6. Transient response calculation on a physical basis#
The operator realizes the direct temporal integration of a transitory linear mechanical problem of the form:
where matrices \(\mathrm{M}\mathrm{,}\mathrm{C}\mathrm{,}\mathrm{K}\) are the real assembled matrices of the finite element problem (respectively) of mass, damping, and stiffness of the system.
The \({\alpha }_{i}\) are functions of time (cf. DEFI_FONCTION [U4.31.02]) and the \({\mathrm{F}}_{i}\) are assembled vectors resulting from imposed force loadings (cf. AFFE_CHAR_MECA [U4.44.01]); they can be provided directly in the form of assembled vectors or in the form of loads that will be assembled in the algorithm.
Solution \((\mathrm{X}\mathrm{,}\dot{\mathrm{X}}\mathrm{,}\ddot{\mathrm{X}})\) is calculated on a \({t}_{i}\) time discretization of the study interval specified by the user.
6.1. Operand MODELE#
◊ MODELE = mo
Name of the model whose elements are the subject of dynamic calculation.
This operand is mandatory when applying a charge-type excitation with the keyword EXCIT (cf. [§4.7]).
6.2. Operand CHAM_MATER#
◊ CHAM_MATER = chmat
Name of the material field assigned on the mo model, required when applying a charge-type excitation with the EXCIT keyword.
6.3. Operand CARA_ELEM#
◊ CARA_ELEM = character
Name of the characteristics of the elements of beams, shells, etc., required when applying a load-type excitation with the keyword EXCIT.
6.4. System matrices#
♦ MATR_MASS = m
Assembled matrix concept of type Matr_ass_ DEPL_R corresponding to the mass matrix of the system.
♦ MATR_RIGI = k
Assembled matrix concept of type Matr_asse_ DEPL_R corresponding to the stiffness matrix of the system.
◊ MATR_AMOR = c
Assembled matrix concept of type Matr_asse_ DEPL_R corresponding to the system’s damping matrix.
Notes:
The three matrices must be based on the same numbering and be built with the same storage mode. This is also true of a damping matrix constructed as a linear combination of stiffness and mass matrices by the Rayleigh method: using the full mass matrix matrix to build the damping matrix and the mass diagonal matrix (explicit schemes such as DIFF_CENTRE or ADAPT) for time integration can lead to numerical instability.
6.5. Integration diagrams. Keyword SCHEMA_TEMPS#
Under this keyword you can fill in an integration diagram with, possibly, its parameters. The available schemas are to be declared under the SCHEMA operand.
6.5.1. Operand SCHEMA#
|'NEWMARK'
Implicit integration diagram of type NEWMARK. This is the default schema for physical-based transient analysis.
The integration parameters \(\beta\) and \(\gamma\) can be specified:
◊ BETA = beta
Value of the \(\beta\) parameter for the NEWMARK method. By default \(\beta =0.25\).
◊ GAMMA = range
Value of the \(\gamma\) parameter for the NEWMARK method. By default \(\gamma =0.5\).
See [R5.05.02] for the choice of other values.
|'WILSON'
Implicit integration diagram of type WILSON. With this diagram we can fill in:
◊ THETA = th
Value of the \(\theta\) parameter for the WILSON method. By default \(\theta \mathrm{=}\mathrm{1,4}\).
This scheme should not be used when non-zero displacements are imposed using an assembled vector. See [R5.05.02].
| 'DIFF_CENTER'
Explicit integration diagram by centered differences. The use of this schema imposes some usage restrictions listed in [§6.14]. The theoretical description of the diagram is given in [R5.05.02].
| 'ADAPT_ORDER2'
Explicit integration diagram with adaptive time steps, variant of the centered differences schema. The use of this schema imposes some usage restrictions listed in [§6.14] (see [R5.05.02]).
Note:
You cannot use the**explicit schemes*( DIFF_CENTRE , ADAPT_ORDRE2 ) with the **explicit schemas (* , ) with **plate and shell elements.*
6.6. Keyword ETAT_INIT#
This feature allows a continuation of a transitory calculation, taking as initial state a result obtained by a previous calculation with DYNA_LINE_TRAN. It also allows you to define initial conditions such as fields at the nodes.
Note:
For higher-order schemas ( NEWMARK or WILSON ), initial acceleration ( acce_init ) plays an important role in initializing the schema.
