Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
1/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R3.01 booklet: General references
Document: R3.01.01
Functions of form and points of integration
finite elements
Summary:
One describes the geometry and the topology of the elements established in Code_Aster, the expression of the functions
of form and the various families of points of integration and the associated weights are detailed.
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
2/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
3/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
1 Introduction
In Code_Aster, one calls “finite element”, a triplet (phenomenon, modeling, type of mesh). There is
three main phenomena: MECHANICS, THERMICS and ACCOUSTICS.
There are many modelings; for example, for the MECHANICAL phenomenon: 3D, C_PLAN,
D_PLAN
, AXIS, DKT, POU_D_E,…
For a given modeling (for example 3D) of a phenomenon (for example MECHANICAL), there exists
in general several finite elements: an element by type of mesh supported: HEXA8, HEXA20,
PENTA6
,…
With final, there thus exists of very many finite elements (more than 500 in July 2004).
On the other hand, the types of mesh are them numbers some reduced: POI1, SEG2, SEG3, SEG4, TRIA3,
TRIA6, TRIA7, QUAD4, QUAD8, HEXA8, HEXA20,…, TETRA4, TETRA10.
In general, each finite element, to carry out its elementary calculations, uses the concepts of function
of interpolation (or function of form) and of diagram of integration. In general also, these functions of
form and these diagrams of integration are defined on an element known as “of reference” whose geometry is
defined in a frame of reference often called: (
,) the passage of the element of
reference to the real element is done thanks to a geometrical transformation which uses the same ones
functions of interpolation. The element is then known as “isoparametric”. These concepts are very well
explained in [bib1].
The high number of finite elements of the code combined with the restricted number of the types of mesh, conduit
with the fact that there are several finite elements being based on the same type of mesh; for example it
quadrilateral with 8 nodes called QUAD8 supports more than 60 different finite elements.
In theory, each finite element can choose its functions of interpolation and its diagrams of integration
as it hears it. But in practice, almost all finite elements being based on the same type
of mesh, use the same element of reference, the same functions of form and the same ones
diagrams of integration. The goal of this document is to describe these various elements of reference
For each element of reference (called in the continuation of document ELREFE), one will indicate:
· the mesh support, the number of the nodes, their local classification and their co-ordinates,
· algebraical expressions of the functions of form and their derivative first (and sometimes
seconds)
· families of points of integration which one will name. For each family, one will give it
a number of points, their co-ordinates and their “weights” of integration. The sum of these weights,
must give the “volume” of the element of reference. For example, the “volume” of the quadrangle of
reference (- 1 <=
<= +1, 1 < < +1) is worth: 4.
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
4/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
2
Linear elements: SE2, SE3 and SE4
SE2: segment with 2 nodes
a number of nodes
: 2
a number of nodes nodes
: 2
SE3: segment with 3 nodes
a number of nodes
: 3
a number of nodes nodes
: 2
X
N1 - 1.0
N2 1.0
N3 0.0
N1
N3
N2
X
SE4: segment with 4 nodes
a number of nodes
: 4
a number of nodes nodes
: 2
X
N1 - 1.0
N2 1.0
N3 - 1./3.
N4 +1./3.
N1
N3
N2
X
N4
functions of form of the segment with 2 nodes:
W X
X
W X
X
1
2
0 5 1
0 5 1
()
. (
)
()
. (
)
=
-
=
+
functions of form of the segment with 3 nodes:
)
1
) (
1
(
)
(
)
1
(
5
.
0
)
(
)
1
(
5
.
0
)
(
3
2
1
X
X
X
W
X
X
X
W
X
X
X
W
-
+
=
+
=
-
-
=
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
5/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
functions of form of the segment with 4 nodes:
()
(
)
(
)
()
(
)
()
(
) (
)
()
(
) (
)
+
+
-
-
=
-
+
-
=
+
-
+
-
=
-
+
-
=
3
1
1
1
27
16
3
1
1
1
27
16
3
1
3
1
1
9
16
3
/
1
3
1
1
9
16
4
3
2
1
X
X
X
X
W
X
X
X
X
W
X
X
X
X
W
X
X
X
X
W
Nb of pts
of intégr.
