Isoparametric elements
For elements with complicated geometrical shape like curved sides or surfaces in 3D it
is advantageous to transform the geometry from cartesian to natural, curvilinear
η
coordinates ζ, η.
1
η
ζ
1
-1
ζ
y
-1
x
Fig.5-1 Transformation of isoparametric element from cartesian to natural coordinates
The transformation relations x = x(ζ, η), y = y(ζ, η), can be formulated in a similar form
as the displacement approximation.
If the displacements 8are approximated as
8
u ( , )   Ni ( , ) . ui
i 1
v( , )   Ni ( , ) . vi
i 1
and the coordinates transformed according to
8
x( , )   Nˆ i ( , ) . xi
i 1
8
y( , )   Nˆ i ( , ) . yi
i 1
Then, in case of Ni  Nˆ i the element is called isoparametric. The reason is that we
need the same number of deformation parameters ui, vi to describe displacement field
as the number of nodal coordinates xi, yi to describe element geometry.
Shape functions of isoparametric elements can be described in a systematic way as
families of certain type and we shall mention two of such types - families with linear
and quadratic shape functions.
ISOPARAMETRIC ELEMENTS WITH LINEAR SHAPE FUNCTIONS
1D elements
Linear shape functions expressed in natural coordinate over a unit element
Nζ1 = ( 1 – ζ ) / 2 ,
Nζ2 = ( 1 + ζ ) / 2
are plotted in Fig.5-2
Nζ1
-1
Nζ2
0
+1
ζ
Fig.5-2 Linear shape functions
We note that they are the shape functions of LINK1 and LINK8 elements, described
previously.
2D elements - bilinear element
v3
4
u3
v4
v2
η
ζ
u4
3
η
u2
ζ
v1
y
x
u1
1
2
Fig.5-3 Bilinear element in cartesian and natural coordinates
Bilinear element in Fig.5-3 has shape functions generated by multiplying linear
expressions in ζ and η direction. Due to multiplication, the product is not a linear
function: N1 = Nζ1 . Nη1 = (1 - ζ).(1 - η ) / 4
N2 = Nζ2 . Nη1 = (1 + ζ).(1 - η ) / 4
N3 = Nζ2 . Nη2 = (1 + ζ).(1 + η ) / 4
N4 = Nζ1 . Nη2 = (1 - ζ).(1 + η ) / 4
One of the shape functions is shown in Fig.5-4a). Besides the basic isoparametric
functions, quadratic „extra“ shape functions are sometimes used to improve the quality
Nηex = 1 – η2 ,
of bilinear element. Nζex = 1 – ζ2 ,
Nζex
N3
η
η
ζ
a)
Nηex
η
ζ
b)
ζ
c)
Fig.5-4 Isoparametric a) and extra b), c) shape functions of bilinear element
In ANSYS, this quadrilateral element can be found under the names PLANE42, or
PLANE182. As mentioned, triangular form of this element can be used, too.
Examples 5.1 and 5.2 show improved quality of the bilinear element with extra shapes in
comparison to a standard bilinear, and especially to triangular element.
3D elements with linear functions in ζ, η and ξ
Eight shape functions of the hexaedral element according to Fig.5-6 a) are created by
mutual multiplication of linear functions in ζ, η and ξ :
N1 = Nζ1 . Nη1 . Nξ1 = (1 - ζ).(1 - η ).(1 - ξ) / 8
N2 = Nζ2 . Nη1 . Nξ1 = (1 + ζ).(1 - η ).(1 - ξ) / 8
N3 = Nζ2 . Nη2 . Nξ1 = (1 + ζ).(1 + η ).(1 - ξ) / 8
N4 = Nζ1 . Nη2 . Nξ1 = (1 - ζ).(1 + η ).(1 - ξ) / 8
(5.23)
N5 = Nζ1 . Nη1 . Nξ2 = (1 - ζ).(1 - η ).(1 + ξ) / 8
N6 = Nζ2 . Nη1 . Nξ2 = (1 + ζ).(1 - η ).(1 + ξ) / 8
N7 = Nζ2 . Nη2 . Nξ2 = (1 + ζ).(1 + η ).(1 + ξ) / 8
N8 = Nζ1 . Nη2 . Nξ2 = (1 - ζ).(1 + η ).(1 + ξ) / 8
They are often complemented by extra quadratic shape functions, improving the
element properties :
Nζex = 1 – ζ2 ,
Nηex = 1 – η2 ,
Nξex = 1 – ξ2 .
