Functionsand formulas

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Outline

1. Array and parameters

2. Functions

3. Formulas

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3

1. Array and parameters

Array – A list of real numbers

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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

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6

2. Functions

Function – Definition by (𝑥𝑖,𝑦𝑖):

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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

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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

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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

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