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
Organization (S): EDF-R & D/AMA
Handbook of Référence
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.
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
Count
matters
1 Introduction ............................................................................................................................................ 3
2 linear elements: SE2, SE3 and SE4 ............................................................................................. 4
The 3 surface elements ...................................................................................................................... 5
3.1 Triangles: ELREFE TR3, TR6, TR7 ................................................................................................ 5
3.2 Quadrangles: ELREFE QU4, QU8, QU9 ........................................................................................ 10
The 4 voluminal elements .................................................................................................................... 13
4.1 Tetrahedrons: ELREFE TE4, T10 .................................................................................................... 13
4.2 Pentahedrons: ELREFE PE6, P15 ................................................................................................... 15
4.3 Hexahedrons: ELREFE HE8, H20, H27 ............................................................................................ 18
4.4 Pyramids: ELREFE PY5, P13 ..................................................................................................... 22
5 Bibliography ........................................................................................................................................ 26
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
1 Introduction
In Code_Aster, one calls “finite element”, a triplet (phenomenon, modeling, type of mesh). There is
three principal phenomena: MECANIQUE, THERMIQUE and ACOUSTIQUE.
There are many modelings; for example, for phenomenon MECANIQUE: 3D, C_PLAN,
D_PLAN, AXIS, DKT, POU_D_E,…
For a given modeling (for example 3D) of a phenomenon (for example MECANIQUE), 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.
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
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
X
N3
N2
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
X
N3
N4
N2
functions of form of the segment with 2 nodes:
W (X) = 0 5
. 1
(- X)
W (X) = 0 5
. 1
(+ X
1
2
)
functions of form of the segment with 3 nodes:
W (X) = - 5
.
0
1
(- X) X
W (X) =
5
.
0
1
(+ X) X
W (X) =
1
(+ X 1
) (- X)
1
2
3
Handbook of Référence
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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
functions of form of the segment with 4 nodes:
1
W (X) = 16 (1 - X)
1
X + (X - 1/)
3
9
3
w2 (X) = - 16 (1+ X) 1
1
- X
X +
9
3
3
3
W (X) = 16 (X -)
1 (X +)
1
1 X -
27
3
w4 (X) = - 16 (X -)
1 (X +)
1
1 X +
27
3
Nb of pts
Not
X
Weight
of intégr.
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
N6
N5
N7
N1
N4
N2
Handbook of Référence
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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
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
