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Titrate:
Operator THER_NON_LINE


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

Handbook of Utilization
U4.5- booklet: Methods of resolution
Document: U4.54.02

Operator THER_NON_LINE

1 Goal

To calculate the thermal response with nonlinearities of behaviors and boundary conditions.

The equation of heat is solved in evolutionary mode (except if no list of moment is provided, only
the stationary regime is then calculated). Nonthe linearities come is behavior
(characteristics of material depend on the temperature), that is to say boundary conditions
(radiation in infinite medium, nonlinear flow). A formulation in enthalpy was selected in order to
to more easily take into account the phase shifts of material.

Evolutionary calculation can be initialized, at the first moment in three different ways (key word
TEMP_INIT):

· by a constant temperature,
· by a field of temperature, definite, or extracted as a preliminary from a preceding calculation,
· by a stationary calculation.

This operator also allows to solve the problems of drying (nonlinear) while solving
the equation of the heat where the water C concentration is comparable at a temperature, for
resolution. Thermal conductivity is in this case the coefficient of diffusion, nonlinear out of C and
function, possibly, of a temperature calculated as a preliminary.

To model the hydration of the concrete, the operator also allows to add a term function source
variable of hydration to the equation of heat. This term is then given by an equation
of evolution where the temperature intervenes.

The concept produced by operator THER_NON_LINE is of evol_ther type as for an analysis
linear by THER_LINEAIRE [U4.54.01].

When a calculation of sensitivity of the result compared to a parameter is required, there is production
of as many structures of data of the evol_ther type of necessary parameters.
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2 Syntax
temper
[evol_ther]
=
THER_NON_LINE


(reuse = temper,



MODELE
=
Mo,








[model]


CHAM_MATER
=
chmat,
[cham_mater]


EXCIT =_F (








CHARGE
=
tank,
[load]







FONC_MULT
=
fonc,
[function]






),


TEMP_INIT =_F (







/STATIONARY = “YES”,
[DEFAUT]
/
VALE =
tinit,
[R]
/
CHAM_NO =
tinit,
[cham_no_TEMP_R]








/
EVOL_THER
=
temp,
[evol_ther]









NUME_INIT
= nuini_evol, [I]







),


SENSIBILITE =_F (
.
.
.
to see
[U4.50.02]
.
.
.







),

INCREMENT =_F (








LIST_INST
=
litps,
[listr8]








NUME_INIT
=/0,
/
nuini,
[I]








NUME_FIN
=
nufin,
[I]







),


COMP_THER_NL =_F (







RELATION
=
/
“THER_NL”,
[DEFAUT]
/
“THER_HYDR”, [txm]
/
“SECH_GRANGER”,
[txm]
/
“SECH_MENSI”,
[txm]
/
“SECH_BAZANT”,
[txm]
/
“SECH_NAPPE”,
[txm]







/ALL = “YES”,
[txm]








/ |
GROUP_MA = l_grmail,
[l_gr_ma]









|
MAILLE
=
l_maille,
[l_ma]







),


EVOL_THER_SECH = resuther,
[evol_ther]
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NEWTON =_F (








REAC_ITER
=/0,
[DEFAUT]














/it,


[I]








RESI_LINE_RELA=/1.E-3, [DEFECT]
/reslin, [R]








ITER_LINE_MAXI=/0,
[DEFAUT]














/iterl,

[R]







),


CONVERGENCE=_F (








RESI_GLOB_RELA=/1.E-6, [DEFECT]














/testr,

[R]








RESI_GLOB_MAXI=/testl, [R]








ITER_GLOB_MAXI=
/10, [DEFAUT]














/iterl,

[R]







),


PARM_THETA
=
/
theta,
[R]









/0.57,





[DEFAUT]


SOLVEUR =_F ([U4.50.01])


ARCHIVAGE
=_F
(/LIST_ARCH
= l_arch
, [listis]









/PAS_ARCH = ipas, [I]









/LIST_INST
= l_inst
, [listr8]









/INST = inst, [R]









PRECISION
=/
10.- 3, [DEFAUT]
/
prec
[R]









CRITERE =/
“RELATIF”,
[DEFAUT]
/
“ABSOLU”,









CHAM_EXCLU
= l_cham
, [l_Kn]








),


OPTION
=
| “FLUX_ELGA_TEMP”,



[l_Kn]









| “FLUX_ELNO_TEMP”,


TITER
= title,
[l_Kn]

);

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3 Operands

3.1 Operand
MODELE

MODELE = Mo

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

3.2 Operand
CHAM_MATER

CHAM_MATER = chmat

Name of the affected material field on the model.

