On use of Mixed Quadrature Rule for Numerical Integration over a Triangular Domain

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1 International Journal of Pure and Applied Mathematical Sciences. ISSN Volume 9, Number 2 (26), pp Research India Publications On use of Mixed Quadrature Rule for Numerical Integration over a Triangular Domain Dwiti Krushna Behera and Rajani Ballav Dash Department of Mathematics Ravenshaw University, Cuttack-7533, Odisha, India. dwiti78@gmail.com, rajani_bdash@rediffmail.com Abstract In this paper, we introduce a mixed quadrature of Fejer s second rule and Gaussian rule for numerical integration over the standard triangular surface: {(x, y) x, y, x + y } in the Cartesian two-dimensional (x, y) space. Mathematical transformation from (x, y) space to (ξ, η) space maps the standard triangle in (x, y) space to a standard 2-square in (ξ, η) space: {(ξ, η) ξ, η }. The relative efficiencies of this rule have been numerically verified on test integrals. Asymptotic error estimate of this rule has been determined. Keywords: Fejer s second quadrature rule, Gaussian rule, mixed quadrature rule, finite element Method(FEM), extended numerical integration. 2 Mathematics Subject Classification: 65D3, 65D32 I. INTRODUCTION The integration theory extends from real line to the plane and three dimensional spaces by the introduction of multiple integrals. In practice, most of integrals those can t be evaluated either analytically or through the evaluation process are very lengthy and tedious. In probabilistic estimation, the finite element method(fem) is also a well known method for evaluating of integrals over arbitrary-shaped domain Ω. The triangular and tetrahedral elements are very widely used in FEM. Also, FEM are used in calculation of mass shell, fluid and mass flows across a surface, electric charge distribution over a surface, plate bending, heat conduction over a plate and

2 92 Dwiti Krushna Behera and Rajani Ballav Dash similar problems in other areas of engineering which are very difficult to analyze using analytic techniques but these problems can be solved by using FEM. The basic problem of integration of an arbitrary function of two variables over the surface of a triangle was first introduced by Hammer et. al. [] and Hammer and Stroud [2, 3]. In connection of FEM, the triangular elements provide a tremendous results. Cowper [4] provided a table of Gaussian quadrature formula for symmetrically placed integration points. Lyness and Jespersen [5] made an elaborate study of symmetric quadrature rules by formulating the problem in polar coordinates. Lannoy [6] discussed the numerical error in integration rule [4]. Laurie [7] derived 7-point integration rule and discussed the numerical error in integrating some functions. Laursen and Gellert [8] gave a detailed table of symmetric integration formulae and suggested some new higher-order formulae of precision up to degree. Lether [9], Hillion[2] and Lague and Baldur [3] considered the product formulae derivable from one-dimensional Gaussian quadrature rules. Reddy [4 ], and Reddy and Shippy [8]derived 3-, 4-, 6 and 7-point formulae which give improved accuracy. The formulation of mixed quadrature rules was first coined by R. N. Das and G. Pradhan [9]. Many author s [2-2, 24] have produced different mixed quadrature rules. Very recently D. K. Behera et. al. [2-2] formed a mixed quadrature rule blending Fejer s second rule and Gaussian rule for evaluating integrals of both real and analytic functions, which leads a excellent result. In this paper, we get motivation for successfully forming a mixed quadrature rule of Fejer s second rule and Gaussian rule over a triangular domain. The mixed quadrature rule over a triangular domain so found has been tested and compared with its constituent rules by computing numerically three tested integrals. II. FORMULATION OF INTEGRALS OVER A TRIANGULAR AREA The finite-element method for two-dimensional problems with triangular elements requires the numerical integration of shape functions on a triangle. Since an affine transformation makes it possible to transform any triangle into the two-dimensional standard triangle T with co-ordinates (, ), (, ) and (, ) in Cartesian frame, we have to consider just the numerical integration on T. The numerical integration of an arbitrary function f over the triangle T is given by I = T f(x, y) x y dxdy = dx f(x, y)dy = dy f(x, y)dx. (2.) It is now required to find the value of the integral by a quadrature formula: N I= m= C m f(x m, y m ). (2.2) Where C m are the weights associated with specific points (x m, y m ) and N is the number of pivotal points related to the required precision.