Handbook of Référence
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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
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
Handbook of Référence
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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
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
(- - 1
) (- 1
(
2 - -))
1 - 1
(
4 - -)
1 - 1
(
4 - -)
- 1
(- 2)
- 1+
4
0
- 1
(- 2)
0
- 1+
4
4 1
(- -)
1
(
4 - 2 -)
-
4
4
4
4
4 1
(- -)
-
4
1
(
4 - - 2)
derived seconds from the triangle with 6 nodes:
{2
2
NR/
}
{2N/
}
{2
2
NR/
}
4 4 4
4 0 0
0 0 4
- 8 - 4 0
0 4 0
0 - 4 - 8
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Functions of form of the elements
Date:
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Author (S):
J. PELLET, X. DESROCHES Key
:
R3.01.01-D Page
: 9/26
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}
1 - (
3 +) + (
2 2
2
+) + 7 - 3 (+)
(- 1+ 2 + 3 - 3 (+))
(1
- + 2 + 3 - 3 (+))
4 1
(- - 4 + 3 (+))
4 (2
- + (
3 +))
4 1
(- 4 - + 3 (+))
27 1
(- -)
derived first from the triangle with 7 nodes:
{NR/
}
{NR/
}
2
- 3 + 4 + 7 - 6 - 3
2
- 3 + 7 + 4 - 6 - 3
2
- 1+ 4 + 3 - 6 - 3
3 1
(- - 2)
3 1
(- 2 -)
2
- 1+ 3 + 4 - 6 - 3
1
(
4 - 2 - 4 + 6 + 3 2
)
4 (4
- + 3 + 6)
4 (2
- + 6 + 3)
4 (2
- + 3 + 6)
4 (4
- + 6 + 3)
(
4 - 1 - 4 - 2 + 6 + 3 2
)
27 1
(- 2 -)
27 1
(- - 2)
derived seconds from the triangle with 7 nodes:
{2
2
NR/
}
{2N/
}
{2
2
NR/
}
4 -
6
7 -
6 -
6
4 -
6
4 -
6
3 -
6 -
6
-
6
-
6
3 -
6 -
6
4 -
6
(
4 - 2 + 6)
(
4 4
- + 6 + 6)
24
24
(
4 2
- + 6 + 6)
24
24
(
4 4
- + 6 + 6)
(
4 - 2 + 6)
-
54
27 1
(- 2 - 2)
-
54
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Functions of form of the elements
Date:
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Author (S):
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:
R3.01.01-D Page
: 10/26
3.2 Quadrangles
:
ELREFE QU4, QU8, QU9
N4
N7
N3
N8
N6
N9
N1
N5
N2
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
Handbook of Référence
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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
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/
}
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
(-)/4
{2
2
NR/
}
{2N/
}
{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/
}
1
(- 1
) (-) (1
- - -)/4
1
(-) (2 +)/4
1
(-) (+ 2)/4
1
(+ 1
) (-) (- 1+ -)/4
1
(-) (2 -)/4
- 1
(+) (- 2)/4
1
(+ 1
) (+) (1
- + +)/4
1
(+) (2 +)/4
1
(+) (+ 2)/4
1
(- 1
) (+) (- 1 - +)/4
- 1
(+) (2
- +)/4
1
(-) (
- + 2)/4
1
(-) 2 1
(-)/2
- 1
(-)
- 1
(
2
-)/2
1
(+ 1
) (-) 2
/2
1
(
2
-)/2
- 1
(+)
1
(-) 2 1
(+)/2
- 1
(+)
1
(
2
-)/2
1
(- 1
) (-) 2
/2
- 1
(
2
-)/2
- 1
(-)
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Functions of form of the elements
Date:
15/09/05
Author (S):
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:
R3.01.01-D Page
: 12/26
derived seconds from the quadrangle with 8 nodes:
{2
2
NR/
}
{2N/
}
{2
2
NR/
}
1
(-)/2
1
(- 2 - 2)/4
1
(-)/2
1
(-)/2
- 1
(+ 2 - 2)/4
1
(+)/2
1
(+)/2
1
(+ 2 + 2)/4
1
(+)/2
1
(+)/2
- 1
(- 2 + 2)/4
1
(-)/2
- 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/
}
(-) (
1 -)
1/4
(2 -)
1 (-)
1/4
(-) (
1 2 -)
1/4
(+) (
1 -)
1/4
(2 +)
1 (-)
1/4
(+) (
1 2 -)
1/4
(+) (
1 +)
1/4
(2 +)
1 (+)
1/4
(+) (
1 2 +)
1/4
(-) (
1 +)
1/4
(2 -)
1 (+)
