The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 399

where

N 1 (ξ ) = (ξ − ξ 2 )(ξ − ξ 3 )

1
2 ξ(ξ − (ξ 1 − ξ 2 )(ξ 1 − ξ 3 ) =

1 )

(ξ ) = (ξ − ξ

)(ξ − ξ )

1 3

N 2

) =− (ξ +

1 )(ξ − 1 ) (6)

(ξ − ξ )(ξ − ξ

2 1 2 3

(ξ ) = (ξ − ξ

)(ξ − ξ ) 1

2 ξ(ξ +

1 2

N 3

) =

1 )

(ξ − ξ )(ξ − ξ

3 1 3 2

Then the weighting factors are

1 1

1
2

1
3

W ( 3 )
1

= N 1 (ξ ) d ξ =

1 ξ(ξ −

1 ) d ξ =

− 1 −

1 1

4
3

W ( 3 )
2

= N 2 (ξ ) d ξ =

1 − (ξ +

1 )(ξ − 1 ) d ξ = (7)

− 1 −

1 1

1
2

1
3

W ( 3 )
3

= N 3 (ξ ) d ξ =

1 ξ(ξ +

1 ) d ξ =

− 1 −

Finally,

1

3

1
3 [ F

W ( 3 )
i

F (ξ ) d ξ ≈ F (ξ i ) = ( − 1 ) + 4 F ( 0 ) + F ( 1 ) ] (8)

− 1

i = 1

which is Simpson's rule.
These same results are listed in Table 6.6.

6.6.2 Gaussian Quadrature
In Gaussian quadrature, the integration points are not fixed at the outset but are chosen
to achieve the best accuracy. Since this provides better accuracy than the evenly spaced
integration points of Newton-Cotes quadrature, Gaussian quadrature is the more popular
method of integration. Return to the one-dimensional case. For Gaussian quadrature, it is
again assumed that the integral can be approximated as a weighted sum of values of F (ξ ).

i

If n integration points are used and the integration interval is [ − 1 , 1],

1

W ( n )
1

W ( n )
2

W ( n )
n

F (ξ ) d ξ ≈ F (ξ ) + F (ξ ) +···+ F (ξ )

1 2 n

− 1

n

W ( n )
i

= F (ξ ) (6.116)

i

i = 1

In this formulation, both W ( n )
i

and ξ

i are unknowns to be determined. For n integration

ξ i and n unknowns W ( n )
i

points, there will be n unknowns .

As in the development of the Newton-Cotes formula, use Lagrangian interpolation where

F (ξ ) is approximated by p (ξ ) such that

n

p (ξ ) = N i (ξ ) F (ξ ) (6.117)

i

i = 1

where ξ i are still unknown. For the determination of ξ i ,define a function

χ(ξ) = (ξ − ξ )(ξ − ξ ) ··· (ξ − ξ )

1 2 n