期刊名称：International Journal of Engineering and Computer Science
印刷版ISSN：2319-7242
出版年度：2013
卷号：2
期号：8
页码：2576-2610
出版社：IJECS
摘要：This paper presents a numerical integration formula for the evaluation of II ( f ) f (x, y)dxdy , wheref C() and is any polygonal domain in 2 . That is a domain with boundary composed of piecewisestraight lines. We then express Mn pQMnP T f f fN n n p121 0( ) ( ) ( )3in which N P is a polygonaldomain of N oriented edges l (k i 1, i 1,2,3,..., N), ik with end points ( , ), i i x y ( , ) k k x y and( , ) ( , ) 1 1 1 1 N N x y x y . We have also assumed that N P can be discretised into a set of M triangles, n T andeach triangle n T is further discretised into three special quadrilaterals ( 0,1, 2) 3 Q p n p which are obtainedby joining the centroid to the midpoint of its sides. We choose xyn pqr T T an arbitrary triangle with vertices(x , y ), p, q, r in Cartesian space (x, y).We have shown that an efficient formula for this purpose isgiven by( ) ( ) (4 ) ( ( , ), ( , )) ,31( ) ( ) f c f x u v y u v ddee eSTn pqr where,( , ) ( ) ( ) , ( , ) ( )1( )3( )1( )2( )1( ) z u v z z z u z z v z x y e e e e e e ( , , ), 1,2,3 ( , , ),( , , ),( , , ) ( )3( )2( )1 p q r q r p r p qe e e z z z e z z z z z z z z z( ) 48 , pqr n c area of T T u [1/ 3,1/ 2,0,0][M ,M ,M ,M ] 1 2 3 4 , [1/ 3,0,0,1/ 2][ , , , ] , 1 2 3 4T v M M M M ( , ) (1 )(1 ) 4,{( , ), 1,2,3,4} {(1,1), (1,1), (1,1), (1,1)} M M andS {( ,) 1 , 1}is the standard 2- square in ( ,) space. Using Gauss Legendre Quadrature Rules oforder 5(5)40, we obtain the weight coefficients and sampling points which can be used for any polygonaldomain, N P or n T or m Q (m 3n 2,3n 1,3n). Boundary integration methods are also proposed whichare helpful in verifying the application of derived formulas to compute some typical integrals