3.3
Key word EXCIT

EXCIT
=

Key word factor allowing to define several loadings. For each occurrence of the key word
factor, one defines a load possibly multiplied by a function of time.

3.3.1 Operand
CHARGE



CHARGE = tank

Concept of the type charges produces by AFFE_CHAR_THER or AFFE_CHAR_THER_F
[U4.44.02].

Important remark:

For each occurrence of the key word factor EXCIT the various concepts tank
used must be built on the same model Mo.

3.3.2 Operand
FONC_MULT



FONC_MULT = fonc

Multiplicative coefficient function of the time (concept of the function type) applied to
charge.

Important remark:

The concomitant use of FONC_MULT with a load containing of
thermal loadings depending on the temperature is prohibited; i.e.
for loadings of the type ECHANGE_, RAYONNEMENT or FLUNL.

3.4 Word
key
TEMP_INIT

TEMP_INIT = litps

Allows to define the initial field from which evolutionary calculation is carried out. The initial field is
affected of the sequence number 0 and the initial moment takes as value the first reality of the list
from moment litps.

Note:

If key word TEMP_INIT misses, one carries out only stationary calculation at the moment
defined under key word INCREMENT.
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3.4.1 Operand
STATIONNAIRE

/
STATIONNAIRE
=
“OUI”

The initial value is then the result of a preliminary stationary calculation. This calculation takes
in account boundary conditions defined under key word CHARGE.

3.4.2 Operand
VALE

/
VALE
=
tinit

The initial value of temperature is taken constant on all the structure.

3.4.3 Operand
CHAM_NO

/
CHAM_NO
=
tinit

The initial value is defined by a cham_no_TEMP_R (result of the operators
AFFE_CHAM_NO [U4.44.11] or RECU_CHAMP [U4.71.01]).

3.4.4 Operand
EVOL_THER



/
EVOL_THER = temp

The initial value is extracted from a structure of data of the evol_ther type.

3.4.5 Operand
NUME_INIT




NUME_INIT = nuini_evol

Sequence number of the field to be extracted from this structure of data.
The initial field is stored in the structure of data temper result under
sequence number 0.

Note:

Attention, it acts of the sequence number in the structure of data read in recovery by
preceding key word EVOL_THER. If this structure of data were calculated with
a list of moments different from that used under the key word factor INCREMENT of
the current resolution, it is imperative to inform NUME_INIT under INCREMENT,
the same value of sequence number corresponding to physical moments
different. If the two lists of moments are identical, one can
to exempt to inform the same NUME_INIT twice, under ETAT_INIT and under
INCREMENT.

3.5 Word
key
SENSIBILITE

SENSIBILITE = sensitive parameter list

Activate the calculation of derived from the field of temperature compared to a significant parameter
problem.

The document [U4.50.01] specifies the operation of the key word.

3.6 Word
key
INCREMENT

INCREMENT =

Allows to define the moments of calculation which determine the intervals of time taken to integrate
the differential equation.

Note:

If key word INCREMENT misses, one creates a list of moments reduced to the only reality 0. and
a stationary calculation is carried out.
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3.6.1 Operand
LIST_INST



LIST_INST = litps

List moments produced by DEFI_LIST_REEL [U4.34.01].

3.6.2 Operand
NUME_INIT



NUME_INIT =/0
/
nuini

Index of the moment of starting calculation in the list litps.

If NUME_INIT misses and if evol_ther is present under TEMP_INIT, then
nuini = nuini_evol.

3.6.3 Operand
NUME_FIN



NUME_FIN = nufin

Index of the moment of final calculation in the list litps.

The moments of calculation are those defined in the concept litps taken between the nuini and it
nufin number of moment. Thus the first step of time is defined between the moment
correspondent with nuini and that corresponding to nuini + 1. Calculation, stationary,
when it is asked, is made at the moment corresponding to nuini.

