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Organization (S): EDF-R & D/AMA, SINETICS

Handbook of Utilization
U4.5- booklet: Methods of resolution
Document: U4.53.02

Operator DYNA_LINE_TRAN

1 Goal
To calculate the transitory dynamic response to an unspecified temporal excitation.

The temporal loading must be given in the form of a linear combination of vectors forces
assembled constant in time. They can be provided directly in the form of vectors
assembled or in the form of loads which will be assembled in the algorithm.

Only the coefficients of the linear combination are a function of time.

The implicit methods of integration available are Wilson-theta and NEWMARK, and the methods
of integration explicit available are the diagram with the differences centered, and a version with step
adaptive of this same diagram.

Moments of filing can be specified.

Product a concept result of the dyna_trans type.
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2 Syntax


dyn [dyna_trans] = DYNA_LINE_TRAN


(reuse
= dyn,

MODELE = Mo, [model]

CHAM_MATER
=
chmat,
[cham_mater]



CARA_ELEM

=
carac,
[cara_elem]



MODE_STAT

=
modestat,
[mode_static]



MATR_MASS

=
m,
[matr_asse_DEPL_R]



MATR_RIGI

=
K,
[matr_asse_DEPL_R]



MATR_AMOR

=
C,
[matr_asse_DEPL_R]



/NEWMARK
=
_F (
ALPHA
=/0.25, [DEFAUT]
/
Al,
[R]










DELTA =
/0.5,
[DEFAUT]
/
of,
[R]


),




/WILSON = (THETA
=/1.4, [DEFAUT]
/
HT,
[R]
),
/
DIFF_CENTER
=
_F (),
/
ADAPT

=
_F (),



/ETAT_INIT = _F (






/DYNA_TRANS = Dy, [dyna_trans]



















/NUME_INIT = nuini,
[I]









/INST_INIT = to,

[R]











PRECISION =














/1.E-03, [DEFAUT]
/
prec
,
[R]










CRITERE =
/
“RELATIF”,
[DEFAUT]












/“ABSOLU”,
/

DEPL_INIT
=
depl,


[cham_no_DEPL_R]







VITE_INIT = quickly, [cham_no_DEPL_R]








),




EXCIT =_F
(/CHARGE = chi,

[char_meca]










FONC_MULT = fi,
[function]









/VECT_ASSE = goes,

[cham_no_DEPL_R]











/COEF_MULT
=
I, [R]
/
FONC_MULT
=
fi,
[function]
/
DEPL
=
depl,
[function]
VITE
=
quickly,
[function]
ACCE
=
acce,
[function]








MULT_APPUI =/“YES”,
/
“NON”,
[DEFAUT]








DIRECTION
= (d1, d2, d3), [l_R]








NOEUD
=
lno,
[l_noeud]









GROUP_NO =
lgrno, [l_gr_noeud]







),




AMOR_MODAL = _F (









AMOR_REDUIT= l_amor, [l_R]









MODE_MECA = mode, [mode_meca]









NB_MODE =/
nbmode,
[I]
/
9999, [DEFAUT]







),
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SENSIBILITE =
_F (see [U4.50.02]),




SOLVEUR =_F (see the document [U4.50.01]),



INCREMENT =_F (/LIST_INST = litps,
[listr8]
/
FONC_INST
=
fonc
, [function]
/
PAS






=
dt,
[R]











INST_INIT = Ti, [R]











/INST_FIN=
tf,
[R]
/
NUME_FIN=
nufin,
[I]









PAS_CALCUL
=
/
1,
[DEFAUT]
/
ipas,
[I]









VITE_MIN =/“MAXIMUM”,
/“NORM”,
[DEFAUT]









COEF_MULT_PAS =/cmp,
[R]
/
1.1,
[DEFAUT]









COEF_DIVI_PAS =/cdp, [R]
/
1.3334,
[DEFAUT]









PAS_LIMI_RELA =/plr,
[R]
/
1.D-06,
[DEFAUT]









NB_POIN_PERIODE =/Np, [I]

/
50,
[DEFAUT]









NMAX_ITER_PAS = nip, [I]
16,
[DEFAUT]








),


SENSIBILITE = (





… to see [U4.50.02]….







