TriangleElement_Seepage - CVEN4304 Structural Analysis...

CVEN4304 Structural Analysis & Finite Elements
Triangle Element
Chongmin Song
School of Civil and Environmental Engineering
University of New South Wales

Introduction
Triangle elements are the most versatile in mesh generation.
Description of mesh (as in 1D
case)
Global nodal number
Element number
Local nodal number
( counter-clockwise )
Element connectivity table
Nodal functions
One element
Nodal hydraulic head:
h
(
e
)
1
, h
(
e
)
2
, h
(
e
)
3
Nodal flux:
Q
(
e
)
1
, Q
(
e
)
2
, Q
(
e
)
3

Shape function
Linear interpolation of the nodal values
h
(
x
,
z
) =
a
0
+
a
1
x
+
a
2
z
At the 3 nodes of element



1 x
(
e
)
1
z
(
e
)
1
1 x
(
e
)
2
z
(
e
)
2
1 x
(
e
)
3
z
(
e
)
3






a
0
a
1
a
2



=





h
(
e
)
1
h
(
e
)
2
h
(
e
)
3





Determinant of the coefficient matrix
2 A
(
e
)
= (
x
(
e
)
2
z
(
e
)
3
-
x
(
e
)
3
z
(
e
)
2
) + (
x
(
e
)
3
z
(
e
)
1
-
x
(
e
)
1
z
(
e
)
3
) + (
x
(
e
)
1
z
(
e
)
2
-
x
(
e
)
2
z
(
e
)
1
)
The solution for constants a
0
, a
1
and a
2
a
0
=
1
2 A
(
e
)
((
x
(
e
)
2
z
(
e
)
3
-
x
(
e
)
3
z
(
e
)
2
)
h
(
e
)
1
+ (
x
(
e
)
3
z
(
e
)
1
-
x
(
e
)
1
z
(
e
)
3
)
h
(
e
)
2
+ (
x
(
e
)
1
z
(
e
)
2
-
x
(
e
)
2
z
(
e
)
1
)
h
(
e
)
3
)
a
1
=
1
2 A
(
e
)
((
z
(
e
)
2
-
z
(
e
)
3
)
h
(
e
)
1
+ (
z
(
e
)
3
-
z
(
e
)
1
)
h
(
e
)
2
+ (
z
(
e
)
1
-
z
(
e
)
2
)
h
(
e
)
3
)
a
2
=
1
2 A
(
e
)
((
x
(
e
)
3
-
x
(
e
)
2
)
h
(
e
)
1
+ (
x
(
e
)
1
-
x
(
e
)
3
)
h
(
e
)
2
+ (
x
(
e
)
2
-
x
(
e
)
1
)
h
(
e
)
3
)

Shape function (cont’ed)
Hydraulic head within element
h
(
x
,
z
) =
N
(
e
)
1
(
x
,
z
)
h
(
e
)
1
+
N
(
e
)
2
(
x
,
z
)
h
(
e
)
2
+
N
(
e
)
3
(
x
,
z
)
h
(
e
)
3
= [
N
(
e
)
(
x
,
z
)] {
h
(
e
)
}
Shape functions
[
N
(
e
)
(
x
,
z
)] = [
N
(
e
)
1
(
x
,
z
)
N
(
e
)
2
(
x
,
z
)
N
(
e
)
3
(
x
,
z
)
]
where
N
(
e
)
1
(
x
,
z
) =
1
2 A
(
e
)
((
x
(
e
)
2
z
(
e
)
3
-
x
(
e
)
3
z
(
e
)
2
) + (
z
(
e
)
2
-
z
(
e
)
3
)
x
+ (
x
(
e
)
3
-
x
(
e
)
2
)
z
)
N
(
e
)
2
(
x
,
z
) =
1
2 A
(
e
)
((
x
(
e
)
3
z
(
e
)
1
-
x
(
e
)
1
z
(
e
)
3
) + (
z
(
e
)
3
-
z
(
e
)
1
)
x
+ (
x
(
e
)
1
-
x
(
e
)
3
)
z
)
N
(
e
)
3
(
x
,
z
) =
1
2 A
(
e
)
((
x
(
e
)
1
z
(
e
)
2
-
x
(
e
)
2
z
(
e
)
1
) + (
z
(
e
)
1
-
z
(
e
)
2
)
x
+ (
x
(
e
)
2
-
x
(
e
)
1
)
z
)
Note that the area of an element A
(
e
)
must not be zero!

Shape function (cont’ed)