6.6.1. Operands RESULTAT#
♦/RESULTAT = dy
Dyna_trans-type concept derived from a previous calculation with DYNA_LINE_TRAN, and defining the initial conditions for the new calculation.
6.6.2. Operands DEPL/VITE/ACCE#
/DEPL = do
Concept corresponding to the initial movements (field with nodes of magnitude DEPL_R).
Concept corresponding to initial speeds (field at knots of magnitude DEPL_R).
ACCE = AO
Concept corresponding to initial accelerations (field at nodes of magnitude DEPL_R).
If the keyword is present, the acceleration field entered is used to initialize the various integration schemes in time according to the algorithms described in document [R5.05.02].
If it is absent, an initial acceleration is calculated by the following formula:
\(\mathit{M.ao}\mathrm{=}\mathit{Fext}(t\mathrm{=}\text{to})–\mathit{C.vo}–\mathit{K.xo}\)
R Important note:
When the initial state of the dynamic system is defined by fields of DEPL, VITE, and/or ACCE, the components of these fields that were not explicitly filled in when the fields were created are considered null during the transient dynamic calculation.
6.6.3. Operands NUME_ORDRE/INST_INIT#
◊/NUME_ORDRE = knot
nuord refers to the archive number of the previous calculation to be extracted and taken as the initial state in the case of a recovery.
/INST_INIT = to
Instant of the previous calculation to be extracted and taken as the initial state in the case of a recovery.
In the absence of NUME_ORDRE and INST_INIT, the recovery time is taken to be equal to the last archived previous calculation time.
6.6.4. Operand CRITERE#
◊ CRITERE =
Indicates how precisely the search for the moment should be done:
“RELATIF”: search interval [(1-prec).instant, (1+prec).instant]
“ABSOLU”: search interval [instant-prec, instant+prec]
The default value for the search criteria is” RELATIF “.
6.6.5. Operand PRECISION#
◊ PRECISION =/1.E-06 [DEFAUT]
/prec [R]
Indicates how precisely the search for the moment should be done.
6.7. Keyword EXCIT#
◊ EXCIT =
Operand allowing to define several space-time excitations. Either by indicating an assembled vector corresponding to a loading, or loads that will lead to the calculation and assembly of a second member. The assembled vector may be associated with a time evolution function or a constant multiplier coefficient.
The total load is the sum of the loads defined by all the occurrences of the EXCIT keyword (see [§4.7.2]).
6.7.1. Operands VECT_ASSE/CHARGE#
♦/VECT_ASSE = vecti
Assembled vector corresponding to a loading (cham_no_ DEPL_R concept).
Multiplicative coefficient of the assembled vector vecti.
/FONC_MULT = \({\alpha }_{i}\)
See [§4.7.2].
/CHARGE = chi
chi is the load possibly including the evolution of a temperature field specified by the \(i\) th occurrence of EXCIT.
See [§4.7.2].
6.7.2. Operand FONC_MULT#
◊ FONC_MULT = :math:`{\alpha }_{i}`
\({\alpha }_{i}\) is the function of the multiplicative time of the assembled vector or of the load specified in the \(i\) th occurrence of EXCIT.
The loading \(\text{ch}\) and the boundary conditions for \(n\) occurrences of the keyword factor EXCIT are:
The temperature field (s) are not multiplied by \({\alpha }_{i}\) in thermomechanical analysis.
Important note:
The boundary conditions of the non-zero imposed displacement type can be imposed with an assembled vector or a load **; it is then imperative to use the Newmark diagram.*
6.7.3. Operands MULTI_APPUI/ACCE/VITE/DEPL//DIRECTION/GROUP_NO/MODE_STAT#
In the case of multi-press excitation (MULT_APPUI = “OUI”), the other operands have exactly the same meaning as in the keyword factor EXCIT of the operator DYNA_TRAN_MODAL [U4.53.21].
6.8. Keyword EXCIT_RESU#
Keyword used to define several loading complements in the form of a transitory evolution of vectors assembled second members.
6.9. Keyword AMOR_MODAL#
This keyword makes it possible to take into account depreciation equivalent to modal damping broken down on a basis of pre-calculated modes in the form of a mode_meca concept. This damping is generally taken into account in the dynamic equilibrium equation as a force correcting the second member \(-C\dot{X}\).
