Code_Aster
®
Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
1/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Organization (S):
EDF/MTI/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.80
Methods of piloting of the loading
Summary:
This document describes the methods of piloting of the loading available in Code_Aster (by a degree of
freedom, by length of arc, increment of deformation and elastic prediction). They introduce one
additional unknown factor, intensity on behalf controllable of the loading, and an additional equation,
stress of piloting. These methods make it possible in particular to calculate the response of a structure which
would as well have instabilities, of origins geometrical (buckling) as material (softening).
Code_Aster
®
Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
2/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
1
Principle of the methods of piloting of the loading
In a general way, the functionalities of piloting available in Code_Aster allow
to determine the intensity of part of the loading to satisfy a stress relating to
displacements. Their employment is limited to simulations for which time does not play of role
physics, which excludes a priori the dynamic or viscous problems. One can distinguish three
ranges of use which answer as many methods of piloting (key word factor
PILOTING
):
·
control physical forces by the displacement of a point of the structure (for example for
to adjust the intensity of the force exerted on a cable so that its arrow reaches a value
data): piloting per degree of freedom imposed (
TYPE: “DDL_IMPO”
);
·
follow-up of geometrical instabilities (buckling), the response of the structure being able exhiber of
“soft” snap-back: piloting by length of arc (
TYPE: “LONG_ARC”
);
·
follow-up of instabilities material (in the presence of laws of lenitive behavior), the answer
structure being able exhiber of the “brutal” snap-back: piloting by the elastic prediction
or more generally by the increment of deformation (
TYPE: 'PRED_ELAS
).
More precisely, the methods of piloting available in Code_Aster rest on both
following ideas. On the one hand, it is considered that the loading (external forces and displacements
imposed) additivement breaks up into two terms, one known and the other whose only direction is
known, its intensity
becoming a new unknown factor of the problem:
F
F
F
U
U
U
ext.
ext.
cst
ext.
pilo
imp
imp
cst
imp
pilo
=
+
=
+
éq 1-1
In addition, in order to be able to solve the problem, one associates a new equation to him which relates to
displacements and which depends on the increment of time: it is the stress of piloting, which is expressed
by:
()
()
P
P
U
0
=
=
with
0
éq 1-2
where
is indirectly an user datum which is expressed via the pitch of current time
T
and one
coefficient of piloting (
COEF_MULT
) by
=
T
COEF_ MULT
. The condition
()
P 0
=
0
is necessary
in order to obtain an increment of all the more small displacement as the pitch of time is small.
Finally, the unknown factors of the problem become the increment of displacements
U
U U
= -
-
, them
multipliers of Lagrange
associated the boundary conditions and the intensity of the controlled loading
, baptized
ETA_PILOTAGE
. The nonlinear system to solve is written henceforth:
(
)
()
F
U
B
F
F
B U
U
U
U
int
ext.
cst
ext.
pilo
imp
cst
imp
pilo
;
P
T
!
+
=
+
=
+
=
éq 1-3
Code_Aster
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Version
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Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
3/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Notice 1:
At present, loadings following (i.e which depend on displacements) and them
conditions of Dirichlet of the `type
DIDI
'are not controllable. Nothing is opposed so that they it
become in a later version of the code.
Notice 2:
The loading does not depend directly any more on time but results from the resolution of all it
nonlinear system [éq 1.3]. That implies that the controlled share of the loading does not have
to depend on physical time, contrary to a thermal loading, for example, but
corresponds to an effort which one adjusts to satisfy a kinematic stress
additional.
2
Resolution of the total system
The introduction of a new equation does not disturb in addition to measurement the method of resolution of
nonlinear system. Indeed, one proceeds as in [R5.03.01] by a linearization of the equations of
[éq 1.3] bearing on the interior forces and the conditions of Dirichlet:
()
()
F
U
U
B
B
0
U
F
F
U
B
U
B U
F
U
int
K
ext.
cst
int
imp
cst
R
ext.
pilo
imp
pilo
R
T
cst
pilo
N
N
N
N
T
T
=
-
-
-
+
“
#
%
“
#
$
%
$
“# %
éq 2-1
One can now express the corrections of displacements
U
and of multipliers of Lagrange
according to
with the help of the resolution of the linear system [éq 2-1] compared to each one of
two second members:
U
U
U
U
K
R
U
K
R
cst
cst
pilo
pilo
cst
cst
T
cst
pilo
pilo
T
pilo
=
+
=
=
-
-
where
and
1
1
éq 2-2
One can now substitute the correction of displacement
U
according to its expression
[éq 2-2] in the equation of control of the piloting of the system [éq 1.3]; it results a scalar equation from it
in
:
()
(
)
~
P
P
=
+
+
=
déf.
