Range Of Step Sizes Research Articles

An Energy-Based General Method for the Optimum Synthesis of Mechanisms

The aim of the present paper is to set forth a method for the optimum synthesis of planar mechanisms. The method in question can be applied in the case of any mechanism and kinematic synthesis (function, path generation, rigid-body guidance, or combination of these). The mechanisms are discretized with bar-finite elements that facilitate the computation of the geometric matrix, which is a stiffness matrix. The error function is based on the elastic energy accumulated by the mechanism when it is compelled to fulfill exactly the synthesis data. Thus during the iterative process the elements of the mechanism may be considered deformable. The energy computation for each synthesis datum requires the solution of the nonlinear equilibrium position. This problem is solved with the help of the geometric matrix and the force vector of the deformed system. The minimization of the error function is based on Newton’s method, with a semianalytic approach. Analytic and finite-difference concepts are used together in the calculation of the gradient vector and of the second-derivative matrix. The method has proved very stable for a wide range of step sizes. There is convergence to a minimum even when the start mechanism is far from a solution. Examples with different mechanisms and syntheses are also provided.

Modelling tropospheric ducting effects on VHF/UHF propagation

The application of the parabolic equation method to range-independent and range-dependent tropospheric propagation modelling problems is demonstrated. The parabolic equation is an approximation to the Helmholtz wave equation which allows progressive calculation of the propagated electromagnetic field as solution is stepped out in range. The solution method described is derived from a technique developed by R.H. Hardin and F.D. Tappert (1973) for application to underwater acoustics problems. This split step Fourier solution involves the discretization of the field with respect to height and the use of the fast Fourier transform at successive range points. The solution is stable, with errors being dependent on wavenumber, range step size, and the gradients of refractive index with respect to height and range. Advantages of this method over other methods used to evaluate the effects of ducting layers on tropospheric propagation are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Moments of an uncertain linear oscillator: Errors due to time-discretized approximations to white-noise excitation

Several first- and second-order moments are studied for an underdamped single-degree-of-freedom linear oscillator with arbitrarily large stiffness uncertainty. General analytical relations for the moments are found and numerically integrated in a time-discretized framework when the excitation is either mean-zero Gaussian white noise, or one of two discretized approximations to white noise: a stepwise constant load or a discrete impulse load (band-limited white noise). Several aspects of moment computation are addressed. These include: the effect of oscillator uncertainty on the range of step sizes over which either discretized load model approximates white noise; the relative convergence rates of the moments computed using the discretized load models to the true moments; and the behavior of the approximate moments within each step. This latter topic is enlightening since the manner in which step-to-step error accumulation occurs becomes clear.

An efficient enhancement of finite-difference implementations for solving parabolic equations

The parabolic approximation method is widely recognized as useful for accurately analyzing and predicting sound transmission intensity in diverse ocean environments. One reason for its attractiveness is that solutions are marched in range, thereby avoiding the large internal storage requirements when using the full wave equation. Present finite-difference implementations employ a range step size that is prescribed by either the user or the code and that remains fixed for the duration of the computation. An algorithm is presented in which the range step is adaptively selected by a procedure within a version of the implicit finite-difference (IFD) implementation of the parabolic approximation. An error indicator is computed at each range step, and its value is compared to an error tolerance window that is readily specified by the user. If the error indicator falls outside this window, a new range step size is computed and used until the error indicator again leaves the tolerance window. Furthermore, for a given tolerance, this algorithm generates a range step size that is optimal in a specified sense and that often leads to large decreases in run time. Additional related modifications to the IFD implementations are discussed. Several examples are presented that illustrate the efficacy of the enhanced algorithm.

An efficient enhancement of finite‐difference implementations for solving parabolic equations

The parabolic approximation method is widely recognized as useful for accurately analyzing and predicting sound transmission intensity in diverse ocean environments. One reason for its attractiveness is that solutions are marched in range, thereby avoiding the large internal storage requirements when using the full wave equation. Present finite‐difference implementations employ a range step size that is prescribed by either the user or the code and that remains fixed for the duration of the computation. An algorithm is presented in which the range step is adaptively selected by a procedure within a version of the implicit finite‐difference (IFD) implementation of the parabolic approximation. An error indicator is computed at each range step, and its value is compared to an error tolerance window that is readily specified by the user. If the error indicator falls outside this window, a new range step size is computed and used until the error indicator again leaves the tolerance window. Furthermore, for a given tolerance, this algorithm generates a range step size that is optimal in a specified sense and that often leads to large decreases in run time. Additional related modifications to the IFD implementations are discussed. Several examples are presented that illustrate the efficacy of the enhanced algorithm.

