Dr. B. Venkatesh currently serves as Associate Professor and Head of the Department of Mathematics at the Amrita School of Engineering, Bengaluru Campus. He was conferred with a Ph. D. in Mathematics, Central College, Bangalore University, Bengaluru. Prior to joining Amrita, he served as Senior Lecturer and Lecturer at The Oxford College of Engineering and BTL Institute of Technology respectively. He has a teaching experience of 19 years.

Qualification

Degree

University

Year

Ph. D

Bangalore University

2006

M.Phil

Bangalore University

1998

M.Sc

Bangalore University

1996

Publications

Publication Type: Journal Article

Year of Publication

Publication Type

Title

2015

Journal Article

T. M. Mamatha and Venkatesh, B., “Gauss quadrature rules for numerical integration over a standard tetrahedral element by decomposing into hexahedral elements”, Applied Mathematics and Computation, vol. 271, pp. 1062–1070, 2015.[Abstract]

In recent years hexahedral elements have gained more importance than compared to tetrahedral elements (e.g. importance in the study of aero-acoustic equations using hexahedral elements to check the computational efficiency between tetrahedral and hexahedral elements). Also among the various integration schemes, Gauss Legendre quadrature which can evaluate exactly the (2n−1)th order polynomial with n-Gaussian points is most commonly used in view of the accuracy and efficiency of calculations. In this paper, we present a Gauss quadrature method for numerical integration over a standard tetrahedral element T[0,1]3 by decomposing into hexahedral elements H[−1,1]3. The method can be used for computing integrals of smooth functions, as well as functions with end-point singularities. The performance of the method is demonstrated with several numerical examples. By the proposed method, with less number of divisions we are obtaining the exact solutions with minimum errors and number of computations is reduced drastically. We have evaluated the aspect ratio value of each hexahedral element which is in the range 1–5, as per the element quality check these elements can be used for mesh generation in FEM.

H. T. Rathod, Venkatesh, B., T, S. K., and M, M. T., “Numerical integration over polygonal domains using convex quadrangulation and gauss legendre quadrature rules”, Inernational journal of engineering and computer science, vol. 2, no. 8, pp. 2576–2610, 2013.

2011

Journal Article

H. Ta Rathod, Venkatesh, B., and .V.Nagaraja, K., “Gauss Legendre - Gauss Jacobi quadrature rules over a Tetrahedral region”, International Journal of Mathematical Analysis, vol. 5, pp. 189-198, 2011.[Abstract]

This paper presents a Gaussian quadrature method for the evaluation of the triple integral I = ∫∫∫/T f (x, y, z) dxdydz, where f (x, y, z) is an analytic function in x, y, z and T refers to the standard tetrahedral region:{(x, y, z) | 0 ≤ x, y, z ≤1, x + y + z ≤1} in three space (x, y, z). Mathematical transformation from (x, y, z) space to (u, v, w) space maps the standard tetrahedron T in (x, y, z) space to a standard 1-cube: {(u,v,w) / 0 ≤ u, v, w ≤1} in (u, v, w) space. Then we use the product of Gauss-Legendre and Gauss-Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T.

H. T. Rathod, Venkatesh, B., Nagabhushan, C. S., and Hariprasad, A. S., “On quintic splines with applications to quadrature over curved domains”, International electronic engineering mathematical society, vol. 6, pp. 126–146, 2011.

2010

Journal Article

H. T. Rathod, Nagabhushan, C. S., Venkatesh, B., and , “On quintic splines with applications to function and integrable function approximations”, International electronic engineering mathematical society, vol. 4, pp. 73–103, 2010.

2010

Journal Article

H. T. Rathod, Hariprasad, A. S., Venkatesh, B., and Nagabhushan, C. S., “The use of quintic splines for high accracy function and integrable function approximations”, International electronic engineering mathematical society, vol. 4, pp. 104–128, 2010.