Not
X
Weight
1 1
0.0
2.0
2 1
0.577350269189626
1.0
2
- 0.577350269189626
1.0
3 1
- 0.774596669241
0.55555…
2
0
0.88888…
3
0.774596669241
0.55555…
4 1
0.339981043584856
0.652145154862546
2
- 0.339981043584856
0.652145154862546
3
0.861136311594053
0.347854845137454
4
- 0.861136311594053
0.347854845137454
3
Surface elements
3.1 Triangles
:
ELREFE TR3, TR6, TR7
N3
N1
N2
N5
N6
N4
N7
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
6/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Co-ordinates of the nodes:
N1
0.0
0.0
N2
1.0
0.0
N3
0.0
1.0
N4
0.5
0.0
N5
0.5
0.5
N6
0.0
0.5
N7
1/3
1/3
Family Not
Weight
FPG1 1 1/3 1/3 1/2
FPG3 1 1/6 1/6 1/6
2 2/3
1/6 1/6
3 1/6
2/3 1/6
FPG4 1 1/5 1/5
25/(24 * 4)
2 3/5
1/5
25/(24 * 4)
3 1/5
3/5
25/(24 * 4)
4 1/3
1/3
- 27/(24 * 4)
FPG6 1
B
B
P2
2
1 2 B
B
P2
3
B
1 2 B
P2
4
has
1 2 A
P1
5 A has P1
6
1 2 A
has
P1
COT3 1 1/2 1/2 1/6
2 0 1/2 1/6
3 1/2 0 1/6
With
P1 = 0.11169079483905,
P2 = 0.0549758718227661
,
To = 0.445948490915965,
B = 0.091576213509771
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
7/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Family Not
Weight
FPG7 1 1/3 1/3 9/80
2 A A P1
3
1-2A A P1
4 A 1-2A P1
5 B B P2
6
1-2B B P2
7 B 1-2B P2
With
To = 0.470142064105115
B = 0.101286507323456
P1 = 0.066197076394253
P2 = 0.062969590272413
Family Not
Weight
FPG12 1
With
With
P1
2
1-2A A P1
3 A 1-2A P1
4 B B P2
5
1-2B B P2
6 B 1-2B P2
7 C D P3
8 D C P3
9
1-C-D
C P3
10
1-C-D D P3
11 C 1-C-D P3
12 D 1-C-D P3
With
To = 0.063089014491502
B = 0.249286745170910
C = 0.310352451033785
D = 0.053145049844816
P1 = 0.025422453185103
P2 = 0.058393137863189
P3 = 0.041425537809187
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
8/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
TR3: triangle with 3 nodes
a number of nodes
: 3
a number of nodes nodes
: 3
functions of form and derived first of the triangle with 3 nodes:
{}
NR
{
}
/
NR
{
}
/
NR
-
-
1
1
-
1
-
1
0
0
1
TR6: triangle with 6 nodes
a number of nodes
: 6
a number of nodes nodes
: 3
functions of form, derived first of the triangle with 6 nodes:
{}
NR
{
}
/
NR
{
}
/
NR
))
1
(
2
1
) (
1
(
-
-
-
-
-
-
)
1
(
4
1
-
-
-
)
1
(
4
1
-
-
-
)
2
1
(
-
-
4
1
+
-
0
)
2
1
(
-
-
0
4
1
+
-
)
1
(
4
-
-
)
2
1
(
4
-
-
4
-
4
4
4
)
1
(
4
-
-
4
-
)
2
1
(
4
-
-
derived seconds from the triangle with 6 nodes:
{
}
2
2
/
NR
{
}
/
2
NR
{
}
2
2
/
NR
4 4 4
4 0 0
0 0 4
- 8 - 4 0
0 4 0
0 - 4 - 8
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
9/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
TR7: triangle with 7 nodes
a number of nodes
: 7
a number of nodes nodes
: 3
functions of form of the triangle with 7 nodes:
{}
NR
)
(
3
7
)
(
2
)
(
3
1
2
2
+
-
+
+
+
+
-
))
(
3
3
2
1
(
+
-
+
+
-
))
(
3
3
2
1
(
+
-
+
+
-
))
(
3
4
1
(
4
+
+
-
-
))
(
3
2
(
4
+
+
-
))
(
3
4
1
(
4
+
+
-
-
)
1
(
27
-
-
derived first from the triangle with 7 nodes:
{
}
/
NR
{
}
/
NR
2
3
6
7
