192 Applied calculus of variations for engineers
We also require that these functions at a certain node point reduce to zero
at the other two nodes. This is called the Kronecker property and presented as
N
i
=
1atnodei,
0atnode= i.
Furthermore, a shape function is zero along the edge opposite to the particular
node at which the shape function is non-zero.
The solution for the nodes of a particular triangular element e can be ex-
pressed in matrix form as
u
e
=
⎡
⎣
u
1
u
2
u
3
⎤
⎦
=
⎡
⎣
1 x
1
y
1
1 x
2
y
2
1 x
3
y
3
⎤
⎦
⎡
⎣
a
b
c
⎤
⎦
.
This system of equations is solved for the unknown coefficients that produce
the shape functions
⎡
⎣
a
b
c
⎤
⎦
=
⎡
⎣
1 x
1
y
1
1 x
2
y
2
1 x
3
y
3
⎤
⎦
−1
⎡
⎣
u
1
u
2
u
3
⎤
⎦
=
⎡
⎣
N
1,1
N
1,2
N
1,3
N
2,1
N
2,2
N
2,3
N
3,1
N
3,2
N
3,3
⎤
⎦
⎡
⎣
u
1
u
2
u
3
⎤
⎦
.
By substituting into the matrix form of the bilinear interpolation