)



ARCHIVAGE
=_F (/LIST_ARCH = list
[listis]
/
PAS_ARCH


=
ipa
[I]









CHAM_EXCLU = ( | “DEPL”,















| “VITE”,















| “ACCE”













),




TITER
=
titrate,
[l_Kn]



INFO =/1,






/2,



)
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3
Equations of behavior under transitory excitation

The operator carries out the direct temporal integration of a transitory linear mechanical problem of
form:

MX & + Cx & + Kx = (T) F (X)
I
I

I

O matrices M, C
, K
are the assembled real matrices of the problem finite elements
(respectively) of mass, damping and rigidity of the system.
I are functions of time (cf DEFI_FONCTION [U4.31.02]) and Fi are vectors
assembled resulting from loadings in imposed force (cf AFFE_CHAR_MECA [U4.44.01]); they can
to be provided directly in the form of assembled vectors or loads which will be
assemblies in the algorithm.

The solution (X, X,
& X&) is calculated on a temporal discretization Ti of the specified interval of study
by the user.

4 Operands
4.1 Operand
MODELE


MODELE = Mo

Name of the model whose elements are the subject of dynamic calculation.

This operand is obligatory when one applies an excitation of the type charges with the key word
EXCIT (cf [§4.8]).

4.2 Operand
CHAM_MATER


CHAM_MATER = chmat

Name of the affected material field on the model Mo, necessary when one is applied
excitation of the type charges with key word EXCIT.

4.3 Operand
CARA_ELEM


CARA_ELEM = carac

Name of the characteristics of the elements of beam, hull etc, necessary when one applies
an excitation of the type charges with key word EXCIT.

4.4 Operand
MODE_STAT

MODE_STAT = modestat

Name of the static mode necessary in the case of a seismic calculation with excitations multi-supports
[R4.05.01].
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4.5
Matrices of the problem

MATR_MASS = m

Concept stamps assembled of matr_asse_DEPL_R type corresponding to the matrix of mass
system.

MATR_RIGI = K

Concept stamps assembled of matr_asse_DEPL_R type corresponding to the matrix of rigidity
system.


MATR_AMOR = C

Concept stamps assembled of matr_asse_DEPL_R type corresponding to the matrix
of damping of the system.

The three matrices must be based on same classification and be built with the same mode
of storage.

4.6 Diagrams
of integration

/“WILSON”

Implicit diagram of integration of WILSON type.
Key word factor allowing to specify the parameters of integration.



THETA = HT

Value of the parameter for the method of WILSON.
By defect = 1 4
.
This diagram should not be used when one imposes nonnull displacements by the intermediary
of an assembled vector.
See [R5.05.02].

/
“NEWMARK”

Implicit diagram of integration of type NEWMARK.
Key word factor allowing to specify the parameters of integration and.



ALPHA = Al

Value of the parameter for the method of NEWMARK. By defect = 0.25.



DELTA = of

Value of the parameter for the method of NEWMARK. By defect = 0.5.

See [R5.05.02].

/“DIFF_CENTER”

Diagram of integration clarifies by centered differences. The use of this diagram imposes
certain enumerated restrictions of use on [§6.3]. The theoretical description of the diagram
is made in [bib 2].

/“ADAPT”
Diagram of integration clarifies with step of adaptive time, alternative of the diagram of the differences
centered. The use of this diagram imposes certain restrictions of use enumerated on
[§6.3] (see [bib 2]).

4.7 Key word
ETAT_INIT

This functionality allows a continuation of a transitory calculation, by taking as initial state one
result obtained by a preceding calculation with DYNA_LINE_TRAN. It also makes it possible to define
initial conditions of fields type to the nodes.
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4.7.1 Operands
DYNA_TRANS

/DYNA_TRANS = Dy

Concept of the dyna_trans type resulting from a preceding calculation with DYNA_LINE_TRAN, and
defining the initial conditions for new calculation.

4.7.2 Operands
DEPL_INIT/VITE_INIT

/DEPL_INIT = C

Concept of the cham_no_DEPL_R type, displacements initial.



VITE_INIT = vo

Concept of the cham_no_DEPL_R type, initial speeds.