3 On use of Mixed Quadrature Rule for Numerical Integration over a Triangular Domain 93 The double integral over the triangle surface of equation (2.) can be transformed to the standard square {(u, v) u, v } by substitution x = u and y = ( u)v We have I = Where J(u, v) = x f(x, y) dydx = f(x(u, v), y(u, v)) x y u x u y v v = u. J dudv. (2.3) From equation (2.3), we have I = f(u, ( u)v)( u)dudv. (2.4) The integral I of equation (2.4) can be transformed further into an integral over the standard 2-square: {(ξ, η) ξ, η } by substitution u = +ξ Then clearly the determinant of the Jacobian and the differential area are (u, v) = u v u v = (ξ, η ) ξ η η ξ 2 () = 2 4 (u, v) 2, v = +η 2. (2.5) dudv = dξdη = dξdη (2.6) (ξ, η ) 4 Now on using equations (2.5) and (2.6) in equation(2.4) we have I = x f(x, y) dydx = f(u, ( u)v)( u) dudv = f ( +ξ, ( ξ)(+η) 2 4 ) ( ξ ) dξdη (2.7) 8 Equation(2.7) represents an integral over the surface of standard 2-square: {(ξ, η) ξ, η }. Efficiently quadrature coefficients are readily obtained [25]. From equation(2.7), we can write: I = f(x(ξ, η), y(ξ, η)) n n ( ξ 8 ) dξdη. I = ( ξ i i= j= w i w j f(x(ξ i, η j ), y(ξ i, η j ) ). (2.8) 8 ) Where ξ i, η j are Gaussian points in the ξ, η directions respectively, and w i, w j are the corresponding weights.

4 94 Dwiti Krushna Behera and Rajani Ballav Dash We can write equation(2.8) as: N=n n I = k= C k f(x k, y k ). (2.9) Where C k, x k, y k are obtained from the relation C k = ( ξ i 8 ) w iw j x k = +ξ i 2 y k = ( ξ i )(+η j) 4 } Where k =,2,, n i =,2,, n j =,2,, n (2.) The weighting coefficients C k and sampling points (x k, y k ) of various orders can now be easily computed by formulae (2.9) and (2.). We have tabulated a sample of these weight coefficients and sampling points in Table-. III. MIXED QUADRATURE OF FEJER S 2 ND RULE AND GAUSS- LEGENDRE RULE IN ONE VARIABLE The mixed quadrature (I mix ) of Fejer s 2 nd rule and Gauss-Legendre rule in one variable developed by D. K. Behera et.al.[2] is given below: I mix = f(x)dx 225 [ 896f ( 3 2 ) 375f ( 3 5 ) + 52f ( 2 ) + 64f() + 52f ( 2 ) 375f ( 3 ) + 896f ( 3 ) 5 2 ] (3.) Where w = , w 2 = , w 3 = , w 4 = , w 5 = , w 6 = , w 7 = Applying the mixed rule(3.) to double integral we have I mix = f(x(ξ, η), y(ξ, η)) ( ξ 8 ) dξdη I mix k= C k f(x ij, y ij ) = k= C k f(x k, y k ) (3.2) Where C k = C ij = ( ξ i 8 ) w iw j x k = x ij and y k = y ij } (3.3)

