Biomedical Engineering Reference
In-Depth Information
(a) Use the program to test whether
n
N
i
=
1,
i
=
1
and
n
n
∂
N
i
∂ξ
∂
N
i
∂η
=
=
0,
i
=
1
i
=
1
for a number of combinations of (
ξ
,
η
).
(b) Extend the program to calculate the Jacobian matrix:
∂
x
∂ξ
,
∂
x
∂η
x
,
ξ
=
∂
y
∂ξ
∂
y
∂η
and the Jacobian determinant
j
=
).
(c) Determine the Jacobian determinant in the following points:
det(
x
,
ξ
(
ξ
1
,
η
1
)
=
(0, 0)
(
ξ
2
,
η
2
)
=
(0.5, 0.5)
(
ξ
3
,
η
3
)
=
(1, 0)
(
ξ
4
,
η
4
)
=
(1,
−
1)
for both elements which are shown in the figure. What can you con-
clude from this?
17.2 Consider the element that is given in the figure below. The element shape
functions of this element are derived after degeneration of a 4-noded
quadrilateral element by coalescence of two nodes in the same way as
discussed in Section
17.3
.
y
3
(0, 1)
1
(0, 0)
2
x
(
x
1
, 0)
(a) Give analytical expressions for the shape functions
N
i
(
ξ
,
η
).
(b) Compute the Jacobian determinant for
ξ
=
η
=
0.
(c)
Plot the result as a function of
x
1
on the interval [
−
2,
+
2] and
comment on the result.
