Trapezoidal Integration Research Articles

Realization Using the γ-Operator

Trapezoidal integration implemented by the so-called γ-operator provides better approximation than rectangular integration associated with the familiar δ-operator, particularly at low-sampling rates. This note shows how the γ-domain transfer function can be realized without an iterative solution.

An implicit multi-time step integration method for structural dynamics problems

A multi-time step integration algorithm is developed based on the trapezoidal rule time integration method for finite element equations of motion. This algorithm uses nodal groups to partition the mesh into subdomains that are updated with different time steps. A␣stability analysis of the method shows that the scheme retains the unconditionally stable behavior of the trapezoidal rule and conserves the same pseudo energy as the parent algorithm. Several numerical examples are used to verify the stability of the method and to investigate the accuracy of the scheme.

Asymptotic expansions for trapezoidal type product integration rules

This paper is concerned with the numerical integration of functions by piecewise polynomial product integration rules followed by application of extrapolation procedures. The studied rules can be considered as generalizations of the conventional trapezoidal rule. Euler-MacLaurin type asymptotic expansions are obtained with only even powers. Furthermore, numerical examples are given in order to show the effectiveness of these methods and a comparison with rules of similar characteristics is also made.

Simulation of lossy multiconductor transmission lines using backward Euler integration

This paper presents a fixed time-step backward Euler integration algorithm for the time-domain analysis of linear circuits containing multiconductor lossy transmission lines. The new method requires the frequency derivatives of the circuit equations at a frequency point on the positive real axis that is related to the integration time step. The transient response is obtained efficiently from a single inversion of the circuit matrix. Although it is not as accurate as trapezoidal integration, the backward Euler integration is stable and gives smooth low-pass approximations for the actual response. In this paper, it is also shown that in case of impulse function excitation, the backward Euler results are equivalent to the shifted moments of the impulse response with an appropriate scaling of the frequency-dependent variables of the circuit. Numerical examples are presented for verification of the proposed method.

Ronchi test with a square grid

We use a square grid in the Ronchi test. This grid allows processing of both the X and the Y directions when calculating optical path difference. We use trapezoidal integration to analyze the new patterns, since it does not have the smoothing drawback at the edges of the wave front.

Electromagnetic transient simulation of power systems using root-matching techniques

For electromagnetic transient simulations the method of substituting the trapezoidal integrator, developed by Dommel, is generally applied. The trapezoidal rule is based on a truncated Taylor series and therefore contains truncation errors. These truncation errors cause numerical oscillations when the time step is large relative to some of the time constants in the network. This leaves the user unsure whether observed oscillations are numerical or true artefacts of the system simulated. Various techniques have been developed to reduce the numerical oscillations, each with strengths and drawbacks. An exponential form of the difference equation is developed, using root-matching techniques and demonstrated on a simplified power system. This technique eliminates the truncation error and provides highly efficient and accurate time domain regardless of the time step used. This exponential form of difference equation generates a solution at each time point that is exact for the step response and a good approximation for an arbitrary forcing function.

Filling The Gap Between The Bilinear and The Backward-Difference Transforms: An Interactive Design Approach

A class of stable, first order, s-to-z transforms is presented. The transforms are obtained by interpolating the traditional trapezoidal and rectangular integration rules. An interactive design approach is introduced by varying one parameter. The sampling rate would be another parameter that may be varied to obtain the lowest sampling rate for the desired response.

Nonlinear System Response for Impulsive Parametric Input

In engineering applications when the intensity of external forces depends on the response of the system, the input is called parametric. In this paper dynamical systems subjected to a parametric deterministic impulse are dealt with. Particular attention has been devoted to the evaluation of the discontinuity of the response when the parametric impulse occurs. The usual forward difference and trapezoidal integration schemes have been shown to provide only approximated solutions of the jump of the response; hence, the exact solution has been pursued and presented under the form of a numerical series. The impulse is represented throughout the paper by means of a classical Dirac’s delta function; however, a new model of Dirac’s delta is presented and adopted in order to validate the results provided by the numerical series.

View factors for an infinite rectangular duct enclosing a transparent medium by the discrete-ordinates method

The purpose of this study is to compute the view factors for an infinite rectangular system by the discrete-ordinates method. The effects of number of spatial divisions, number of discrete-ordinate directions, and the values of the weighting factors on the accuracy of the solution are studied. If shading appears, the accuracy of the solution depends both on the number of discrete ordinate directions and number of spatial divisions. Three different weighting factor sets, equal weighting factors for each direction cosine, Chevyshef quadratures and weighting factors from the trapezoidal integration rule are tested. The results are not sensitive to the choice of three different weighting factor sets.