4
3
-
-
+
+
-
2
3
6
4
7
3
-
-
+
+
-
2
3
6
3
4
1
-
-
+
+
-
)
2
1
(
3
-
-
)
2
1
(
3
-
-
2
3
6
4
3
1
-
-
+
+
-
)
3
6
4
2
1
(
4
2
+
+
-
-
)
6
3
4
(
4
+
+
-
)
3
6
2
(
4
+
+
-
)
6
3
2
(
4
+
+
-
)
3
6
4
(
4
+
+
-
)
3
6
2
4
1
(
4
2
+
+
-
-
-
)
2
1
(
27
-
-
)
2
1
(
27
-
-
derived seconds from the triangle with 7 nodes:
{
}
2
2
/
NR
{
}
/
2
NR
{
}
2
2
/
NR
6
4
-
6
6
7
-
-
6
4
-
6
4
-
6
6
3
-
-
6
-
6
-
6
6
3
-
-
6
4
-
)
6
2
(
4
+
-
)
6
6
4
(
4
+
+
-
24
24
)
6
6
2
(
4
+
+
-
24
24
)
6
6
4
(
4
+
+
-
)
6
2
(
4
+
-
54
-
)
2
2
1
(
27
-
-
54
-
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
10/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
3.2 Quadrangles
:
ELREFE QU4, QU8, QU9
N4
N1
N2
N3
N8
N5
N6
N7
N9
Co-ordinates of the nodes:
N1 - 1.0 - 1.0
N2 1.0 - 1.0
N3 1.0 1.0
N4 - 1.0 1.0
N5 0.0 - 1.0
N6 1.0 0.0
N7 0.0 1.0
N8 - 1.0 0.0
N9 0.0 0.0
Family Not
Weight
FPG1 1
0
0
4
FPG4 1
- has
- has 1.0
2
has
- has
1.0
3
has
has
1.0
4 - has
has
1.0
has = 1/3
FPG9 1
- has
- has 25/81
2
has
- has
25/81
3
has
has
25/81
4 - has
has
25/81
5.0.0 - has
40/81
6
has
0.0
40/81
7 0.0
has
40/81
8 - has 0.0
40/81
9.0.0.0.0
64/81
a= 0.774596669241483
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
11/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
QU4: quadrangle with 4 nodes
a number of nodes
: 4
a number of nodes nodes
: 4
functions of form, derived first and seconds of the quadrangle with 4 nodes:
{}
NR
{
}
/
NR
{
}
/
NR
4
/
)
1
) (
1
(
-
-
4
/
)
1
(
-
-
4
/
)
1
(
-
-
4
/
)
1
) (
1
(
-
+
4
/
)
1
(
-
4
/
)
1
(
+
-
4
/
)
1
) (
1
(
+
+
4
/
)
1
(
+
4
/
)
1
(
+
4
/
)
1
) (
1
(
+
-
4
/
)
1
(
+
-
4
/
)
1
(
-
{
}
2
2
/
NR
{
}
/
2
NR
{
}
2
2
/
NR
0 1/4 0
0 - 1/4 0
0 1/4 0
0 - 1/4 0
QU8: quadrangle with 8 nodes
a number of nodes
: 8
a number of nodes nodes
: 4
functions of form and derived first of the quadrangle with 8 nodes:
{}
NR
{
}
/
NR
{
}
/
NR
4
/
)
1
) (
1
) (
1
(
-
-
-
-
-
4
/
)
2
) (
1
(
+
-
4
/
)
2
) (
1
(
+
-
4
/
)
1
) (
1
) (
1
(
-
+
-
-
+
4
/
)
2
) (
1
(
-
-
4
/
)
2
) (
1
(
-
+
-
4
/
)
1
) (
1
) (
1
(
+
+
-
+
+
4
/
)
2
) (
1
(
+
+
4
/
)
2
) (
1
(
+
+
4
/
)
1
) (
1
) (
1
(
+
-
-
+
-
4
/
)
2
) (
1
(
+
-
+
-
4
/
)
2
) (
1
(
+
-
-
2
/
)
1
(
)
1
(
2
-
-
)
1
(
-
-
2
/
)
1
(
2
-
-
2
/
)
1
) (
1
(
2
-
+
2
/
)
1
(
2
-
)
1
(
+
-
2
/
)
1
(
)
1
(
2
+
-
)
1
(
+
-
2
/
)
1
(
2
-
2
/
)
1
) (
1
(
2
-
-
2
/
)
1
(
2
-
-
)
1
(
-
-
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
12/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
derived seconds from the quadrangle with 8 nodes:
{
}
2
2
/
NR
{
}
/
2
NR
{
}
2
2
/
NR
2
/
)
1
(
-
4
/
)
2
2
1
(
-
-
2
/
)
1
(
-
2
/
)
1
(
-
4
/
)
2
2
1
(
-
+
-
2
/
)
1
(
+
2
/
)
1
(
+
4
/
)
2
2
1
(
+
+
2
/
)
1
(
+
2
/
)
1
(
+
4
/
)
2
2
1
(
+
-
-
2
/
)
1
(
-
+
- 1
0
0
-
-
- 1
-
- 1
-
0
0
+
- 1
QU9: quadrangle with 9 nodes
a number of nodes
: 9
a number of nodes nodes
: 4
functions of form and derived first of the quadrangle with 9 nodes:
