Code_Aster
®
Version
4.0
Titrate:
Integration clarifies relations of nonlinear behavior
Date:
22/04/98
Author (S):
P. GEYER, E. LORENTZ, C. VOGEL
Key:
R5.03.14-A
Page:
1/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-42/98/006/A
Organization (S):
EDF/RNE/EMMA, IMA/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.14
Integration clarifies relations of behavior
nonlinear
Summary
This document not describes a method of integration explicit for the resolution of problems with behavior
linear used in elastoviscoplasticity (in the operator
STAT_NON_LINE
[R5.03.01]).
The numerical method presented is that of Runge_Kutta of command 2 with adaptive pitch.
Code_Aster
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Version
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Titrate:
Integration clarifies relations of nonlinear behavior
Date:
22/04/98
Author (S):
P. GEYER, E. LORENTZ, C. VOGEL
Key:
R5.03.14-A
Page:
2/6
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R5.03 booklet: Nonlinear mechanics
HT-42/98/006/A
Contents
1 Introduction ............................................................................................................................................ 3
2 general Notations ............................................................................................................................... 3
3 Integration clarifies of an incremental relation of behavior ....................................................... 4
3.1 Integration ........................................................................................................................... 4 drank
3.2 General step ......................................................................................................................... 4
3.3 Use of a method clarifies .................................................................................................. 4
3.4 Control pitch of time ............................................................................................................... 5
3.5 Influence on the total stage ............................................................................................................ 5
3.6 Establishment of a new model of behavior by the method clarifies ............................. 6
4 Bibliography .......................................................................................................................................... 6
Code_Aster
®
Version
4.0
Titrate:
Integration clarifies relations of nonlinear behavior
Date:
22/04/98
Author (S):
P. GEYER, E. LORENTZ, C. VOGEL
Key:
R5.03.14-A
Page:
3/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-42/98/006/A
1 Introduction
In the operator
STAT_NON_LINE
[U4.32.01] is introduced a method of integration explicit of type
Runge_Kutta of command 2 (option
“RUNGE_KUTTA_2”
operand
RESO_INTE
key word factor
CONVERGENCE
) for the incrémentaux problems of nonlinear behavior (operand
RELATION
key word factor
COMP_INCR
).
For more details to consult the document [U4.32.01] user's manual. Relations concerned
currently are:
VISCOCHAB:
Élasto-viscoplastic behavior of Chaboche,
V_ENDO_CHAB:
Viscoplastic behavior with damage of Chaboche
POLY_CFC:
Polycrystalline élasto-viscoplastic behavior for materials of
cubic structure with centered faces.
The relation
VISCOCHAB
can also be integrated by the implicit method (cf [R5.03.11] while
the 2 other relations mentioned are accessible only by the explicit method presented hereafter.
This type of integration makes it possible to very easily establish a new model of behavior [bib2].
One describes the calculation of the stress field starting from an increment of deformation given while following
evolution of the internal variables.
2 Notations
general
All the quantities evaluated at the previous moment are subscripted by
-
.
Quantities evaluated at the moment
T
-
+
T
are not subscripted.
The increments are indicated by
. One has as follows:
Q
=
Q T
-
+
T
(
)
=
Q T
-
()
+
Q
=
Q
-
+
Q
.
tensor of the stresses.
increment of deformation.
With
tensor of elasticity.
Y
internal variables
µ
, E, v
moduli of the isotropic elasticity.
T
temperature.
ij
total deflection.
ij
p
plastic deformation.
ij
HT
thermal deformation.
Code_Aster
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Version
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Titrate:
Integration clarifies relations of nonlinear behavior
Date:
22/04/98
Author (S):
P. GEYER, E. LORENTZ, C. VOGEL
Key:
R5.03.14-A
Page:
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R5.03 booklet: Nonlinear mechanics
HT-42/98/006/A
3 Integration clarifies of a relation of behavior
incremental
3.1
Integration drank
In a structural analysis, the writing of the equilibrium equations is translated in integral form and
require the calculation of the interior efforts. It is thus necessary to calculate the stress field in
each point of integration; in the case of linear elasticity, this calculation is elementary but, in
situations where the representation of the behavior of material requires the use of a model not
linear, the aforementioned is more complex and it is necessary, moreover, to follow the evolution of the internal variables
([cf R5.03.01]).
The nonlinear behavior of material also modifies the strategy of calculation since one adopts,
in the majority of the cases, an incremental step in time. The aforementioned consists in discretizing
the interval of study in increments for better following nonthe linearity of the response of the structure.
3.2 Step
general
One places oneself at one moment
T
-
where the structure is in balance and where one knows, at each point of Gauss,
values of the internal variables
Y
-
. For the models élasto (visco) plastic, these internal variables
allow to calculate the stress
ij
with the law of Hooke (plastic deformation
ijp
is often
present among the internal variables):
ij
=
=
=
=
WITH (T)
[
ij
(U) -
ij
p
-
ij
HT
(T)]
To pass to the following increment (urgent
T
-
+
T
), the stress should be estimated
ij
(T
-
+
T)
, associated
by the relation of behavior, with the increment of displacement
U
K
proposed by the iteration
K
of
the total stage. It is thus enough to follow the evolution of the internal variables
Y
by solving the system
differential equations:
Dy/dt = F (Y,
K
, T)
with the initial conditions
Y (T
-
)
=
Y
-
.