U
U
U
cst
pilo
N
éq 2-3
Code_Aster
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Version
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Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
4/12
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
The method of solution of this equation depends on nature on control on piloting adopted cf [§3].
Finally, it any more but does not remain to reactualize the unknown factors displacements and multipliers of
Lagrange:
U
U
U
U
cst
pilo
cst
pilo
N
N
N
N
+
+
=
+
+
=
+
+
1
1
éq 2-4
Notice 3:
During iterations of Newton, it can happen that the equation [éq 2-3] does not admit
solution, without in so far as there is an error of use. In this case, one decides then
to determine
as the value which minimizes
~
P
, provided that one checks
()
~
P
>
;
as the increment of displacement is all the more small as
is small, such a condition
impose coarsely that the increment of displacement is at least also large (without entering
in precise mathematical definitions) that that prescribed by the equation of piloting
()
~
P
=
. Moreover, one imposes has minimum an additional iteration of Newton, in order to
to check, with convergence, not only the equilibrium equations and the conditions of Dirichlet,
but also the equation of piloting.
Notice 4:
There is no linearization compared to the variable of piloting
. This way, one preserves
all the methodology of reactualization of the tangent operator already implemented for
calculations without piloting. Moreover, the structure “bandages” tangent matrix is preserved.
Notice 5:
This mechanism of resolution is incompatible with the use of linear search. In fact, it
would be possible in the presence of a function of linear piloting P, to see Shi and Crisfield [bib4],
but it is not true any more in the general case. This is why simultaneous use of search
linear and of piloting is prohibited.
3
Equation of control of piloting
3.1
Piloting by control of a degree of freedom of displacements:
DDL_IMPO
For this first type of piloting, the function P is restricted to extract a degree of freedom from the increment from
displacement. In particular, it is thus about a linear function:
()
P
U
L
U
L
=
=
with
node N, ddl I
0
0
1
0
0
!
!
éq 3.1-1
Code_Aster
®
Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
5/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
where the nodal vector
L
is null everywhere except for the degree of freedom being extracted where it is worth 1. The equation
[éq 2-3] is reduced then also to a linear equation which leads to:
=
-
-
L
U
L U
L U
cst
pilo
N
éq 3.1-2
It will be noted that there is no solution when the correction of controlled displacement
U
pilo
does not allow
to adjust the degree of freedom required, which can arrive if, by error, one locks the degree of freedom in
question.
3.2
Piloting by length of arc:
LONG_ARC
Another form of piloting very largely used consists in controlling the standard of the increment of
displacement (compared to certain nodes and certain components): one speaks then about piloting by
length of arc, to see Bonnet and Wood [bib1]. More precisely, the function P is expressed by:
()
P
U
U
U L
U
L
=
=
éq 3.2-1
where, again, the nodal vector
L
allows to select the degrees of freedom employed for calculation
standard (it is worth 1 for the selected ddl, 0 elsewhere). In this case, the equation of piloting is reduced
with a quadratic equation:
[
]
(
)
[
]
(
) (
)
[
]
U
L U
U
U
L U
U
U
L
U
U
pilo
pilo
cst
pilo
cst
cst
+
+
+
+
+
-
=
2
2
2
0
N
N
N
éq 3.2-2
This equation can not admit a solution. In this case, one chooses the value
who minimizes it
polynomial [éq 3.2-2]. One checks then well
()
~
P
>
. In the contrary case, it admits two roots
(or a double root). One chooses that of both which minimizes the angle formed by
U
front
and
U
N
+
1
(where
U
front
is the increment of solution displacement of the pitch of preceding time), i.e. that which
maximize the cosine of this angle whose expression is:
(
) (
)
cos
,
U
U
U
U
U
U
U
U
U
U
front
cst
pilo
front
cst
pilo
front
N
N
N
+
=
+
+
+
+
1
éq 3.2-3
3.3
Piloting by the increment of deformation:
PRED_ELAS
The last two modes of piloting, controls by increment of deformation and control per prediction
rubber band, cf Lorentz and Badel [bib3], are activated by the same key word
PRED_ELAS
. In fact, the second
depends explicitly on the law of behavior and is established only for certain laws (
ENDO_LOCAL
and
BETON_ENDO_LOCAL
); when it is available, it is employed. For the other laws, it is piloting
by increment of deformation which is activated. It is probable that this mechanism evolves/moves within the framework of
version 6 of Code_Aster and that these two modes of piloting are activables independently one
other.