5 On use of Mixed Quadrature Rule for Numerical Integration over a Triangular Domain 95 The weighting coefficients C k and sampling points (x k, y k ) of various orders can be easily computed using the equation (2.). NUMERICAL VERIFICATIONS Integrals Exact Value Approximate Values Error y I = (x + y) 2dxdy y I 2 = (x + y) 2dxdy x I 3 = e y2 cos(xy)dxdy V. CONCLUSIONS From the numerical verification we conclude that the mixed quadratute rule used in this paper gives better result than those obtained in the previous papers [6, 22, 24 ] on integration of real functions over triangles. REFERENCES [] P. C. Hammer, O. J. Marlowe and A. H. Stroud, Numerical integration over simplexes and cones, Math Tables other Aids Computation,, 3-36 (956). [2] P. C. Hammer and A. H. Stroud, Numerical integration over simplexes, Math Tables other Aids Computation,, (956). [3] P. C. Hammer and A. H. Stroud, Numerical evaluation of multiple integrals, Math Tables other Aids Computation, 2, (958). [4] G. R. Cowper, Gaussian quadrature formulas for triangles, Int. Num. Meth. Engg., 7, (973). [5] J. N. Lyness and D. Jespersen, Moderate degree symmetric quadrature rules for the triangle, J. Inst. Math. Applic., 5, 9-32 (975). [6] F. G. Lannoy, Triangular Finite Elements and Numerical Integration, Computer Struct, 7, 63 (977). [7] D. P. Laurie, Automatic numerical integration over a triangle, CSIR Spec. Rep. WISK 273, National Institute for Mathematical Sciences, Pretoria (977).

6 96 Dwiti Krushna Behera and Rajani Ballav Dash [8] M. E. Laursen and M. Gellert, Some criteria for numerically integrated matrices and quadrature formulas for triangles, Int. J. Num. Meth. Engg, 2, (978). [9] F. G. Lether, Computation of double integrals over a triangle, J. Comp. Applic. Math., 2, (976). [] J. Stoer and R.Bulirsch, Introduction to Numerical Analysis, 3 rd ed.(springer International Edition). [] Kendall E. Atkinson, An Introduction to numerical Analysis, 2 nd ed.(john Wiley). [2] P. Hillion, Numerical integration on a triangle, Int. J. Num. Meth. Engg.,, (977). [3] G. Lague and R. Baldur, Extended numerical integration method for triangular surfaces, Int. J. Num. Meth. Engg.,, (977). [4] C. T. Reddy, Improved three point integration schemes for triangular finite elements, Int. J. Num. Meth. Engg., 2, (978). [5] H. T. Rathod and M. S. Karim, An Explicit Integration Scheme based on Recursion for Curved Triangular Finite Elements, Computer Struct, 8, 43-76(22). [6] H. T. Rathod et. al., Gauss-Legendre Quadrature over Triangles, J. Indian Inst. Sci., 84, 83 88(24). [7] H. T. Rathod and M. S. Karim, An explicit integration scheme based on recursion for curved triangular finite elements, Computers Struct., 8, 43-76(22). [8] C. T. Reddy and D. J. Shippy, Alternative integration formulae for triangular finite elements, Int. J. Num. Meth. Engg., 7, 33-39(98). [9] R. N. Das and G. Pradhan, A mixed quadrature rule for approximate evalution of real definite integrals, Int. J. Math. Educ. Sci. and Technology, 27(2), (996). [2] D. K. Behera, A. K. Sethi and R. B. Dash, An Open type Mixed Quadrature Rule using Fejer and Gaussian Quadrature Rules, American International Journal of Research in Science, Technology, Engineering & Mathematics, 9(3), (25). [2] D. K. Behera and R. B. Dash, A Mixed Quadrature Rule for Numerical Integration of Analytic Functions by using Fejer and Gaussian Quadrature Rules, Bulletin of Pure and Applied Sciences, 34E(2), 6-67 (25). [22] K. T. Shivaram, Generalised Gaussian Quadrature over a triangle, American Journal of Engineering Research, 2(9), (23. [23] F. Hussain and M. S. Karim, Accurate Evalution Schemes for Triangular Domain Integrals, IOSR Journal of Mechanical and Civil Engineering, 2(6), 38-5(22). [24] S. R. Jena and R. B. Dash, Mixed Quadrature of real definite Integral over triangles, Pacific-Asian Journal of Mathematics, 3(-2), 9-24(29). [25] M. Abramowicz and I. A. Stegum(eds), Handbook of Mathematical Functions, Dover (964).

Short Communication. Gauss Legendre quadrature over a triangle

Short Communication. Gauss Legendre quadrature over a triangle J. Indian Inst. Sci., Sept. Oct. 4, 4, 3 Indian Institute of Science. Short Communication Gauss Legendre quadrature over a triangle H. T. RATHOD *, K. V. NAGARAJA, B. VENKATESUDU 3 AND N. L. RAMESH 4 Department