{}
NR
{
}
/
NR
{
}
/
NR
4
/
)
1
) (
1
(
-
-
4
/
)
1
(
)
1
2
(
-
-
4
/
)
1
2
) (
1
(
-
-
4
/
)
1
) (
1
(
-
+
4
/
)
1
(
)
1
2
(
-
+
4
/
)
1
2
) (
1
(
-
+
4
/
)
1
) (
1
(
+
+
4
/
)
1
(
)
1
2
(
+
+
4
/
)
1
2
) (
1
(
+
+
4
/
)
1
) (
1
(
+
-
4
/
)
1
(
)
1
2
(
+
-
4
/
)
1
2
) (
1
(
+
-
2
/
)
1
(
)
1
(
2
-
-
)
1
(
-
-
2
/
)
1
2
) (
1
(
2
-
-
2
/
)
1
) (
1
(
2
-
+
2
/
)
1
) (
1
2
(
2
-
+
)
1
(
+
-
2
/
)
1
(
)
1
(
2
+
-
)
1
(
+
-
2
/
)
1
2
) (
1
(
2
+
-
2
/
)
1
) (
1
(
2
-
-
2
/
)
1
) (
1
2
(
2
-
-
)
1
(
-
-
)
1
) (
1
(
2
2
-
-
)
1
(
2
2
-
-
)
1
(
2
2
-
-
derived seconds from the quadrangle with 9 nodes:
{
}
2
2
/
NR
{
}
/
2
NR
{
}
2
2
/
NR
2
/
)
1
(
-
4
/
)
2
/
1
) (
2
/
1
(
-
-
2
/
)
1
(
-
2
/
)
1
(
-
4
/
)
2
/
1
) (
2
/
1
(
-
+
2
/
)
1
(
+
2
/
)
1
(
+
4
/
)
2
/
1
) (
2
/
1
(
+
+
2
/
)
1
(
+
2
/
)
1
(
+
4
/
)
2
/
1
) (
2
/
1
(
+
-
2
/
)
1
(
-
)
1
(
-
-
)
1
2
(
-
-
2
1
-
2
1
-
)
1
2
(
+
-
)
1
(
+
-
)
1
(
+
-
)
1
2
(
+
-
2
1
-
2
1
-
)
1
2
(
-
-
)
1
(
-
-
)
1
(
2
2
-
-
4
)
1
(
2
2
-
-
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
13/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
4
Voluminal elements
4.1 Tetrahedrons
:
ELREFE TE4, T10
N1
N2
N3
N4
X
y
Z
N7
N8
N9
N5
N6
N10
X
y
Z
N1 0. 1. 0.
N2 0. 0. 1.
N3 0. 0. 0.
N4 1. 0. 0.
N5 0. 0.5 0.5
N6 0. 0. 0.5
N7 0. 0.5 0.
N8 0.5.0.5 0.
N9 0.5 0. 0.5
N10 0.5 0. 0.
Functions of form:
Formulate with 4 nodes
(
)
(
)
(
)
(
)
W X y Z
y
W X y Z
Z
W X y Z
X y Z
W X y Z
X
1
2
3
4
1
,
,
,
,
=
=
=
- - -
=
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
14/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Formulate with 10 nodes
(
)
(
)
(
) (
)
(
)
W
y
y
W
Z
Z
W
X y Z
X
y
Z
W
X
X
W
y Z
1
2
3
4
5
2
1
2
1
1
1 2
2
2
2
1
4
=
-
=
-
=
- - -
-
-
-
=
-
=
(
)
(
)
(
)
W
Z
X y Z
W
y
X y Z
W
X y
W
X Z
W
X
X y Z
6
7
8
9
10
4 1
4 1
4
4
4 1
=
- - -
=
- - -
=
=
=
- - -
Formulate numerical integration:
Formulate at 4 points, of command 2 in
X y Z
,
: (FPG4)
Not
X
y
Z
Weight
1
has
has
has
1/24
2
has
has
B
1/24
3
has
B
has
1/24
4
B
has
has
1/24
with:
20
5
3
5
20
5
5
+
=
-
=
B
has
Formulate at 5 points, of command 3 in
X y Z
,
: (FPG5)
Not
X
y
Z
Weight
1
has
has
has
- 2/15
2
B
B
B
3/40
3
B
B
C
3/40
4
B
C
B
3/40
5
C
B
B
3/40
with:
has
B
C
=
=
=
0 25
1
6
0 5
.
.
Formulate at 15 points, of command 5 in
X y Z
,
: (FPG15)
Not
X
y
Z
Weight
1
has
has
has
8/405
2
3
4
5
B
1
B
1
B
1
C
1
B
1
B
1
C
1
B
1
B
1
C
1
B
1
B
1
2.665 14 15
226 800
-
6
7
8
9
B
2
B
2
B
2
C
2
B
2
B
2
C
2
B
2
B
2
C
2
B
2
B
2
2.665 14 15
226 800
+
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
15/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
10
11
12
13
14
15
D
D
E
D
E
E
D
E
D
E
D
E
E
D
D
E
E
D
5
567
with:
has
= 0 25
.
B
B
1
2
7
15
34
7
15
34
=
+
=
-
C
C
1
2
13 3 15
34
13 3 15
34
=
-
=
+
D
E
=
-
=
+
5
15
20
5
15
20
4.2 Pentahedrons
:
ELREFE PE6, P15
Z
N2
N1
N3
N4
N5
N6
X
y
N7
N8
N9
N10
N11
N12
N13
N14
N15
X
y
Z
N1 - 1. 1. 0.
