A General and Effective Numerical Integration Method to Evaluate Triple Integrals Using Generalized Gaussian Quadrature

Jayan, Sarada; Nagaraja, K.V.

doi:10.1016/j.proeng.2015.11.457

Cited by 11 publications

(5 citation statements)

References 10 publications

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“…Let f(x) � 􏽢 g(x) − 􏽢 p(x) which can be further composed into f(x 1 , x 2 , x 3 ); then numerical integration can be approximated to equation ( 10) by applying the generalized Gaussian quadrature to evaluate the nodes and weights for the product of the polynomial and logarithmic functions [27]. e generalized Gaussian quadrature formula has been proven to give better results for integration over three-dimensional regions particularly in common applications in science and engineering [27].…”

Section: Optimisation Methodmentioning

confidence: 99%

Demonstration of Smart Railway Level Crossing Design and Validation Using Data from Metro Rail, South Africa

Tshaai

Mishra

Pidanič

2022

Journal of Advanced Transportation

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Long waiting time at railway level crossings poses a risk on the safety and affects capacity of rail and road traffic. However, in most cases, the long closing time can be prevented by reducing the time lost at a railway level crossing. The emphasis of this study is to present a numerical optimisation algorithm to reduce the time lost per train trip at a railway level crossing. Thus, attributes with the highest impact on the railway level crossing closing time were extracted from the data analysis of rail-road level crossings on the southern corridor of the Western Cape metro rail. Powell’s optimisation algorithm was formulated on the minimisation of the time lost at the railway level crossing per trip. Thus, time lost is constrained by the technical and train’s traction constraints. The upper and lower bounds of Powell’s algorithm were defined by the threshold closing time in addition to the actual and expected probability density functions. The algorithm was implemented in Matlab. Furthermore, the algorithm was trained on 8000 data sets and tested on 2000 data sets. The developed algorithm proved to be effective and robust in comparison to the current state of railway level crossings under study. Thus, the algorithm was validated to reduce the time lost at the railway level crossing by at least 50%.

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Section: Optimisation Methodmentioning

confidence: 99%

Demonstration of Smart Railway Level Crossing Design and Validation Using Data from Metro Rail, South Africa

Tshaai

Mishra

Pidanič

2022

Journal of Advanced Transportation

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“…The above equation ( 33) can be numerically calculated using the triple gauss integration method (Jayan and Nagaraja, 2015). To improve the accuracy, six integration points were chosen.…”

Section: Finite Element Methods Formulationmentioning

confidence: 99%

Finite element method and analytical analysis of static and dynamic pull-in instability of a functionally graded microplate

Pazhouheshgar

Haghighatfar

Moghanian

2020

Journal of Vibration and Control

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The static and dynamic pull-in phenomenon of a functionally graded Al/Al2O3 microplate, considering the damping coefficient and fringing field effects, has been analyzed because of its crucial effect in micro-electromechanical systems application, especially in microswitches. The nonlinear equation of motion of functionally graded microplate has been derived using Hamilton’s principle, and solved analytically. Furthermore, a finite element code has been developed to solve the problem. Comparing the theoretical and numerical results for specific boundary conditions demonstrates that the numerical solution predicts the pull-in phenomenon with the least errors; and it can be used for various material power laws, damping coefficients, and initial gaps between the microplate and the substrate. The numerical results for various boundary conditions demonstrate that by increasing the damping coefficient, the dynamic pull-in voltage is also increased, and pull-in time will slow down. Moreover, the effect of power law and applied voltage on the pull-in instability is investigated.

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“…Where we note that the values converge vertically towards the value of 0.2779460489911 as well as matching the values of integration in the last two rows when and by using the two methods R(MSM) and R(SMM). Therefore, it is possible to say that the value of integration is correct for thirteen decimal places when applying these two methods We can't use the fundamental calculation theorems to evaluate this integral, therefore, we can replace it by the approximation methods R(MSM) and R(SMM) where we obtain the results listed in tables (9) and (10) respectively. Where we note that the values converge vertically towards the value of 0.357703307442 as well as matching the values of integration in the last two rows when and by using the two methods R(MSM) and R(SMM).…”

Section: From Table (2): Whenmentioning

confidence: 99%

“…In 2015, Aljassas [10] introduced a numerical method () RM RMM to calculating triple integrals with continuous integrands by using Romberg acceleration with Mid-point rule on the three dimensions when the number of divisions on the interior dimension is equal to the number of divisions on the middle dimension, but both of them are deferent from the number of divisions on the exterior dimension and she got a high accuracy in the results in a little sub-intervals relatively and a short time. Also in 2015, Sarada et al [9] use the generalized Gaussian Quadrature to evaluate triple integral and got a good results. In this research we presented two theories with their proves to derive two new numerical methods to evaluate the triple integrals with continuous integrands and their correction terms, this two methods product from applied two rules of Newton-Cotes (Mid-point and Simpson) the first method by using Mid-point rule on the interior dimension X, Simpson's rule on the middle dimension Y and Mid-point rule on the exterior dimension Z, the second method by using Mid-point rule on both two dimensions of interior X and middle Y, while Simpson's rule used on the exterior dimension Z which denoted by symbols MSM and SMM respectively with the same number of divisions on the three dimensions.…”

Section: Introductionmentioning

confidence: 99%

Driving Two Numerical Methods to Evaluate the Triple Integrals R(MSM),R(SMM) and Compare Between Them

Aljassas¹,

Alsharify²,

Al–Karamy³

2018

IJET

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The main objective of present work is the conclusion of two new numerical methods to evaluate the triple integrals with continuous integrands and their partial derivatives are continuous too, the first method through using Mid-point rule on the interior dimension X, Simpson's rule on the middle dimension Y and Mid-point rule on the exterior dimension Z, which denoted by the symbol MSM. The second method by using Mid-point rule on both two dimensions of interior X and middle dimension Y and Simpson's rule on the exterior dimension Z which denoted by SMM, where the number of divisions on the three dimensions are equals. We have concluded two theorems with their proves to find their rules and the correction terms that we found it and to improve the results we us ed Romberg acceleration which denoted by R(MSM),R(SMM) where we got high accuracy in the results by little sub-intervals relatively and short time.

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A General and Effective Numerical Integration Method to Evaluate Triple Integrals Using Generalized Gaussian Quadrature

Cited by 11 publications

References 10 publications

Demonstration of Smart Railway Level Crossing Design and Validation Using Data from Metro Rail, South Africa

Demonstration of Smart Railway Level Crossing Design and Validation Using Data from Metro Rail, South Africa

Finite element method and analytical analysis of static and dynamic pull-in instability of a functionally graded microplate

Driving Two Numerical Methods to Evaluate the Triple Integrals R(MSM),R(SMM) and Compare Between Them

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