• Journal of the Korean Society for Precision Engineering
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• v.12 no.5
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• pp.57-68
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• 1995

A general procedure for the numerical solution of coupled, nonlinear, differential two-point boundary-value problems, solutions of which are crucial to the controller design, has been developed and demonstrated. A fixed-end-points, free-terminal-time, optimal-control problem, which is derived from Pontryagin's Maximum Principle, is solved by an extension of Davidenko's method, a differential form of Newton's method, for algebraic root finding. By a discretization process like finite differences, the differential equations are converted to a nonlinear algebraic system. Davidenko's method reconverts this into a pseudo-time-dependent set of implicitly coupled ODEs suitable for solution by modern, high-performance solvers. Another important advantage of Davidenko's method related to the time-optimal problem is that the terminal time can be computed by treating this unkown as an additional variable and sup- plying the Hamiltonian at the terminal time as an additional equation. Davidenko's method uas used to produce optimal trajectories of a single-degree-of-freedom problem. This numerical method provides switching times for open-loop control, minimized terminal time and optimal input torque sequences. This numerical technique could easily be adapted to the multi-point boundary-value problems.

• The Journal of Korean Institute of Electromagnetic Engineering and Science
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• v.14 no.1
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• pp.1-7
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• 2003

Leaky modes on a circular dielectric rod are investigated from the precisely determined normalized complex propagation constants using Davidenko's complex root finding technique. Below the cutoff frequency of the guided mode, distinct frequency regions that have unique properties are observed, such as nonphysical region, antenna mode region, reactive mode region, and spectral gap region.

• KIEE International Transactions on Electrophysics and Applications
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• v.3C no.5
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• pp.199-206
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• 2003

Numerical solutions to complex characteristic equations are quite often required to solve electromagnetic wave problems. In general, two traditional complex root search algorithms, the Newton-Raphson method and the Muller method, are used to produce such solutions. However, when utilizing these two methods, the choice of the initial iteration value is very sensitive, otherwise, the iteration can fail to converge into a solution. Thus, as an alternative approach, where the selection of the initial iteration value is more relaxed and the computation speed is high, Davidenko's method is used to determine accurate complex propagation constants for leaky circular symmetric modes in circular dielectric rod waveguides. Based on a precise determination of the complex propagation constants, the leaky mode characteristics of several lower-order circular symmetric modes are then numerically analyzed. In addition, no modification of the characteristic equation is required for the application of Davidenko's method.

• Journal of electromagnetic engineering and science
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• v.5 no.2
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• pp.72-79
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• 2005

The leaky dispersion characteristics of a circular dielectric rod were investigated using Davidenko's method for several lower-order transverse magnetic(TM) modes. The normalized complex propagation constants were precisely determined and their tolerances below $10^{-10}$ compared with zero for both real and imaginary parts. It was also checked whether the normalized complex propagation constants obtained represented forward leaky waves. The leaky modes existing below the cutoff frequency of the guided mode were classified as a nonphysical mode, reactive mode, antenna mode, and spectral gap based on a precise determination of the complex propagation constants. Finally, the effects of the dielectric constant and radius of the dielectric rod on the leaky dispersion characteristics were also considered.

• Structural Engineering and Mechanics
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• v.36 no.5
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• pp.529-544
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• 2010

Nonlinear equations of structures are generally solved numerically by the iterative solution of linear equations. However, this iterative procedure diverges when the tangent stiffness is ill-conditioned which occurs near limit points. In other words, a major challenge with simple iterative methods is failure caused by a singular or near singular Jacobian matrix. In this paper, using the Newton-Raphson algorithm based on Davidenko's equations, the iterations can traverse the limit point without difficulty. It is argued that the propose algorithm may be both more computationally efficient and more robust compared to the other algorithm when tracing path through severe nonlinearities such as those associated with structural collapse. Two frames are analyzed using the proposed algorithm and the results are compared with the previous methods. The ability of the proposed method, particularly for tracing the limit points, is demonstrated by those numerical examples.

• Proceedings of the Korea Electromagnetic Engineering Society Conference
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• 2002.11a
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• pp.356-360
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• 2002

Leaky modes on a circular dielectric rod are investigated from the precisely determined normalized complex propagation constants using Davidenko's complex root finding technique. Below the cutoff frequency of the guided mode, distinct frequency regions that have unique properties are observed, such as nonphysical region, antenna mode region, reactive mode region, and spectral gap region. The effects of tro design parameters, dielectric constants and the radius of the rod, to the leaky mode characteristics are also considered.