Note:
This way of introducing modal damping into a problem calculated on a physical basis can reduce the stability properties of patterns over time. In particular for the integration diagram “NEWMARK”, it can lead to a reduction in the time step compared to the time step without damping in order to avoid numerical differences.
6.9.1. Operands MODE_MECA/AMOR_REDUIT/NB_MODE#
♦ MODE_MECA = fashion
♦ AMOR_REDUIT = l_love
◊ NB_MODE = nbmode
The mode_meca concept mode (entered by operand MODE_MECA) represents the pre-calculated mode base on which modal damping is broken down. This database must have the same numbering profile as that of the dynamic system defined by the parameters of the keyword SOLVEUR [§4.11]. It is possible to truncate the modal base to a number of modes defined by NB_MODE. Otherwise, we take all the modes from the modal base.
Modal depreciation in reduced form is given in the form of a list of reals whose number of terms is less than or equal to the number of modes taken into account. If the number of terms in the list is strictly smaller, this list is extended with the value of its last term until its size reaches the number of calculated modes.
6.10. Keyword ENERGIE#
◊ ENERGIE = _F (...)
This keyword allows you to activate the calculation of the energy balance, its display during calculation and its storage in the name table PARA_CALC. The energy balance can be retrieved from this table using the RECU_TABLE [U4.71.02] command.
6.11. Keyword INCREMENT#
Keyword factor defining the moments of calculation.
6.11.1. Operands LIST_INST/PAS#
For Newmark and Wilson diagrams:
♦/LIST_INST = l_temp
Listr8 real estate list concept.
List of reals defining the moments \({t}_{i}\) for calculating the solution
For patterns of centered differences and with adaptive time steps:
/PAS = dt
Refers to the time step used by the algorithm. This keyword is mandatory for the centered differences schema and for the adaptive schema and not available for the Newmark and Wilson schemas.
For the adaptive schema, it refers to both the initial time step and the maximum time step used by the algorithm.
This parameter should be low enough:
to allow the calculation of static phases (which always use the maximum step),
to start the algorithm correctly.
However, it must be high enough so as not to penalize the entire calculation.
6.11.2. Operands INST_INIT/INST_FIN/NUME_FIN#
For patterns of centered differences and with adaptive time steps:
◊ INST_INIT = you
In case of resumption, the keyword ETAT_INIT [§4.6] is used: under this keyword, the initial instant is retrieved with the operand INST_INIT or taken equal to the last archived previous calculation moment.
The INST_INIT operand under INCREMENT should therefore only be used if there is no repeat of a previous calculation.
Time at which the transitory calculation ends. Mandatory for centered difference schemes and with adaptive time steps.
/NUME_FIN = nufin
End of calculation time number in LIST_INST (only for Newmark and Wilson diagrams).
If INST_INIT is not present, the initial instant is zero.
Operands VITE_MIN/COEF_MULT_PAS/COEF_DIV_PAS/PAS_LIMI_RELA//NB_POIN_PERIODE/NMAX_ITER_PAS/PAS_MINI ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~
These operands only concern the adaptive time step schema.
◊ VITE_MIN =/'NORM' [DEFAUT]
/”MAXI”
Reference speed calculation method used to assess apparent frequency.
When the denominator of apparent frequency \(({x}_{n}\mathrm{-}{x}_{n\mathrm{-}1})\) becomes low, the apparent frequency can become very high, leading to unwarranted refinement of the time step. To remedy this, the algorithm uses the following criterion for each degree of freedom \(i\):
\({v}_{\mathit{min}}^{i}\) can be calculated in two different ways depending on the value of VITE_MIN:
“ NORM “: \({v}_{\mathrm{min}}^{i}({t}_{n})=\mathrm{Max}(\frac{\mathrm{Max}({\dot{x}}_{n+1/2}^{k},{\dot{x}}_{n+1/2}^{l})}{100},{10}^{-15}{\mathrm{ms}}^{-1})\) where \(k\) and \(l\) are the degrees of freedom of the same nature as the degree of freedom \(i\) closest to \(i\) in the numbering (\(\mathit{DX}\) or \(\mathit{DY}\) or \(\mathit{DZ}\)…).
“ MAXI “: \({v}_{\mathrm{min}}^{i}({t}_{n})=\underset{0<{t}_{p}<{t}_{n}}{\mathrm{Max}}(\frac{\mid {v}^{i}({t}_{p})\mid }{100},{10}^{-15}{\mathrm{ms}}^{-1})\) for the degree of freedom \(i\).