3.7 Word
key
COMP_THER_NL

COMP_THER_NL
=

The resolution of drying was added in Code_Aster because of analogy of the equations of
the thermics and of drying. That supposes to assimilate the variable of calculation of drying,
water concentration, with a variable of the type “TEMP” during the resolution.
By defect, the resolution carried out will be nonlinear thermics. This key word factor allows
thus to distinguish the resolution of drying from thermics. Moreover, the equation of drying is
characterized by a coefficient of diffusion which can be expressed in various forms. This key word
factor also makes it possible to choose one of the equations of the drying, defined by the expression of sound
coefficient of diffusion, available in Aster. To carry out a nonlinear calculation of thermics,
this key word becomes optional, and the concept of behavior is transparent for the user.

Note:

If key word COMP_THER_NL misses, a nonlinear calculation of thermics is carried out
“standard”.

3.7.1 Operand
RELATION

RELATION: /

“THER_NL”
[DEFAUT]
/

“THER_HYDR”
/

“SECH_GRANGER”
/

“SECH_MENSI”
/

“SECH_BAZANT”
/

“SECH_NAPPE”

The syntax and the processing of this key word are similar to the use of the key words
COMP_INCR and COMP_ELAS of operator STAT_NON_LINE.

/
“THER_NL”

Standard nonlinear thermal resolution.

Supported modelings:

· continuous mediums 3D: 3D
· continuous mediums 2D: 2D, AXIS
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/
“THER_HYDR”

Resolution of the equation of heat with an additional source term: Q &


Q is the heat of hydration, presumedly constant. The variable of hydration is
solution of the nonlinear law of evolution, solved simultaneously with the problem of thermics:

&
E
= () -
With
E
RT

One will refer to the documentation of operator DEFI_MATERIAU for the significance of
various parameters.

Supported modelings:

· continuous mediums 3D: 3D
· continuous mediums 2D: 2D, AXIS

/
“SECH_GRANGER”

C
Resolution of the drying characterized by the equation
-
[
Div (
D C, T) C]

= 0
T
It is the equation of nonlinear heat where the variable of drying C holds the role of
temperature. The choice of the relation of behavior makes it possible to define the coefficient of
diffusion
(
D C, T) according to various usual forms of the literature. The formulation of
Granger of the coefficient of diffusion is given by the expression:

T
Q 1
1
D C T = A
BC
S
(,)
exp (
)
exp-
-

T
R
0
T T0



One will refer to the documentation of operator DEFI_MATERIAU for the significance of
various parameters. In the case of the use of this law SECH_GRANGER, it is
necessary to ensure itself of coherence enters the material used and the law of behavior:
i.e. that key word SECH_GRANGER indeed was indicated in DEFI_MATERIAU for
the material used.

Supported modelings:

· continuous mediums 3D: 3D
· continuous mediums 2D: 2D, AXIS

As the resolution of drying is carried out by an operator of thermics, them
supported modelings are thermal modelings, but which do not have whereas one
conceptual value of a geometrical nature.

/
“SECH_MENSI”

Resolution of the drying characterized by the law of MENSI.
In the case of the use of this law SECH_MENSI, it is necessary to be ensured of
coherence enters the material used and the law of behavior: i.e. that the key word
SECH_MENSI indeed was indicated in DEFI_MATERIAU for material used.
Supported modelings: analog with SECH_GRANGER.

/
“SECH_BAZANT”

Resolution of the drying characterized by the law of BAZANT.
In the case of the use of this law SECH_BAZANT, it is necessary to be ensured of
coherence enters the material used and the law of behavior: i.e. that the key word
SECH_BAZANT indeed was indicated in DEFI_MATERIAU for material used.
Supported modelings: analog with SECH_GRANGER.
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/
“SECH_NAPPE”

Resolution of drying with a coefficient of diffusion defined by a Aster tablecloth.
In the case of the use of this law SECH_NAPPE, it is necessary to be ensured of
coherence enters the material used and the law of behavior: i.e. that the key word
SECH_NAPPE indeed was indicated in DEFI_MATERIAU for material used.
Supported modelings: analog with SECH_GRANGER.

