
Computational mechanics 197
the form:
B(x, y)=
⎡
⎣
∂N
1
∂x
∂N
2
∂x
∂N
3
∂x
∂N
1
∂y
∂N
2
∂y
∂N
3
∂y
⎤
⎦
.
Since the shape functions are defined in terms of the parametric coordinates,
the derivatives of the local shape functions are computed by using the chain
rule as
∂N
i
∂v
=
∂N
i
∂x
∂x
∂v
+
∂N
i
∂y
∂y
∂v
and
∂N
i
∂w
=
∂N
i
∂x
∂x
∂w
+
∂N
i
∂y
∂y
∂w
.
These relations may be gathered as
∂N
i
∂v
∂N
i
∂w
=
∂x
∂v
∂y
∂v
∂x
∂w
∂y
∂w
∂N
i
∂x
∂N
i
∂y
.
The first term on the right-hand side is
∂x
∂v
∂y
∂v
∂x
∂w
∂y
∂w
= J,
as we found it earlier. Hence
∂N
i
∂v
∂N
i
∂w
= J
∂N
i
∂x
∂N
i
∂y
and
∂N
i
∂x
∂N
i
∂y
= J
−1
∂N
i
∂v
∂N
i
∂w
.
The inverse of the Jacobian matrix may be computed by
J
−1
=
adj(J)
det(J)
.
This equation clarifies the earlier warning comment about