(untitled) (International Journal of Engineering and Computer Science, 2017)
This paper presents a numerical integration formula for the evaluation of òò W IIW ( f ) = f (x, y)dxdy , where f ÎC(W) and W is any polygonal domain in 2 Â . That is a domain with boundary composed of piecewise straight lines. We then express å å å = = = ÷ ÷ ø ö ç ç è æ II = II = II - M n p Q M n P T f f f N n n p 1 2 1 0 ( ) ( ) ( ) 3 in which PN is a polygonal domain 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 = N+1 N+1 x y x y . We have also assumed that PN can be discretised into a set of M triangles, Tn and each triangle Tn is further discretised into three special quadrilaterals ( 0,1, 2) Q3n- p p = which are obtained by joining the centroid to the midpoint of its sides. We choose xy Tn = Tpqr an arbitrary triangle with vertices ((x , y ), a = p,q,r) a a in Cartesian space (x, y). We have shown that an efficient formula for this purpose is given by ( ) ( ) (4 ) ( ( , ), ( , )) , 3 1 ( ) ( ) f c x h f x u v y u v dxdh e e e S T pqr n ÷ ø ö ç è æ II = òò + + å= 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 q e e e z z z e = = z z z z z z z z z ( ) 48 , pqr area of Tn c = 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 4 T v = M M M M = (x,h) = (1+ xx )(1+hh ) 4,(x ,h ), b = 1,2,3,4 = (-1,-1),(1,-1),(1,1),(-1,1) M b M b b b b b and S = (x, h) -1£ x, h £1 is the standard 2- square in (x,h) space. Using Gauss Legendre Quadrature Rules of order 5(5)40, we obtain the weight coefficients and sampling points which can be used for any polygonal domain, W = PN or Tn or Qm ( m = 3n - 2,3n -1,3n). Boundary integration methods are also proposed which are helpful in verifying the application of derived formulas to compute some typical integrals