Function definitions
Function are particularly useful for the application of boundary conditions, but they can also
be used for other definitions. The definition of a function is extremely versatile and may be
accomplished in various manners. Any number of different function definitions are allowed, and the
respective section always starts with FUNCT, followed by a number, which is referred in other sections,
e.g., FUNCT1:, FUNCT2:, etc.
Two different applications for functions exist: One case is a spatial and temporal variation of a condition. The other one is a function of some (result or internal) variable, which is mainly used in combination with materials.
Functions for spatial and/or temporal variation
These functions allow to define spatial and temporal conditions in terms of mathematical functions, e.g., in sine form etc.
The respective input line for defining a symbolic function is of the following form in the easiest case:
FUNCT<n>:
- SYMBOLIC_FUNCTION_OF_SPACE_TIME|SYMBOLIC_FUNCTION_OF_TIME: <functiondefinition>
or, if one needs a function expression for several components of a vector (here for 3 components, only available for SYMBOLIC_FUNCTION_OF_SPACE_TIME):
FUNCT<n>:
- COMPONENT: 0
SYMBOLIC_FUNCTION_OF_SPACE_TIME: <functiondefinition>
- COMPONENT: 1
SYMBOLIC_FUNCTION_OF_SPACE_TIME: <functiondefinition>
- COMPONENT: 2
SYMBOLIC_FUNCTION_OF_SPACE_TIME: <functiondefinition>
As the name says, functiondefinition can be a function \(f\)
of space and time, that is, \(f(x,y,z,t)\), or
only of time, \(f(t)\)
4C has its own function parser, which recognises variables, algebraic terms, brackets and a number of functions, namely the following: \(\mathrm{acos}, \mathrm{asin}, \mathrm{atan}\), \(\cos, \sin, \tan\), \(\cosh, \sinh, \tanh\), \(\exp, \log, \log10\), \(\mathrm{sqrt}\), \(\mathrm{heaviside}\), \(\mathrm{fabs}, \mathrm{atan2}\). Additionally, the constant \(pi\) is known.
If the function cannot easily be given as a symbolic expression, the function may depend on additional user defined variables, which are given in the following way:
# multi-linear representation
FUNCT<n>:
- SYMBOLIC_FUNCTION_OF_SPACE_TIME: <>
- VARIABLE: <num>
NAME: <varname>
TYPE: linearinterpolation
NUMPOINTS: <num_points>
TIMES: [ <time_1>, ... , <time_num_points> ]
VALUES: [ <val_1>, ... , <val_num_points> ]
# further variables possible
#
# concatenating several functions
FUNCT<n>:
- VARIABLE: <num>
NAME: <varname>
TYPE: multifunction
NUMPOINTS: <num_points>
TIMES: [ <time_1>, ... , <time_num_points> ]
DESCRIPTION: [ <expression_1>, ... , <expression_num_points-1> ]
# since expressions are defined between the time values, only n-1 expressions are to be defined
#
# fourier interpolation
FUNCT<n>:
- VARIABLE: <num>
NAME: <varname>
TYPE: fourierinterpolation
NUMPOINTS: <num_points>
BYNUM:
TIMERANGE: [ <start_time>, <end_time> ]
VALUES: [ <val_1>, ... , <val_num_points> ]
PERIODIC:
T1: <time_1>
T2: <time_2>
where
expression is simply a symbolic expression similar to the function definition itself. That is,
FUNCT1:
- SYMBOLIC-FUNCTION_OF_SPACE_TIME: 10*myvar
- VARIABLE: 0
NAME: myvar
TYPE: expression
DESCRIPTION: 5*t
is completely equivalent to
FUNCT1:
- SYMBOLIC-FUNCTION_OF_SPACE_TIME: 50*t
linearinterpolation defines a table with a given number of sampling points leading to multilinear expression. For example, one can define an amplitude curve with several hold states as done in example prestress_new.4C.yaml:
FUNCT1: - COMPONENT: 0 SYMBOLIC_FUNCTION_OF_SPACE_TIME: a - VARIABLE: 0 NAME: a TYPE: linearinterpolation NUMPOINTS: 5 TIMES: - 0 - 1 - 4 - 5 - 7 VALUES: - 0 - 1 - 1 - 0.5 - 0.5
multifunction is similar to expression, but a number of symbolic functions can be entered, which are only valid in a specific time range. therefore, one has to define one point in time more than functions. As an example, see the trapezoidal temperature progression in tsi_locsys_monolithic.4C.yaml:
FUNCT2: - SYMBOLIC_FUNCTION_OF_TIME: v - VARIABLE: 0 NAME: v TYPE: multifunction NUMPOINTS: 5 TIMES: - 0 - 2 - 3 - 4 - 5 DESCRIPTION: - 293.15-(293.15-40.0)*t/2.0 - '40.0' - 40+(293.15-40.0)*(t-3) - '293.15'