4.7.3 Operands
NUME_INIT/INST_INIT

/NUME_INIT = nuini

nuini indicates the number of filing of preceding calculation to extract and take as state
initial in the case of a recovery.


/INST_INIT = to

Moment of preceding calculation to in the case of extract and take as initial state one
recovery.

In the absence of NUME_INIT and INST_INIT, the moment of recovery is taken equal to the last
moment of filed preceding calculation.

4.7.4 Operand
CRITERE

CRITERE =

Indicate with which precision the search of the moment must be done:

“RELATIF”: interval of search [(1-prec) .instant, (1+prec) .instant]
“ABSOLU”: interval of search [moment-prec, instant+prec]

The default value of the search key is “RELATIF”.

4.7.5 Operand
PRECISION

PRECISION =/1.E-03
[DEFAUT]
/
prec [R]

Indicate with which precision the search of the moment must be done.

4.8 Key word
EXCIT

EXCIT =

Operand allowing to define several space-time excitations. Maybe by indicating one
vector assembled correspondent with a loading, is loads which will lead to calculation and to
assembly of a second member. The assembled vector can be associated a function
of temporal evolution or a constant multiplying coefficient.

The total loading is the sum of the loadings defined by all the occurrences of the key word
EXCIT (cf [§4.8.2]).
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4.8.1 Operands
VECT_ASSE/LOAD

/VECT_ASSE = vecti

Vector assembled correspondent with a loading (concept of the cham_no_DEPL_R type).



/COEF_MULT = Ci

Multiplicative coefficient of the vector assembled vecti.

/
FONC_MULT
= I

See [§4.8.2].


/CHARGE = chi

chi is the loading possibly comprising the evolution of a field of temperature
specified by the ième occurrence of EXCIT.

See [§4.8.2].

4.8.2 Operand
FONC_MULT

FONC_MULT = I

I is the multiplicative function of the time of the assembled vector or the loading specified with
ième occurrence of EXCIT.

The CH loading and boundary conditions for N occurrences of the key word factor EXCIT
are:

N
CH (T) = (T) CH
I
I
I =1

The fields of temperature are not multiplied by I in thermomechanical analysis.

Important remark:

The boundary conditions of the displacement type imposed not no one can be imposed
with an assembled vector or a load; it is then necessary to use it imperatively
diagram of Newmark.

4.8.3 Operands MULTI_APPUI/ACCE/QUICKLY/DEPL/DIRECTION/NODE/
GROUP_NO

In the case of an excitation multi-supports (MULT_APPUI = “OUI”), the other operands have
exactly same significance as in the key word factor EXCIT of the operator
DYNA_TRAN_MODAL [U4.53.21].

4.9 Key word
AMOR_MODAL

This key word makes it possible to take into account a damping equivalent to modal damping
broken up on a basis of modes precalculated in the form of concept of the mode_meca type. This
damping is taken overall into account in the dynamic equilibrium equation like one
correct force with the second member - CX &.
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4.9.1 Operands
MODE_MECA/AMOR_REDUIT/NB_MODE

MODE_MECA
= mode
AMOR_REDUIT = l_amor

NB_MODE = nbmode

The concept mode of the mode_meca type (entered by operand MODE_MECA) represents the base of
modes precalculated on which one breaks up modal damping. This base must
imperatively to have the same profile of classification as that of the dynamic system defined by
parameters of key word SOLVEUR [§4.10]. It be possible to truncate the modal base with one
a number of modes defined by NB_MODE. Failing this, one takes all the modes of the modal base.

Modal depreciation in reduced form is given in the form of a list of realities of which
the number of terms is lower or equal to the number of modes taken into account. If the number of
terms of the list is strictly lower, one extends this list with the value of its last term
until its size reaches the number of calculated modes.

4.10 Key word
SENSIBILITE


SENSIBILITE = _F (…)

Activate the calculation of derived from the field from displacement, speed and acceleration compared to one
significant parameter of the problem.

The document [U4.50.02] specifies the operation of the key word.

4.11 Key word
SOLVEUR

The syntax of this key word common to several commands is described in the document [U4.50.01].

4.12 Key word
INCREMENT

Key word factor defining the moments of calculation.

