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Code_Aster
®
Version
3
Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
X. DESROCHES
Key:
R3.06.03-B
Page:
1/10
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
Document: R3.06.03
Calculation of the stresses to the nodes by local smoothing
Summary:
One presents a local method of calculation of stresses at the nodes starting from the stresses at the points of
GAUSS. It is used in the options
SIGM_ELNO_DEPL
and
SIEF_ELNO_ELGA
control
CALC_ELEM
[U4.61.02].
This method is summarized to calculate the stresses at the tops of an element by multiplying the stresses
at the points of GAUSS by a matrix of smoothing, constant for each type of element.
For the isoparametric elements of degree 2, the stresses with the nodes mediums are obtained by
average of the values of the stresses at the 2 tops of the edge.
This method of smoothing has two advantages:
·
the nodal stresses obtained have a command of precision moreover than by direct calculation with the nodes,
·
the method is inexpensive in time CPU.
This method was generalized:
·
with calculations of the deformations (option
EPSI_ELNO_DEPL
) and of the variables intern (option
VARI_ELNO_ELGA
) with the nodes in mechanics,
·
with the calculation of flows (option
FLUX_ELNO_TEMP
) with the nodes in thermics.
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Code_Aster
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Version
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
X. DESROCHES
Key:
R3.06.03-B
Page:
2/10
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
1 Preliminaries
This method is based on the observation [bib1] which it exists points where the calculation of the stresses, with
to leave displacements in a primal formulation in displacements, is more precise.
In the case of isoparametric finite elements of command 2 (
SEG3
in 1D,
QUAD8
and
QUAD9
in 2D,
HEXA20
in 3D), one shows that the points of GAUSS of the formula of quadrature to 2
N
points (
N
: dimension of
space) are such as one can hope, without that being formally shown, for the calculation of



the same command of precision as for the calculation of the field of displacement
U
.
The idea of the method is to calculate for each element the stresses
!
with the nodes from
K
at the points of GAUSS, these last being calculated on each element by the formula:



K
K
ik
I
NR
I
D
B U
=
=
=
D B U
1
0
where:
D
is the matrix of elasticity,
B
K
is the matrix connecting the deformations to displacements at the point of GAUSS K,
U
are nodal displacements.
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Code_Aster
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Version
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
X. DESROCHES
Key:
R3.06.03-B
Page:
3/10
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
2
Local method of minimization by least squares
Generally, one wishes to approximate, within the meaning of least squares, a function
()
X y
,
by a polynomial function:
()
!
,
,
,



X y
X y has
ij
I
I
I
p
I
Q
=
=
=
0
0
The problem amounts finding the coefficients
has
ij
who minimize the functional calculus:
(
)
=
-



!
2
dx Dy
Values of the function
are known here only at the points of Gauss:
(
)
K
K
K
X
y
=
,
The minimum will be reached if and only if:
has
I
p
J
Q
ij
=
=…
=…
0
0
0
,
,
Within the framework of the finite element method in displacement, the function of smoothing is written:
()
()
!
,
,
!






X y
NR X y
I
I
I
N
=
=
1
where:
NR
I
is related to form associated with the node
I
,
!
I
is the value of the stress to the node
I
sought,
N
the number of nodes retained for smoothing.
One must thus solve the system:
!
,
I
I
N
=
=…
0
1
éq 2-1
One can choose between two methods of local smoothing: continuous smoothing or discrete smoothing.
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Code_Aster
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
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Key:
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Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
3
Methods of local smoothing (ref. [bib2] and [bib3])
3.1
Continuous local smoothing
This type of smoothing led to solve the system [éq 2-1] with
(
)
(
)
=
-
=
-
!
!
2
2
E
I I
E
NR
.
What leads to
M
F
E
E
!



====
with:
() ()
(
)
()
(
)
M
NR NR dx Dy
NR
NR
J
F
NR
dx Dy
NR
J
ije
I
J
E
I
K
K
npg
J
K
K
K
IE
I
I
K
npg
K
K
K
K
E
=
=
=
=
=
=
1
1
det
det
and



where
K
are the points of GAUSS in the element of reference
(
)
det J
K
the jacobien of the geometrical transformation enters the element of reference and
the element running at the point
K
.
K
: the weight associated with the point
K



K
: the stress at the point
K
()
NR
I
K
: the value of the function of form in the element of reference to the point
K
smoothed stresses
^
^
3
^
2
^
1
^
4
4
1
2
3
4
1
2
3
direct calculation of the stresses



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Code_Aster
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Version
3
Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
X. DESROCHES
Key:
R3.06.03-B
Page:
5/10
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
Note:
·
If spaces of interpolation of



and of
!



are the same ones, one has obviously



=
!
.
In practice, one retains for space of
!