Photon Correlation Spectroscopy of Polydisperse Samples: II. Optimal choice of histogram parameters with exponential sampling

The use of a histogram model of the distribution function of decay rates in the analysis of light-scattering data from polydisperse samples is investigated. Histograms having a wide range of step sizes and number of steps have been fitted to computer-generated data of various polydispersities and noise levels. These histograms have been compared with the original distributions to determine the optimum histogram parameters in each case. The parameter ymax, which determines step size, decreases linearly with increasing ln (noise) but only at about half the rate predicted by the theory of McWhirter and Pike. ymax also decreases with increasing polydispersity. The number of steps is determined, as expected, by the step size and the need for the histogram to span fully that part of the distribution function which contributes to the autocorrelation function at a level above noise.

Computer simulation of rapidly responding forced exponential systems

Two common problems in computer simulations are the decisions to ignore or include a particular element of a system under study in a model and the choice of an appropriate integration algorithm. To examine aspects of these problems, a simple exponential system is considered in which a large simulation error is induced by a rather small truncation error. The effect of computational precision, step size and hardware selection on this error is examined at standard and extended precisions over a range of step sizes and on a variety of computers. For this model, simulation accuracy is an exponential function of the number of bits in the mantissa of the computer word. Optimal step size is a function of accuracy required and precision used; a trade-off between truncation and round-off errors becomes important as accuracy requirements increase. Machine selection is important primarily in economic terms if the required precision is available. We conclude that the effect on a simulation of small terms such as truncation errors can be unexpectedly large, that solutions should always be checked, and that high precision and wide dynamic range are important to the successful computer simulation of models such as that examined.

A discussion on finite‐difference solutions to the parabolic wave equation

The application of numerical ordinary differential equation (ODE) methods for solving the parabolic wave equation has been presented at past symposia. These techniques were shown to be an efficient alternative to the usual solution via the split‐step Fourier algorithm due to their stability and capability for a large step size in range. In this presentation, a solution based on finite differences is discussed. All three of the above techniques have been used to solve a number of representative, range‐dependent, as well as range‐independent, problems. The merits and disadvantages of each will be contrasted. [Work support jointly by NAVMAT and NAVSEA.]

Applications of a full diffraction theory numerical simulation of sound propagation in the ocean

A review of this numerical technique and its physical basis is presented. A comparison of the technique and the parabolic approximation is made with particular attention paid to broad beam errors in the two approaches. The propagation is simulated in stepwise fashion with the range step sizes being limited by the Rayleigh length. The Rayleigh length is defined for a variety of different sound velocity profiles. The sensitivity of the technique to step size and sampling rates is examined by varying these parameters. The theory is extended to include smooth boundaries, sloping boundaries, and Snell's law bending in range (as well as depth). SOFAR channel, surface, duct, split beam shadow zone and sloping bottom are among the examples presented. [Work partially supported by the Office of Naval Research.]

Detecting the Presence of Speech Using ADPCM Coding

When speech is coded using a differential pulse-code modulation system with an adaptive quantizer, the digital code words exhibit considerable variation among all quantization levels during both voiced and unvoiced speech intervals. However, because of limits on the range of step sizes, during silent intervals the code words vary only slightly among the smallest quantization steps. Based on this principle, a simple algorithm for locating the beginning and end of a speech utterance has been developed. This algorithm has been tested in computer simulations and has been constructed with standard integrated circuit technology.

The refractivity intercept and the specific refraction equation of Newton. II. The electronic interpretation of the refractivity intercept and of the specific refraction equations of Newton, Eykman and Lorentz-Lorenz

We present a novel numerical method and algorithm for the solution of the 3D axially symmetric time-dependent Schrödinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the split-operator approximation of the exponential operator with the implicit equations of second order cylindrical 2D Crank–Nicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full Crank–Nicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the split-operator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a few-cycle laser pulse with linear polarization.