Game Development Reference

In-Depth Information

r

r
+
s
+
t
,
ξ
2
=

s

r
+
s
+
t
,and

Figure 10.3.
Barycentric coordinates of a triangle, where
ξ
1
=

t

r
+
s
+
t
. This representation can be extended for tetrahedral elements. Barycentric

coordinates can be described from areas (left) or from distances (right).

ξ
3
=

obtained by linear interpolations of the tetrahedron nodal displacements. We can

write this as

ˆ

u
=
ξ
1
u
1
+
ξ
2
u
2
+
ξ
3
u
3
+
ξ
4
u
4
,
(10.2)

where
ξ
i
are the shape functions and
u
i
are the displacements of the four nodes of

the tetrahedron. We can rewrite this using the components of each displacement as

⎡

⎣

⎤

⎦

u
1
x

u
1
y

u
1
z

u
2
x

u
2
y

u
2
z

u
3
x

u
3
y

u
3
z

u
4
x

u
4
y

u
4
z

⎡

⎤

ξ
1

00
ξ
2

00
ξ
3

00
ξ
4

00

u
=

⎣

⎦

ξ
1

00
ξ
2

00
ξ
3

00
ξ
4

0

0

00
ξ
1

00
ξ
2

00
ξ
3

00
ξ
4

or simply

u
=
H
e
u,

(10.3)

where
H
e
is a 3

12 matrix containing the shape functions.

Let us now describe how to relate the displacements to the strains. From the

definition of the linear strain, we can explicitly write Equation (10.1) as

×

⎡

2
∂u
x

∂x

⎤

2
∂u
x

∂x

2
∂u
x

∂x

+
∂u
y

1

+
∂u
x

1

1

+
∂u
z

⎣

2
∂u
y

∂x

∂y
2
∂u
y

∂y

∂y
2
∂u
y

∂z

∂y

⎦

+
∂u
y

1

+
∂u
x

+
∂u
z

ε

=

,

∂x

2
∂u
z

∂y

∂z
2
∂u
z

∂z

2
∂u
z

∂z

∂z

+
∂u
y

1

+
∂u
x

1

+
∂u
z

∂x

∂y

∂z