1/4
(-) (
1 2 +)
1/4
1
(
2
-) (-)
1/2
- (-)
1
1
(
2
-) (2 -)
1/2
(+ 1
) (
1
2
-)/2
(2 + 1
) (
1
2
-)/2
- (+)
1
1
(
2
-) (+)
1/2
- (+)
1
1
(
2
-) (2 +)
1/2
(- 1
) (
1
2
-)/2
(2 - 1
) (
1
2
-)/2
- (-)
1
1
(
2
- 1
) (
2
-)
- 2 1
(
2
-)
- 2 1
(
2
-)
derived seconds from the quadrangle with 9 nodes:
{2
2
NR/
}
{2N/
}
{2
2
NR/
}
(-)
1/2
(- 1/) (
2 - 1/)
2/4
(-)
1/2
(-)
1/2
(+1/) (
2 - 1/)
2/4
(+)
1/2
(+)
1/2
(+1/) (
2 +1/)
2/4
(+)
1/2
(+)
1/2
(- 1/) (
2 +1/)
2/4
(-)
1/2
- (-)
1
- (2 -)
1
2
1 -
2
1 -
- (2 +)
1
- (+)
1
- (+)
1
- (2 +)
1
2
1 -
2
1 -
- (2 -)
1
- (-)
1
- 1
(
2
2
-)
4
- 1
(
2
2
-)
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Functions of form of the elements
Date:
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Author (S):
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:
R3.01.01-D Page
: 13/26
4
Voluminal elements
4.1 Tetrahedrons
:
ELREFE TE4, T10
Z
N2
N5
N6
N9
N3
N7
N1
N10
y
N8
N4
X
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
w1 (X, y, Z) = y
w2 (X, y, Z) = Z
W3 (X, y, Z) = 1 - X - y - Z
W
4 (X, y, Z) = X
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Functions of form of the elements
Date:
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Author (S):
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:
R3.01.01-D Page
: 14/26
Formulate with 10 nodes
W
= y
W
= 4 Z
6
(1 - X - y - Z)
1
(2y -) 1
W
= Z (2z -)
W
= 4 y
7
(1 - X - y - Z)
2
1
W
W
= 4 X y
3
= (1 - X - y - Z) (1 - 2x - 2y - 2z)
8
W
= X (2x -)
W
= 4 X Z
4
1
9
W
= 4 y Z
W
= 4 X
10
(1 - X - y - Z)
5
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
5 - 5
5 + 3 5
with: has =
B =
20
20
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
1
with: has
= 0 2
. 5
B =
C = 0 5
.
6
Formulate at 15 points, of command 5 in X, y, Z: (FPG15)
Not
X
y
Z
Weight
1
has
has
has
8/405
2
B
1
b1
b1
3
B
2.665 - 14 15
1
b1
c1
4
b1
c1
b1
226 800
5
c1
b1
b1
6
B
2
b2
b2
7
B
2.665 + 14 15
2
b2
c2
8
b2
c2
b2
226 800
9
c2
b2
b2
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
10
D
D
E
11
D
E
D
12
E
D
D
5
13
D
E
E
14
567
E
D
E
15
E
E
D
with:
has = 0 2
. 5
7 + 15
13 - 3 15
5 - 15
b1 =
C
=
D =
34
1
34
20
7 - 15
13 + 3 15
5 + 15
b2 =
C
=
E =
34
2
34
20
4.2 Pentahedrons
:
ELREFE PE6, P15
Z
N2
N11
N8
N7
N3
N1
N9
N5
N12
N10
y
N14
N13
N6
N15
N4
X
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.
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
Functions of form:
Formulate with 6 nodes
1
1
W = y
W
=
y (X +)
1
(1 - X)
1
2
4
2
1
1
W = Z
W
=
Z (X +)
2
(1 - X)
1
2
5
2
1
1
W =
w6 =
(1 - y - Z) (X +)
3
(1 - y - Z) (1 - X)
1
2
2
Formulate with 15 nodes
W
= y (1 - X)
W
= 2y
9
(1 - y - Z) (1 - X)
1
(2y - 2 - X)/2
W
= Z (1 - X) (2z - 2 - X)
2
2
/2
W
= y
10
(1 - X)
W
= (X -)
3
1 (1 - y - Z) (X + 2y + 2z)/2
W
= Z
2
11
(1 - X)
W
= y (1+ X)
4
(2y - 2+ X)/2
2
w12 = (1 - y - Z) (1 - X)
W
= Z (1+ X) (2z - 2 + X)
5
/2
W
= 2yz (1+ X)
13
W
= (- X -)
6
1 (1 - y - Z) (- X + 2y + 2z)/2
W
= 2z
14
(1 - y - Z) (1+ X)
W
= 2yz (1 - X)
7
W
= 2y
15
(1 - y - Z) (1+ X)
W
= 2z
8
(1 - y - Z) (1 - X)
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
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
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
9
1 × 80
2
-
has
has
155+ 15
3
-
1-2a
has
×
4
-
has
1-2a
c1 2 400
5
-
B
B
155 - 15
6
-
1-2b
B
C ×
7
1
-
B
1-2b
2 400
8 0.