The accurate evaluation of a particular Fermi-Dirac integral

In this paper, we provide a simple and very accurate numerical method to evaluate a particular two-dimensional Fermi-Dirac integral that arises in the plasma transport theory. The proposed method involves suitable coordinate and variable transformations which eliminate variable limits of integration. The resulting inner integral is evaluated by a trapezoidal integration with pole correction. The outer integral is evaluated by the IMT scheme. We provide reasonable error bounds. The results agree very well with the results of Fullerton and Rinker, who employ a combination of quadrature, asymptotic methods and Chebyshev fitting.

Digital simulation of electromagnetic transients

This paper presents the development of a new time domain simulation program for the computation of electromagnetic transients over a wide frequency spectrum. The application of the program to investigate the Subsynchronous Resonance phenomenon is given. The proposed technique depends on applying the implicit trapezoidal integration method to the differential equations simultaneously to get algebraic equations which are solved simultaneously using the Newton-Raphson method. The sparsity technique and factorization have been adopted to reduce computing time and storage. The IEEE Benchmark model is used to apply the proposed technique. The results of the new technique have been compared with the results of both the EMTP technique and the Runge-Kutta-Merson method. The comparison shows the superiority of the proposed technique.

Efficient Evaluation of Convolution Integrals Arising in Fdtd Formulations of Electromagnetic Dispersive Media

The Trapezoidal Recursive Convolution (TRC) scheme was previously used to model Nth order Lorentz type dispersive media. In this paper, the full derivation of this quasi-trapezoidal-based algorithm is presented and the derivation is expanded to include the Nth order Debye type dispersion as well as Sellmeyer's dispersion equation. In addition, the case of general convolution integrals is considered where any arbitrary integrand or the integral itself is represented as a sum of exponential functions, i.e. Prony's method. The technique is compared to several previously published schemes and it is shown that its performance equals or exceeds various other methods in terms of accuracy, robustness, and computational efficiency. A comparison to the exact application of trapezoidal numerical integration is made and it is shown that, for time increments encountered in typical FDTD analyses, the truncation error due to applying the quasi-trapezoidal approximation is negligible. Finally, it is shown how the skin effect phenomenon, as it applies to multiconductor transmission lines, can be modeled using a rational function approximation to the frequency dependency of the line resistance. This model is obtained by using Levy's method to curve fit the line resistance directly in the frequency domain and then the convolution integral is formulated in a form amenable to the TRC algorithm.

Moment Analysis of Tracer Experiments

For the estimation of moments from pulse-tracer experiments, prior workers have used trapezoidal rule integration as well as a generalized moment of inertia approach. In this technical note, these two methods are compared, and it is shown that the latter method produces biased estimators of the mean, particularly the variance of the residence time distribution, while the trapezoidal rule is substantially less biased under usual conditions. Therefore, the trapezoidal rule should be applied for the analysis of such data.

Integral evaluation of the modal phase coefficients in curvature sensing: Albrecht’s cubatures

The modal coefficients describing a given wave front as a linear combination of a set of orthogonal functions can be determined independently of each other by a suitable integration of the signal provided by a curvature sensor. This fact allows us to avoid the use of matrix inversion routines and overcomes some problems related to the modeling error arising in conventional modal fits. Several procedures for evaluating these integrals from a discrete set of intensity measurements are compared. The results show that, for a given number of sampling points, the combination of an Albrecht’s cubature in the inner pupil region with a composite trapezoidal integration of the edge signal can give more accurate results with smaller noise propagators than other methods also analyzed.

Exactitud del ELlSA para el diagnóstico de fascioliasis bovina: análisis del área bajo la curva del receptor operador (ROC)

The enzyme-linked immuno-sorbent assay (ELISA) test for the serodiagnosis of bovine fascioliasis was standardized and evaluated. Three kinds of antigen were employed: crude somatic (SAg), total excretorykecretory (T-EISAg) and partially purified excretoty/secretory (PP-EISAg). Thirty positive samples were obtained from naturally infected canle, parasitological confirmation being positive due to the finding of adult flukes in bile ducts during post-mottem examination. The infected cattle came from regions lying between 1,800 and 2,400 meters above sea level, which are considered to be endemic for this disease. Forty-five negative samples were obtained from cattle living in non-endemic areas, whose age, breeding and growth conditions had been recorded since bitth to the actual moment at which the blood samples were taken. The receiver operator curve (ROC) was used for the evaluation of the ELlSA test, following trapezoidal integration methodology, in order to identify parasitized bovines from healthy ones. The three antigens employed -somatic (SAg), total excretory/secretoy (T-E/SAg) and partially purified excretory/secretory (PP-E/SAg)- can be used for performing fascioliasis cerodiagnosis by ELlSA test because al1 of them differentiate positive from negative samples.