4.12.1 Operands LIST_INST/FONC_INST/NOT

·
For the diagrams of Newmark and Wilson:

/LIST_INST = l_temp

Concept lists realities of the listr8 type.

List realities defining the moments Ti of calculation of the solution


/FONC_INST = fonc

Concept of function the defining type of an unspecified variable with constant step
evolution in time

·
For the diagrams of the centered differences and with step in adaptive time:

/PAS = dt

Indicate the step of time used by the algorithm. This key word is obligatory for
diagram of the centered differences and for the diagram adaptive and nonavailable for
diagrams of Newmark and Wilson.

For the adaptive diagram, it indicates at the same time the step of initial time and the step of time
maximum used by the algorithm.
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This parameter must be sufficiently weak:

·
to allow the calculation of the static phases (which always use the step
maximum),
·
to start the algorithm correctly.

It must however be sufficiently high not to penalize the whole of calculation.

4.12.2 Operand PAS_CALCUL



PAS_CALCUL = ipas

Entirety usable for the diagrams Newmark and Wilson, and defining the periodicity of
calculation of the solution: one carries out calculation all the urgent ipas of the function of time
fonc or of the list of moments l_temp.

4.12.3 Operands INST_INIT/INST_FIN/NUME_FIN

For the diagrams of the centered differences and with step in adaptive time:


INST_INIT = Ti

In the event of recovery one uses key word ETAT_INIT [§4.7]: under this key word, the initial moment is recovered
with operand INST_INIT or taken equal to the last moment of filed preceding calculation.

Operand INST_INIT under INCREMENT must thus be used only if it did not begin again there of one
preceding calculation.


/INST_FIN = tf

Moment of end of transitory calculation. Obligatory for the diagrams of the differences
centered and with step of adaptive time.

/
NUME_FIN
=
nufin

Number of the moment of end of calculation in LIST_INST (only for
diagrams of Newmark and Wilson).

4.12.4 Operands VITE_MIN/COEF_MULT_PAS/COEF_DIV_PAS/PAS_LIMI_RELA/
NB_POIN_PERIODE/NMAX_ITER_PAS

These operands relate to only the diagram with step of adaptive time.

VITE_MIN
=/
“NORM”
[DEFAUT]
/
“MAXI”

Method of calculation the speed of reference used to evaluate the apparent frequency.

When the denominator of the apparent frequency (xn - xn-1) becomes weak, the frequency
connect can become very high, which leads to an unjustified refinement of the step of time.
To cure it, the algorithm uses the following criterion for each ddl I:

xi - xi
I
I
N
n-1
1
X
& - X

&
VI
F
N
N
=
- 1
T
min
APn

2

VI
T

min
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imin can be calculated in two ways different according to the value from VITE_MIN:

Max (xk
&
, xl

n+/
1 2 &n+/)

1 2

I
- 15
- 1
“NORM”: v min (T) = Max
, 10
ms
N
where K and L are the degrees of
100




freedom of comparable nature that the ddl I closest to I in classification (DX or DY or
DZ…).

VI (T


p)
I
- 15
- 1
“MAXI”: v
T) = Max
, 10
ms
min (N

for the ddl I.
<
0 T <t
100

p
N


Can be used if the order of magnitude speed does not vary too much in the course of time.

COEF_MULT_PAS = cmp

Coefficient of déraffinement of the step of time (>1) when the error is sufficiently weak:

0 7
. 5
T <
since more than 5 consecutive steps T
= min (cmp T, T
N
)
Nf
n+1
N
max
APn

with T
T
max =
initial

Its default value (cmp = 1.1) guarantees stability and precision, but it can in general be
increased (with more up to 1.3) to accelerate integration.

COEF_DIVI_PAS = cdp

Coefficient of refinement of the step of time (>1) when the error is higher than 1, that it
iteration count maximum (NMAX_ITER_PAS) is reached and only the step of times
minimal is not reached:

1
T
T >
, Niter < Niter
T > plr * T
T
N
N
=

Nf
max and
N
initial
N
APn
cdp

Its default value is 1.3334, that is to say a reduction of a factor 0,75.