1/3
1/3 C 9
2 × 80
9
0.
has
has
155+ 15
10
0.
1-2a
has
C ×
11
0.
has
1-2a
2
2 400
12
0.
B
B
155 - 15
13
0.
1-2b
B
C ×
14
0.
2
B
1-2b
2 400
15
1/3 1/3
C
9
1 × 80
16
has
has
155+ 15
17
1-2a
has
C ×
18
has
1-2a
1
2 400
19
B
B
155 - 15
20
1-2b
B
C ×
21
1
B
1-2b
2 400
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
with:
3
5
8
=
c1 =
c2 =
5
9
9
6 + 15
6 - 15
has =
B =
21
21
4.3 Hexahedrons
:
ELREFE HE8, H20, H27
Z
N5
N8
N26
N6
N25
N7
N27
y
N24
N1
N22
N4
N23
N21
N14
X
N2
N3
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.
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
Functions of form:
Formulate with 8 nodes
1
1
W
=
(1 - X)
W
=
(1 - X)
5
(1 - y) (1+ Z)
1
(1 - y) (1 - Z)
8
8
1
1
W
=
(1+ X)
W
=
(1+ X)
6
(1 - y) (1+ Z)
2
(1 - y) (1 - Z)
8
8
1
1
W
=
(1+ X)
W
=
(1+ X)
7
(1+ y) (1+ Z)
3
(1+ y) (1 - Z)
8
8
1
1
W
=
(1 - X)
W
=
(1 - X)
8
(1+ y) (1+ Z)
4
(1+ y) (1 - Z)
8
8
Formulate with 20 nodes
1
1
W
=
(1 - X)
W
2
11
=
(1 - X) (1+ y) (1-z)
1
(1 - y) (1 - Z) (- 2 - X - y - Z)
8
4
1
1
W
=
(1+ X)
W
2
12
=
(1 - y) (1 - X) (1-z)
2
(1 - y) (1 - Z) (- 2+ X - y - Z)
8
4
1
1
W
=
(1+ X)
W
2
13
=
(1-z) (1 - X) (1 - y)
3
(1+ y) (1 - Z) (- 2+ X + y - Z)
8
4
1
1
W
=
(1 - X)
W
2
14
=
(1-z) (1+ X) (1 - y)
4
(1+ y) (1 - Z) (- 2 - X + y - Z)
8
4
1
1
W
=
(1 - X)
W
2
15
=
(1-z) (1+ X) (1+ y)
5
(1 - y) (1+ Z) (- 2 - X - y + Z)
8
4
1
1
W
=
(1+ X)
W
2
16
=
(1-z) (1 - X) (1+ y)
6
(1 - y) (1+ Z) (- 2+ X - y + Z)
8
4
1
1
W
=
(1+ X)
(
Z)
W
2
17
=
(1 - X) (1 - y) (1+z)
7
(1+ y) (1+ Z) - 2+ X + y +
8
4
1
1
W
=
(1 - X)
W
2
18
=
(1 - y) (1+ X) (1+z)
8
(1+ y) (1+ Z) (- 2 - X + y + Z)
8
4
1
1
W
2
W
2
19
=
(1 - X) (1+ y) (1+z)
9
=
(1 - X) (1 - y) (1-z)
4
4
1
1
W
2
W
2
20
=
(1 - y) (1 - X) (1+z)
10
=
(1 - y) (1+ X) (1-z)
4
4
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
Formulate with 27 nodes
1
1
W
=
X (X -)
1 y
W
=
X (X +)
1 y
2
15
(y +) 1 (1 - Z)
1
(y -) 1 Z (Z -) 1
8
4
1
1
W
=
X (X +)
1 y
W
=
X (X -)
1 y
2
16
(y +) 1 (1 - Z)
2
(y -) 1 Z (Z -) 1
8
4
1
1
W
=
X (X +)
1 y
W
2
17
=
(1 - X) y (y) 1 Z (z+)
3
(y +) 1 Z (Z -) 1
1
8
4
1
1
W
=
X (X -)
1 y
W
=
X (X +)
2
18
1 (1 - y) Z (Z +)
4
(y +) 1 Z (Z -) 1
1
8
4
1
1
W
=
X (X +)
1 y
W
2
19
=
(1 - X) y (y+) 1 Z (z+)
5
(y -) 1 Z (Z +) 1
1
8
4
1
1
W
=
X (X +)
1 y
W
=
X (X -)
2
20
1 (1 - y) Z (Z +)
6
(y -) 1 Z (Z +) 1
1
8
4
1
1
W
=
X (X +)
1 y
W
2
2
21
=
(1 - X) (1 - y) Z (Z)
7
(y +) 1 Z (Z +) 1
1
8
2
1
1
W
=
X (X -)
1 y
W
2
2
22
=
(1 - X) y (y) 1 (1-z)
8
(y +) 1 Z (Z +) 1
8
2
1
1
W
2
W
=
X (X +)
2
2
23
1 (1 - y) (1 - Z)
9
=
(1 - X) y (y) 1 Z (Z) 1
4
2
1
1
W
