Gauss Legendre - Gauss Jacobi quadrature rules over a Tetrahedral region

This paper presents a Gaussian quadrature method for the evaluation of the triple integral I = â«â«â«/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 â¤1, x + y + z â¤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 â¤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

“Women in the democratic framework of electoral politics”

Women have the right to participate in political processes that affect the progress of the nation, state, society, their family and individual personality. It is observed that increase in women’s political participation and leadership leads to development of the society and also affects the performance of political process and tends to move towards more inclusiveness, responsible, egalitarian, and democratic in nature

Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface

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: View the MathML source Turn MathJax on 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

Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

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

On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface

This paper first presents a Gauss Legendre quadrature rule for the evaluation of I = ∫ ∫T f (x, y) d x d y, 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 ∫ ∫S f (x (ξ, η), y (ξ, η)) frac(∂ (x, y), ∂ (ξ, η)) d ξ d η 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 TiC (i = 1, 2, 3) 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 TiC and the linearity property of integrals leads to the result:I = underover(∑, i = 1, 3) ∫ ∫TiC f (x, y) d x d y = frac(1, 3) ∫ ∫T G (X, Y) d X d Y,where G (X, Y) = ∑i = 1n × n f (xiC (X, Y), yiC (X, Y)) and x = xiC (X, Y) and y = yiC (X, Y) refer to affine transformations which map each TiC into T the standard triangular surface. We then write ∫ ∫T G (X, Y) d X d Y = ∫ ∫S G (X (ξ, η), Y (ξ, η)) frac(∂ (X, Y), ∂ (ξ, η)) d ξ d η and a composite rule of integration is thus obtained. We next 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 / (2 n2) 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:∫ ∫T f (x, y) d x d y = underover(∑, i = 1, n × n) ∫ ∫Ti f (x, y) d x d y = frac(1, n2) ∫ ∫T H (X, Y) d X d Y,where H (X, Y) = ∑i = 1n × n f (xi (X, Y), yi (X, Y)) and x = xi (X, Y), y = yi (X, Y) refer to affine transformations which map each Ti in (x, y) space into T a standard triangular surface T in the (x, y) space. We can now apply the two rules earlier derived to the integral ∫ ∫T H (X, Y) d X d Y, this amounts to application of composite numerical integration of T into n2 and 3n2 triangles of equal sizes respectively. We can now apply the rules, which are derived earlier to the evaluation of the integral, ∫ ∫T f (x, y) d x d y and each of these procedures converges to the exact value of the integral ∫ ∫T f (x, y) d x d y for sufficiently large value of n and the convergence is much faster for higher order rules. We have demonstrated this aspect by applying the above composite integration method to some typical integrals. © 2007 Elsevier Inc. All rights reserved

On the application of two Gauss-Legendre quadrature rules for composite numerical integration over a tetrahedral region

In this paper we first present a Gauss-Legendre quadrature rule for the evaluation of I = ∫ ∫ T ∫ f (x, y, z) d x d y d z, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z){divides}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 {Mathematical expression} over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ){divides} -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 T i c (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 T i c and the linearity property of integrals leads to the result:I = underover(∑, i = 1, 4) ∫ ∫ Tic ∫ f (x, y, z) d x d y d z = frac(1, 4) ∫ ∫ T ∫ G (X, Y, Z) d X d Y d Z,where{Mathematical expression}refer to an affine transformations which map each T i c into the standard tetrahedral region T. We then write{Mathematical expression}and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p 3 tetrahedra T i (i = 1(1)p 3) each of which has volume equal to 1/(6p 3) units. We have again shown that the use of affine transformations over each T i and the use of linearity property of integrals leads to the result:{Mathematical expression}where{Mathematical expression}refer to the affine transformations which map each T i in (x (α,p), y (α,p), z (α,p)) space into a standard tetrahedron T in the (X, Y, Z) space. We can now apply the two rules earlier derived to the integral ∫ ∫ T ∫ H (X, Y, Z) d X d Y d Z, this amounts to the application of composite numerical integration of T into p 3 and 4p 3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals. © 2006 Elsevier Inc. All rights reserved

Data Aware Routing in Wireless Sensor Networks

The wireless sensor network (WSN) has set of wireless intelligent sensor nodes with high speed network. Nodes are deployed randomly in a surge of unanticipated applications. The routing is one of the most important challenges in WSNs for data transmission over the sensor nodes. The paper proposes the data aware routing in WSNs, which incorporates energy efficient routing of data. The objective of the proposed work is to improve the performance of network in terms of energy consumption and throughput. The simulation results show that the proposed approach perform better in-terms of utilization of minimum energy, efficient for cluster formation, and reduce communication overhead in WSNs

Gauss Legendre quadrature over a triangle

This paper presents a Gauss Legendre quadrature method for numerical integration over the standard triangular surface: (x, y) | 0 â x, y â 1, x + y â 1 in the Cartesian two-dimensional (x, y) space. Mathematical transformation from (x, y) space to (ξ, η) space map the standard triangle in (x, y) space to a standard 2-square in (ξ, η) space: (ξ, η)|-l â ξ, η â 1. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points and yields results which are accurate and reliable. Results obtained with new formulae are compared with the existing formulae. © Indian Institute of Science