Can be used if the order of magnitude of the speed does not vary too much over time.
◊ COEF_MULT_PAS = cmp
Time-step deraffination coefficient (\(>1\)) when the error is low enough:
\(\Delta {t}_{n}<\frac{0.75}{{\mathit{Nf}}_{{\mathit{AP}}_{n}}}\) for more than 5 consecutive steps \(\mathrm{\Rightarrow }\Delta {t}_{n+1}\mathrm{=}\mathit{min}(\mathit{cmp}\Delta {t}_{n},\Delta {t}_{\mathit{max}})\)
with \(\Delta {t}_{\mathit{max}}\mathrm{=}\Delta {t}_{\mathit{initial}}\)
Its default value (\(\mathit{cmp}\mathrm{=}1.1\)) guarantees stability and precision, but it can generally be increased (up to \(1.3\) at most) to speed up integration.
◊ COEF_DIVI_PAS = cdp
Time-step refinement coefficient (\(>1\)) when the error is greater than 1, the maximum number of iterations (NMAX_ITER_PAS) is not reached, and the minimum time step is not reached:
\(\Delta {t}_{n}>\frac{1}{{\mathit{Nf}}_{{\mathit{AP}}_{n}}},\mathit{Niter}<{\mathit{Niter}}_{\mathit{max}}\) and \(\Delta {t}_{n}>\mathit{plr}\mathrm{\ast }\Delta {t}_{\mathit{initial}}\mathrm{\Rightarrow }\Delta {t}_{n}\mathrm{=}\frac{\Delta {t}_{n}}{\mathit{cdp}}\)
Its value by default is \(1.3334\), which is a reduction by a factor of \(\mathrm{0,75}\).
◊ PAS_LIMI_RELA = PLR
Coefficient applied to the initial time step to define the refinement limit and therefore the minimum time step:
\(\Delta {t}_{\mathit{min}}\mathrm{=}\mathit{plr}\mathrm{\ast }\Delta {t}_{\mathit{initial}}\)
◊ NB_POIN_PERIODE = N
Number of points per apparent period. It is this parameter that determines the accuracy of the calculation. It must be at least equal to 20; its default value (50) guarantees satisfactory precision (of the order of 1 to 2%) in most cases.
◊ NMAX_ITER_PAS
Maximum number of time step reductions per calculation step:
if \(\mathit{err}>1\) and \({N}_{\mathit{iter}}<{N}_{\mathit{iter}}\mathit{max}\): \(\Delta {t}_{n}\mathrm{=}\mathit{cdp}\mathrm{\ast }\Delta {t}_{n}\)
It is by default equal to 16, which limits the step reduction coefficient to \({(1\mathrm{/}\mathrm{1,33})}^{16}\mathrm{=}{10}^{\mathrm{-}2}\) per iteration. NMAX_ITER_PAS maybe:
increased to allow the time step to fall more sharply,
decreased if the time step seems excessively refined.
◊ PAS_MINI = dtmin
Minimum value of the time step. If the conditions for reducing the time step are met, the current time step can then decrease up to this limit value.
If the user does not give a value to this optional parameter, then the code will calculate the minimum time step starting from PAS_LIMI_RELA.
6.12. Keyword ARCHIVAGE#
◊ ARCHIVAGE =
Keyword factor defining archiving. In the absence of this keyword factor, all time steps are archived.
Regardless of the archiving option chosen, we archive the last step of time and all associated fields to allow a possible continuation.
6.12.1. Operands LIST_INST/INST#
◊/LIST_INST = list
List of reals defining the calculation times for which the solution must be archived in the dyna_tran result concept.
◊/INST
Computing moments for which the solution must be archived in the resultdyna_tran concept.
6.12.2. Operand PAS_ARCH#
/PAS_ARCH = ipa
Integer defining the archiving periodicity of the transitory calculation solution in the dyna_trans result concept.
If ipa = 5 we archive every 5 calculation steps.
Note:
PAS_ARCHet LIST_INST/INSTsont both taken into account in the keyword ARCHIVAGE, you must be careful to set PAS_ARCHà a value greater than the number of calculation steps if you want to drive only by LIST_INST/INST.
