Gauss-Legendre Quadrature Integration Research Articles

The Effect of Initial Intensity of Light Sources on the Fraunhofer Diffraction Pattern for Circular Apperture Using the Gauss-Legendre 4 Point Quadrature Integration Numerical Method: Scientific and Islamic Integration-Interconnection

Visualization of Fraunhofer diffraction pattern of circular apperture has been obtained using the 4 point Gauss-Legendre Quadrature integration method. The 4 point Gauss-Legendre quadrature integration method is used to solve the integral function of intensity. The intensity function is used to display the intensity distribution of the Fraunhofer diffraction pattern of circular apperture in a visual form through a computer programming simulation based on GUI (Graphycal User Interface) MATLAB. Simulation results show changes in initial intensity affect diffraction patterns.

Non-stationary response of variable-mass Duffing oscillator with mass disturbance modeled as Gaussian white noise

This paper investigates the probability density evolution process of a variable-mass Duffing oscillator with mass disturbance modeled as Gaussian white noise. For small mass disturbance, the transition probability density function (TPDF) solution of the variable-mass system is calculated by using path integration method. The corresponding Fokker–Planck–Kolmogorov (FPK) equation of the approximate mass disturbance system are derived. The solution procedure based on Gauss–Legendre quadrature integration rule is formulated to obtain and study the probabilistic solutions of the strongly and weekly nonlinear variable-mass system. Different values of excitation intensity are used to examine the effectiveness of the proposed method. Compared with the Monte Carlo simulation results, good agreement is achieved with the path integration method. For large mass disturbance, the probability density evolution process are numerically calculated via the fourth order Runge–Kutta algorithm.

Comprehensive gravitational modeling of the vertical cylindrical prism by Gauss–Legendre quadrature integration

Comprehensive gravitational modeling of the vertical cylindrical prism by Gauss–Legendre quadrature integration

Probabilistic solution of non-linear random ship roll motion by path integration

This paper investigates the probability density function (PDF) of non-linear random ship roll motion using a previously developed path integration method. The mathematical model of ship rolling motion consists of a linear-plus-cubic damping and a non-linear restoring moment in the form of odd-order polynomials up to fifth-order terms. In the path integration method, the interpolation scheme is based on the Gauss–Legendre quadrature integration rule and the short-time transition probability density function is formulated by short-time Gaussian approximation. The present work extends the path integration method to the case of non-linear random ship roll motion. Different values of non-linearity coefficient and excitation intensity are used to examine the effectiveness of the path integration method. Numerical analysis shows that the results of the path integration method agree well with the simulation results, even in the tail region. The path integration method is effective and it is simply implemented in the examined cases. Due to the presence of non-linear damping terms and non-linear restoring moment terms, the PDFs of roll angle and angular velocity exhibit highly non-Gaussian behaviors.

Spherical prism magnetic effects by Gauss-Legendre quadrature integration

Regional spherical coordinate observations of the Earth's crustal magnetic field components are becoming increasingly available from shipborne, airborne, and satellite surveys. In assessing the geological significance of these data, theoretical anomalous magnetic fields from geologic models in spherical coordinates need to be evaluated. This study explicitly develops the elegant Gauss—Legendre quadrature formulation for numerically modelling the complete magnetic effects (i.e. potential, vector and tensor gradient fields) of the spherical prism. We also use these results to demonstrate the magnetic effects for the crustal prism and to investigate the crustal magnetic effects at satellite altitudes for a large region of the Middle East centred on Iran.

Spherical prism gravity effects by Gauss-Legendre quadrature integration

SUMMARY Regional spherical coordinate observations of the Earth’s crustal magnetic field components are becoming increasingly available from shipborne, airborne, and satellite surveys. In assessing the geological significance of these data, theoretical anomalous magnetic fields from geologic models in spherical coordinates need to be evaluated. This study explicitly develops the elegant Gauss‐Legendre quadrature formulation for numerically modelling the complete magnetic effects (i.e. potential, vector and tensor gradient fields) of the spherical prism. We also use these results to demonstrate the magnetic effects for the crustal prism and to investigate the crustal magnetic effects at satellite altitudes for a large region of the Middle East centred on Iran.

Acoustic discharge measurements for the performance testing of low-head hydroelectric turbines under disturbed flow conditions

Field performance testing of a low-head hydroelectric turbine is essential to evaluate the efficiency and economics of an operation. For low-head hydroelectric turbines, it is difficult to accurately measure the discharge through a unit. Transit-time velocity measurement technology has recently been used to develop, in a laboratory setting, a unique traversing acoustic discharge meter for low-head hydroelectric applications. This technology was recently combined with Gauss-Legendre quadrature integration as an alternative method of measuring the flow through a low-head hydroelectric turbine. However, laboratory testing of this technology has only dealt with undisturbed or ideal flow conditions. Additional physical modeling has been performed to compare the relative accuracy of the continuous traversing acoustic discharge meter with that of a multilevel Gauss-Legendre quadrature integration in disturbed or nonideal flow conditions. The data indicate that while Gauss-Legendre quadrature may provide more accurate estimates in ideal flow conditions, the continuous traversing acoustic discharge meter is better suited to disturbed flow condition because it can better resolve an intricate velocity profile. The accuracy of this instrumentation is sensitive to relatively large scale vorticity rotating in the plane of the acoustic transducers, although accuracies within 2% are still attainable, which is better than the conventional velocity-area method. Key words: acoustic discharge measurement, disturbed flow, turbine, performance testing.

