Numerical Extrapolation Research Articles

Torsional Line Splittings in the (ν 4+ν 8)−ν 4 Hot Transitions of C 2H 6 between 1400 and 1510 cm −1

A detailed investigation of the high-resolution infrared spectrum of ethane revealed the occurrence of features belonging to the hot perpendicular system (ν 4+ν 8)−ν 4 between 1400 and 1510 cm −1. Transition lines of the subbranches with K″Δ K from −7 to 4, exhibiting torsional splittings of several tenths of a cm −1, were observed and measured in this region. The observed line splittings are strongly influenced by the interaction between the ν 4+ν 8 and 2ν 4+ν 12 states and change with the values of K″Δ K, depending on the zero-order energy separation of the interacting levels. We found by numerical extrapolation that splittings still occur far from resonance, showing that the intrinsic torsional splittings of the combining states ν 4+ν 8 and ν 4 are quite different. We determined the intrinsic torsional splitting of ν 4+ν 8 to be less than 0.083 cm −1, compared with 0.236 cm −1 estimated for the ν 4 state. This result is in agreement with the expected effects of torsional Coriolis and head–tail coupling and is consistent with previous observations on vibrationally degenerate states of ethane-like molecules.

Fracture mechanics based fatigue analysis of steel bridge decks by two-level cracked models

An alternative fatigue assessment is introduced for orthotropic highway bridge decks, based on fracture mechanics. Crack growth simulation is carried out by the numerical integration of the Paris formula, using K factors obtained from two-level cracked models of the bridge deck. Prior to the simulation, unit wheels are applied to the 3D-shell model of the deck, and the translations of characteristic points are transferred to the 2D plane strain or plane stress finite element sub-models of fatigue-sensitive areas. The K factor is computed by numerical extrapolation, introducing cracks in the sub-models. Fixed values of input parameters (initial- and critical crack length, crack growth rate data, and the error of K factor computation) are used for single simulations as well as parametric studies for sensitivity analysis. Reliability assessment is performed by repeated crack growth analysis, using histograms to generate input parameters by Monte-Carlo simulation at the start of each “virtual experiment”.

Magnetohydrodynamic models of solar coronal magnetic fields

We present some results concerning the possibility of determining the structure of solar active regions using measurements of the vector magnetic field on the Sun surface as boundary conditions for the new numerical extrapolation codes. From these computations the main features of these configurations, as shear and twist (which are particular forms of magnetic helicity), are then used as ingredients to define model problems and solved for the magnetohydrodynamic (MHD) analysis of solar eruptive phenomena, in which ejection (or redistribution) of helicity occurs.

Trotter-number scaling in Monte Carlo studies of the small polaron

A path-integral quantum Monte Carlo method is applied to the two-dimensional single-polaron Holstein model. Simulation data are shown to exhibit -scaling with the number of imaginary-time slices M used in the Trotter decomposition. Numerical extrapolation to yields very accurate estimation of polaron energetic characteristics. Kinetic, potential and total energy are calculated as functions of the electron - phonon coupling strength for different phonon frequencies. The small-polaron regime is found in the strong-coupling limit for all of the frequencies studied. The transition to the polaron state is sharp in the adiabatic regime but broadens as the frequency increases. In the adiabatic limit the polaron forms at .

RECONSTRUCTING THE SOLAR CORONAL MAGNETIC FIELD AS A FORCE-FREE MAGNETIC FIELD

We present some preliminary results on different mathematical problems encountered in attempts to reconstruct the coronal magnetic field, assumed to be in a force-free state, from its values in the photosphere. We discuss the formulations associated with these problems, and some new numerical methods that can be used to get their approximate solutions. Both the linear constant-α and the nonlinear cases are considered. We also discuss the possible use of dynamical 3D MHD codes to construct approximate solutions of the equilibrium force-free equations, which are needed for testing numerical extrapolation schemes.

Characteristics of the Armstrong-Frederick type plasticity models

Abstract The ability to model inelastic deformation is of practical interest in many design applications, with the current emphasis on lighter weight structures and components. Incremental plasticity algorithms are often employed to model the plastic deformation. Some of the questions raised with both theoretical and engineering ramifications include, how are the material constants determined and how sensitive is the model to the choice of the constants? The characteristics of a category of Armstrong-Frederick hardening rules are examined, invoking both theoretical and experimental considerations. Depending on the complexity of the model, phenomena such as ratchetting rate decay can be modeled with these algorithms. The model is shown to be insensitive to the yield stress chosen. Numerical extrapolation of the cyclic stress-strain curve is shown to enhance modeling without any penalty. The additional material constants employed to model ratchetting have a minimal influence on either proportional or nonproportional balanced loadings, hence these terms can be considered to be decoupled. Behavior of the predicted deformation becomes asymptotic for all loadings investigated with approximately ten terms in the backstress expansion. All these factors indicate the engineering potential of the Armstrong-Frederick type models, which include the Chaboche and Ohno-Wang enhancements. Furthermore, the basic structure of the model is amenable to incorporation of other material behaviors such as non-Masing, cyclic hardening or softening, and nonproportional hardening.