Extension of Berenger'sPML boundary condition in matchinglossy medium and evanescent waves

The authors have found that the generalised perfectly matched layer can be derived using wave impedance and effective permittivity concepts. This approach uses a trapezoidal integration scheme which leads to significantly better performance in matching propagating waves and a slight improvement in absorption of evanescent waves, in a lossy medium.

Reevaluation of the hadronic contribution to alpha (MZ2).

We reevaluate the hadronic part of the electromagnetic vacuum expectation value using the standard dispersion integral approach that utilizes the hadronic cross section measured in $\ee$ experiments as input. Previous analyses are based upon point-by-point trapezoidal integration which does not treat experimental errors in an optimal way. We use a technique that weights the experimental inputs by their stated uncertainties, includes correlations, and incorporates some refinements. We find the five-flavor hadronic contribution to the fractional change in the electromagnetic coupling constant at $q^2=M_Z^2$, $\Delta\alpha(MZ)$, to be $0.02752\pm0.00046$, which leads to a value of the electromagnetic coupling constant, $\alpha^{-1}(M_Z^2) = 128.96\pm0.06$. [This is an updated version of SLAC-PUB-6710 (hep-ph/9411353) which fixes a small bias in the fitting procedure (1/3 of the change) and incorporates a new and precise cross section measurement near charm threshold (2/3 of the change).]

Effect Of Rotary Disturbances On Aircraft Response

A numerical model is introduced to study the effect of rotary disturbances on aircraft response. The model follows the technique proposed by the senior author for solving the coupled equations of motion of an aircraft. The general small-disturbance equations of motion are obtained as a system of linearized first-order differential equations. These equations are written in the form of a state equation. The latter equation is integrated by trapezoidal integration, which is stable for all increment sizes. The proposed numerical model simulates the aircraft response due to rotary disturbances that arise from variations in loads applied on the aircraft. Computer-generated illustrations of these responses are presented for initial pitching velocity disturbance, rolling disturbance, and yawing disturbance during straight flight. For an initial longitudinal rotary disturbance, the responses exhibit overshoots, short-period critically damped modes, and low-amplitude long-period oscillation modes for the symmetrical perturbations of the aircraft. On the other hand, a lateral initial disturbance causes heavily damped short period oscillation modes, exponentially decaying or increasing modes, and also critically damped modes for asymmetrical motion. Based on such responses, it is possible to extract the aircraft stability criteria.

Investigation of phaeochromocytoma.