2008

Journal Article

H. Ta Rathod, .V.Nagaraja, K., Naidu, VaKesavulu, and Venkatesh, B., “The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements”, Finite Elements in Analysis and Design, vol. 44, pp. 920-932, 2008.[Abstract]

H. T. Rathod, .V.Nagaraja, K., and Venkatesh, B., “On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface”, Applied mathematics and computation, vol. 190, pp. 21–39, 2007.[Abstract]

This paper first presents a Gauss Legendre quadrature rule for the evaluation of View the MathML source, where f(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)|0⩽x,y⩽1,x+y⩽1} in the two space (x,y). We transform this integral into an equivalent integral View the MathML source where S is the 2-square in (ξ, η ) space: {(ξ,η)|-1⩽ξ,η⩽1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules with new weight coefficients and new sampling points. Then a second Gauss Legendre quadrature rule of composite type is obtained. This rule is derived by discretising T into three new triangles View the MathML source of equal size which are obtained by joining centroid of T , C=(1/3,1/3) to the three vertices of T . By use of affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result:
More »»

2007

Journal Article

H. T. Rathod, .V.Nagaraja, K., and Venkatesh, B., “Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space”, Applied Mathematics and Computation, vol. 191, pp. 397–409, 2007.[Abstract]

In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z ) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the triple integrals of trivariate polynomials over an arbitrary tetrahedron as a special case. It is also demonstrated that certain integrals which are nonpolynomial functions of trivariates x,y,z can be computed by the proposed method. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al. [H.T. Rathod, K.V. Nagaraja, B. Venkatesudu, N.L. Ramesh, Gauss Legendre Quadrature over a Triangle, J. Indian Inst. Sci. 84 (2004) 183–188] to evaluate the typical integrals governed by the proposed method.
More »»

2007

Journal Article

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region”, Applied mathematics and computation, vol. 189, pp. 131–162, 2007.[Abstract]

In this paper we first present a Gauss–Legendre quadrature rule for the evaluation of View the MathML source, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z)∣0 ⩽ x, y, z ⩽ 1, x + y + z ⩽ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ ) to change the integral into an equivalent integral View the MathML source over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ)∣ −1 ⩽ ξ, η, ζ ⩽ 1}. We then apply the one-dimensional Gauss–Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss–Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra View the MathML source (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T . By use of the affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result:
More »»

2007

Journal Article

H. T. Rathod, Venkatesh, B., and Nagaraja, K. V., “Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface”, Applied Mathematics and Computation, vol. 188, no. 1, pp. 865–876, 2007.[Abstract]

This paper first presents a Gauss Legendre quadrature method for numerical integration of View the MathML source, where f(x, y) is an analytic function in x, y and T is the standard triangular surface: {(x, y)∣0 ⩽ x, y ⩽ 1, x + y ⩽ 1} in the Cartesian two dimensional (x, y) space. We then use a transformation x = x(ξ, η), y = y(ξ, η ) to change the integral I to an equivalent integral View the MathML source, where S is now the 2-square in (ξ, η) space: {(ξ, η)∣ − 1 ⩽ ξ, η ⩽ 1}. We then apply the one dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti (i = 1(1)n2) each of which has an area equal to 1/(2n2) units. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result:
where View the MathML source and x = xi(X, Y) and y = yi(X, Y) refer to affine transformations which map each Ti in (x, y) space into a standard triangular surface T in (X, Y) space. We can now apply Gauss Legendre quadrature formulas which are derived earlier for I to evaluate the integral View the MathML source. We observe that the above procedure which clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integral View the MathML source, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We have demonstrated this aspect by applying the above explained Composite Numerical Integration method to some typical integrals.
More »»

2007

Journal Article

H. T. Rathod, Venkatesh, B., Nagaraja, K. V., and Islam, M. Shafiqul, “Gauss Legendre–Gauss Jacobi quadrature rules over a tetrahedral region”, Applied mathematics and computation, vol. 190, pp. 186–194, 2007.[Abstract]

This paper presents a Gaussian Quadrature method for the evaluation of the triple integral View the MathML source, where f(x,y,z) is an analytic function in x, y, z and T refers to the standard tetrahedral region: {(x,y,z)|0⩽x,y,z⩽1,x+y+z⩽1} in three space (x,y,z). Mathematical transformation from (x,y,z) space to (U,V,W) space map the standard tetrahedron T in (x,y,z) space to a standard 1-cube: {(U,V,W)/0⩽U,V,W⩽1} in (U,V,W) space. Then we use the product of Gauss Legendre and Gauss Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T.
More »»

2006

Journal Article

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “Gauss legendre quadrature formulas over a tetrahedron”, Numerical Methods for Partial Differential Equations, vol. 22, pp. 197–219, 2006.[Abstract]

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a tetrahedral region”, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 7, no. 6, pp. 445–459, 2006.[Abstract]

