshev polinomials,on the Picard iterationand on the Lyapunov-Floquet transfonnation(transfonnationL-F). It is applied to the dynamicalanalysis and control of linear periodic systems.For the control, two projectsto eliminatethe chaoticbehaviorof time periodic systemsare applied.The first one, uses the feedbackcontroldesignbased on the L-F transfonnation,and the controller'sobjectiveis to drive the orbit of the systemto an equilibriumpoint or a periodicorbit. fu the secondone, the non-lineal control for bifurcations used, and the objective,in this case, is to modify (to put back or to eliminate) the characteristicsof a bifurcation along its route to chaos. As example, the technique for dynamical analysis and control was applied, successfully, to a pendulum with parametric excitement, the Duffing's oscillator,the Rõssler's systemand the inverteddoublependulum The methodwas, also, to be shownsatisfactoryin the analysisand controlof a monorailnon-idealsystem / Doutorado / Mecanica dos Sólidos e Projeto Mecanico / Doutor em Engenharia Mecânica

Wind power is a rapidly increasing resource of electrical power world-wide. With the increasing number of wind turbines installed one major concern is the noise they generate. Sometimes already built wind turbines have to be put down or down-regulated, when certain noise levels are exceeded, resulting in economical and environmental losses. Therefore, accurate sound propagation calculations would be beneficial already in a planning stage of a wind farm. A model that can account for varying wind speeds and complex terrains could therefore be of great importance when future wind farms are planned. In this report an extended version of the classical wave equation that allows for variations in wind speed and terrain is derived which can be used to solve complex terrain and wind settings. The equation are solved with the use of Fourier transforms and Chebyshev polynomials and a numerical code is developed. The numerical code is evaluated against test cases where analytical and simple solutions exist. Tests with no wind for both totally free propagation and with a ground surface is evaluated in both 2D and 3D settings. For these simple cases the developed code shows good agreement to analytical solutions if the computational domain is sufficiently large. More advanced test cases with wind and terrain is not evaluated in this report and needs further validation. If the sound pressure needs to be calculated for a large area, and if the frequency is high, the developed model has problems regarding computational time and memory. These problems could be solved by further development of the numerical code or by using other solution methods.

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Orientador: Vanessa Avansini Botta Pirani / Resumo: O presente trabalho tem como objetivo principal estudar o comportamento dos zeros de alguns tipos de polinômios auto-recíprocos gerados a partir de polinômios quaseortogonais de Chebyshev de ordens um e dois. Os zeros dos polinômios auto-recíprocos que construímos estão ligados aos zeros de polinômios quase-ortogonais. Os polinômios quaseortogonais podem ser obtidos a partir de uma sequência de polinômios ortogonais. Neste trabalho, usaremos os polinômios de Chebyshev para obter polinômios quase-ortogonais e usaremos resultados sobre o comportamento de zeros desses polinômios para obter informações sobre o comportamento dos zeros de polinômios auto-recíprocos. / Abstract: The main objective of this work is to study the behavior of the zeros of some classes of self-reciprocal polynomials related to Chebyshev quasi-orthogonal polynomials of order one and two. The zeros of self-reciprocal polynomials are linked to the zeros of quasiorthogonal polynomials, which can be obtained from a sequence of orthogonal polynomials. In this work we use the Chebyshev polynomials to obtain classes of quasi-orthogonal polynomials and from results on the behavior of their zeros, we obtain information about the zeros of some classes of self-reciprocal polynomials. / Mestre