N2 - 1. 0. 1.
N3 - 1. 0. 0.
N4 1.
1. 0.
N5 1.
0. 1.
N6 1.
0. 0.
N7 - 1. 0.5.0.5.
N8 - 1. 0. 0.5.
N9 - 1. 0.5 0.
N10 0.
1. 0.
N11 0.
0. 1.
N12 0.
0. 0.
N13 1.
0.5 0.5
N14 1.
0. 0.5
N15 1.
0.5 0.
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
16/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Functions of form:
Formulate with 6 nodes
(
)
(
)
(
) (
)
X
Z
y
W
X
Z
W
X
y
W
-
-
-
=
-
=
-
=
1
1
2
1
1
2
1
1
2
1
3
2
1
(
)
(
)
(
)
(
)
W
y X
W
Z X
W
y Z X
4
5
6
1
2
1
1
2
1
1
2 1
1
=
+
=
+
=
- -
+
Formulate with 15 nodes
(
)
(
)
(
) (
)
(
)
(
) (
)
(
)
(
)
(
) (
)
(
)
(
) (
)
(
)
(
)
(
)
W
y
X
y
X
W
Z
X
Z
X
W
X
y Z X
y
Z
W
y
X
y
X
W
Z
X
Z
X
W
X
y Z
X
y
Z
W
yz
X
W
Z
y Z
X
1
2
3
4
5
6
7
8
1
2
2
2
1
2
2
2
1 1
2
2
2
1
2
2
2
1
2
2
2
1 1
2
2
2
2
1
2 1
1
=
-
- -
=
-
- -
=
-
- -
+
+
=
+
- +
=
+
- +
=
- -
- -
- +
+
=
-
=
- -
-
/
/
/
/
/
/
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
W
y
y Z
X
W
y
X
W
Z
X
W
y Z
X
W
yz
X
W
Z
y Z
X
W
y
y Z
X
9
10
2
11
2
12
2
13
14
15
2 1
1
1
1
1
1
2
1
2 1
1
2 1
1
=
- -
-
=
-
=
-
=
- -
-
=
+
=
- -
+
=
- -
+
Formulas of numerical integration at 6 points (command 3 in
X
, command 2 in
y
and
Z
) (FPG6)
Not
X
y
Z
Weight
1
- 3 3
0.5.0.5 1/6
2
- 3 3
0. 0.5 1/6
3
- 3 3
0.5 0.
1/6
4
3 3
0.5.0.5 1/6
5
3 3
0. 0.5 1/6
6
3 3
0.5 0.
1/6
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
17/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Formulate numerical integration at 8 points: (FPG8)
2 points of Gauss in
X
(command 3).
4 points of Hammer in
y
and
Z
(command 3).
Not
X
y
Z
Weight
1 - has
1/3
1/3 - 27/96
2 - has
0.6
0.2 25/96
3 - has
0.2
0.6 25/96
4 - has
0.2
0.2 25/96
5 +a
1/3
1/3 - 27/96
6 +a
0.6
0.2 25/96
7 +a
0.2
0.6 25/96
8 +a
0.2
0.2 25/96
With A = 0.577350269189626
Formulate numerical integration at 21 points: (FPG21)
3 points of Gauss in
X
(command 5).
7 points of Hammer in
y
and
Z
(command 5 in
y
and
Z
).
Not
X
y
Z
Weight
1
-
1/3 1/3
C
1
980
×
2
3
4
-
-
-
has
1-2a
has
has
has
1-2a
C
1
155
15
2 400
×
+
5
6
7
-
-
-
B
1-2b
B
B
B
1-2b
C
1
155
15
2 400
×
-
8 0.
1/3
1/3
C
2
980
×
9
10
11
0.
0.
0.
has
1-2a
has
has
has
1-2a
C
2
155
15
2 400
×
+
12
13
14
0.
0.
0.
B
1-2b
B
B
B
1-2b
C
2
155
15
2 400
×
-
15
1/3 1/3
C
1
980
×
16
17
18
has
1-2a
has
has
has
1-2a
C
1
155
15
2 400
×
+
19
20
21
B
1-2b
B
B
B
1-2b
C
1
155
15
2 400
×
-
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
18/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
with:
=
3
5
C
1
5
9
=
C
2
8
9
=
has
=
+
6
15
21
B
=
-
6
15
21
4.3 Hexahedrons
:
ELREFE HE8, H20, H27
y
Z
X
N2
N3
N4
N8
N14
N5
N6
N25
N26
N21
N24
N27
N7
N22
N23
N1
X
y
Z
N1 - 1.
- 1.
- 1.
N2 1.
- 1.
- 1.
N3 1.
1.
- 1.