In this paper, we first present a Gauss Legendre Quadrature rule for the evaluation of I = ∫∫∫T f(x,y,z) dxdydz , where f(x,y,z) is an analytic function in x,y,z and T is the standard tetrahedral region: {(x,y,z) |0 ≤ x,y,z ≤ 1,x + y + z ≤ 1} in three space ( x,y,z) . We then use the transformations x = x(ξ,η,ζ), y = y (ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral I into an equivalent integral I = ∫− 1 1∫− 1 1∫− 1 1f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) dξ dη dζ over the standard 2-cube in (ξ,η,ζ) space: {(ξ,η,ζ) | − 1 ≤ ξ,η,ζ ≤ 1} . We then apply the one-dimensional Gauss Legendre Quadrature rule in ξ , η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss Legendre Quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra Ti c ( i = 1,2,3,4) of equal size, which are obtained by joining centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. Use of the affine transformations defined over each Ti c and the linearity property of integrals leads to the result:
More »»

2006

Journal Article

H. T. Rathod, Venkatesh, B., and Nagaraja, K. V., “On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a tetrahedral region 2006”, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 7, pp. 445–459, 2006.

2005

Journal Article

H. T. Rathod, Venkatesh, B., and Nagaraja, K. V., “Symmetric Gauss Legendre quadrature rules for composite numerical integration over a tetrahedral region”, Journal of Bulletin of Mathematics, vol. 24, pp. 51–79, 2005.

2005

Journal Article

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “Gauss Legendre Quadrature Formulae for Tetrahedra”, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 6, no. 3, pp. 179–186, 2005.[Abstract]

In this paper we consider the Gauss Legendre quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)| 0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: {(ξ, η, ζ)| − 1 ≤ ξ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight co-efficients and sampling points. The effectiveness of the formulae is demonstrated by applying them to the integration of three nonpolynomial and three polynomial functions.
More »»

Dr. B. Venkatesh currently serves as Associate Professor and Head of the Department of Mathematics at the Amrita School of Engineering, Bengaluru Campus. He was conferred with a Ph. D. in Mathematics, Central College, Bangalore University, Bengaluru. Prior to joining Amrita, he served as Senior Lecturer and Lecturer at The Oxford College of Engineering and BTL Institute of Technology respectively. He has a teaching experience of 19 years.

## Qualification

## Publications

In recent years hexahedral elements have gained more importance than compared to tetrahedral elements (e.g. importance in the study of aero-acoustic equations using hexahedral elements to check the computational efficiency between tetrahedral and hexahedral elements). Also among the various integration schemes, Gauss Legendre quadrature which can evaluate exactly the (2n−1)th order polynomial with n-Gaussian points is most commonly used in view of the accuracy and efficiency of calculations. In this paper, we present a Gauss quadrature method for numerical integration over a standard tetrahedral element T[0,1]3 by decomposing into hexahedral elements H[−1,1]3. The method can be used for computing integrals of smooth functions, as well as functions with end-point singularities. The performance of the method is demonstrated with several numerical examples. By the proposed method, with less number of divisions we are obtaining the exact solutions with minimum errors and number of computations is reduced drastically. We have evaluated the aspect ratio value of each hexahedral element which is in the range 1–5, as per the element quality check these elements can be used for mesh generation in FEM.

More »»This paper presents a Gaussian quadrature method for the evaluation of the triple integral I = ∫∫∫/T f (x, y, z) dxdydz, where f (x, y, z) is an analytic function in x, y, z and T refers to the standard tetrahedral region:{(x, y, z) | 0 ≤ x, y, z ≤1, x + y + z ≤1} in three space (x, y, z). Mathematical transformation from (x, y, z) space to (u, v, w) space maps the standard tetrahedron T in (x, y, z) space to a standard 1-cube: {(u,v,w) / 0 ≤ u, v, w ≤1} in (u, v, w) space. Then we use the product of Gauss-Legendre and Gauss-Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T.

More »»This paper is concerned with curved boundary triangular elements having one curved side and two straight sides. The curved elements considered here are the 6-node (quadratic), 10-node (cubic), 15-node (quartic) and 21-node (quintic) triangular elements. On using the isoparametric coordinate transformation, these curved triangles in the global (x, y) coordinate system are mapped into a standard triangle: { (ξ, η) / 0 ≤ ξ, η ≤ 1, ξ + η ≤ 1 } in the local coordinate system (ξ, η). Under this transformation curved boundary of these triangular elements is implicitly replaced by quadratic, cubic, quartic and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in quartic and quintic arcs in such a way that each arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides. © 2008 Elsevier B.V. All rights reserved.