Solving multivariable mathematical models by the quadrature and cubature methods

AbstractThe utilization and generalization of quadrature and cubature approximations for numerical solution of mathematical models of multivariable transport processes involving integral, differential, and integro‐differential operators, and for numerical interpolation and extrapolation, are presented. The methodology for determination of the quadrature and cubature weights for composite operators is developed to accommodate for general functional representations. Application of these methods is demonstrated by solving two‐dimensional steady‐state and one‐dimensional transient‐state problems. The solutions are compared with exact‐analytical solutions to evaluate the performance of these methods. It is demonstrated that the quadrature and cubature approximations are simple and universal; i.e., the same formula is applicable irrespective of the order of accuracy of the numerical approximation, the type of linear operator, and the number of temporal and/or spatial variables. Since the quadrature and cubature methods can produce solutions with sufficient accuracy even when using fewer discrete points, both the programming task and computational effort are reduced considerably. Therefore, the quadrature and cubature methods appear to be very practical in solving the mathematical models of a variety of transport processes. © 1994 John Wiley & Sons, Inc.

Seismological aspects of the 1989–1990 eruption at Redoubt Volcano, Alaska: the Materials Failure Forecast Method (FFM) with RSAM and SSAM seismic data

Seismic activity during the December 1989 to April 1990 eruption of Redoubt Volcano, Alaska, has been tracked by the Alaska Volcano Observatory using Real-time Seismic Amplitude Measurement (RSAM) and Seismic Spectral Amplitude Measurement (SSAM) systems with up to five stations. Data consist of 10-minute averages of the absolute amplitudes of seismometer output. These data have been used to test in hindsight the Materials Failure Forecast Method (FFM), which attempts to define the time of eruption with a time series for precursory phenomena whose rate of change accelerates measurably before events. Practical application of the method emphasizes inverse-rate plots, both for early detection of signal emergent from background, and for event forecasting. Eruption windows are determined by graphical or numerical extrapolation of the inverse-rate trend and by intersection of a data envelope, reflecting data scatter and consistency, with an empirical critical rate near the time of eruption. Prior to the dome-destroying eruption of January 2, rate changes were of sufficient consistency, duration, and intensity for a qualitative or quantitative FFM predictive analysis, using either RSAM or SSAM. Signal-to-noise ratio was high for both RSAM or SSAM data sets, and FFM analyses could have provided useful support for decision making. Following January 2, in association with a rapid succession of dome-collapse eruptions, signal strength diminished and RSAM signal-to-noise ratio declined. The most distinct patterns on RSAM reflected noise rather than signal, and forecasting exclusively based for RSAM would have been misleading. By eliminating the noise in excluded frequencies, an enhanced signal-to-noise ratio was generally produced by SSAM for banded frequencies near 2 Hz. After January 2, only SSAM exhibited signal-to-noise ratios suitable for FFM analysis. Inverse-SSAM plots could have been informative for early detection of precursory long-period seismicity. The prospect of combining the FFM approach with spectral amplitudes in selected frequencies offers considerable promise.

Finite-element computation of perturbation energies for the two-electron atom.

Perturbation energies ${\mathit{E}}_{2}$, ${\mathit{E}}_{3}$, and ${\mathit{E}}_{4}$ in the 1/Z expansion for the ground-state two-electron atom are calculated using accurate representations of the partial waves L=0,1,2,. . . of the first- and second-order eigenfunctions. These are obtained by combining L-Hylleraas basis functions with a one-dimensional finite-element procedure. The contribution to ${\mathit{E}}_{2}$, ${\mathit{E}}_{3}$, and ${\mathit{E}}_{4}$, made by partial waves L\ensuremath{\gg}1, can, with a surprising accuracy, be covered by numerical extrapolation with the help of a Schwartz-type expression in powers of 1/(L+1/2).

Investigation of flux formulae in transonic shock wave/turbulent boundary layer interaction

AbstractThe aim of the present study is to examine the accuracy and improvement of various numerical methods in the solution of the transonic shock/turbulent boundary layer interaction problem and to show that a significant source of numerical inaccuracies in turbulent flows is not only the inadequacy of the turbulence model but also the numerical discretization. Comparisons between a Riemann solver and a flux‐vector‐splitting method as well as between various numerical high‐order extrapolation schemes with corresponding experimental results are presented.