3.7.2 Operands
ALL/GROUP_MA/MESH

/ALL = “YES”


/
GROUP_MA = l_grmail


/
MAILLE =
l_maille

Specify the meshs to which the relation of behavior is applied. In way
analog with mechanics, one can use several different laws of drying, applied
with groups of complementary meshs. On the other hand, thermal cannot be it
mixed with drying. Behavior “THER_NL” is necessarily applied to all
grid, option TOUT: “OUI”, default option, which is in fact, in the general case,
transparent for the user.

3.8 Operand
EVOL_THER_SECH

EVOL_THER_SECH=
resuther

This operand is specific to the resolution of drying. Drying is solved after a calculation
preliminary thermics in the general case, (calculation not coupled but chained thermal/drying), it
thermal field intervening like auxiliary variable, allowing to calculate the coefficient of
diffusion of certain laws. It is an input datum of the calculation of the drying, which must be one
structure of data of the evol_ther type. This key word is obligatory only for the laws
“SECH_GRANGER” and “SECH_NAPPE”, whose coefficient of diffusion depends on the temperature.
The structure of data of here well informed thermal evolution will have been obtained by an execution
the preceding one of an operator of thermics, linear or not.

3.9 Word
key
NEWTON

NEWTON =

Specify the characteristics of the method of resolution of the nonlinear problem (method of
NEWTON-RAPHSON).

3.9.1 Operand
REAC_ITER



REAC_ITER
=/0 [DEFAUT]








/it

The matrix used for the total iterations of the method is the tangent matrix which
is revalued at the beginning of each increment of time and all the it iterations of
NEWTON for an increment of time given (precisely to the iterations of number it,
2it, 3it…). Thus with the first iteration of NEWTON, one does not reassemble the matrix
tangent that if it is worth 1: if not one keeps the matrix used in the phase of
prediction. By convention if it is worth 0 the matrix is not revalued during all the step
time.
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3.9.2 Operand
RESI_LINE_RELA/ITER_LINE_MAXI




RESI_LINE_RELA =/1.E-3
[DEFAUT]









/reslin



ITER_LINE_MAXI
=
/0 [DEFAUT]









/itlin

In fact the parameters of linear search make it possible to ensure the best
convergence of the method of NEWTON (Cf. [R5.03.01] for more details). One gives
the maximum iteration count itelin to carry out (the default value 0 indicates that
one does not make linear search) and the precision reslin to reach to realize
convergence of linear search.

Note:

It is not necessary to specify a precision very nor an iteration count
raised, practice showing that 2 or 3 iterations of linear search are
sufficient. One can thus be satisfied to ask for 3 iterations with
precision by defect.

3.10 Word
key
CONVERGENCE

CONVERGENCE:

Allows to define the values associated with the criteria with convergence:

If none of the two operands following is present, then all occurs like if:
RESI_GLOB_RELA = 1.E-6.

3.10.1 Operand RESI_GLOB_RELA



RESI_GLOB_RELA =/1.e-6








/testr

The algorithm continues the external iterations as long as the relative residue is higher than
testr.
1/2
1/2









(2
F N)

(S 2
I
I)
testr


>
I =1,…, Nb ddl

i=1,…, Nb ddl

where Fi indicates the residue and If the thermal loading, index N indicates the nth one
iteration.

3.10.2 Operand RESI_GLOB_MAXI



RESI_GLOB_MAXI =/
1.e-3









/testl

The algorithm continues the external iterations as long as the absolute residue is higher than
testl.
N
max
I
F > testl
I =1,…, Nb ddl

where Fi indicates the residue, index N indicates the nth iteration.

3.10.3 Operand ITER_GLOB_MAXI



ITER_GLOB_MAXI =/10








/iterl

The algorithm continues the iterations as long as their number is lower than iterl.
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3.11 Operand
PARM_THETA

PARM_THETA = theta

The argument theta is the parameter of the théta-method applied to the evolutionary problem. It must
to lie between 0 (explicit method) and 1 (completely implicit method). In the absence, word
key, the value used is theta=0.57, a little higher than theta=0.5 corresponds to the diagram
of Crank-Nicholson. The incidence of the choice of theta on the stability of the method is detailed
in [R5.02.02].

