Civil Engineering Reference
In-Depth Information
and we observe that, owing to the symmetry of the integration limits, all odd
powers integrate to zero. Also note that we assume here that
n
is an even integer.
The approximation to the integral per Equation 6.99 is
m
W
1
a
0
+
a
n
r
1
a
2
r
1
+
a
3
r
1
+···+
I
≈
W
i
f
(
r
i
)
=
a
1
r
1
+
W
2
a
0
+
a
n
r
2
i
=
1
a
2
r
2
+
a
3
r
2
+···+
+
a
1
r
2
+
W
3
a
0
+
a
n
r
3
a
2
r
3
+
a
3
r
3
+···+
+
a
1
r
3
+
(6.101)
.
W
m
a
0
+
a
n
r
m
a
2
r
m
+
a
3
r
m
+···+
+
a
1
r
m
+
Comparing Equations 6.100 and 6.101 in terms of the coefficients
a
j
of the poly-
nomial, the approximation of Equation 6.101 becomes exact if
m
W
i
=
2
i
=
1
m
W
i
r
i
=
0
i
=
1
m
2
3
W
i
r
i
=
i
=
1
m
W
i
r
i
(6.102)
=
0
i
=
1
m
2
5
W
i
r
i
=
i
=
1
.
m
W
i
r
n
−
1
=
0
i
i
=
1
m
2
W
i
r
i
=
n
+
1
i
=
1
where
m
is the number of sampling (Gauss) integration points.
Equation 6.102 represents
n
equations in
2
m
+
1
unknowns. The unknowns
are the weighting factors
W
i
,
the sampling point values
r
i
,
and most trouble-
some, the
number
of sampling points
m
. While we do not go into the complete
theory of Gaussian quadrature, we illustrate by example how the sampling points
and weights can be determined using both the equations and logic. First, note that
the equations corresponding to odd powers of the polynomial indicate a zero
summation. Second, note that the first equation is applicable regardless of the
order of the polynomial; that is, the weighting factors must sum to the value of
2 if exactness is to be achieved.
