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Functionsand formulas
code_aster, salome_meca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html)
Outline
1. Array and parameters
2. Functions
3. Formulas
GNU FDL licence | code_aster, salome_meca course material
3
1. Array and parameters
Array – A list of real numbers
GNU FDL licence | code_aster, salome_meca course material
Usages:
Time discretization of non-linear or dynamic calculation, function definition, …
Define by command DEFI_LIST_REEL
Example: ListR = DEFI_LIST_REEL(VALE=(1.,3.,5.,7.,9.,11.))
ListR = DEFI_LIST_REEL(VALE=range(1.,13.,2.))
ListR = DEFI_LIST_REEL ( DEBUT =1., INTERVALLE = ( _F ( JUSQU_A = 5., NOMBRE = 2,),
_F ( JUSQU_A = 11., PAS=2.,),))
4
1. Array and parameters
Parameters in code_aster
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Parameter names are determined by code_aster, NOT users
List of the most useful parameters
ABSC_CURV Curvilinear abscissa EPSI Strain
DXDYDZ
Displacement along XDisplacement along YDisplacement along Z
SIGM Stress
DRXDRYDRZ
Rotation around XRotation around YRotation around Z
INST Time
XYZ
Coordinate XCoordinate YCoordinate Z
TEMP Temperature
Outline
1. Array and parameters
2. Functions
3. Formulas
GNU FDL licence | code_aster, salome_meca course material
6
2. Functions
Function – Definition by (𝑥𝑖,𝑦𝑖):
GNU FDL licence | code_aster, salome_meca course material
Parameter 𝑥𝑖
Abscissas values are strictly increasing: 𝑥0 < 𝑥1 < 𝑥2 < 𝑥3 < 𝑥4 < 𝑥5 < ⋯
Function values 𝑦𝑖
7
2. Functions
Function – Mathematical sense
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One value 𝑦𝑖 for one abscissa 𝑥𝑖:
Several values → NOT a function
Function has a domain definition
Function values can be real or
complex, but parameter must be real
Domain definition
8
2. Functions
Function – Trois extension possibilities
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EXCLU – no extension (by default)
CONSTANT – extension by last value
LINEAR – extension by linear extrapolation
Keyword for
extension on left:
PROL_GAUCHE
Keyword for
extension on left:
PROL_DROITE
9
2. Functions
Function - Defined by command DEFI_FONCTION
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Example
Definition with values
fonc = DEFI_FONCTION ( NOM_PARA = 'INST',
VALE=( 0. , 2. ,
1. ,3. ,
2. ,4. ,
3. ,3. ,)
PROL_GAUCHE = 'EXCLU',
PROL_DROITE = 'CONSTANT‘ )
10
2. Functions
Function - Defined by command DEFI_FONCTION
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Example
Definition with arrays ABSC = DEFI_LIST_REEL ( VALE = ( 0., 1., 2., 3., ) )
ORDO = DEFI_LIST_REEL ( VALE = ( 2., 3., 4., 3., ) )
fonc = DEFI_FONCTION ( NOM_PARA = 'INST',
VALE_PARA = ABSC,
VALE_FONC = ORDO,
PROL_GAUCHE = 'EXCLU',
PROL_DROITE = 'CONSTANT‘ )
11
2. Functions
Function – Interpolation between points
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linear interpolation (by default): INTERPOL = ‘LIN’
Logarithmic interpolation: INTERPOL = ‘LOG’
12
2. Functions
Function - Usages
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Where using functions ?
Boundary conditions or loads (AFFE_CHAR_MECA_F command for instance)
Behavior law: traction curve for elastoplastic, laws depending on temperature
Multiplicative functions for boundary conditions or loads
Examples:
FORC = AFFE_CHAR_MECA ( … FORCE_CONTOUR = _F ( GROUP_MA='haut', FY=1.,),);
RAMPE = DEFI_FONCTION ( NOM_PARA = 'INST', VALE = ( 0.0, 0.0, 1000.,1000., ),);
RESU = STAT_NON_LINE ( … EXCIT=( _F ( CHARGE = FORC,
FONC_MULT = RAMPE,),), … )
Constant force 𝑓
Function 𝑔 𝑡 = 𝑡
Final load𝑓 ∗ 𝑔(𝑡)
13
2. Functions
Function – Other related commands
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LIRE_FONCTION
Read function from file
INFO_FONCTION
Get information about function (maximum, rms, etc.)
IMPR_FONCTION
Print a function (for XmGrace software for instance)
RECU_FONCTION
Create function from results or from field
Outline
1. Array and parameters
2. Functions
3. Formulas
GNU FDL licence | code_aster, salome_meca course material
15
3. Formulas
FORMULA – Defined by a mathematical function
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Using Python as a calculator
Main functions of Python math module (imported by default in Code_Aster)
For more information: http://docs.python.org/tut/tut.html,
http://docs.python.org/lib/module-math.html
Python function by command FORMULE
Python evaluation
SIa = FORMULE ( NOM_PARA = 'X', VALE = ‘sin(X)’ )
X = SIa (1.57)
Function of function
SIb = FORMULE ( NOM_PARA = 'X', VALE = ‘X*SIa(X)’ )
16
3. Formulas
FORMULA – Defined by a mathematical function
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Python function by command FORMULE
High-level function
def HEAVISIDE(x) :
if x<0. : return 0.
If x>=0. : return 1.
F_HVS = FORMULE( NOM_PARA = 'INST', VALE = 'HEAVISIDE(INST) ' )
Function with several parameters
from math import pi
OMEGA = 30.
NAP = FORMULE ( NOM_PARA = ( ‘OMEGA ', 'FREQ'),
VALE = '(1./((2.*pi*FREQ)**2 - OMEGA**2 )**2 )’ )
0. si 1.
0. si 0.
x
<x=xHEAVISIDE
17
3. Formulas
FORMULA – Transform into code_aster function
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Command CALC_FONC_INTERP
Example
SI = FORMULE ( NOM_PARA = 'INST', VALE = 'sin(INST)‘ )
DEPI = 2.*pi
LI1 = DEFI_LIST_REEL ( DEBUT = 0, INTERVALLE =_F ( JUSQU_A = DEPI,
NOMBRE = 200 ),)
F_SI = CALC_FONC_INTERP ( FONCTION = SI,
LIST_PARA = LI1,
PROL_GAUCHE = 'EXCLU',
PROL_DROITE = 'CONSTANT‘ )
Outline
1. Array and parameters
2. Functions
3. Formulas
GNU FDL licence | code_aster, salome_meca course material
Remarks
19
Remarks
Functions or formulas?
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Function are tabulated
faster when using in low-level Fortran (behavior law for instance)
Formulas are « exact »: more precise
See zzzz100a test-case for several examples
NAPPE
Tabulated (discrete) function depending on two parameters
Function of function
End of presentation
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