Reference deconvolution. Elimination of distortions arising from reference line truncation

Reference deconvolution (G. A. Morris; J. Magn. Reson. 80, 547 (1988)) or reference lineshape adjustment ( J. M. Wouters and G. A. Petersson, J. Magn. Reson. 28,81 (1977) ; J. M. Wouters, G. A. Petersson, W. C. Agosta, F. H. Field, W. A. Gibbons, H. Wyssbrod, and D. Cowburn, J. Magn. Reson. 28, 93 (1977) ) relies on extracting the signal of a known reference material from an experimental spectrum and using a comparison between the experimental reference signal and that predicted by theory to correct instrumental contributions to the full experimental spectrum. Truncation of the reference line can produce unwanted artifacts and distortions in the final spectrum. The source of these anomalies usually lies in the dispersion-mode rather than in the absorption-mode data, as discontinuities are formed when the dispersion “wings” are truncated. These discontinuities can be reduced by a numerical extrapolation covering the lost region; this is sufficient to provide adequate results in most cases, but fails where spectra have baseline errors. A method of recreating the full dispersion signal directly from the corresponding adsorption data, based on the Hilbert transform relationship between the real and imaginary parts of a Fourier transform, is demonstrated here. This method results in the removal of truncation distortions from the deconvoluted spectrum, thus extending the range of application of the reference deconvolution technique.

The suitability of copper/lithium alloys for application as wall material in tokamaks

In Cu/Li alloys of low lithium content, the lithium component segregates to the surface, forming a protective overlayer. When applied as wall material in a tokamak, the sputtering of the high- Z copper component can be reduced by a considerable amount. To learn about the behavior of the Cu/Li binary system under particle bombardment, detailed investigations have been carried out during the last few years. The experiments show that wall temperatures above 400 ° C are necessary to achieve a distinct lithium coating effect. The protective layer cannot be maintained permanently under low-energy light-ion irradiation and needs a periodic conditioning procedure. However, a numerical extrapolation of measured layer lifetimes particle flux densities relevant to fusion reactors shows that it is worthwhile to consider a test of Cu/Li alloys as wall material in a tokamak.

Anglesite (PbSO 4) solubility in acetate solutions: The determination of stability constants for lead acetate complexes to 85°C

Anglesite solubilities were determined at 25, 40, 55, 70, and 85°C in solutions with equal molal concentrations of acetic acid and sodium acetate (each ranging from 0.001 to 0.25 m) and at pH values near 4.6. Experiments were conducted using 1000 ml, externally heated glass reaction vessels, and lead concentrations were measured using atomic absorption spectrophotometry. For run conditions with free acetate concentrations less than about 0.01 m, measured a anglesite solubilities are less than or equal to 100 mg/l. Above free acetate concentrations of 0.01 m, measured anglesite solubilities rise sharply to concentrations near 1000 mg/l. Extracted from these solubility data, by a numerical successive linear extrapolation method, were stoichiometries and thermodynamic stability constants (ionic strength = 0) for the complexes Pb(CH 3COO) + and Pb(CH 3COO) 2. Log K 1 and log K 2 values for the formation reactions Pb + + CH 3COO − = PbCH 3COO + and Pb 2+ + 2CH 3COO − = Pb(CH 3COO) 2 respectively are log K 1: (2.4±0.1, 25°C), (2.4±0.1, 40°C), (2.5±0.1, 55°C), (2.6±0.1, 70°C), (2.7±0.2, 85°C) and log K 2: (3.4±0.4, 25°C), (3.6±0.2, 40°C), (3.8 ±0.2, 55°C), (4.0±0.2, 70°C), (4.3±0.2, 85°C). Lead hydrolysis was found to be significant in those experiments conducted at 70 and 85°C. However, hydrolyzed lead was not detected at 25, 40, and 55°C. PbOH + as percent of total observed lead ranged from 0.2 to 24% and 2 to 86% in those experiments at 70 and 85°C, respectively. Calculations of lead speciation in acetate solutions show that the equivalence point for PbCH 3COO + and Pb(CH 3COO) 2 shifts to lower acetate concentrations from 25 to 85°C. These speciation calculations also suggest that lead acetate complexes could be significant in high acetate formation waters. These waters typically contain at least 1,000 ppm acetate (0.017 molal as acetic acid). Furthermore, if these waters are not too acid and if concentrations of competing ligands are low, lead acetate complexes should be the dominant lead species.

ChemInform Abstract: Kinetic and Thermodynamic Analysis of Hydrogen Evolution at Nickel Electrodes.

AbstractiR‐corrected steady‐state polarization curves for H2 evolution in alkaline solution at crystalline and amorphous Ni electrodes (Ni78Si8B14) are obtained using an experimental procedure consisting of a potentiostatic computer‐assisted interruption technique and numerical extrapolation of potential transients.