Besides the hexaedral shape, other „degenerated“ element shapes ACCORDING TO
Fig5-6 b),c),d) can be obtained. In some FE systems such elements are classified as
special types, in ANSYS they are all seen as degenerated shapes of the same
element SOLID45 or SOLID 185.
y
x
a)
b)
c)
d)
z
Fig.5-6 Eight node hexaedral element and its degenerated shapes
The shape 5-6 a) is used to create mapped meshes, which are saving computer time
and memory, but are difficult to prepare. The shape 5-6 d) is used for fully
automatic free meshing, and the intermediate shapes b), c) can be used for transition
between these two mesh types.
ISOPARAMETRIC ELEMENTS WITH QUADRATIC SHAPE FUNCTIONS
1D elements
Quadratic shape functions expressed in natural coordinate over an unit element
2
Nζ1 = ζ .( ζ – 1 ) / 2 ,
2
Nζ2 = ( 1 + ζ ).( 1 - ζ ) ,
2
Nζ3 = ζ .( 1 + ζ ) / 2 .
are plotted in Fig.5-7
v3
ζ
u3
2
N1ζ
2
N2ζ
2
N3ζ
v2
v1
y
u2
u1
-1
0
+1
ζ
x
Fig.5-7 Quadratic 1D element and its shape functions
2D quadratic elements
By a systematic evolution of quadratic elements, similar to the evolution of linear
elements we obtain 9-node plane element according to Fig.5-8 a). Its nine shape
functions are obtained by multiplication of qudratic functions from the preceding
paragraph: N1 = 2Nζ1 .2Nη1, N2 = 2Nζ2 .2Nη1, N3 = 2Nζ3 .2Nη1,
N4 = 2Nζ3 .2Nη2,
N5 = 2Nζ3 .2Nη3,
N6 = 2Nζ2 .2Nη3,
N7 = 2Nζ1 .2Nη3,
N8 = 2Nζ1 .2Nη2,
N9 = 2Nζ2 .2Nη2,
Practical usage of this element is limited and the 8-node version according to Fig.5-8
b) prevails. In ANSYS, degenerated triangular form with curved sides (Fig. 5-8 c) is
included in the same element type PLANE82, or PLANE183.
6
5
7
4
9
5
7
5
4
4
3
6
8
2
y
3
1
x
6
a)
8
2
1
3
b)
Fig.5-8 2D quadratic elements
2
1
c)
3D quadratic elements
In a 3D space, internal face and body nodes are always excluded and the hexaedral
form of element has then 20 nodes with 60 degrees of freedom (Fig.5-9a). In ANSYS
this element can be found under the name SOLID95, or SOLID186 and also its other
geometrical forms (Figs.5-9b,c,d) belong to the same element type. Like the linear 3D
elements, the shape 5-9 a) is used for mapped meshes, 5-6 d) for fully automatic free
meshing, and the intermediate shapes b), c) can be used for transition between these
two mesh types.
y
x
a)
b)
c)
d)
z
Fig.5-9 Twenty-node hexaedral element and its degenerated shapes
Discussion of element quality
The question which element to use is always open and will be hardly ever finally
answered. It always depends on circumstances and only general advice can be
given. For regular shapes, mapped hexaedral mesh always leads to more effective
computational models, but its preparation may be an exhausting man-controlled
task. Fully automatic free meshing by tetraedral elements can be used for easy
model preparation even for complicated shapes, such models then need more
computational power to solve. Quadratic 10-node tetraedral elements should be
preferred in such cases over linear 4-node tetraedra. Some insight into the
behaviour and quality of 3D elements gives the Example 5.3.