Least-Squares, Moment-Based, and Hybrid Polynomializations of Drag Forces

In order to carry out nonlinear dynamic analyses of fixed offshore structures in the frequency domain, polynomial approximations of the distributed drag term, u∣u∣, and inundation drag term, ηu∣u∣, of the Morison forces are studied. The methods of least-squares and moment-based approximations are considered for cases with and without current. Numerical results and analytical expressions of the polynomial coefficients are presented for the cubic approximations of u∣u∣ and quartic approximations of ηu∣u∣. The curve shapes, first four central moments, and probability density functions of the different approximations are evaluated and compared with the corresponding exact solutions. For the nonmonotonic inundation drag term with the current effect included, a hybrid polynomialization, based on the least-squares approximation for the odd-order polynomial coefficients and the moment-based approximation for the even-order coefficients, is proposed.     

Analytical Solution for Normal Irrigation Distribution Parameters

The model most widely used to represent sprinkler irrigation distribution parameters is based on numerical solutions to the normal cumulative probability density function. For most practical irrigation design and management applications, numerical solutions are too laborious. One other study reported analytical approximations for several irrigation distribution parameters derived from the normal model. The estimation error resulting from those approximations were variable over the operational range of irrigation uniformity and irrigation adequacy and were quite high in some ranges. In this note, more accurate analytical approximations are presented for the distribution coefficient, the application efficiency, the water requirement efficiency, the deficiently irrigated volume, and the average deficit over the deficiently irrigated area. On average, over the entire operational range of irrigation uniformity and irrigation adequacy, the new approximations are about an order of magnitude more accurate than the previous approximations and introduce negligible error for most practical applications.     

Linear Analysis of Shallow Water Wave Propagation in Open Channels

Flood wave movement in an open channel can be treated as disturbances imposed at the upstream and downstream boundaries of a channel to an initially steady uniform flow. Linearized, cross sectionally integrated continuity and momentum equations are introduced to describe one-dimensional, unsteady, gradually varied flow in open channels. The Laplace transform method is adopted to obtain first-order analytical spatio-temporal expressions of upstream and downstream channel response functions. These expressions facilitate a critical comparison among different wave approximations in terms of their mathematical properties and physical characteristics. One or two families of characteristic waves, parameterized by an attenuation factor and a wave celerity, are found for various wave approximations. The effects of the downstream boundary condition on different wave approximations are discussed and compared. Wave translation, attenuation, reflection, distortion, and configuration from the analyses are further investigated and interpreted; thus, the differences and similarities in the propagating mechanism among the various wave approximations are revealed.     

Approximating Wetted Perimeter in Power-Law Cross Section

The wetted perimeter is an open-channel parameter that influences surface drag and also influences infiltration in furrows and unlined canals. Various approximations to the wetted perimeter are examined for channel cross sections in which the top width is a power law of depth. Exact solutions are available only for select values of the exponent. Interpolation between exact values, partial sums of series approximations, and numerical integration are compared. A simple nonlinear interpolation scheme or a three-term summation yields results accurate to within 1% of the range of typical channel geometries. Common numerical approximations are shown to yield errors in excess of a range of from 3 to 10%.     

Simple Approximations for Improving Second-Order Reliability Estimates

Simple correction factors are given for improving the second-order reliability estimates. The correction factors are applied to Hohenbichler and Rackwitz's formula. The factors are obtained based on the main curvature of the limit state function at the design point. The proposed approximations can also be applied in the point-fitting approach for second-order reliability analysis. The accuracy of the proposed approximations is illustrated through examples.     

New Approximations for Reliability Integrals

A new asymptotic expansion is applied to approximate reliability integrals. The asymptotic approximation reduces the problem of evaluating a multidimensional probability integral to solving an unconstrained minimization problem. Approximations are developed in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of the second-order reliability method integrals. In many cases, it may be computationally expensive to transform to normal variables, and an approximation using the probability distribution for the original variables can be used. Examples are presented illustrating the accuracy of the approximations, and results are compared with some existing approximations of reliability integrals.     

Overview no. 41 The interactions of composition and stress in crystalline solids

The thermodynamics of stressed crystals that can change phase and composition is examined with particular attention to hypotheses used and approximations made. Bulk and surface conditions are obtained and for each of them practical expressions are given in terms of experimentally measurable quantities. The concept of open-system elastic constants leads to the reformulation of internal elastochemical equilibrium problems into purely elastic problems, whose solutions are then used to compute the composition distribution. The atmosphere around a dislocation in a cubic crystal is one of several examples that are completely worked out. The effects of vacancies, and their equilibrium within a solid and near surfaces are critically examined and previous formulas are found to be first order approximations. Consequences of the boundary equations that govern phase changes are studied with several examples. Finally, problems connected with diffusional kinetics and diffusional creep are discussed.     