Attenuation of complex water‐bottom multiples by wave‐equation‐based prediction and subtraction

Multiple reflections that are generated by the water bottom in marine seismic data can be predicted by a combination of numerical wave extrapolation through the water layer and estimation of the water‐bottom reflectivity. Attenuation of the multiples occurs when the predicted wave field is subtracted from the original record. I derive the expressions needed for prediction of the multiples, following the ideas of Morley, in a form that can be used to estimate the reflectivity of a water bottom that has a complex shape and has a reflectivity that varies with lateral position, frequency, and reflection angle. The specific form of the operations needed for prediction is implemented without assumptions about the simplicity or flatness of the water bottom. The derivation implies that the recorded wave field may be interpreted as both an upgoing and a downgoing wave. This interpretation is correct except for a simple surface ghost present in both interpretations. Application of the predictive method to data collected over a hard, complex water bottom demonstrates that it effectively attenuates water‐bottom and peg‐leg multiples even when the water bottom is complex and changes in character within the span of a shot gather. The same data were processed with various combinations of predictive attenuation, prestack automatic gain control, prestack moveout discrimination, and stacking. The multiple attenuation achieved with the predictive method alone was greater than that achieved by moveout discrimination alone; the net attenuation achieved by sequential application of all the methods was approximately additive.

Strategic behaviour: Some experimental evidence

A test to detect the presence of strategic bias in responses to direct willingness to pay questioning is developed and applied in an experimental setting involving three separate groups of individuals. The Smith auction, a demand revealing process, is also tested using the same format. Under direct questioning, significant over-stating behaviour was observed whilst the bids made in the Smith auctions were predominantly under-statements. Whilst the Smith auction mean bids appear to be most suited to the task of numerical extrapolation, the direct questioning process is cheaper to implement. It is concluded that a combination of the two questioning techniques, involving the adjustment of direct questioning bids on the basis of information generated by a Smith auction carried out using a sub-sample of respondents, offers the most potential for estimating values in a real-world case.

Efficient numerical techniques for perturbative lattice gauge theory computations

We discuss a set of methods and numerical tools, which are useful for a computer based approach to perturbative calculations in lattice gauge theory. The topics considered include the automatic generation of gluon vertex programs, a derivation of the Faddeev-Popov determinant on lattices with boundary, the use of a partially finite lattice with twisted boundary conditions as an infrared cutoff without zero modes, and finally the numerical extrapolation of lattice Feynman diagrams to the continuum limit. As an illustration of the methods we describe their implementation in the computation of the on-shell improved lattice action at weak coupling.

Temperature Dependences of the Nonradiative Multiphonon Carrier Capture and Ejection Properties of Deep Traps in Semiconductors. II. Interpretation and Extrapolation of Capture Data

AbstractOn the basis of our semiempirical theory of nonradiative multiphonon (NMP) carrier capture processes a numerical analysis and extrapolation of experimental data on electron capture coefficients of the third acceptor level of copper and the second acceptor level of gold in germanium is made. The associated configuration coordinate diagrams are determined. From the correlation between NMP electron and hole capture properties follows that the NMP mechanism for hole capture is comparatively little effective (at usual temperatures). The activation of NMP carrier capture by thermal motion of the lattice is found to be more pronounced than suggested by Sinyavskii and Kovarskii. the uncertainties inherent to extrapolations up to very high temperatures are shown analytically to be much larger than expected by Henry and Lang. The high‐temperature capture cross‐sections of the traps under study are estimated to lie within the range 10−16 to 10−14 cm2. A modification is presented of the analytical apparatus adequate for numerical applications in the ranges of low and intermediate temperatures.

A guide to the literature for undergraduate and graduate courses in numerical mathematics

This report contains a compilation of selected references which have been useful in the preparation of lectures and in assigning outside reading for the numerical mathematics courses taught in the Department of Computer Science at Southern Methodist University. Introductory remarks describe the environment in which the courses are taught as well as criteria for reference selection. The references are divided up into eleven sections:1. General Texts; 2. Numerical Approximation, Interpolation, and Extrapolation; 3. Numerical Integration and Differentiation; 4. Numerical Linear Algebra; 5. Numerical Solution of Differential Equations; 6. Software Influences; 7. Symbolic and Algebraic Manipulation; 8. Computational Complexity; 9. Hardware Influences; 10. Computer arithmetic; 11. Recommended Periodicals. In addition, many of the references are placed in one or more of three categories: generally-recognized classics in a subject area; relatively applied presentations; works containing extensive bibliographical material. Brief descriptions of the courses which utilize the references are included along with a list of sources for research and technical reports.

The critical exponents of Ising model systems with pure three-body interactions. III

For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<Tc) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '=2/3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu =2/3, eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.