Math Expressions¶

The math expression parser allows to use simple mathematical expressions in combination with if statements inside of the xml files of openCFS. For this user-defined variables can be used

<domain>
  <variableList>
    <var name="A" value="1e6"/>
    <var name="B" value="1500"/>
  </variableList>
</domain>

The registered variables get defined in the global scope and can be used in any expression, which gets parsed by the mathParser. Note however, that for establishing user-defined variables computed variables such as t can not be used. Using user-defined variables for defining an additional user-defined variable however, works.

<domain>
  <variableList>
    <var name="C" value=A*B/>
  </variableList>
</domain>

In addition, the following pre-defined variables can be found in openCFS:

t - Current time value, only available for transient simulations
dt - Time step size, only available for transient simulations
f - Current frequency value, only available for harmonic simulations
step - Current time/frequency step, available for harmonic and transient simulations
x,y,z - Coordinates
r,phi,z - Cylindrical coordinates

Defining loads¶

Math expressions can be used to define a synthetic load. For the following example a transient acoustic simulation (Singlefield -> AcousticPDE) is considered. The load is impressed onto a certain node, and defined as a sinus function

 <bcsAndLoads>
    <pressure name="excite" value="2*sin(2*pi*100*t)"/>
 </bcsAndLoads>

In this example a sinus load with 100 Hz and an amplitude of 2 is defined. $t$ denotes the control variable. A different possibility for defining such a synthetic load is by using one of the custom functions of openCFS:

spike(duration, timeval) - Generates a short spike signal of a given duration starting from 0.
fadeIn(duration, mode, timeval) - This function is used to perform a fadein of an excitation. The returned value is between zero and one and has to be used as envelope of the excitation signal (mode: see detailed explanation below)
sinBurst(freq, nperiods, nfadein, nfadeout, timeval) - Creates data for a sine burst signal. A sine burst signal denotes a sine that will be active during a certain amount of time until it will be deactivated. Additionally, a fade-in/out can be applied, according to the sin^2 method of the fadeIn to reduce the higher harmonics of the signal.
squareBurst(freq, nperiods, bipolar, pwidth, rtime, timeval) - Create data for a square burst signal, which can either be unipolar or bipolar.
gauss(mue, sigma, normval, timeval) - Generates a Gaussian curve, which is either normalized to a maximum value of 1 or to having an area of 1.
locCoord2D(coordsysid, component, x, y) - This function returns for a point in a 2D cartesian coordinate system one component of a local coordinate system, as defined in the < coordSysList >-section of the < domain >-element (for details, see 4.4). This can be used,e.g. to define loads / dirichlet boundary conditions depending on polar / cylindrical coordinates.
locCoord3D(coordsysid, component, x, y,z) - Similar to the function locCoord2D, this method can be used for local 3-dimensional coordinate systems (e.g. cylindrical or spherical).
sample1D(filename, sampleval, interpolation) - This function can be used to read in and interpolate values of a 2-column formatted text file. This can be used e.g. to prescribe space/time/frequency-dependent boundary or load conditions.
if(condition, trueval, elseval) - if ... then ... else ...
sin(2 * pi * frequency * timeStep)
cos(2 * pi * frequency * timeStep)
tan(2 * pi * frequency * timeStep)
asin(2 * pi * frequency * timeStep)
acos(2 * pi * frequency * timeStep)
atan(2 * pi * frequency * timeStep)
sinh(2 * pi * frequency * timeStep)
cosh(2 * pi * frequency * timeStep)
tanh(2 * pi * frequency * timeStep)
asinh(2 * pi * frequency * timeStep)
acosh(2 * pi * frequency * timeStep)
atanh(2 * pi * frequency * timeStep)
log2( X ) - logarithm to the base 2
log10( X ) - logarithm to the base 10
log( X ) - logarithm to the base 10
ln( n ) - logarithm to base e (2.71828...)
exp( X ) - e raised to the power of x
sqrt( X ) - square root of a value
sign( X ) - sign function -1 if x < 0; 1 if x > 0
rint( X ) - round to nearest integer
abs( X ) - absolute value
min( X ) - min of all arguments
max( X ) - max of all arguments
sum( X ) - sum of all arguments
avg( X ) - mean value of all arguemnts

Additionally, the following built-in operators and constants are available:

= - Assigment
and - logical and
or - logical or
xor - logical xor
<= or le * - less or equal
>= or ge * - greater or equal
!= - not equal
== - equal
> or gt * - greater than
< ot lt * - less then
+ - addition
- - subtraction
* - multiplication
/ - division
^ - raise to the power of
pi or _pi - 3.1415926..
_e - Eulerian number e = 2.7182818..

Defining fields¶

Math expressions can also be used for defining e.g flow, or temperature fields.

 <flowList>
  <flow name="backward_Z"> 
   <comp dof="x" value="0"/>
   <comp dof="y" value="120000/60*2*pi*z"/> 
   <comp dof="z" value="-120000/60*2*pi*y"/>
  </flow>
 </flowList>

This can also be used for defining blending functions, which e.g. necessary for the usage of non-conforming interfaces in combination with aeroacoustic source terms. For more details look into Singlefield -> AcousticPDE

Detailed explanations and examples¶

Following we will show two different examples for such custom functions:

Spike¶

spike(duration, timeval)

duration -> Duration of spike in s.
timeval -> Time parameter at which function gets evaluated. Usually the time variable t is used.