3.12 Word
key
SOLVEUR

SOLVEUR =

This key word factor is optional: it makes it possible to choose another solvor of resolution of system.

This operand is common to the whole of the total commands [U4.50.01].

3.13 Word
key
ARCHIVAGE

ARCHIVAGE
=

This key word is optional: by defect, the whole of the computed fields for all the calculated steps
is filed in the concept result resulting from the command. It is used for tocker certain numbers
of command in a structure of data result and/or to exclude from storage certain fields.

Note:

In the event of stop of calculation for lack of time CPU, steps of time previously
calculated are backed up in the base.

3.13.1 Operand LIST_ARCH

Concept of the type lists of entirety created by command DEFI_LIST_ENTI [U4.34.02] describing the list
sequence numbers having to be stored in the structure of data result.

3.13.2 Operand PAS_ARCH

Whole value giving the value of step of filing: one will store the multiple sequence numbers of
value ipas as well as the last actually calculated sequence number.

3.13.3 Operand CHAM_EXCLU

List texts indicating the fields excluded from filing. The list of the possible fields is
described in the documents on the concepts result [U5.01].

3.14 Operand
OPTION

OPTION = “FLUX_ELGA_TEMP”

This option carries out the calculation of the heat flow at the points of integration from
temperature.

OPTION = “FLUX_ELNO_TEMP”

This option carries out the calculation of the heat flow to the nodes starting from the temperature. Calculation
precondition of “FLUX_ELGA_TEMP” is not obligatory.

3.15 Operand
TITER

TITER
=
titrate

Titrate that one wants to give to the result stored in temper, structure of data of the type
evol_ther [U4.03.01].
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4 Modeling

The problems of nonlinear thermics can be dealt with with models using the elements
stop 3D, 2D or AXIS described in the documents [U3.22.01], [U3.23.01] and [U3.23.02] and [U3.24.01].

5 Example

One defined below the principal commands used to carry out a calculation of thermics
non-linear transient. The example indicates how to continue calculation by enriching the concept
result and how to specify the “initial” field.

LR8 = DEFI_LIST_REEL (BEGINNING = 0.D0,








INTERVALLE =_F (
JUSQU_A=5.e-3, NOMBRE= 10),








INTERVALLE =_F (
JUSQU_A=5.e-2, NOMBRE= 9),








INTERVALLE =_F (
JUSQU_A=4.e-0, NOMBRE= 79),








INTERVALLE =_F (
JUSQU_A=6.e-0, NOMBRE= 20),








)

conduc =
DEFI_FONCTION (
NOM_PARA
=
“TEMP”,
VALE =_F (


0.0
,
210.0,













660.0, 210.0,

660.01,

95.0,
1200.00,

95.0
)
PROL_DROIT
=
“LINEAIRE”,
PROL_GAUCHE =
“LINEAIRE”,








)

enthal =
DEFI_FONCTION
(
NOM_PARA
=
“TEMP”,
VALE =_F (



0.0
,
0.0,


660.0
,
1.980E9,

660.01,
3.060E9,
1200.00,
4.451E9
)
PROL_DROIT
=
“LINEAIRE”,
PROL_GAUCHE =
“LINEAIRE”,








)

aluminum = DEFI_MATERIAU
(THER_NL =_F (LAMBDA
= conduc,
BETA =
enthal
)
)

temple
= THER_NON_LINE (MODELE=moth, CHAM_MATER=chmat,
TEMP_INIT
=_F (VALE
=
20.0
),
INCREMENT
=_F (LIST_INST
=
lr8),
EXCIT
=_F (CHARGE
=
chth
),
CONVERGENCE =_F (RESI_GLOB_RELA
=1.E-6,

ITER_GLOB_MAXI
=10
),








)

temple = THER_NON_LINE
(reuse = temple,
MODELE=moth
,
CHAM_MATER=chmat,
TEMP_INIT
=_F (EVOL_THER
=
temple,
NUME_INIT
=
20),
INCREMENT
=_F (LIST_INST
=
lr8),
EXCIT
=_F (CHARGE
=
chth
),
CONVERGENCE =_F (RESI_GLOB_RELA
=
1.E-6,

ITER_GLOB_MAXI=
10)








)


FIN ();
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