a space smaller than that where is defined



by the finite element.
·
One sees the link between the approximation at the points of GAUSS of



where



thus converge
better and this process of smoothing whose justification is on the contrary continuous.
·
The way in which



is calculated at the points of GAUSS does not intervene. Generalization
with the nonlinear problems is thus obvious, although it cannot concern same
justification.
This method is however not adopted because it requires a resolution of system linear for
each calculation of
!
.
3.2
Discrete local smoothing
In this case, the functional calculus
is replaced by the summation:
() ()
(
)
()
()
~
!
!
=
-
=
-




=
=
=












K
K
K
npg
K
I
I
N
I
K
K
npg
NR
2
1
1
2
1
4
1
2
3
smoothed stresses
stresses at the points of GAUSS
1
4
2
3
IV
III
II
I
^
4
^
1
^
2
^
3
^
The system to be solved is written there still:
~
!
I
=
0
that is to say:
() ()
() ()
NR
NR
NR
I
I
K
J
K
J
I
K
K
K
npg
J
K
npg
!






=
=
=
1
1
maybe in matric form:
{
} {
}
M
P
!






node
GAUSS
=
Matrices
M
and
P
being then independent of the current element
E
.
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Code_Aster
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Version
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
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Key:
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Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
They can thus be calculated once and for all on the element of reference.
Note:
·
This method is more economic than the preceding one and gives results
comparable [bib2],
·
There still, manner thus
K
is calculated in each point of GAUSS is indifferent
(since the number of points of GAUSS used for the calculation of
and that of
!
is the same one). One will be able to thus use this method into nonlinear (option
SIEF_ELNO_ELGA
).
4
Application of the method to calculation of the stresses to
nodes for various elements
The local smoothing adopted in Code_Aster is the discrete local smoothing [§2.2], which makes it possible to avoid it
calculation of integrals on the element.
On all the elements of continuous medium 2D and 3D, one chose a space of smoothing resting on
functions of form relating to the nodes of the element.
The method thus makes it possible to obtain the stresses at the tops. In the case of elements of command 2,
one calculates the stresses with the nodes mediums by taking the average value of the two nodes
“framing” the node medium considered.
One gives hereafter the matrices of passage allowing to calculate the stresses with the nodes
nodes starting from the stresses at the points of GAUSS. These matrices can be square or
rectangular. Indeed, matrices of passage
MR. P
-
1
are calculated once and for all with
the initialization of each type of finite element (in
AFFE_MODELE
). Two types of matrices exist:
·
matrices
M
-
1
P
square, which is to be used when the number of points of GAUSS used
for the calculation of the stresses at the points of GAUSS
K
is identical to the number of nodes
nodes,
·
matrices
M
-
1
P
rectangular, which is to be used when the number of points of GAUSS
of
K
is different (in general higher) nodes nodes.
4.1
Square matrices of passage
These matrices are used by all the elements for the option
SIGM_ELNO_DEPL
. The option calculates in
first stresses in a number of points of GAUSS equal to the number of nodes. Then them
matrices
M
-
1
P
(given afterwards) are used to calculate the stresses with the nodes. These matrices
are also used for the option
SIEF_ELNO_ELGA
, in the elements for which the number
points of GAUSS of the calculation of
SIEF_ELGA
(in
STAT_NON_LINE
) is equal to the number of
nodes. They are the elements:
·
in 2D: QUAD4, TRIA6, under-integrated QUAD8,
·
in 3D: TETRA4, PENTA6, HEXA8, PYRAM5 and under-integrated HEXA20.
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Code_Aster
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Version
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
X. DESROCHES
Key:
R3.06.03-B
Page:
7/10
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
4.1.1 Square matrices of passage for the elements 2D
4.1.1.1 Triangles
MR. P
-
=
-
-
-
-
-
-




1
1
3
5
1
1
1
5
1
1
1
5
4.1.1.2 Quadrangles
MR. P
-
=
+
-
-
-
-
+
-
-
-
-
+
-
-
-
-
+
















1
1
3
2
1
2
1
3
2
1
2
1
2
1
3
2
1
2
1
3
2
1
3
2
1
2
1
3
2
1
2
1
2
1
3
2
1
2
1
3
2
4.1.2 Square matrices of passage for the elements 3D
4.1.2.1 Tetrahedrons
MR. P
-
= -
-
-
-
-






= -
= +
1
1
1
1
1
1
5
5
20
5 3 5
20
B has
has
has
has
has
has
has
has
has
has
has
has
has
has
has
has
has
has
B
4.1.2.2 Pentahedrons
-
-
-
1
-
-
1 1
-
1
-
1
-
-
1
-
1 1
-
1
-
1
-
-
1 1
-
1
-
1
-
-
1
-
1 1
-
1
-
-
-
-
M
-
1
P
=
=
+
3 1
2
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Code_Aster
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
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Key:
R3.06.03-B
Page:
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Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
4.1.2.3 Hexahedrons
(
)
(
)
(
)
MR. P
-
=




















