Numerical Integration Over Polygonal Domains using Convex ... - Ijecs

x
y
x
y
x
y
(1)
(1)
(2)
(2)
(3)
(3)
( , )
( , )
( , )
( , )
( , )
( , )
1
x
p
xq
xr
1
y
p
yq
yr
1
xq
xr
x
p
1
yq
yr
y
p
1
xr
x
p
xq
1
yr
y
p
yq
with ( , )
, (
, )
as given in eqns. (19) and (24).
(26)
(27)
(28)
6 Numerical Integration Formulas
6.1 Numerical integration over an Arbitrary Triangle PQR
We could use either of the formulas in eqn.(16) or eqns.(26-28). We prefer to use eqn.(25), since it requires
(1) (1)
the computation of just one set of ( u,
v)
( u , v ) for all the three quadrilaterals. The transformation
formulas of eqns. (26-28) are easy to implement as a computer code, since the coordinates of PQR
are to
( e)
( e)
(1) (1)
be used in cyclic permutation in ( x , y ), e 1,2,3)
. Note that in ( x , y )
T
T
T
( x
p,
y
p
) , ( xq,
yq
) , ( xr
, yr
) respectively. In
T
T
T
( x
q,
yq
) , ( xr
, yr
) , ( xp,
y
p
) respectively and in
(2) (2)
( x , y )
(3) (3)
( x , y )
H. T. Rathod a IJECS Volume 2 Issue 8 August, 2013 Page No.2576-2610 Page 2582
T
T
T
the coefficients of w , u,
v are
the coefficients of w , u,
v are
the coefficients of w , u,
v are
T
T
T
( x
r
, yr
) , ( x
p,
y
p
) , ( xq,
yq
) respectively. We can use Gauss Legendre quadrature rule to evaluate eqn.(25).
The resulting numerical integration formula can be written as
PQR
NN
3
( e)
( N ) ( N ) ( e)
( N ) ( N )
f ( x ( U
k
, Vk
), y ( U
k
, Vk
))
( N )
I I ( f ) 2
W
(29)
pqr
k 1
k
e1
The weights and sampling points in the above formula satisfy the relation
N)
( N)
( N)
( N)
( N)
( N)
( N)
( N)
( N)
( N)
( W , U , V ) (4 s s ) w w 96, u(
s , s ), v(
s , s
( ( N)
)
k
k
k
i
j
i
j
k 1 ,2,3,......
N N , i , j 1,2,3 ,...... N
(30)
and
( N)
( N)
( N)
( N)
( N)
( N)
u(
s , s ) (1 s )(1 s ) 12 (1 s )(1 s ) 8,
i
j
i
j
i
j
( N)
( N)
( N)
( N)
( N)
( N)
v(
si
, s
j
) (1 si
)(1 s
j
) 12 (1 si
)(1 s
j
) 8
(31)
( N ) ( N )
for a N- point Gauss Legendre rule of order N with (
wn , sn
), n 1, 2,3,......
N as the weights and
sampling points respectively.
( N ) ( N ) ( N )
2
We can compute the arrays ( Wk , U
k
, Vk
), k 1, 2,3,...... N for any available Gauss Legendre
quadrature rule of order N. We have listed a code to compute the arrays
( N ) ( N ) ( N )
2
( Wk , U
k
, Vk
, k 1, 2,3,...... N ) for N = 5, 10, 15, 20, 25, 30, 35, 40. This is necessary since explicit list
( N ) ( N ) ( N )
of ( Wk , U
k
, Vk
, N = 5, 10, 15, 20, 25, 30, 35, 40)
will generate a large amount of values, viz: 25, 100,
( N ) ( N ) ( N )
225, 400, 625, 900, 1225, and 1600 for each of W
k
, U
k
, Vk
. The computer code will be simple with
( N ) ( N )
few statements and it requires the input values of (
w , s , n 1,2,..., N),
N 5,10,15, 20,25,30,35,40
6.2 Composite Integration over a polygonal Domain P N
i
j
i
j
n n
.
We now consider the evaluation of II
( f ) f ( x,
y)
dxdy , where f C()
and is any polygonal
2
domain in . That is a domain with boundary composed of piecewise straight lines. We then write
M
M
2
P
( f )
T
( f )
Q
( f )
(32) in which, we define
N
n
3 n p
n1
n1
p0
P is a polygonal domain of N oriented edges ( k i 1,
i 1,2,3,..., N),
with end points x , y ), x , y )
N
l ik
(
i i
(
k k