background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
1/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Organization (S):
EDF/AMA















Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
Document: R3.06.06



Functions of form and points of integration
elements pyramid at square base




Summary:

The free maillor of CASTEM 2000 creating under certain conditions of the meshs of pyramidal form at base
quadrangular, the associated finite elements were established in Code_Aster.

These elements have the characteristic to have rational functions of form, although guaranteeing a connection
continuous with the conventional tetrahedrons or hexahedrons.

The expression of the functions of form and the formulas of numerical integration were communicated to us by
ECA/DMT [bib1] and is thus those used by CASTEM 2000.
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
2/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
1 General information
Two new finite elements pyramid at square base were established in Code_Aster for
modelings three-dimensional mechanics and thermics:
·
pyramid with 5 nodes,
·
pyramid with 13 nodes.
The functions of form associated with these elements are rational functions which make it possible to have
a continuous connection
()
C
0
between these elements and conventional tetrahedrons and hexahedrons.
For example, the function of form associated with a node with the base with the pyramid is the product of
equations of the plans passing by the other nodes, divided by the distance to the base of the pyramid.
On a triangular face of the pyramid containing this node, the distance to the axis is simplified with
the equation of the plan of the opposite face: the expression of the function form is then that of the triangles
traditional.
The functions of form are not derivable at the top of the pyramid. Integration by points of
Gauss cannot thus be exact even for the element of reference.
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
3/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
2
Pyramid at square base
2.1 Denominations
The names of the finite elements are coded and respect following conventions
·
the characters in position 1 to 4 indicate the modelized phenomenon:
MECA
: mechanics
THER
: thermics
·
the character in position 5 is _,
·
starting from character 6, the name of the mesh support:
PYRAM5
: pyramid at square base with 5 nodes,
PYRAM13
: pyramid at square base with 13 nodes.
Example:
MECA_PYRAM5
: pyramid at square base with 5 nodes in mechanics.

2.2
Geometry, topology and functions of form
2.2.1 Pyramid with 5 nodes
N2
N3
N5
Z
N4
N1
X
y
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.
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
4/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Functions of form:
(
) (
)
(
)
(
) (
)
(
)
(
) (
)
(
)
(
) (
)
(
)
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
= - + + -
- - + -
-
= - - + -
- + -
-
=
+ + -
- + -
-
=
+ + -
- + + -
-
=
Formulate numerical integration at 5 points:
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
1 initialized family:
1
era
family: formulate at 5 points.
2.2.2 Pyramid with 13 nodes
N2
N3
N5
Z
N4
N1
X
y
N12
N7
N6
N9
N8
N13
N10
N11
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
5/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
The square base is consisted mesh QUAD8 (NR
1
NR
2
NR
3
NR
4
NR
6
NR
7
NR
8
NR
9
) 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
Functions of form:

(
) (
)
(
)
(
)
(
) (
) (
)
(
)
(
) (
)
(
)
(
)
(
) (
) (
)
(
)
(
)
(
) (
) (
)
(
)
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
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
6/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/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: formulate at 6 points

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
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
7/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Formulate at 27 points:
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
background image
Code_Aster
®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. PELLET
Key
:
R3.06.06-B
Page
:
8/8
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/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

Note:
It proved in practice, in particular for the thermal elements, that the formula
of integration at 6 points was not satisfactory. Only the formula at 27 points is thus used.

1 initialized family:
1
era
family: formulate at 27 points.




3 Bibliography
[1]
F. DUBON: “Formulation of an element pyramid at square base”. Report/ratio DEMT B4/310
(SMTS/LAMS/84-144). Commissariat à l' Énergie Atomique.