PAS_LIMI_RELA = plr

Coefficient applied to the step of initial time to define the limit of refinement and thus the step of
minimal time:

T
= plr T
min
* initial

NB_POIN_PERIODE = NR

A number of points per apparent period. It is this parameter which fixes the precision of calculation. It must
to be at least equal to 20; its default value (50) guarantees a satisfactory precision (command
of 1à 2%) in the majority of the cases.

NMAX_ITER_PAS

A maximum number of reductions of the step of time per step of calculation:

if err > 1 and Niter < Niter max: tn = cdp * tn

It is by defect equal to 16, which limits the coefficient of reduction of the step to (1/1,33) 16 = 10­2 by
iteration. NMAX_ITER_PAS can be:

·
increased to allow the step time to fall in a more brutal way,
·
decreased if the step of time seems excessively refined.
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4.13 Operands
SENSIBLITE


SENSIBILITE =

Activate the calculation of derived from the field from displacement, speed and acceleration compared to
a significant parameter of the problem.
The document [U4.50.01] specifies the operation of the key word.

4.14 Key word
ARCHIVAGE

ARCHIVAGE =

Key word factor defining filing. In the absence of this key word factor, all steps of time
are filed.

Whatever the option of filing chosen, one files the last step of time and all the fields
associated to allow a possible continuation.

4.14.1 Operand LIST_ARCH

/LIST_ARCH = l_arch

List entireties defining the moments of calculation for which the solution must be filed
in the concept dyna_tran result.

4.14.2 Operand PAS_ARCH


/PAS_ARCH = ipa

Entirety defining the periodicity of filing of the solution of transitory calculation in the concept
dyna_trans result.

If ipa = 5 one files all the 5 steps of calculation.

This operand is obligatory for the step of adaptive time, if the key word factor ARCHIVAGE is
present. This entirety then makes it possible to calculate the interval between two moments of filing in the concept
result, equal to PAS_ARCH * PAS. With this convention, the step of filing is always higher or
equal to the maximum step used by calculation. For one moment of filing given, one seeks the moment of
calculation nearest, which will be the actually filed moment.

4.14.3 Operand CHAM_EXCLU

CHAM_EXCLU = (
I “DEPL”,







I “QUICKLY”,







I “ACCE”,






)

Allows to exclude filing from one or more fields among “DEPL”, “VITE” and “ACCE”.

This exclusion is ignored for the last moment of calculation: the three fields are necessary
for a POURSUITE.

4.15 Operand
TITER

TITER = title

Titrate structure of data result [U4.03.01].

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5 Concept
product

dyn is a produced concept of dyna_trans type which will contain from 1 to 3 fields with the nodes for
each filed moment.

These cham_no has as a reference symbol:

DEPL: displacement
VITE: speed
ACCE: acceleration

6 Phase
of execution

6.1 Classification

The operator checks that the assembled matrices are built on the same classification.

6.2 Continuation

In the event of continuation, one checks:

·
the existence in the concept of the dyna_trans type provided moment (or sequence number)
who is used as initial conditions,
·
for this moment of the existence of fields “DEPL”, “VITE” and “ACCE”.

6.3 Production run for the diagrams of the centered differences and
adaptive

The use of these diagrams imposes certain restrictions of use:

·
these two diagrams require the use of a matrix of diagonal mass. A test checks
that the matrix of mass was created with option “MASS_MECA_DIAG” of
CALC_MATR_ELEM. In addition, the matrix of mass must be stored in line of sky,
·
there should not be other boundary conditions only blocked ddls.
A test checks that there are no boundary conditions of the connections type between ddls.
It is not either possible to impose nonnull displacements via one
assembled vector,
·
for the diagram of the centered differences, one makes sure that the step of selected time checks them
stability conditions:
1
K
dt < 0 0
, 5/F
front F
II
= max

EC. and K and m
max
max
II
II
1 I
nddl 2
diagonal terms of
mii
matrices of stiffness and mass.

7 Bibliography

[1]
BATHE K.J. : Finite Element Procedures in Analysis engineering. Prentice-Hall, 1982.
[2]
LIGHT A.C.: Introduction of the explicit diagrams “centered differences” and “not of time
adaptive “in operator DYNA_LINE_TRAN of Code_Aster. Note EDF HP51/97/067/A
1997.
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