=
X (X +)
2
W
2
2
24
=
(1 - X) y (y+) 1 (1-z)
10
1 (1 - y) Z (Z -)
1
4
2
1
1
W
2
W
=
X (X -)
2
2
25
1 (1 - y) (1 - Z)
11
=
(1 - X) y (y+) 1 Z (Z) 1
4
2
1
1
W
=
X (X -)
2
W
2
2
26
=
(1 - X) (1 - y) Z (z+)
12
1 (1 - y) Z (Z -)
1
1
4
2
1
W
2
2
2
27
= (1 - X) (1 - y) (1 - Z)
W
=
X (X -)
1 y
2
13
(y -) 1 (1 - Z)
4
1
W
=
X (X +)
1 y
2
14
(y -) 1 (1 - Z)
4
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.
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
Formulate quadrature of Gauss at 3 points in each direction (command 5): (FPG27)
Not
X
y
Z
Weight
1
-
-
-
c31
2
-
-
0.
c2 C
1
2
3
-
-
c31
4
-
0.
-
c2 C
1
2
5
-
0. 0.
C c2
1 2
6
-
0.
c2 C
1
2
7
-
-
c31
8
-
0.
c2 C
1
2
9
-
c3
1
10 0.
-
-
c2 C
1
2
11 0.
-
0.
C c2
1 2
12 0.
-
c2 C
1
2
13 0.
0.
-
C c2
1 2
14 0.
0. 0.
c32
15 0.
0.
C c2
1 2
16 0.
-
c2 C
1
2
17 0.
0.
C c2
1 2
18
0.
c2 C
1
2
19
-
-
c31
20
-
0.
c2 C
1
2
21
-
c31
22
0.
-
c2 C
1
2
23
0. 0.
C c2
1 2
24
0.
c2 C
1
2
25
-
c31
26
0.
c2 C
1
2
27
c31
with:
3
5
8
=
c1 =
c2 =
5
9
9
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
4.4 Pyramids
:
ELREFE PY5, P13
Z
N5
N12
N3
N13
N8
N7
N11
N10
N4
N2
y
N9
N6
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.
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
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
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
Functions of form:
Formulate with 5 nodes
(- 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 = 1 - Z
5
Formulate with 13 nodes
(- 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
R3.01 booklet: General references
HT-66/05/002/A
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
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 at 5 points (FPG5):
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
Formulate numerical integration at 6 points (FPG6):
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
R3.01 booklet: General references
HT-66/05/002/A
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
Formulate numerical integration at 27 points (FPG27):
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
R3.01 booklet: General references
HT-66/05/002/A
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
with:
a1 = 0.788073483
b6 = 0.499369002
b1 = 0.848418011
c8 = 0.478508449
c1 = 0.652816472
d12 = 0.032303742
d1 = 1.106412899
5 Bibliography
[1]
DHATT G., TOUZOT G.: A presentation of the finite element method 2nd edition.
Editor: MALOINE S.A. Année 984
Handbook of Référence
R3.01 booklet: General references
HT-66/05/002/A
Outline document