SummaryOBJECTIVE Fasting is known to clearly increase both spontaneous and GHRH‐stimulated GH secretion in normal subjects and this effect is likely to be due to hypothalamic mechanism(s). Our aim was to clarify the effect of a 3 or 4‐day fast, on the GH response to GHRH alone or combined with arginine, an amino acid probably acting via inhibition of hypothalamic somatostatin release.DESIGN Two tests with GHRH (1 μg/kg i.v.), administered either alone or in combination with arginine (ARG, 0.5 g/kg i.v.) were performed, in a randomized order at least 3 days apart. In obese women the two tests were repeated after a 3 or 4‐day fast.PATIENTS Seven obese women (06, aged 17–54 years, BMI 42.4 ± 3.6 kg/m2, waist‐hip ratio (WHR) 0.85±0.01) and ten healthy women, as control subjects (CS, aged 20–44 years, BMI 23.1 ±1.1 kg/m2, WHR 0.79 ± 0.01) were studied.MEASUREMENTS Serum GH and IGF‐I levels were measured by radioimmunoassay. The GH secretory responses were expressed either as absolute values (mU/l) or as areas under the curve (AUC, mU/l/h) calculated by trapezoidal integration. IGF‐I concentrations were expressed as absolute values (μg/l) with reference to a pure recombinant IGF‐I preparation. Results are expressed as mean ± SEM.RESULTS Basal GH and IGF‐l levels in OB were lower than in CS (0.8 ± 0.2 vs 4.8±1.0 mU/I, P<0.0001 and 120.1 ± 21.4 vs 188.7 ± 13.1 μg/l, P <0.02, respectively). The GHRH‐induced GH rise in OB was lower (P <0.00001) than in CS (AUC 340.2 ± 81.0 vs 2125.0 ± 199.6 mU/l/h). ARG increased the GHRH‐induced GH rise in both groups, but in OB the GH response to ARG + GHRH (1458.4 ± 439.0 mU/l/h, P < 0.03 vs GHRH alone) remained lower (P <0.0001) than in CS (6396.2 ± 772.2 mU/l/h, P < 0.01 vs GHRH alone). In spite of a reduction in body weight and IGF‐I, Insulin and glucose levels, in OB fasting failed to modify both the basal GH levels and the somatotroph responsiveness to GHRH when administered either alone or combined with ARG. An increase in free fatty acids (FFA) was also found after fasting.CONCLUSIONS The results of this study demonstrate that in obesity the somatotroph hyporesponsiveness to GHRH, either alone or combined with arginine, is not improved by short‐term fasting. As fasting is considered a CNS mediated stimulus to GH secretion, its ineffectiveness in obesity does not support a hypothalamic pathogenesis and suggests that long standing metabolic alterations, such as hyperinsulinaemia and/or elevated free fatty acids, could play a major role in causing GH insufficiency in obese patients.

Short-term fasting in obesity fails to restore the blunted GH responsiveness to GH-releasing hormone alone or combined with arginine.

Fasting is known to clearly increase both spontaneous and GHRH-stimulated GH secretion in normal subjects and this effect is likely to be due to hypothalamic mechanism(s). Our aim was to clarify the effect of a 3 or 4-day fast, on the GH response to GHRH alone or combined with arginine, an amino acid probably acting via inhibition of hypothalamic somatostatin release. Two tests with GHRH (1 microgram/kg i.v.), administered either alone or in combination with arginine (ARG, 0.5 g/kg i.v.) were performed, in a randomized order at least 3 days apart. In obese women the two tests were repeated after a 3 or 4-day fast. Seven obese women (OB, aged 17-54 years, BMI 42.4 +/- 3.6 kg/m2, waist-hip ratio (WHR) 0.85 +/- 0.01) and ten healthy women, as control subjects (CS, aged 20-44 years, BMI 23.1 +/- 1.1 kg/m2, WHR 0.79 +/- 0.01) were studied. Serum GH and IGF-I levels were measured by radioimmunoassay. The GH secretory responses were expressed either as absolute values (mU/l) or as areas under the curve (AUC, mU/l/h) calculated by trapezoidal integration. IGF-I concentrations were expressed as absolute values (microgram/l) with reference to a pure recombinant IGF-I preparation. Results are expressed as mean +/- SEM. Basal GH and IGF-I levels in OB were lower than in CS (0.8 +/- 0.2 vs 4.8 +/- 1.0 mU/l, P < 0.0001 and 120.1 +/- 21.4 vs 188.7 +/- 13.1 micrograms/l, P < 0.02, respectively). The GHRH-induced GH rise in OB was lower (P < 0.00001) than in CS (AUC 340.2 +/- 81.0 vs 2125.0 +/- 199.6 mU/l/h). ARG increased the GHRH-induced GH rise in both groups, but in OB the GH response to ARG+GHRH (1458.4 +/- 439.0 mU/l/h, P < 0.03 vs GHRH alone) remained lower (P < 0.0001) than in CS (6396.2 +/- 772.2 mU/l/h, P < 0.01 vs GHRH alone). In spite of a reduction in body weight and IGF-I, insulin and glucose levels, in OB fasting failed to modify both the basal GH levels and the somatotroph responsiveness to GHRH when administered either alone or combined with ARG. An increase in free fatty acids (FFA) was also found after fasting. The results of this study demonstrate that in obesity the somatotroph hyporesponsiveness to GHRH, either alone or combined with arginine, is not improved by short-term fasting. As fasting is considered a CNS mediated stimulus to GH secretion, its ineffectiveness in obesity does not support a hypothalamic pathogenesis and suggests that long standing metabolic alterations, such as hyperinsulinaemia and/or elevated free fatty acids, could play a major role in causing GH insufficiency in obese patients.