N4 - 1.
1.
- 1.
N5 - 1.
- 1.
1.
N6 1.
- 1.
1.
N7 1.
1.
1.
N8
- 1.
1.
1.
N9 0.
- 1.
- 1.
N10 1.
0.
- 1.
N11 0.
1.
- 1.
N12 - 1.
0.
- 1.
N13 - 1.
- 1.
0.
N14 1.
- 1.
0.
N15 1.
1.
0.
N16
- 1.
1.
0.
N17 0.
- 1.
1.
N18 1.
0.
1.
N19 0.
1.
1.
N20
- 1.
0.
1.
N21 0.
0.
- 1.
N22 0.
- 1.
0.
N23 1.
0.
0.
N24 0.
1.
0.
N25
- 1.
0.
0.
N26 0.
0.
1.
N27 0.
0.
0.
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
19/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Functions of form:
Formulate with 8 nodes
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
W
X
y
Z
W
X
y
Z
W
X
y
Z
W
X
y
Z
1
2
3
4
1
8 1
1
1
1
8 1
1
1
1
8 1
1
1
1
8 1
1
1
=
-
-
-
=
+
-
-
=
+
+
-
=
-
+
-
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
W
X
y
Z
W
X
y
Z
W
X
y
Z
W
X
y
Z
5
6
7
8
1
8 1
1
1
1
8 1
1
1
1
8 1
1
1
1
8 1
1
1
=
-
-
+
=
+
-
+
=
+
+
+
=
-
+
+
Formulate with 20 nodes
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
W
X
y
Z
X y Z
W
X
y
Z
X y Z
W
X
y
Z
X y Z
W
X
y
Z
X y Z
W
X
y
Z
X y Z
W
X
y
Z
X y Z
W
X
y
Z
X y
1
2
3
4
5
6
7
1
8 1
1
1
2
1
8 1
1
1
2
1
8 1
1
1
2
1
8 1
1
1
2
1
8 1
1
1
2
1
8 1
1
1
2
1
8 1
1
1
2
=
-
-
-
- - - -
=
+
-
-
- + - -
=
+
+
-
- + + -
=
-
+
-
- - + -
=
-
-
+
- - - +
=
+
-
+
- + - +
=
+
+
+
- + + +
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
Z
W
X
y
Z
X y Z
W
X
y
Z
W
y
X
Z
8
9
2
10
2
1
8 1
1
1
2
1
4 1
1
1
1
4 1
1
1
=
-
+
+
- - + +
=
-
-
-
=
-
+
-
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
) (
)
W
X
y
Z
W
y
X
Z
W
Z
X
y
W
Z
X
y
W
Z
X
y
W
Z
X
y
W
X
y
Z
W
y
X
Z
W
X
y
Z
W
y
X
Z
11
2
12
2
13
2
14
2
15
2
16
2
17
2
18
2
19
2
20
2
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
1
4 1
1
1
=
-
+
-
=
-
-
-
=
-
-
-
=
-
+
-
=
-
+
+
=
-
-
+
=
-
-
+
=
-
+
+
=
-
+
+
=
-
-
+
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
20/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Formulate with 27 nodes
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X X
y y
Z Z
W
X
y y
Z Z
1
2
3
4
5
6
7
8
9
2
1
8
1
1
1
1
8
1
1
1
1
8
1
1
1
1
8
1
1
1
1
8
1
1
1
1
8
1
1
1
1
8
1
1
1
1
8
1
1
1
1
4 1
1
1
=
-
-
-
=
+
-
-
=
+
+
-
=
-
+
-
=
+
-
+
=
+
-
+
=
+
+
+
=
-
+
+
=
-
-
-
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
W
X X
y
Z Z
W
X
y y
Z Z
W
X X
y
Z Z
W
X X
y y
Z
W
X X
y y
Z
10
2
11
2
12
2
13
2
14
2
1
4
1 1
1
1
4 1
1
1
1
4
1 1
1
1
4
1
1 1
1
4
1
1 1
=
+
-
-
=
-
+
-
=
-
-
-
=
-
-
-
=
+
-
-
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
W
X X
y y
Z
W
X X
y y
Z
W
X
y y
Z Z
W
X X
y
Z Z
W
X
y y
Z Z
W
X X
y
Z Z
W
X
y
Z Z
W
X
y y
Z
W
X X
y
Z
15
2
16
2
17
2
18
2
19
2
20
2
21
2
2
22
2
2
23
2
2
1
4
1
1 1
1
4
1
1 1
1
4 1
1
1
1
4
1 1
1
1
4 1
1
1
1
4
1 1
1
1
2 1
1
1
1
2 1
1 1
1
2
1 1
1
=
+
+
-
=
-
+
-
=
-
-
+
=
+
-
+
=
-
+
+
=
-
-
+
=
-
-
-
=
-
-
-
=
+
-
-
(
)
(
)
(
)
(
)
(
) (
)
(
) (
)
(
)
(
) (
) (
)
W
X
y y
Z
W
X X
y
Z
W
X
y
Z Z
W
X
y
Z
24
2
2
25
2
2
26
2
2
27
2
2
2
1
2 1
1 1
1
2
1 1
1
1
2 1
1
1
1
1
1
=
-
+
-
=
-
-
-
=
-
-
+
=
-
-
-
Formulate quadrature of Gauss at 2 points in each direction (command 3) (FPG8)
Not
X
y
Z
Weight
1
- 3 3
- 3 3
- 3 3
1.