Example for spike signal: spike(1,t-2)

spike

fadeIn¶

fadeIn(duration, mode, timeval)

duration -> The amount of time for which the fade in operation is performed in s.
mode -> Three different implementation exist for the envelope of the fade in process:
```
         mode = 1 (using a sin^2 formulation) 
         mode = 2 (using an exponential formulation exp)
         mode = 3 (using a squared exponential formulation exp^2)
```
- timeval -> Time parameter at which function gets evaluated. Usually the time variable t is used.

Comparison of 3 fadeIn signals: sin 2 -based (red), exp-based (green) and exp 2 -based (blue):

fadein

sinBurst¶

sinBurst(freq, nperiods, nfadein, nfadeout, timeval)

freq -> Frequency of the sine pulse.
nperiods -> Total number of periods of the pulse.
nfadein -> Specifies the number of periods that will be used for the transient process. This will realize a kind of ”soft” transient process that will reduce the higher harmonics of the resulting signal.
nfadeout -> Specifies the number of periods that will be used for the decay process. This will realize a kind of ”soft” decay process that will reduce the higher harmonics of the resulting signal.
timeval -> Time parameter at which function gets evaluated. Usually the time variable t is used.

Sine burst without fade-In: sinBurst(1,6,0,0,t):

sinburst

Sine burst with 2 periods of fade-in/out: sinBurst(1,6,2,2,t)

sinburst2

squareBurst¶

squareBurst(freq, nperdios, bipolar, pwidth, rtime, timeval)

freq -> Frequency of the rectangular signal.
nperiods -> Total number of periods of the pulse.
bipolar -> Switch for generating bipolar / unipolar signal; 0: unipolar signal (0 ≤ y ≤ 1); 1: bipolar signal (−1 ≤ y ≤ 1)
pwidth -> Pulse width of the signal in percent (0 ≤ pw ≤ 100).
rtime -> Defines the rise time of the square burst. It denotes the time in seconds that the pulse will take to change from low to high level.
timeval -> Time parameter at which function gets evaluated. Usually the time variable t is used.

squareburst

gauss¶

gauss(mue, sigma, normval, timeval)

mue -> Mean value of Gauss curve.
sigma -> Variance of Gauss curve.
normval -> Type of normalization: 0: Maximum of curve gets normalized to 1; 1: Area of curve gets normalized to 1
timeval -> Time parameter at which function gets evaluated. Usually the time variable t is used.

locCoord2¶

locCoord2D(coordsysid, component, x, y)

coordsysid Id -> string of the referred coordinate system.
component -> Coordinate component of the local coordinate system, which can be either 1 or 2 in 2D. For a polar coordinate system, 1=r-direction and 2=phi-direction.
x -> x-coordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by x.
y -> y-coordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by y.

In the following example, a cylindrical coordinate system with id mid is defined. An inhomogeneous Dirichlet boundary condition for an electrostatic analysis is applied, with a value varying with the angle phi (second component in a polar coordinate system).

<domain>
  <coordSysList>
    <polar id="mid">
      <origin x="0" y="0"/>
      <rAxis x="1" y="0"/>
    </polar>
  </coordSysList>
</domain>
....
<potential name=".." value="locCoord(’mid’,2,x,y)"/>

locCoord3¶

locCoord3D(coordsysid, component, x, y,z)

coordsysid Id -> string of the referred coordinate system.
component -> Coordinate component of the local coordinate system, which can be either 1, 2 or in 3D. For a cylindrical coordinate system, 1=r-direction, 2=phi-direction and 3=z-direction.
x -> x-coordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by x.
y -> y-coordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by y.
z -> z-coordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by z.

sample1D¶

sample1D(filename, sampleval, interpolation)

filename is the name of the file to be read (any path in the filename is taken relative to the location of the .xml-file). The file must consist of exactly 2 columns, which are separated by whitespace (space, tab).
sampleval denotes the sample value (value of the 1st column ), for which a value in the 2nd column is seeked. In most cases either the current time t or a coordinate component (x, y, z) may be used.

interpolation selects the one of 3 different methods for interpolating the sampled values:

0: Nearest neighbor is taken (i.e. no interpolation at all)
1: Linear interpolation
2: Cubic spline interpolation

We can prescribe boundary conditions with values read in from sample files. The notation is as follows:

<bcsAndLoads>
  <potential name="transducer" value="sample1D('x_amplitude.dat',x,2)" phase="sample1D('x_phase.dat',x,2)"/>
</bcsAndLoads>

Sample data (2 nd column) is read in from the files x amplitude.dat and x phase.dat depending on the current x-coordinate and gets interpolated using cubic spline interpolation. Please note the type of quotes used!

if statement:¶

 <bcsAndLoads>
    <pressure name="excite" value="(t lt 1)? (cos(2*pi*(t-0.5))+1)*sin(2*pi*t): 0")"/>
 </bcsAndLoads>

Furthermore, it is possible to superpose multiple different custom functions, with e.g. if statements. In the following example we superpose a time depending if statement with a spatial if statement

 <bcsAndLoads>
    <pressure name="exciteSurface" value="((t lt 1)? (cos(2*pi*(t-0.5))+1)*sin(2*pi*t): 0"))*((x lt 1)? (1 : 0))"/>
 </bcsAndLoads>

Hint: With multiple if statements it is especially important to look after the positioning of braces.