Classically, one adopts a linear interpolation of displacement and temperature (known for
analyzes uncoupled) with the course the interval from time
[T
-
, T
=
T
-
+
T]
, which leads to:
T
-
, T
[]
T (
) = T
-
+
- T
-
T [T - T
-
]
ijk
ij
ijk
K
ij
U
T
T
U
U
U
() =
(
) + -
[
(
+
) -
(
)]
-
-
-
-
3.3
Use of an explicit method
One of the techniques simplest to implement to solve these differential equations
is the use of explicit methods. So that they numerically remain effective, it is
essential to associate an automatic control of pitch to them to preserve a good compromise cost
precision. Methods of explicit and encased Runge and Kutta [bib1], [bib2] are undoubtedly them
simplest diagrams of integration respecting these criteria. Their principle is to associate two
diagrams of integration of a different nature to control the pitch of time according to a precision
required. According to the command of integration chosen, several algorithms are available and the diagram
simplest is a method of command two. The calculation of the variables intern at the moment
T + H
is not
function that values of their derivatives
()
Dy
dt
F Y T
=
,
according to the following diagram:
Y
T
+
H
=
Y
(2)
if the criterion of precision is satisfied (cf [§3.4])
Y
(2)
=
Y + H2 [F (Y, T) + F (Y
(1)
, T + H)]
with
Y
(1)
=
Y + H F (Y, T)
Code_Aster
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Version
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Titrate:
Integration clarifies relations of nonlinear behavior
Date:
22/04/98
Author (S):
P. GEYER, E. LORENTZ, C. VOGEL
Key:
R5.03.14-A
Page:
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Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-42/98/006/A
The difference enters
()
Y
2
(diagram of command 2) and
()
Y
1
(diagram of command 1, Euler) provides an estimate
error of integration and makes it possible to control the size of the pitch of time
H
who is initialized with
T
I
for
first attempt. Thus, the method remains effective if the behavior remains elastic during
the increment and one naturally have under local pitches independent of the total increment for
to integrate, with a better precision, the evolution of the variables intern at the points of Gauss where not
linearity of the behavior is most significant.
3.4
Control pitch of time
The strategy of the control of the pitch is defined on the standard basis of the difference between the two methods
of integration:
|| Y
(2)
-
Y
(1)
||
and of the precision required by the user
(key word:
RESI_INTE_RELA
).
The criterion selected is as follows, where one notes
Y
=
(y
1
, y
2
,…, y
NR
)
:
Y (T)
=
sup
J
=
1, NR
| y
J
(2)
-
y
J
(1)
|
max [
, | y
J
(T)|]
<
The parameter
is fixed at 0,001. Precision of desired integration
must be coherent with
level of precision necessary for the total stage. For the models elastoviscoplastic, a value
conventional this parameter is 0,001. For materials with a low viscosity and for
elastoplastic models, the value of this parameter is a little lower (0,0001 or 0,00001).
If the criterion is not checked, the pitch of time is redécoupé according to a discovery method. When it
no time becomes too weak (H < 1.10
20
), calculation is stopped with an error message.
3.5
Influence on the total stage
In its current version, the method does not provide a tangent matrix; its evaluation would be possible
by a technique of disturbances. One uses for the moment the elastic matrix for total balance,
what implies a more significant number of increments so that the total resolution is facilitated.
To mitigate these disadvantages, one can suggest the possibility of developing at the total level one
diagram of the type BFGS [bib3]. Another simpler solution of development consists, instead of
to consider the increment of displacement for the first iteration at the moment running (suggested with
stamp elastic), to initialize the increment of displacement sought for the first iteration
U
1
with
to leave the value
U
N
-
, solution with the preceding increment (cf appears new page).
The key word
PREDICTION: “EXTRAPOL”
(under NEWTON) allows to initialize with the increment
converged of the preceding pitch (balanced by the report/ratio of the pitches of time). This estimate is projected
on the field of displacements kinematically acceptable so that the final solution checks well
boundary conditions of Dirichlet.
Code_Aster
®
Version
4.0
Titrate:
Integration clarifies relations of nonlinear behavior
Date:
22/04/98
Author (S):
P. GEYER, E. LORENTZ, C. VOGEL
Key:
R5.03.14-A
Page:
6/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-42/98/006/A
U
U
N
-
U
1
U
N
Loading
U
1
-
U
1
*
3.6 Establishment of a new model of behavior by
explicit method
To establish a model of behavior, it is enough to modify the 2 routines of shunting
NMCOMP
(routine General of the environment
PLASTI
) and
RDIF01
(routine suitable for explicit integration) and
to write 2 additional routines:
·
XXXMAT
: recovery of the data material,
·
RKDXXX
: writing of the equations constitutive of the model (expression of the derivatives temporal
internal variables; this requires the preliminary calculation of the plastic multiplier using
the equation of consistency for the elastoplastic models).
The update of the catalogs material only concerns
DEFI_MATERIAU
and
STAT_NON_LINE
.
4 Bibliography
[1]
CROUZEIX Mr., MIGNOT A.L.: “Numerical Analysis of the differential equations”, Masson,
1989.
[2]
CAILLETAUD G., PILVIN P.: “Use of polycrystalline models for the caclul by
finite elements ", European Review of the finite elements, vol.3, n°4, pp. 515-541, 1994.
[3]
MATTHIES H., STRANG G., Int. J. Numer. Meth. Eng., vol.4, pp. 1613-1626, 1979.