Code_Aster
®
Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
6/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Piloting by increment of deformation consist in requiring that the increment of deformation of the pitch
current remains close in direction to the deformation at the beginning of the pitch to time, and this for at least one
not Gauss of the structure. It is required thus qualitatively that minimum a point of the structure has
preserve the mode of deformation which it had as a preliminary (for example, traction in a direction
data). Mathematically, one can give an account of this requirement with the help of the choice of
following function of piloting:
()
P
Max
U
B U
B
U
=
=
=
-
-
-
-
G
G
G
G
G
G
G
G
with
and
éq 3.3-1
where the index
G
sweep the points of Gauss of the structure and where deformation in a point of Gauss
deduced from the nodal vector of displacements via the symmetrical use of the matrices “left the gradient
functions of form “
B
G
(not to be confused with the matrix of the conditions of Dirichlet).
control piloting according to
is written then:
()
(
)
()
(
)
~
P
max
()
()
L
()
()
=
+
=
=
+
=
-
-
-
-
G
G
G
G
G
G
G
N
G
G
G
G
With
With
With
With
G
0
1
0
1
“#
$$
%
$$
with
B
U
U
B
U
cst
pilo
éq 3.3-2
Such a function is convex and linear per pieces. It generally admits no, one or two
solutions, cf [Figure 3.3-a]. When it does not admit solutions, one chooses like previously
value
who minimizes
()
~
P
; it meets the condition
()
min ~P
>
. When it admits two
solutions, one chooses that which leads to
()
~
U
nearest to
U
-
.
Code_Aster
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Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
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Key:
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Page:
7/12
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Notice 6:
In the presence of great deformations, one can generalize the function of piloting [éq 3.3-1] in
employing deformations of Green - Lagrange (Lagrangian measurement of the deformations in
initial configuration):
()
(
)
P
Max
T
U
E
E
E
E
F F Id
F Id
U
E E E
=
=
-
=
+
= -
-
-
-
G
G
G
G
with
1
2
éq 3.3-3
However, one would not lead any more like previously to a function closely connected per pieces. For y
to cure, one decides to carry out a linearization of
E
G
compared to
U
.
P
one has then
expression similar to [éq 3.3-2] with:
(
)
[
]
[
]
With
With
G
G
G
G
G
N
G
G
G
G
()
T
()
T
sym
sym
0
1
=
+
=
-
-
-
-
E
E
F
U
U
E
E
F
U
cst
pilo
éq 3.3-4
where
G
U
indicate the gradient (not symmetrized) of displacements
U
evaluated at the point of Gauss of index
G
.
()
~
P
()
~
P
()
~
P
Appear 3.3-a: the various cases of figure for the equation [éq 3.3-2]: two, one or no solutions
To solve the equation [éq 3.3-2], one proposes the algorithm presented in the table [An1-1], [§5]. It is
based on the construction of encased intervals: the terminals of the last of them are the solutions of
the equation and, as announced previously, one that which chooses leads to
()
~
U
nearest to
U
-
. This algorithm, rapid, are based on the resolution of G scalar equations linear, where
G
indicate
the total number of points of Gauss. The algorithm can end prematurely when one of
intervals is empty, which means that the equation [éq 3.3-2] does not admit solutions.
When there are not solutions with the equation [éq 3.3-2], it acts then, in accordance with remark 3,
to minimize
()
~
P
, problem which can be still expressed like the minimization of a function
linear with two variables under G forced linear inequalities:
()
()
()
()
min max L
min
L
,
L
,
L
,
G
G
y
G
y
G
G
G
y
y
y
=
-
0
with
éq 3.3-5
Code_Aster
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Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
8/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Here the algorithm of the simplex, cf Bonnans and Al [bib2] are employed, whose application to our problem is
presented in the table [An1-2], [§5].
3.4
Piloting by the elastic prediction:
PRED_ELAS
If piloting by the increment of deformation proves to be sufficient to follow dissipative solutions
in the majority of instabilities materials, the existence of solutions nevertheless is not proven. One him
then prefer a method of piloting based on the elastic prediction for which the existence of
solutions is shown but which, on the other hand, is specific to each law of behavior
(established only for the laws
ENDO_LOCAL
and
BETON_ENDO_LOCAL
). More precisely, when
the law of behavior is controlled by a threshold, one defines P as the maximum on all the points
of Gauss of the value of the function threshold in the case of an elastic test (incremental answer
rubber band of material).
