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Titrate:
Methods of control of the loading
Date:
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Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
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Organization (S): EDF/MTI/MN
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.80
Methods of control of the loading
Summary:
This document describes the methods of control 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,
constraint of control. 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).
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Titrate:
Methods of control of the loading
Date:
21/05/01
Author (S):
E. LORENTZ
Key:
R5.03.80-A
Page:
2/12
1
Principle of the methods of control of the loading
In a general way, the functionalities of control available in Code_Aster allow
to determine the intensity of part of the loading to satisfy a constraint 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 control (key word factor
PILOTAGE):
· 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): control 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: control 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: control by the elastic prediction
or more generally by the increment of deformation (TYPE: 'PRED_ELAS).
More precisely, the methods of control 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
cst
pilo
ext. = Fext + Fext


éq 1-1
U
cst
pilo
imp = U imp +
U
imp


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 constraint of control, which is expressed
by:
(
P U) =
with P (0) = 0
éq 1-2
where is indirectly an user datum which are expressed via the step of current time T and one
coefficient of control (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 step of time is small.
Finally, the unknown factors of the problem become the increment of displacements U = U - U -, them
multipliers of Lagrange associated with the boundary conditions and the intensity with the controlled loading
, baptized ETA_PILOTAGE. The nonlinear system to solve is written henceforth:
F
T
cst
pilo
int (U
; ) B
F
! +
=
ext. + Fext



B U = Ucst
pilo
imp + U imp
éq 1-3


P (U
) =

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Titrate:
Methods of control of the loading
Date:
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Key:
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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 constraint
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:
Fint






(
cst
N
T

N
pilo
U
N)
T

B
U
Fext - Fint (U
) - B
Fext
U
=
+







éq 2-1



cst
pilo
N
B
0

Uimp - B U

Uimp






“$$ #
$
%
“#
$
%
$
“# %
pilo
K
R cst
R
T
One can now express the corrections of displacements U and 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

Ucst

Upilo

Ucst

Upilo
=




where
K - 1 cst
- 1
pilo


T
R
and
KT R

+


=

=
éq 2-2
cst

pilo

cst

pilo

One can now substitute the correction of displacement U according to his expression
[éq 2-2] in the equation of control of the control of the system [éq 1.3]; it results a scalar equation from it
in:
~ (
P)

=
(PUn +Ucst + Upilo) =
éq 2-3
déf.
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The method of solution of this equation depends on nature on control on control adopted cf [§3].
Finally, it any more but does not remain to reactualize the unknown factors displacements and multipliers of
Lagrange:
Un+1 = One + Ucst

+ Upilo



éq 2-4
n+1
N
cst
pilo

= + +



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 control
~ (
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 control.
Notice 4:
There is no linearization compared to the variable of control. This way, one preserves
all the methodology of reactualization of the tangent operator already implemented for
calculations without control. 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 control 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 control is prohibited.
3
Equation of control of control
3.1
Control by control of a degree of freedom of displacements:
DDL_IMPO
For this first type of control, 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
with L = 0
0
1
0

0
!
!
éq 3.1-1



node N, ddl I


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Titrate:
Methods of control of the loading
Date:
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Author (S):
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Key:
R5.03.80-A
Page:
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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 One - L Ucst
=
éq 3.1-2
L Upilo
It will be noted that there is no solution when the correction of displacement controlled Upilo does not allow
to adjust the degree of freedom required, which can arrive if, by error, one blocks the degree of freedom in
question.
3.2
Control by length of arc: LONG_ARC
Another form of control very largely used consists in controlling the standard of the increment of
displacement (compared to certain nodes and certain components): one speaks then about control by
length of arc, to see Bonnet and Wood [bib1]. More precisely, the function P is expressed by:
(
P U) = U = U L U
éq 3.2-1
L
where, again, the nodal vector L makes it possible 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 control is reduced
with a quadratic equation:
[Upilo LUpilo] 2 +2 ([U
N + Ucst) L Upilo]
éq 3.2-2
+ ([U
N + Ucst) L (U
N + Ucst) - 2
] = 0
This equation can not admit a solution. In this case, one chooses the value which 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 Uavant and Un+1
(where Uavant is the increment of solution displacement of the step of preceding time), i.e. that which
maximize the cosine of this angle whose expression is:
(
One + Ucst

+ Upilo

Uavant
Uavant Un+1) (
)
cos
,
=
éq 3.2-3
One + Ucst

+ Upilo

Uavant
3.3
Control by the increment of deformation: PRED_ELAS
The last two modes of control, controls by increment of deformation and control per prediction
rubber band, cf Lorentz and Badel [bib3], are activated by 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 control
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 control are activables independently one
other.
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Control by increment of deformation consist in requiring that the increment of deformation of the step
current remains close in direction to the deformation at the beginning of the step 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 control:
-
(

P U) = Max G
-
G
with G = Bg U and G = Bg U
éq 3.3-1
-
G
G
where the index G sweeps 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 control according to is written then:

