Civil Engineering Reference
In-Depth Information
The volume coordinates are defined as
V
1
V
L
1
=
V
2
V
L
2
=
(6.61)
V
3
V
L
3
=
V
4
V
L
4
=
where
V
is the total volume of the element given by
1
x
1
y
1
z
1
1
6
1
x
2
y
2
z
2
V
=
(6.62)
1
x
3
y
3
z
3
1
x
4
y
4
z
4
As with area coordinates, the volume coordinates are not independent, since
V
1
+
V
2
+
V
3
+
V
4
=
V
(6.63)
Now let us examine the variation of the volume coordinates through the
element. If, for example, point
P
corresponds to node 1, we find
V
1
=
V
,
V
2
=
0
. Consequently
L
1
=
1,
L
2
=
L
3
=
L
4
=
0
at node 1. As
P
moves away from node 1,
V
1
decreases linearly, since the volume of a tetrahe-
dron is directly proportional to its height (the perpendicular distance from
P
to
the plane defined by nodes 2, 3, and 4) and the area of its base (the triangle
formed by nodes 2, 3, and 4). On any plane parallel to the base triangle of nodes
2, 3, 4, the value of
L
1
is constant. Of particular importance is that, if
P
lies in the
plane of nodes 2, 3, 4, the value of
L
1
is zero. Identical observations apply to
volume coordinates
L
2
,
L
3
, and
L
4
. So the volume coordinates satisfy all re-
quired nodal conditions for interpolation functions, and we can express the field
variable as
V
3
=
V
4
=
L
4
4
(6.64)
Explicit representation of the interpolation functions (i.e., the volume co-
ordinates) in terms of global coordinates is, as stated, algebraically complex
but straightforward. Fortunately, such explicit representation is not generally
required, as element formulation can be accomplished using volume coordinates
only. As with area coordinates, integration of functions of volume coordinates
(required in developing element characteristic matrices and load vectors) is rela-
tively simple. Integrals of the form
(
x
,
y
,
z
)
=
L
1
1
+
L
2
2
+
L
3
3
+
L
1
L
2
L
3
L
4
d
V
(6.65)
V