6.12.3. Operand CRITERE#
◊ CRITERE =
Indicates how precisely the search for the moment to be archived must be done:
“RELATIF”: search interval [(1-prec).instant, (1+prec).instant]
“ABSOLU”: search interval [instant-prec, instant+prec]
The default value for the search criteria is” RELATIF “.
6.12.4. Operand PRECISION#
◊ PRECISION = /1.E-06 [DEFAUT]
/prec [R]
Indicates how precisely the search for the moment to be archived should be done.
6.12.5. Operand CHAM_EXCLU#
◊ CHAM_EXCLU = (I 'DEPL',
I “VITE”, I “ACCE”, )
Allows you to exclude the archiving of one or more fields among “DEPL”, “VITE” and “ACCE”.
This exclusion is ignored for the last moment of calculation: all three fields are required for a POURSUITE.
6.13. Keyword OBSERVATION#
◊ OBSERVATION = _F ()
◊ TITRE = title
This keyword makes it possible to post-process certain fields at the nodes (DEPL, VITE, ACCE) on parts of the model at times of a list (called observation) that is generally less refined than the list of archived moments defined in the keyword ARCHIVAGE (where all the fields on the entire model are stored). It is mainly used to save on storage, but also to evaluate fields on reduced parts of the mesh, without the need to post-process after the calculation.
This keyword is repeatable and allows the creation of an observation table that can be extracted using the RECU_TABLE command. Cf. [U4.51.03] (3.22.8) for detailed table content.
You can only use a maximum of 99 occurrences of the OBSERVATIONau keyword.
It is possible to name an occurrence of the observation (column NOM_OBSERVATION) using the TITRE keyword. If not used, column NOM_OBSERVATIONcontient OBSERVATION_xxavec xx varies from 1 to 99.
6.13.1. Operands LIST_INST/INST/PAS_OBSE/OBSE_ETAT_INIT#
◊/LIST_INST = list_r8
/INST = l_r8 /PAS_OBSE = not /OBSE_ETAT_INIT = /” OUI “, [DEFAUT] /” NON “
Cf. [U4.51.03] for detailed syntax.
6.13.2. Operands PRECISION/CRITERE#
◊ PRECISION = prec
◊ CRITERE = /' ABSOLU '
Cf. [U4.71.00] for detailed syntax.
6.13.3. Operands NOM_CHAM/NOM_CMP#
♦ NOM_CHAM = (I 'DEPL',
I “VITE”, I “ACCE”, )
◊ NOM_CMP = nomcmp
Cf. [U4.51.03] for detailed syntax.
6.13.4. Operands TOUT/NOEUD/GROUP_NOEUD/MAILLE/GROUP_MA#
◊/TOUT = 'NON' [DEFAUT]
“OUI” /NOEUD = no [no] /GROUP_NO = grno [grno] /MAILLE = lma [ma] /GROUP_MA = lgrma [grma]
Cf. [U4.51.03] for detailed syntax.
6.13.5. Operands EVAL_CMP/FORMULE/EVAL_CHAM#
◊ EVAL_CMP = /' VALE ', [DEFAUT]
/” FORMULE “
◊ FORMULE = form [aster_formula]
◊ EVAL_CHAM = /” VALE “, [DEFAUT]
/” MIN “, /” MAX “, /” MOY “,
Cf. [U4.51.03] for detailed syntax.
6.14. Control phase#
The use of centered and adaptive difference schemes imposes certain restrictions on use:
these two schemes require the use of a diagonal mass matrix. A test verifies that the mass matrix was created with the CALC_MATR_ELEM “MASS_MECA_DIAG” option. On the other hand, the mass matrix must be stored in a sky line,
there should be no other boundary conditions other than blocked degrees of freedom.
A test verifies that there are no boundary conditions such as links between degrees of freedom.
It is also not possible to impose non-zero displacements by means of an assembled vector,
for the diagram of centered differences, we ensure that the time step chosen verifies the stability conditions:
\(\mathit{dt}<\mathrm{0,05}\mathrm{/}{f}_{\mathit{max}}\) with \({f}_{\mathit{max}}\mathrm{=}\underset{1\mathrm{\le }i\mathrm{\le }\mathit{nddl}}{\mathit{max}}(\frac{1}{2\pi }\sqrt{\frac{{k}_{\mathit{ii}}}{{m}_{\mathit{ii}}}})\) and \({k}_{\mathit{ii}}\) and \({m}_{\mathit{ii}}\) diagonal terms of the stiffness and mass matrices.