2
- 3 3
- 3 3
3 3
1.
3
- 3 3
3 3
- 3 3
1.
4
- 3 3
3 3
+ 3 3
1.
5
3 3
- 3 3
- 3 3
1.
6
3 3
- 3 3
3 3
1.
7
3 3
3 3
- 3 3
1.
8
3 3
3 3
3 3
1.
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
21/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Formulate quadrature of Gauss at 3 points in each direction (command 5): (FPG27)
Not
X
y
Z
Weight
1
-
-
-
C
13
2
-
-
0.
C C
12 2
3
-
-
C
13
4
-
0.
-
C C
12 2
5
-
0. 0.
C C
1 22
6
-
0.
C C
12 2
7
-
-
C
13
8
-
0.
C C
12 2
9
-
C
13
10 0.
-
-
C C
12 2
11 0.
-
0.
C C
1 22
12 0.
-
C C
12 2
13 0.
0.
-
C C
1 22
14 0.
0. 0.
C
23
15 0.
0.
C C
1 22
16 0.
-
C C
12 2
17 0.
0.
C C
1 22
18
0.
C C
12 2
19
-
-
C
13
20
-
0.
C C
12 2
21
-
C
13
22
0.
-
C C
12 2
23
0. 0.
C C
1 22
24
0.
C C
12 2
25
-
C
13
26
0.
C C
12 2
27
C
13
with:
=
3
5
C
1
5
9
=
C
2
8
9
=
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
22/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
4.4 Pyramids
:
ELREFE PY5, P13
N2
N3
N5
Z
N4
N1
X
y
N12
N7
N6
N9
N8
N13
N10
N11
The square base is consisted the quadrangle NR
1
NR
2
NR
3
NR
4
and NR
5
is the node of the pyramid.
X
y
Z
NR
1
1. 0.
0.
NR
2
0. 1.
0.
NR
3
1. 0. 0.
NR
4
0.
1.
0.
NR
5
0. 0.
1.
NR
6
0.5
0.5
0.
NR
7
0.5 0.5
0.
NR
8
0.5
0.5
0.
NR
9
0.5
0.5
0.
NR
10
0.5
0.
0.5
NR
11
0. 0.5
0.5
NR
12
0.5 0. 0.5
NR
13
0.
0.5
0.5
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
23/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Functions of form:
Formulate with 5 nodes
(
) (
)
(
)
(
) (
)
(
)
(
) (
)
(
)
(
) (
)
(
)
W
X y Z
X y Z
Z
W
X y Z
X y Z
Z
W
X y Z
X y Z
Z
W
X y Z
X y Z
Z
W
Z
1
2
3
4
5
1
1
4 1
1
1
4 1
1
1
4 1
1
1
4 1
1
= - + + -
- - + -
-
= - - + -
- + -
-
=
+ + -
- + -
-
=
+ + -
- + + -
-
= -
Formulate with 13 nodes
(
) (
)
(
)
(
)
(
) (
) (
)
(
)
(
) (
)
(
)
(
)
(
) (
) (
)
(
)
(
)
(
) (
) (
)
(
)
W
X y Z
X y Z
X
Z
W
X y Z
X y Z
y
Z
W
X y Z
X y Z
X
Z
W
X y Z
X y Z
y
Z
W
Z Z
W
X y Z
X y Z
X y Z
Z
W
1
2
3
4
5
6
7
1
1
0 5
2 1
1
1
0 5
2 1
1
1
0 5
2 1
1
1
0 5
2 1
2
0 5
1
1
1
2 1
= - + + -
- - + -
-
-
= - - + -
- + -
-
-
=
- + -
+ + -
- -
-
=
+ + -
- + + -
- -
-
=
-
= - - + + -
- - + -
- + -
-
.
.
.
.
.
