Numerical Integration Over Arbitrary Tetrahedral Element by Transforming into Standard 1-cube .

TM Mamatha; B Venkatesh

Author(s)

TM Mamatha Department of Mathematics, Amrita University, India

B Venkatesh Department of Mathematics, Amrita University, India

Abstract

In this paper, we are using two different transformations to transform the arbitrary linear tetrahedron element to a standard 1-Cube element and obtain the numerical integration formulas over arbitrary linear tetrahedron element implementing generalized Gaussian quadrature rules, with minimum computational time and cost. We also obtain the integral value of some functions with singularity over arbitrary linear tetrahedron region, without discretizing the tetrahedral region into P3 tetrahedral regions. It may be noted the computed results are converging faster than the numerical results in referred articles and are exact for up to 15 decimal values with minimum computational time. In a tetrahedral sub-atomic geometry, a focal particle is situated at the middle with four substituents that are situated at the sides of a tetrahedron. The bond edges are cos-1(-?) = 109.4712206...° ˜ 109.5° when each of the four substituents are the same, as in methane (CH4) and in addition its heavier analogs. The impeccably symmetrical tetrahedron has a place with point amass Td, yet most tetrahedral particles have brought down symmetry. Tetrahedral atoms can be chiral. Mathematically the problem is to evaluate the volume integral over a arbitrary tetrahedron transforming the triple integral over arbitrary linear tetrahedron into the integrals over a standard 1-cube using two different parametric transformations.

Publication Details

Page(s) 629-634

DOI DOI not available

Published Journal: Pakistan Journal of Biotechnology, Volume: 15, Issue: 3, Year: 2018

Request the full text from PASTIC

References

Cattani , C.,A.Paoluzzi, 1990.Boundary integration over linear Polyhedral,Computer Aided Design 22 130 -135

Cattani , C.,A.Paoluzzi, 1990.,Symbolic Analysis of linear Polyhedral. Engineering and Computers 6 17 -29

Lee , Y.T.,A.A.G.Requicha, 1982.Algorithms for computing the volume and other integral properties of solids I: known methods and open issues,Commun. ACM 25 635 -641

MA , J., V.,Rokhlin and S.,Wandzura, 1996.Generalized Gaussian Quadrature rules for systems of arbitrary functions,Siam Journal of Numerical Analysis 33 971 -996

Mamatha , T.M.,B.Venkatesh, 2015.Gauss Quadrature rules for numerical integration over a standard tetrahedral element by decomposing into hexahedral elements,Applied Mathematics and Computation 271 1062 -1070

Mamatha , T.M.,B.Venkatesh, 2016.Generalised Gaussian Quadrature rules for numerical integration over tetrahedral Element,Proc. of 1st International conference on Research in Engineering, Computers and Technology (ICRECT 172 -177

Rathod , H.T. and H.S.,GovindaRao, 1996.Integration of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space,Computers and Structures 59 55 -65

Rathod , H. T. and H. S.,GovindaRao, 1998.Integration of tri-variant polynomials over linear polyhedral in Euclidean three-dimensional space,Journal of Australian mathematical Society Series B39 355 -385

Rathod , H.T., B.,Venkatesudu and K.V.,Nagaraja, 2005.,International Journal of Computational Methods in Engineering Science and Mechanics 6 179 -186

Rathod , H.T., B.,Venkatesudu and K.V.,Nagaraja, 2005.Symmetric Gauss Legendre Quadrature formulae for composite numerical integration over a tetrahedral region,The Aligarh Bullettin of Maths 24 1 -2

dral region,International Journal of Applied Mathematics and Computation 190 186 -194 2007

Rathod , H.T., B.,Venkatesudu and K.V.,Nagaraja, 2007.On the application of two Gauss Legendre Quadrature rules for composite numerical integration over a tetrahedral region,Applied Mathematics and Computation 189 131 -162

Rathod , H.T. and B.,Venkatesudu, 2011.Gauss Jacobi Quadrature rules over a tetrahedral region,International Journal of Methematical Analysis 5 189 -198

Stroud A. H . and D.,Secrest,EnglewoodCliffs, 1966., -