The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 398

TABLE 6.6
Weights for Newton-Cotes Quadrature Formulas
Newton-Cotes Integration of F

(ξ ) over Interval [ a, b ]:

b

n

W ( n )
i

F (ξ ) d ξ = F (ξ i )

a

i

n Number of integration points

Weighting Factor

=

W ( n )
i

C ( n )
i

= = ( − )

b a

Upper Bound on Error R n as a

nC ( n )
1

C ( n )
2

C ( n )
3

C ( n )
4

C ( n )
5

C ( n )
6

Function of the Derivative of F

1
2

1
2

10 − 1

3 F ( 2 ) (

2 ( b − a ) r )

3

1
6

4
6

1
6

10 − 3 ( b − a )

5 F ( 4 ) (

r )

1
8

3
8

3
8

1
8

4 10 − 3 ( b − a )

5 F ( 4 ) (

r )

7
90

32
90

12
90

32
90

7
90

10 − 6

7 F ( 6 ) (

5 ( b − a ) r )

19
288

75
288

50
288

50
288

75
288

19
288

10 − 6

7 F ( 6 ) (

6 ( b − a ) r )

∗ F ( n ) (

r ) is the n th derivative of F and r is a point in [ a, b ].

EXAMPLE 6.10 Newton-Cotes Quadrature for n = 2 and n = 3

For n = 2 over [ − 1 , 1], choose equally spaced integration points ξ =− 1 , ξ = 1 to evaluate

1 2

1
− 1 F

(ξ ) d ξ. The interpolation function is

2

p (ξ ) = N i (ξ ) F (ξ ) (1)

i

i = 1

in which, from Eq. (6.64)

(ξ ) = ξ − ξ 2

1
2 (

N 1

ξ 1 − ξ 2 =

1 − ξ)

(2)

N 2 (ξ ) = ξ − ξ 1

1
2 (

ξ 2 − ξ 1 =

1 + ξ)

It follows from Eq. (6.114) that the weighting factors are

1 1

1
2

W ( 2 )
1

= N 1 (ξ ) d ξ =

1 (

1 − ξ) d ξ = 1

− 1 −

(3)

1 1

1
2

W ( 2 )
2

= N 2 (ξ ) d ξ =

1 (

1 + ξ) d ξ = 1

− −

1

Finally, Eq. (6.113) gives the Newton-Cotes quadrature for n = 2as

1

2

W ( 2 )
i

F (ξ ) d ξ ≈ F (ξ i ) = F ( − 1 ) + F ( 1 ) (4)

− 1

i = 1

This is the trapezoidal rule.
For n = 3 over [ − 1 , 1], choose integration points ξ 1 =− 1 , ξ 2 = 0 , ξ 3 = 1 . The interpolating

function is

3

p (ξ ) = N i (ξ ) F (ξ ) (5)

i

i = 1