fourierinterpolation defines a Fourier series with a given number of sampling points. An example is taken from one_d_3_artery_network.4C.yaml
FUNCT1: - SYMBOLIC_FUNCTION_OF_TIME: a - VARIABLE: 0 NAME: a TYPE: fourierinterpolation NUMPOINTS: 60 BYNUM: TIMERANGE: - 0 - 1 VALUES: - 7.83 - 7.72 - 8.05 - 10.65 - 15.11 - 19.91 - 26.54 - 38.11 - 55.31 - 72.69 - 89.48 - 106.85 - 119.6 - 136.97 - 147.09 - 153.04 - 157.64 - 157.81 - 157.53 - 156.7 - 154.33 - 151.61 - 146.49 - 139.93 - 132.96 - 128.34 - 120.9 - 111.45 - 105.15 - 101.88 - 94.05 - 85.02 - 76.14 - 66.87 - 55.42 - 44.79 - 33.34 - 24.12 - 18.71 - 12.33 - 9.78 - 9.21 - 8.48 - 8.59 - 9.36 - 9.97 - 10.7 - 11.99 - 13.25 - 14.18 - 15.21 - 16.1 - 16.25 - 16.74 - 18.47 - 18.46 - 17.77 - 17.76 - 17.95 - 17.96 PERIODIC: T1: 0 T2: 100
Fluid
For fluids some other keywords are available beside the ones given above:
FUNCT1:
- FLUID_FUNCTION: BELTRAMI
c1: <value>
FUNCT2:
- FLUID_FUNCTION: KIMMOIN-RHS
MAT: <num>
ISSTAT: [0|1]
ISSTOKES: [0|1]
FUNCT3:
- FLUID_FUNCTION: KIMMOIN-UP
MAT: <num>
ISSTAT: [0|1]
FUNCT4:
- FLUID_FUNCTION: KIMMOIN-STRESS
MAT: <num>
ISSTAT: [0|1]
AMPLITUDE: <value>
FUNCT5:
- FLUID_FUNCTION: CHANNELWEAKLYCOMPRESSIBLE
FUNCT6:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_FSI_FLUID
MAT_FLUID: <num>
MAT_STRUCT: <num>
FUNCT7:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_FSI_FLUID_FORCE
MAT_FLUID: <num>
MAT_STRUCT: <num>
FUNCT8:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_FSI_FLUID_VISCOSITY
MAT_FLUID: <num>
MAT_STRUCT: <num>
FUNCT9:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_FSI_STRUCTURE
MAT_STRUCT: <num>
FUNCT10:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_FSI_STRUCTURE_FORCE
MAT_STRUCT: <num>
FUNCT11:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_CFD
MAT: <num>
FUNCT12:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_CFD_FORCE
MAT: <num>
FUNCT13:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_ETIENNE_CFD_VISCOSITY
MAT: <num>
FUNCT14:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_MANUFACTUREDFLOW
MAT: <num>
FUNCT15:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_MANUFACTUREDFLOW_FORCE
MAT: <num>
FUNCT16:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_POISEUILLE
MAT: <num>
FUNCT17:
- FLUID_FUNCTION: WEAKLYCOMPRESSIBLE_POISEUILLE_FORCE
MAT: <num>
FUNCT18:
- FLUID_FUNCTION: ZALESAKSDISK
FUNCT19:
- XFLUID_FUNCTION: FORWARDFACINGSTEP
<further_paramters>
FUNCT20:
- XFLUID_FUNCTION: MOVINGLEVELSETCYLINDER
<further_paramters>
FUNCT21:
- XFLUID_FUNCTION: MOVINGLEVELSETTORUS
<further_paramters>
FUNCT22:
- XFLUID_FUNCTION: MOVINGLEVELSETTORUSVELOCITY
<further_paramters>
FUNCT23:
- XFLUID_FUNCTION: MOVINGLEVELSETTORUSSLIPLENGTH
<further_paramters>
FUNCT24:
- XFLUID_FUNCTION: URQUIZABOXFLOW
<further_paramters>
FUNCT25:
- XFLUID_FUNCTION: URQUIZABOXFLOW_TRACTION
<further_paramters>
FUNCT26:
- XFLUID_FUNCTION: URQUIZABOXFLOW_FORCE
<further_paramters>
FUNCT27:
- XFLUID_FUNCTION: TAYLORCOUETTEFLOW
<further_paramters>
FUNCT28:
- XFLUID_FUNCTION: COLLAPSINGWATERCOLUMN
<further_paramters>
FUNCT29:
- XFLUID_FUNCTION: CORRECTIONTERMCHANNELWEAKLYCOMPRESSIBLE
<further_paramters>
Porous materials
Here, we can also consider the keyword POROMULTIPHASESCATRA_FUNCTION, which needs a specific function name as a value.
The parameters of this function is then given after the subsection PARAMS:
FUNCT1:
- POROMULTIPHASESCATRA_FUNCTION: <specific_name>
PARAMS:
<param_name>: value
...
Functions depending on simulation variables
If one needs to define a function for some material behavior, which depends on simulation variables,
the option VARFUNCTION is the one to be used.
Here, it is crucial that the independent variables used in the function must exactly be named as defined in the code for the specific behavior.
The definition of constants is optional. An example is shown in the test file porofluid_pressure_based_2D_quad4.4C.yaml:
FUNCT2:
- VARFUNCTION: 1.0/0.01*tan(0.5*pi*(0.3-S))
FUNCT3:
- VARFUNCTION: S0-2/pi*atan(0.01*dp)
NUMCONSTANTS: 1
CONSTANTS:
S0: 0.3
For this case, the variable S is used in function 2, and dp in function 3. The latter uses the additional constant S0 and the predefined constant pi.