Thus, let us consider that the state of material is described by the deformation
and a whole of variables
interns A. Appelons respectively
()
, has
and
()
With
has
,
stresses and forces
thermodynamic associated
has
. Let us suppose moreover that the laws of evolution of
has
are
controlled by a threshold
(
)
F
,
With
has
and a function of flow
(
)
G,
With
has
in the following way:
(
)
(
)
(
)
&
G
,
F
,
F
,
has
With
has
With
has
With
has
=
=
with
0
0
0
éq 3.4-1
Such a formulation includes the majority of the models of behavior dissipative and independent of
rate loading. The function threshold is worth then for an elastic test:
()
(
)
(
)
F
F With,
,
el
=
-
-
has
has
éq 3.4-2
One simplifies the problem while linearizing
F
el
compared to
in the vicinity of a point
that one will define
later:
()
()
(
)
F
F
F
F
L
def
el
el
----
=
+
+
With
With
éq 3.4-3
Finally, the function of control of piloting is defined like the maximum of
F
L
el
compared to
all points of Gauss G, function which depends only on
U
:
()
(
)
P
Max F
Lel
U
B U
B U
=
+
=
-
G
G
G
G
with
éq 3.4-4
It remains to define
, not in the vicinity of which the linearization is carried out. It is selected like the point
nearest to
-
-
-
-
who realizes:
()
()
(
)
()
-
=
=
+
+
+
=
~
~
arg min ~f
with
and
B U
U
U
U
cst
pilo
G
N
Lel
éq 3.4-5
Code_Aster
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Version
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Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
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Key:
R5.03.80-A
Page:
9/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
A graphic interpretation this choice is given on [Figure 3.4-a]. Finally, the equation of
control piloting is written:
[
]
()
()
(
)
(
)
(
)
max
F
F
F
F
F
()
()
F
()
el
()
Lel
G
G
G
G
N
G
With
With
With
With
With
With
With
With
1
0
0
1
+
=
=
+
+
+
+
-
=
+
-
“#
$$
%
$$
B U
U
U
B
U
cst
pilo
éq 3.4-6
One is thus brought back to a problem identical to that of piloting by the increment of the deformation. One
of course employ the same algorithms of resolution as those presented to [§5].
1
2
()
F
el
=
0
-
1
2
()
F
el
=
0
-
()
~
()
~
Appear 3.4-a: Definition of
according to the relative positions of surface threshold
F
el
and of the deformations
~
4 Bibliography
[1]
Bonnet J. and Wood R.D. [1997]. Nonlinear continuum mechanics for finite element analysis.
Cambridge university near.
[2]
Bonnans J.F., Gilbert J. - C., Lemaréchal C. and Sagastizabal C. [1997]. Optimization
numerical: theoretical and practical aspects. Mathematics and Applications, 27, ED. Springer.
[3]
Lorentz E. and Badel P. [2001]. With load control method for ramming finite element simulations.
Note intern EDF R & D, to appear.
[4]
Shi J. and Crisfield Mr. A. [1995]. Combining arc-length control and line searches in path
following. Com. Num. Meth. Eng. 11, pp. 793-803.
Code_Aster
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Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
10/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Appendix 1: Algorithms of resolution
(1)
Initialization of the interval
]
[
I
0
= - +
(2)
Loop on the points of Gauss
G
(2.1)
Root of the active linear function
G
G
G
tq L (
)
=
(2.2)
Construction of the following interval
(2.2.1)
If the active linear function is increasing
]
]
With
I
I
G
G
G
G
(1)
>
=
-
-
0
1
(2.2.2)
If the active linear function is decreasing
[
[
With
I
I
G
G
G
G
(1)
<
=
+
-
0
1
(2.3)
To stop if the interval is empty
(3)
The solutions are the terminals of the interval
()
()
=
Fr
max L
I
G
G
G
Count An1-1: Algorithm of resolution of the equation refines per pieces
(1)
Initialization with
0
given
(1.1)
Gradient of the function to be minimized
()
G
=
-
0
1
,
(1.2)
Initial Summit
(
)
()
S
0
0
0
0
0
=
=
,
max L
y
y
G
G
with
(1.3)
Activated stress
()
G
y
L
G
0
0
0
0
tq
=
(2)
Browsing of the successive nodes: loop on
S
(2.1)
Definition of a direction of descent
(
)
D
G
G NR
NR NR NR
NR
S
S
S
S
S
S
G
With
S
= -
= -
with
(1)
, 1
(2.2)
Found minimum if
D G
S
=
0
(flat bottom)
(2.3)
Acceptable projection
G
for each stress
G
(
)
=
+
G
G
G
S
S
max
L S
D
0
(2.4)
Effective projection
S
S
G
G
=
min
(2.5)
Found minimum if
S
=
0
(2.5)
Following Summit
S
S
D
S
S
S
S
+
=
+
1
(2.6)
Following activated stress
G
S
+
1
G
S
S
G
S
+
=
+
1
1
tq
Count An1-2: Algorithm of minimization of the function refines per pieces
Code_Aster
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Version
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Titrate:
Methods of piloting of the loading
Date:
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Key:
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Page:
11/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
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Appear graphic An1-a: Illustration of the algorithm of minimization
Code_Aster
®
Version
5.0
Titrate:
Methods of piloting of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
12/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Intentionally white left page.