x
=
nen
X
I
=
1
N
e
I
x
I
,
=
N
e
1
x
1
+
N
e
2
x
2
+
N
e
3
x
3
,
=
(

ξ
)(1

ξ
)
2
x
1
+
(1
+
ξ
)(1

ξ
)
x
2
+
(

1

ξ
)(

ξ
)
2
x
3
.
c) A desirable property of the isoparametric mapping is that it is invertible for all points in the
element. State the mathematical condition that expresses this property.
Solution
(2 points): Invertability of the map over the whole element would be guaranteed if
J
=
det
"
∂
x
∂
ξ
#
,
0
,
∀
ξ
,
(for a system where the number of spatial dimensions
>
1) and, in particular, we would like
J
to
be strictly positive, to prevent elements from being overly distorted.
For 1D, this reduces to
J
=
∂
x
∂ξ
,
0
,
∀
ξ.
d) Consider an element in
physical
space with nodes 1, 2, and 3 at positions
x
1
,
x
2
, and
x
3
, such
that
x
1
<
x
2
<
x
3
. What restriction must be placed on
x
2
, in terms of
x
1
,
x
3
, for the element to be
”wellbehaved”?
Solution
(6 points): First, calculate an expression for
J
:
J
=
det
∂
x
∂
ξ
!
,
=
∂
x
∂ξ
,
=
∂
N
1
∂ξ
x
1
+
∂
N
2
∂ξ
x
2
+
∂
N
3
∂ξ
x
3
,
=

1
+
2
ξ
2
x
1
+
(

2
ξ
)
x
2
+
1
+
2
ξ
2
x
3
.
Next, state the problem we would like to answer. In this case, we want to know where we can place
x
2
in physical space, in terms of
x
1
and
x
3
, so that the element is wellbehaved (i.e.,
J
is positive,
∀
ξ
). To this end, let us first look at how
J
varies over the master element (i.e., how
J
varies with
respect to
ξ
):
∂
J
∂ξ
=
x
1

2
x
2
+
x
3
.