Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 1/8
Organization (S): EDF/AMA
Handbook of Référence
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 traditional 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.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 2/8
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 (C0) between these elements and traditional 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.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 3/8
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 modelled 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
Z
N5
N3
N4
N2
y
N1
X
The square base is consisted the quadrangle N1 N2 N3 N4 and N5 is the node of the pyramid.
X
y
Z
N1 1. 0.
0.
N2 0. 1.
0.
N3 1. 0. 0.
N4 0.
1.
0.
N5 0. 0.
1.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 4/8
Functions of form:
(- X + y + Z -) 1 (- X - y + Z -) 1
w1 =
4 (1 - Z)
(- X - y + Z -) 1 (X - y + Z -) 1
w2 =
4 (1 - Z)
(X + y + Z -) 1 (X - y + Z -) 1
W3 =
4 (1 - Z)
(X + y + Z -) 1 (- X + y + Z -) 1
w4 =
4 (1 - Z)
W = Z
5
Formulate numerical integration at 5 points:
Not X y Z Poids
1 0.5
0.
h1 2/15
2 0.
0.5
h1 2/15
3 0.5
0.
h1 2/15
4 0.
0.5
h1 2/15
5 0.
0.
H2 2/15
with:
h1 = 0.1531754163448146
H2 = 0.6372983346207416
1 initialized family:
1ère family: formulate at 5 points.
2.2.2 Pyramid with 13 nodes
Z
N5
N12
N3
N13
N8
N7
N11
N10
N4
N2
y
N9
N6
N1
X
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 5/8
The square base is consisted mesh QUAD8 (N1 N2 N3 N4 N6 N7 N8 N9) and N5 is the node of
the pyramid.
X
y
Z
N1 1. 0.
0.
N2 0. 1.
0.
N3 1. 0. 0.
N4 0.
1.
0.
N5 0. 0.
1.
N6 0.5
0.5
0.
N7 0.5 0.5
0.
N8 0.5
0.5
0.
N9 0.5
0.5
0.
N10 0.5
0.
0.5
N11 0. 0.5
0.5
N12 0.5 0. 0.5
N13 0.
0.5
0.5
Functions of form:
(- X + y + Z -) 1 (- X - y + Z -) 1 (X - 0. ) 5
w1 =
2 (1 - Z)
(- X - y + Z -) 1 (X - y + Z -) 1 (y - 0. ) 5
w2 =
2 (1 - Z)
(X - y + Z -) 1 (X + y + Z -) 1 (- X - 0. ) 5
W3 =
2 (1 - Z)
(X + y + Z -) 1 (- X + y + Z -) 1 (- y - 0. ) 5
w4 =
2 (1 - Z)
W = 2 Z (Z - 0. )
5
5
(- X + y + Z -) 1 (- X - y + Z -) 1 (X - y + Z -) 1
w6 = -
2 (1 - Z)
(- X - y + Z -) 1 (X - y + Z -) 1 (X + y + Z) 1
w7 = -
2 (1 - Z)
(X - y + Z -) 1 (X + y + Z -) 1 (- X + y + Z -) 1
W = -
8
2 (1 - Z)
(
X + y + Z -)
1 (- X + y + Z -)
1 (- X - y + Z -)
1
W = -
9
2 (1 - Z)
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 6/8
Z (- X + y + Z -)
1 (- X - y + Z -)
1
w10 =
1 - Z
Z (- X - y + Z -)
1 (X - y + Z -)
1
w11 =
1 - Z
Z (X - y + Z -)
1 (X + y + Z -)
1
w12 =
1 - Z
Z (X + y + Z -)
1 (- X + y + Z -)
1
w13 =
1 - Z
Formulate numerical integration: formulate at 6 points
Not X y Z Poids
1 A 0.
h1
p1
2 0.
has
h1
p1
3
0. h1 has
p1
4 0.
has
h1
p1
5 0.
0.
H2
p2
6 0.
0.
h3
p3
with:
p1 = 0.1024890634400000
p2 = 0.1100000000000000
p3 = 0.1467104129066667
= 0.5702963741068025 have
h1 = 0.1666666666666666
H2 = 0.08063183038464675
h3 = 0.6098484849057127
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 7/8
Formulate at 27 points:
Not X y Z Poids
1 0.
0.
1/2 a1
2
B
B
1 (
1
1 - Z)
1/2
b6
1
(- Z)
2
2
3
B
B
- 1 1
(- Z)
1 1
(- Z)
1/2
b6
2
2
4
B
B
- 1 1
(- Z)
- 1 1
(- Z)
1/2
b6
2
2
5
B
B
1 (
1
1 - Z)
1/2
b6
-
1
(- Z)
2
2
6
0.
0.
1 - b1
B
6
2
7
0.
0.
1 + b1
B
6
2
8
C (
Z)
C
1 1 -
0.
(1-c)/2
8
1
9 0.
C (
Z)
C
1 1 -
(1-c)/2
8
1
10
- C (- Z)
C
1 1
0.
(1-c)/2
8
1
11 0.
- C (
)
C
1 1 - Z
(1-c)/2
8
1
12
C (
Z)
C
1 1 -
0.
(1+c)/2
8
1
13 0.
C (
Z)
C
1 1 -
(1+c)/2
8
1
14
- C (
)
C
1 1 - Z
0.
(1+c)/2
8
1
15 0.
- C (
)
C
1 1 - Z
(1+c)/2
8
1
16
d1
D
(
)
1
(- Z)
1 - D/2
1 1
(- Z)
1
d12
2
2
17
D
D
- 1
(
)
1
(- Z)
1 - D/2
1 1
(- Z)
1
d12
2
2
18
D
D
- 1
(
)
1
(- Z)
1 - D/2
- 1 1
(- Z)
1
d12
2
2
19
d1
D
(
)
1
(- Z)
1 - D/2
- 1 1
(- Z)
1
d12
2
2
20
D (
)
1 1 - Z
0. 1/2d12
21 0.
D (
)
1 1 - Z
1/2d12
22
- D (- Z)
1 1
0. 1/2d12
23 0.
- D (
)
1 1 - Z
1/2d12
24
d1
D
D
1
(- Z)
1 1
(- Z)
(1+d)/2
12
1
2
2
25
D
D
- 1
(
)
1
(- Z)
1 + D/2
1 1
(- Z)
1
d12
2
2
26
D
D
- 1
(
)
1
(- Z)
1 + D/2
- 1 1
(- Z)
1
d12
2
2
27
d1
D
(
)
1
(- Z)
1 + D/2
- 1 1
(- Z)
1
d12
2
2
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Functions of form and points of integration of the elements pyramid
Date:
16/02/02
Author (S):
J. Key PELLET
:
R3.06.06-B Page
: 8/8
with:
a1 = 0.788073483
b6 = 0.499369002
b1 = 0.848418011
c8 = 0.478508449
c1 = 0.652816472
d12 = 0.032303742
d1 = 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:
1ère 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.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/02/004/A
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