Bayesian Updating Application into Simulation in the North Edmonton Sanitary Trunk Tunnel Project

Simulation models are built on assumptions, approximations, and estimates. Repetitive long-term projects such as tunnel construction provide opportunities to finetune approximations based on input from actual project progress. Bayesian updating techniques are an effective approach for improving the quality of simulation input and output based on what has already been observed. This paper presents a case study in which Bayesian techniques were applied to a simulation model of an actual tunnel project, the North Edmonton Sanitary Trunk. The study shows that using Bayesian techniques greatly improves the quality of projections. The novelty of this work includes the enhancement of the application of Bayesian updating techniques, the demonstration of simulation applications with a fully monitored tunneling project, and the demonstration of the extent of improvement to planning predictions from the use of actual data and the Bayesian updating techniques.     

Multiple correlation: Exact power and sample size calculations.

This article discusses power and sample size calculations for observational studies in which the values of the independent variables cannot be fixed in advance but are themselves outcomes of the study. It reviews the mathematical framework applicable when a multivariate normal distribution can be assumed and describes a method for calculating exact power and sample sizes using a series expansion for the distribution of the multiple correlation coefficient. A table of exact sample sizes for level .05 tests is provided. Approximations to the exact power are discussed, most notably those of J. Cohen (1977). A rigorous justification of Cohen's approximations is given. Comparisons with exact answers show that the approximations are quite accurate in many situations of practical interest. More extensive tables and a computer program for exact calculations can be obtained from the authors. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Drawdowns due to Intermittent-Pumping Cycles

This paper offers solutions for drawdowns due to intermittent pumping cycles or cyclic pumping, which are high accuracy approximations of the series of Theis functions superimposed in time. The proposed approximation formulas are an improvement over the earlier works. The earlier approximations are valid only if the number of pumping cycles is greater than 10 and involve gamma functions that are less convenient to evaluate than the rational approximation formulas offered in this paper. The proposed approximations are valid for any number of pumping cycles and involve simple functions that can be computed even using a calculator. The drawdown functions are defined for the drawdowns at the end of pumping or shutoff periods. The proposed expressions for these functions are also suitable for the estimation of aquifer parameters by plotting the observed drawdowns on semilogarithmic paper. Procedures for estimation of storage coefficient and head loss at the well from cyclic pumping drawdowns are not available. This paper also offers procedures for the estimation of transmissivity, storage coefficient, and head loss at the pumped well from the observed intermittent (cyclic) pumping drawdowns.     

Presentation rate and order of approximation to English as determinants of short-term memory.

Presented sequentially, to 30 undergraduates, 28 sequences of 10 letters representing 0- and 2nd-order approximations to English at presentation rates of 3 letters/sec, 2 letters/sec, and 1 letter/sec. Groups of 10 Ss received each rate of presentation. In agreement with previous studies, the increased time for input processing available at the slower presentation rates produced greater differentiation in the immediate recall of 0- and 2nd-order approximations. Additional analyses suggest that the increased differentiation was attributable to both a greater increase in the efficiency of processing order information and more rapid processing of individual letters in the 2nd-order approximations. (French summary) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comparing squared multiple correlation coefficients: Examination of a confidence interval and a test significance.

I. Olkin and J. D. Finn (1995) presented 2 methods for comparing squared multiple correlation coefficients for 2 independent samples. In 1 method, the researcher constructs a confidence interval for the difference between 2 population squared coefficients; in the 2nd method, a Fisher-type transformation of the sample squared correlation coefficient is used to obtain a test statistic. Both methods are based on asymptotic theory and use approximations to the sampling variance. The approximations are incorrect when the population multiple correlation coefficient is zero. The 2 procedures were examined for equal and unequal population multiple correlation coefficients in combination with equal and unequal sample sizes. As expected, the procedures were inaccurate when the population multiple correlation coefficients were zero or very small and, in some conditions, were inaccurate when sample sizes and coefficients were unequal. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Vibration of Tensioned Beams with Intermediate Damper. II: Damper near a Support