More »»This paper first presents a Gauss Legendre quadrature rule for the evaluation of View the MathML source, where f(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)|0⩽x,y⩽1,x+y⩽1} in the two space (x,y). We transform this integral into an equivalent integral View the MathML source where S is the 2-square in (ξ, η ) space: {(ξ,η)|-1⩽ξ,η⩽1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules with new weight coefficients and new sampling points. Then a second Gauss Legendre quadrature rule of composite type is obtained. This rule is derived by discretising T into three new triangles View the MathML source of equal size which are obtained by joining centroid of T , C=(1/3,1/3) to the three vertices of T . By use of affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result: More »»

In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z ) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the triple integrals of trivariate polynomials over an arbitrary tetrahedron as a special case. It is also demonstrated that certain integrals which are nonpolynomial functions of trivariates x,y,z can be computed by the proposed method. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al. [H.T. Rathod, K.V. Nagaraja, B. Venkatesudu, N.L. Ramesh, Gauss Legendre Quadrature over a Triangle, J. Indian Inst. Sci. 84 (2004) 183–188] to evaluate the typical integrals governed by the proposed method. More »»

In this paper we first present a Gauss–Legendre quadrature rule for the evaluation of View the MathML source, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z)∣0 ⩽ x, y, z ⩽ 1, x + y + z ⩽ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ ) to change the integral into an equivalent integral View the MathML source over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ)∣ −1 ⩽ ξ, η, ζ ⩽ 1}. We then apply the one-dimensional Gauss–Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss–Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra View the MathML source (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T . By use of the affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result: More »»

This paper first presents a Gauss Legendre quadrature method for numerical integration of View the MathML source, where f(x, y) is an analytic function in x, y and T is the standard triangular surface: {(x, y)∣0 ⩽ x, y ⩽ 1, x + y ⩽ 1} in the Cartesian two dimensional (x, y) space. We then use a transformation x = x(ξ, η), y = y(ξ, η ) to change the integral I to an equivalent integral View the MathML source, where S is now the 2-square in (ξ, η) space: {(ξ, η)∣ − 1 ⩽ ξ, η ⩽ 1}. We then apply the one dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti (i = 1(1)n2) each of which has an area equal to 1/(2n2) units. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result: where View the MathML source and x = xi(X, Y) and y = yi(X, Y) refer to affine transformations which map each Ti in (x, y) space into a standard triangular surface T in (X, Y) space. We can now apply Gauss Legendre quadrature formulas which are derived earlier for I to evaluate the integral View the MathML source. We observe that the above procedure which clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integral View the MathML source, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We have demonstrated this aspect by applying the above explained Composite Numerical Integration method to some typical integrals. More »»

This paper presents a Gaussian Quadrature method for the evaluation of the triple integral View the MathML source, where f(x,y,z) is an analytic function in x, y, z and T refers to the standard tetrahedral region: {(x,y,z)|0⩽x,y,z⩽1,x+y+z⩽1} in three space (x,y,z). Mathematical transformation from (x,y,z) space to (U,V,W) space map the standard tetrahedron T in (x,y,z) space to a standard 1-cube: {(U,V,W)/0⩽U,V,W⩽1} in (U,V,W) space. Then we use the product of Gauss Legendre and Gauss Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T. More »»

In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: {(ξ, η, ζ)| − 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three-dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 More »»

In this paper, we first present a Gauss Legendre Quadrature rule for the evaluation of I = ∫∫∫T f(x,y,z) dxdydz , where f(x,y,z) is an analytic function in x,y,z and T is the standard tetrahedral region: {(x,y,z) |0 ≤ x,y,z ≤ 1,x + y + z ≤ 1} in three space ( x,y,z) . We then use the transformations x = x(ξ,η,ζ), y = y (ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral I into an equivalent integral I = ∫− 1 1∫− 1 1∫− 1 1f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) dξ dη dζ over the standard 2-cube in (ξ,η,ζ) space: {(ξ,η,ζ) | − 1 ≤ ξ,η,ζ ≤ 1} . We then apply the one-dimensional Gauss Legendre Quadrature rule in ξ , η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss Legendre Quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra Ti c ( i = 1,2,3,4) of equal size, which are obtained by joining centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. Use of the affine transformations defined over each Ti c and the linearity property of integrals leads to the result: More »»

In this paper we consider the Gauss Legendre quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)| 0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: {(ξ, η, ζ)| − 1 ≤ ξ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight co-efficients and sampling points. The effectiveness of the formulae is demonstrated by applying them to the integration of three nonpolynomial and three polynomial functions. More »»

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