Antarctic crustal modeling from the spectral correlation of free‐air gravity anomalies with the terrain

We investigated the use of enhanced spectral correlation theory for modeling the crustal features of the Antarctic from regional observations of gravity and terrain. The analysis considered 1°‐gridded free‐air gravity anomalies and topographic rock, ice, and water components for the region south of 60°S. We modeled terrain gravity effects at 150‐km altitude by Gauss‐Legendre quadrature (GLQ) integration assuming densities of 2800 kg/m3 for rock, 900 kg/m3 for ice, and 1030 kg/m3 for seawater. These effects are substantial relative to the free‐air anomalies and must be compensated by the effects of subsurface density variations. Significant terrain‐correlated free‐air anomalies were revealed by the wavenumber correlation spectrum between the free‐air anomalies and the modeled terrain gravity effects, which we interpreted mostly to reflect possible isostatic imbalances of the crust. Subtracting the terrain‐correlated free‐air anomalies from the total free‐air anomalies and topographic gravity effects yielded terrain‐decorrelated free‐air anomalies and the gravity effects of isostatically compensated terrain features, respectively, which are uncorrelated with each other. The compensating effects that annihilate the latter were attributed to undulations of the Moho, which we estimated by inversion using GLQ integration and a mantle‐to‐crust density contrast of 400 kg/m3. The inversion produced a Moho map with nearly 40 km of total relief that agrees very well with deep seismic refraction soundings. For East Antarctica, a bimodal variation in crustal thickness was found: the crustal wedge between the eastern Weddell Sea (≃330°E) and the eastern flank of the Gamburtsev Subglacial Mountains (≃90°E) has a mean thickness of about 37 km, whereas the mean crustal thickness is near 32 km for the northern half of the rest of East Antarctica up to the western flank of the Transantarctic Mountains (≃150°E). For West Antarctica and the oceanic regions, mean crustal thicknesses of about 30 km and 14 km, respectively, are inferred. The terrain‐decorrelated free‐air anomalies may be related to long‐wavelength, large‐amplitude subcrustal density variations and to much shorter‐wavelength, smaller‐amplitude intracrustal density variations.

Gauss-legendre quadrature estimates of discharge for the performance testing of low-head hydroelectric turbines

The performance testing of low-head hydroelectric turbines is necessary for the efficient and economic operation of a generating station. Of the information required for performance testing, discharge is the most difficult to measure accurately. The velocity-area method is typically used to measure discharge through a low-head plant. Herein the combination of acoustic technology and Gauss-Legendre quadrature (GLq) integration is investigated as an alternative method of measuring discharge. Physical hydraulic modeling was performed to examine the accuracy of discharge estimates associated with different GLq sampling strategies. Data obtained from laboratory testing indicate that the accuracy of estimating discharge increases with the number of GLq levels used. Discharge errors of less than 1% were observed (for undisturbed flow conditions) when 8 or more GLq levels were used.Key words: turbine, performance testing, acoustic discharge measurement, Gauss-Legendre quadrature integration.

A comparison of methods for computing the discharge through hydroelectric turbines

The economic and efficient operation of a hydroelectric generating station requires an accurate description of the head-power-discharge relationship for each turbine. Of the information required to determine this relationship, absolute discharge is the most difficult to measure and the most susceptible to measurement error. For low-head hydroelectric plants, the velocity-area method is often used. This method consists of traversing an array of current meters across the intake to obtain a grid of velocity measurements. The location of the metering points is central to the accuracy of the discharge estimate. The codes governing the performance testing of hydraulic turbines require the metering plane to be located in conditions that cannot be met for low-head hydroelectric plants. In spite of this, the codes are still applied. In this paper a densely spaced array of Ott meter data, collected according to the German (DIN) code, are used to examine the velocity profiles at the entrances of low-head hydroelectric turbine. In addition, these data are interpolated using cubic splines to estimate the time-averaged velocities required to compute discharge according to ASME (PTC-18) code. Interpolated data are also used to examine an alternative method of metering low-head plants based on Gauss-Legendre quadrature integration. The velocity profiles observed are significantly different from those assumed by the German and ASME codes to determine metering locations, which raises questions about the applicability of these codes to low-head plants. The results indicate that a Gauss-Legendre quadrature integration of a 4 × 5 grid of metering points estimates discharge more accurately than the German code, which requires a 7 × 12 grid. The suggestion is the location, and not simply the number of metering points, is central to the accuracy of a discharge estimate through a low-head hydroelectric plant.Key words: low-head hydroelectric turbine, performance testing, head-power-discharge relationship, Ott meter, Gauss-Legendre quadrature integration.

A new path integration procedure based on Gauss-Legendre scheme

A new path integration scheme based on the Gauss-Legendre quadrature integration rule is proposed for calculating the probability density of the response of a dynamical system under Gaussian white-noise excitation. The new scheme is capable of producing accurate results of probability density as it evolves with time, including the tail region where the probability level is very low. This low probability region is important for the system reliability estimation.