(
) (
) (
)
(
)
(
) (
) (
)
(
)
(
) (
) (
)
(
)
= - - - + -
- + -
+ + -
-
= -
- + -
+ + -
- + + -
-
= -
+ + -
- + + -
- - + -
-
X y Z
X y Z
X y Z
Z
W
X y Z
X y Z
X y Z
Z
W
X y Z
X y Z
X y Z
Z
1
1
1
2 1
1
1
1
2 1
1
1
1
2 1
8
9
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
24/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
(
) (
)
(
) (
)
(
) (
)
(
) (
)
W
Z
X y Z
X y Z
Z
W
Z
X y Z
X y Z
Z
W
Z X y Z
X y Z
Z
W
Z X y Z
X y Z
Z
10
11
12
13
1
1
1
1
1
1
1
1
1
1
1
1
=
- + + -
- - + -
-
=
- - + -
- + -
-
=
- + -
+ + -
-
=
+ + -
- + + -
-
Formulate numerical integration at 5 points (FPG5):
Not X y Z Weight
1 0.5
0.
H
1
2/15
2 0.
0.5
H
1
2/15
3 0.5
0.
H
1
2/15
4 0.
0.5
H
1
2/15
5 0.
0.
H
2
2/15
with:
H
1
= 0.1531754163448146
H
2
= 0.6372983346207416
Formulate numerical integration at 6 points (FPG6):
Not X y Z Weight
1 A 0.
H
1
p
1
2 0.
has
H
1
p
1
3
0. H has
1
p
1
4 0.
has
H
1
p
1
5 0.
0.
H
2
p
2
6 0.
0.
H
3
p
3
with:
p
1
= 0.1024890634400000
p
2
= 0.1100000000000000
p
3
= 0.1467104129066667
= 0.5702963741068025 have
H
1
= 0.1666666666666666
H
2
= 0.08063183038464675
H
3
= 0.6098484849057127
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
25/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
Formulate numerical integration at 27 points (FPG27):
Not X y Z Weight
1 0.
0.
1/2 A
1
2
(
)
B
Z
1
2 1
B
1
2 1 - Z
(
)
1/2
B
6
3
- B
1
2 1 - Z
(
)
B
1
2 1 - Z
(
)
1/2
B
6
4
- B
1
2 1 - Z
(
)
- B
1
2 1 - Z
(
)
1/2
B
6
5
(
)
B
Z
1
2 1
- B
1
2 1 - Z
(
)
1/2
B
6
6
0.
0.
1
- B
1
2
B
6
7
0.
0.
1
+ B
1
2
B
6
8
(
)
C
Z
1
1
-
0.
(
)
1
2
1
- C/
C
8
9 0.
(
)
C
Z
1
1
-
(
)
1
2
1
- C/
C
8
10
(
)
-
-
C
Z
1
1
0.
(
)
1
2
1
- C/
C
8
11 0.
- C
1
1
- Z
(
)
(
)
1
2
1
- C/
C
8
12
(
)
C
Z
1
1
-
0.
(
)
1
2
1
+ C/
C
8
13 0.
(
)
C
Z
1
1
-
(
)
1
2
1
+ C/
C
8
14
- C
1
1
- Z
(
)
0.
(
)
1
2
1
+ C/
C
8
15 0.
- C
1
1
- Z
(
)
(
)
1
2
1
+ C/
C
8
16
D
1
2 1 - Z
(
)
D
1
2 1 - Z
(
)
1
- D
1
(
)
/2
D
12
17
- D
1
2 1 - Z
(
)
D
1
2 1 - Z
(
)
1
- D
1
(
)
/2
D
12
18
- D
1
2 1 - Z
(
)
- D
1
2 1 - Z
(
)
1
- D
1
(
)
/2
D
12
19
D
1
2 1 - Z
(
)
- D
1
2 1 - Z
(
)
1
- D
1
(
)
/2
D
12
20
D
1
1
- Z
(
)
0. 1/2D
12
21 0.
D
1
1
- Z
(
)
1/2D
12
22
(
)
-
-
D
Z
1
1
0. 1/2D
12
23 0.
- D
1
1
- Z
(
)
1/2D
12
24
D
1
2 1 - Z
(
)
D
1
2 1 - Z
(
)
(
)
1
2
1
+ D/
D
12
25
- D
1
2 1 - Z
(
)
D
1
2 1 - Z
(
)
1
+ D
1
(
)
/2
D
12
26
- D
1
2 1 - Z
(
)
- D
1
2 1 - Z
(
)
1
+ D
1
(
)
/2
D
12
27
D
1
2 1 - Z
(
)
- D
1
2 1 - Z
(
)
1
+ D
1
(
)
/2
D
12
Code_Aster
®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
J. PELLET, X. DESROCHES
Key
:
R3.01.01-D
Page
:
26/26
Manual of Reference
R3.01 booklet: General references
HT-66/05/002/A
with:
has
1
= 0.788073483
B
6
= 0.499369002
B
1
= 0.848418011
C
8
= 0.478508449
C
1
= 0.652816472
D
12
= 0.032303742
D
1
= 1.106412899
5 Bibliography
[1]
DHATT G., TOUZOT G.: A presentation of the finite element method 2
ème
edition.
Editor: MALOINE S.A. Année 984