74 HIGHER-ORDER ELEMENTS
i
=
1,2,3;
j =
2,3,1;
k
=
3,1,2. The area
A
is
Λ
=
ι(Μ
2
-Μι) (3.45)
and represents the area
of
the projection
of
the element
on
the
x,
y
plane.
The transformation
can now be
carried
out as
previously.
The
reason for using ξ
3
as a dependent variable is that it renders
a
simple
expression
for
the interpolation functions but care should be taken
not
to
confuse
it
with an independent variable.
Linear expressions for the coordinates or as u functions are now,
3
η
= η
ι
ξ
ι
+ η
2
ξ
2
+
η
ζ
ξ
ζ
=
Σ
u
i<t>i
(3.46)
X
=
X
1
£l+X
2
£
2
+
*3£3
= Σ
X
i<t>i
i=
1
Note that the interpolation functions are
φ
(
=
ξ^
The homogeneous triangular coordinates have
one
more advan-
tage.
This
is
that the integrals for polynomial
terms can be
carried out
using
a
simple rule,
i.e.
Jf
ξ
'
ΑΆΛΑ
-ν^ϊϋ
1Α
(3
·
47)
This formula greatly simplifies
the
algebra when numerical
in-
tegration
is not
used.
HIGHER-ORDER TRIANGULAR ELEMENTS
The next triangular model expresses
the
function
as a
complete
second degree polynomial, which has a mid-side node (Figure 3.15).
The model satisfies interelement compatibility and gives,
u
=
ΣΦΜ (3-48)
where
^
=
^(2^-1), tf>
4
=
4^
2
Φι
=
ξιΜι-ΙΙ
Φ5
=
4ξ
2
ξ
3
(3.49)
Φ
3
=
ξ
3
(2ζ
3
-1),
φ
6
=
4ξ
3
ξ
1