= +
= +
=
-
= -
1
5 3 3
4
1
3
4
3 1
4
5 3 3
4
B B C B C C D has
B C C D has B B C
C D B C B a.c. B
B a.c. B C D B C
B has C B C B D
C
C B D
C B has C B
D
C C B C B B has
C B B has D
C C B
has
B
C
D
4
VI
VIII
8
7
6
2
1
3
5
IV
III
I
V
II
VII
Z
y
X
Appear 4.1.2.3-a: Classification of the points of GAUSS
on the hexahedron with 8 nodes
4.2
Matrices of passage
MR. P
-
1
rectangular
Into nonlinear for certain types of elements (
TRIA3
,
QUAD8
and
QUAD9
in 2D,
TETRA10
,
PENTA15
and
HEXA20
in 3D), the internal stresses and variables at the points of GAUSS are calculated on one
family of points of richer GAUSS (9 points for the quadrangles, 15 points for the tetrahedrons, 21
points for the pentahedrons, 27 points for the hexahedrons).
Discrete local smoothing is then carried out starting from these fields and made transport with the nodes
to intervene of the matrices different from the preceding ones. They are not square any more, because of dimension
(Nb nodes, Nb points of GAUSS). Matrices of passage
MR. P
-
1
are not calculated
explicitly, in particular
M
is reversed by Aster.
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Code_Aster
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
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Key:
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Page:
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R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
In the particular case of the triangle with 3 nodes, the fields are supposed to be constant by element (one
only point of GAUSS) and:
MR. P
-
=




1
1
1
1
For example, the calculation carried out by the option
SIEF_ELNO_ELGA
is then the following:
If the stresses were calculated (in
STAT_NON_LINE
for example) on a family
having a number of points of GAUSS higher than the number of nodes (for the elements
announced above).
MR. P
-
1
is then rectangular, and
(
)
!
I
I
Nb nodes
K
Nb pts Gauss
ik
K
=
=
=
-
1
1
1
MR. P
.
If not, if the number of points of GAUSS equal to the number of nodes,
(
)
MR. P
-
1
is then square.
One calculates
(
)
!
I
ik
K
=
-
MR. P
1
[§4.1].
5
Other options of calculation using the same method
The method described previously is used in Code_Aster to calculate the deformations, them
internal variables and flows with the nodes.
The list of modelings supporting these options is given below, with the numbers of
elementary routines of terms
TE
agents.
The produced fields are
cham_elem
with the nodes.
5.1 Phenomenon
:
“MECHANICAL”
MODELING
SIGM_ELNO_DEPL
EPSI_ELNO_DEPL
SIEF_ELNO_ELGA
VARI_ELNO_ELGA
AXIS
TE0086
TE0087
TE0098
AXIS_SI
TE0086
TE0087
TE0098
C_PLAN
TE0086
TE0087
TE0098
D_PLAN
TE0086
TE0087
TE0098
D_PLAN_SI
TE0086
TE0087
TE0098
AXIS_FOURIER
TE0116
TE0114
3D
TE0023
TE0025
TE0020
3d_SI
TE0023
TE0025
TE0020
AXIS_META
TE0352
TE0087
TE0098
3d_META
TE0357
TE0025
TE0020
AXIS_INCO
TE0448
TE0447
not disp
PLAN_INCO
TE0448
TE0447
not disp
3d_INCO
TE0454
TE0453
not disp
COQUE_AXIS
TE0230
TE0229
not disp
COQUE_C_PLAN
TE0230
TE0229
not disp
COQUE_D_PLAN
TE0230
TE0229
not disp
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Code_Aster
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Version
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Titrate:
Calculation of the stresses to the nodes by local smoothing
Date:
23/01/97
Author (S):
X. DESROCHES
Key:
R3.06.03-B
Page:
10/10
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HI-75/97/004/A
5.2 Phenomenon
:
“THERMAL”
MODELING
FLUX_ELNO_TEMP
META_ELNO_TEMP
META_INIT_ELNO
AXIS
TE0069
TE0067
TE0320
PLAN
TE0069
TE0067
TE0320
AXIS_FOURIER
TE0265
not disp
not disp
3D
TE0063
TE0064
TE0321
HULL
not disp
not disp
not disp
6
Other methods of smoothing of stresses
There are two other methods of smoothing, relating only to the stresses, used by
estimators of Zhu-Zienkiewicz version 1 and 2 [R4.10.01 §3].
The stress fields to the produced nodes are then
cham_no
.
The corresponding options of calculation are accessible by the control
CALC_ELEM
[U4.61.02].
7 Bibliography
[1]
BARLOW J. - Optimal stress hirings in finite element models - International Newspaper for
Numerical Methods in Engineering Vol.10 p 243 - 251 (1976).
[2]
HINTON E., CAMPBELL JJ. - Total Room and smoothing off discontinuous finite element
functions using has least public gardens method - International Newspaper for Numerical Methods in
Engineering Vol.8 p 461 - 480 (1974).
[3]
HINTON E., SCOTT F.C., RICKETTS R.E. - Local least public gardens stress smoothing for
parabolic isoparametric elements - Int. J. for Num. Meth. in Eng. Flight 9 p 235 - 256 (1975)