-

(0)

G
N
cst
G
With
=
B U
U
-
G (
+
)
~ (

0
1

P)
= max (()
()
G
G
With + G
With) =
with
-
éq 3.3-2
G
“$ #
$
%
$$

()
1
G
pilo
L
B
U
G ()
G
With =
G
-


G
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 which minimizes P ()
; it meets the condition min P ()
>. When it admits two
~
solutions, one chooses that which leads to U ()
nearest to U.
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Notice 6:
In the presence of great deformations, one can generalize the function of control [éq 3.3-1] in
employing deformations of Green - Lagrange (Lagrangian measurement of the deformations in
initial configuration):
-
(
E
1
P U) = Max G eg
with E =
-
-
(T


F F - I)
D
F = Id + U

E = E - E
G
E
2
G
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 has one then
expression similar to [éq 3.3-2] with:

-
E
With
G
(0)
T
N
cst
G
=
sym F
U

U

-
[gg (+)]


E G

-
éq 3.3-4

E
With
G
()
1
T
pilo
G
=
sym

F
U

-
[G
]

eg

where U indicates the gradient (not symmetrized) of displacements U evaluated at the point of Gauss of index
G
G.
~
~
~
P ()

P ()

P ()







Appear 3.3-a: Les 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 indicates
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 Lg ()


min
y
with lg (,
y) = Lg ()
- y
éq 3.3-5

G
, y
G lg (, y) 0
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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
Control by the elastic prediction: PRED_ELAS
If control 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 control 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 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) the constraints and the forces
thermodynamic associated A. Supposons moreover that the laws of evolution of A are
controlled by a threshold F (,
With, has) and a function of flow G (,
With, has) in the following way:
&a = G (,
With, has)
with 0 F

(,
With,) 0 F has (,
With,) = 0 have
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 el () = F (A (, -), -
has
has)
éq 3.4-2
One simplifies the problem by linearizing F el compared to in the vicinity of a point which one will define
later:
With

F
F
F el
el

=
+
+
-
éq 3.4-3
L ()
F ()
(
)
def



With

Finally, the function of control of control is defined like the maximum of F el compared to
L
all points of Gauss G, function which depends only on U:
(
P U) = Max F el

-
L (+ B G U)
with G = BgU
éq 3.4-4
G
It remains to define, not in the vicinity of which the linearization is carried out. It is selected like the point
nearest to - which realizes:
~ (
) with ~ ()

B (-
U
One

Ucst Upilo
el
G
)

=
=
+
+
+
and =
~
arg min fL ()
éq 3.4-5

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A graphic interpretation this choice is given on [Figure 3.4-a]. Finally, the equation of
control control is written:




0
With
()
F
F
With = F el () +
+
(
-
N
cst

Bg (U
U
U))




With

+
+
-
max
1
0


G
With + G
With
=
éq 3.4-6
G
[()
()]
“$ #
$
%
$$
F With

F
()
1

pilo
F el
With =
+
B U
L ()
(G
)




With



One is thus brought back to a problem identical to that of control by the increment of the deformation. One
of course employ the same algorithms of resolution as those presented to [§5].

~ ()

~
2
2
()



-
-
F el () = 0
F el () = 0


1
1
Appear 3.4-a: Définition 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.
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Appendix 1: Algorithms of resolution
(1)
Initialization of the interval
I0 =] - + [

(2)
Loop on the points of Gauss G
(2.1)
Root of the active linear function
G tq Lg (G) =
(2.2)
Construction of the following interval
(2.2.1)
If the active linear function is increasing
With (1)
G
> 0 I G = I g-1] - G]
(2.2.2)
If the active linear function is decreasing
With (1)
G
< 0 I G = I g-1 [G + [

(2.3)
To stop if the interval is empty
(3)
The solutions are the terminals of the interval
Fr (IG) max Lg ()
=
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
S0 = (0
, 0
y)
with
0
y = max Lg (0
)
G
(1.3)
Activated constraint
g0
tq
y0 = Lg
0 (
0)
(2)
Browsing of the successive nodes: loop on S
(2.1)
Definition of a direction of descent
G NR
D
S
(1)
S = G -
NR S
with NR = - A, 1
NR
S
gs
S NR
(
)
S
(2.2)
Found minimum if D S G = 0 (flat bottom)
(2.3)
Acceptable projection G for each constraint G G G =
max

lg (S +Ds) 0
(2.4)
Effective projection S
S = min G
G
(2.5)
Found minimum if S = 0
(2.5)
Following Summit
Ss+1 = S + S
Ds
(2.6)
Following activated constraint gs+1
gs+1 tq S = gs+1
Count An1-2: Algorithm of minimization of the function refines per pieces
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~
P ()

G
L
1 ()
S0
D0
L
4 ()
S1
D
S
1
2
L
3 ()
L
2 ()

2
1
0
Appear graphic An1-a: Illustration of the algorithm of minimization
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