Numerical Integration over arbitrary Tetrahedral Element by transforming into standard 1-Cube

• Published in: IOP conference series. Materials Science and Engineering 2019-11, Vol.577 (1), p.12172
• Main Authors: Mamatha, T M, Venkatesh, B
• Format: Article
• Language: English
• Subjects: Computational efficiency; Computing time; Integrals; Methane; Numerical integration; Quadratures; Singularity (mathematics); Tetrahedra; Transformations (mathematics)
• ISSN: 1757-8981; 1757-899X
• DOI: 10.1088/1757-899X/577/1/012172

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(−⅓) = 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 an arbitrary tetrahedron transforming the triple integral over arbitrary linear tetrahedron into the integrals over a standard 1-cube using two different parametric transformations.