|
| << vorige | volgende >> |
Tijdschrift beschrijving
Alle jaargangen van het bijbehorende tijdschrift
Alle afleveringen van het bijbehorende jaargang
Alle artikelen van de bijbehorende aflevering
Details van artikel 16 van 19 gevonden artikelen
| Titel: |
On the Application of Two Symmetric Gauss Legendre Quadrature Rules for Composite Numerical Integration Over a Tetrahedral Region |
| Auteur: |
Rathod, H. T. Venkatesudu, B. Nagaraja, K. V. |
| Verschenen in: |
International journal for computational methods in engineering science and mechanics |
| Paginering: | Jaargang 7 (2006) nr. 6 pagina's 445-459 |
| Jaar: |
2006-12-01 |
| Inhoud: |
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 = ∫- 11∫- 11∫- 11f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) [image omitted] 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 Tic (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 Tic and the linearity property of integrals leads to the result: [image omitted] where [image omitted] refer to the affine transformations, which map each Tic into the standard tetrahedral region T. We then write [image omitted] and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p3 tetrahedra Ti (i = 1 (1) p3), each of which has volume equal to 1/(6p3) units. We have again shown that the use of affine transformations over each Ti and the use of linearity property of integrals leads to the result: [image omitted] where [image omitted] refer to the affine transformations, which map each Ti 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)dXdYdZ; this amounts to the application of composite numerical integration of T into p3 and 4p3 tetrahedra of equal volume. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals. |
| Uitgever: |
Taylor & Francis |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
Details van artikel 16 van 19 gevonden artikelen
| << vorige | volgende >> |