Analytical solutions are used to investigate the free vibrations of tensioned beams with a viscous damper attached transversely near a support. This problem is of particular relevance for stay-cable vibration suppression, but no restrictions on the level of axial load are introduced, and the results are quite broadly applicable. Characteristic equations for both clamped and pinned supports are rearranged into forms suitable for numerical solution by fixed-point iteration, whereby the complex eigenfrequencies and corresponding damping ratios can be accurately computed within a few iterations. Explicit asymptotic approximations for the complex eigenfrequencies are also obtained, subject to restrictions on the closeness of the eigenfrequencies to their undamped values. These asymptotic approximations are expressed in the same “universal” form identified in previous studies. It is observed that the maximum attainable modal damping ratios and the corresponding optimal values of the damper coefficient can be significantly affected by bending stiffness and by the nature of the support conditions, and a nondimensional parameter grouping is identified that enables an assessment of when bending stiffness should be considered.     

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method. Numerical results demonstrate the good performance of the spectral method.     

Evaluation of the effects of presentation rate and speaking rate on children's auditory perception of two levels of linguistic material

Twenty-four first- and second-order sentential approximations of four words each were presented auditorily to two groups of eight normal children. Eight first- and eight second-order approximations were spoken at three rates: 90 (slow); 140 (normal); and 175 (fast) words/min. The 48 approximations were randomized and rates altered as necessary so that one group heard the material at a slow rate (90 words/min) of presentation and the other group at a normal rate (140 words/min) of presentation. The results indicated that although presentation rate did not significantly effect auditory perception, speaking rate did. It was concluded that a slow speaking rate can be expected to enhance the perception of low-level linguistic material regardless of the presentation rate, whereas a normal speaking rate would be best for higher level linguistic material.     

Calculating Dislocation Spacings in Pile-Ups at Grain Boundaries

Dislocation pile-ups near planar grain boundaries have been analyzed employing full anisotropic elastic solutions, single-crystal anisotropic approximations, and isotropic approximations. The calculations were performed for a series of iron-silicon bicrystal configurations and results from the various methods compared. Three of the bicrystal/slip system combinations exhibited repulsive image forces, allowing elastically self-consistent position calculations. Based on these results, either linear elastic or nonlinear effects can dominate the behavior of these pile-ups, depending on specific boundary conditions.     

New Approach to Satellite Formation-Keeping: Exact Solution to the Full Nonlinear Problem

This paper presents a new, simple, and exact solution to the formation keeping of satellites when the relative distance between the satellites is so large that the linearized relative equations of motion no longer hold. We employ a recently proposed approach, the Udwadia-Kalaba approach, which makes it possible to explicitly obtain the desired control function without making any approximations related to the nonlinearities in the underlying dynamics. We use an inertial frame of reference to describe the motion of a satellite and since no approximations are made, the results obtained apply to situations even when the distance between the satellites is arbitrarily large. The paper deals with a projected circular formation, but the methodology in this paper can be applied to any desired configuration or orbital requirements. Numerical simulations confirm the brevity and the accuracy of the analytical solution to the dynamical control problem developed herein.     

Stress and solubility for solutes with asymmetrical distortion fields

The influence of stress on chemical potential and solubility for solutes with asymmetrical distortion fields are considered. Numerical solutions and series approximations are presented for several examples of different stress states. Both open and closed thermodynamic systems are treated. The calculations are compared with results for hydrogen in crystalline Pd and amorphous Pd alloys, the latter of which involves an asymmetrical distortion field.     

Calculation-experimental technique for predicting the relaxation properties of thin polymer fibers during long-term loading

A technique is developed to predict the strain and the stress in a fiber under combined relaxation and creep conditions during long-term loading. Stepwise exponential approximations of the characteristic segments in a relaxation curve are suggested for the initial stage of stress relaxation, and they are generalized by an adaptive exponential approximation containing at most two approximation parameters.     

Factor space of the Wechsler Intelligence Scale for Children—Third Edition: Critical thoughts and recommendations.

The Wechsler Intelligence Scale for Children—Third Edition (WISC—III) manual incorporates a detailed and careful series of factor analyses. It recommends using approximations of the Verbal Comprehension, Perceptual Organization, Freedom From Distractibility, and Processing Speed factor scores. These approximations are simple sums of the scores of the subtests that load most highly on a factor. These simple sum factor estimates suffer from reduced factorial specificity. The simple estimates share substantially more variance with the factor of General Intelligence, or the g factor, and less variance with the other unrotated factors than the best estimates of the factor. This state of affairs has implications for clinical interpretation of factor scores. Application of the procedures recommended here is practical with the use of a computer and a basic spreadsheet program. (PsycINFO Database Record (c) 2010 APA, all rights reserved)