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 userdefined 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 userdefined variables computed variables such as t
can not be used.
Using userdefined variables for defining an additional userdefined variable however, works.
<domain>
<variableList>
<var name="C" value=A*B/>
</variableList>
</domain>
In addition, the following predefined variables can be found in openCFS:
t
 Current time value, only available for transient simulationsdt
 Time step size, only available for transient simulationsf
 Current frequency value, only available for harmonic simulationsstep
 Current time/frequency step, available for harmonic and transient simulationsx,y,z
 Coordinatesr,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 fadein/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 3dimensional coordinate systems (e.g. cylindrical or spherical).sampled1D(filename, interpolation, sampleval)
 This function can be used to read in and interpolate values of a 2column formatted text file. This can be used e.g. to prescribe space/time/frequencydependent 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 2log10( X )
 logarithm to the base 10log( X )
 logarithm to the base 10ln( n )
 logarithm to base e (2.71828...)exp( X )
 e raised to the power of xsqrt( X )
 square root of a valuesign( X )
 sign function 1 if x < 0; 1 if x > 0rint( X )
 round to nearest integerabs( X )
 absolute valuemin( X )
 min of all argumentsmax( X )
 max of all argumentssum( X )
 sum of all argumentsavg( X )
 mean value of all arguemnts
Additionally, the following builtin operators and constants are available:
=
 Assigmentand
 logical andor
 logical orxor
 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 ofpi 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 nonconforming 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,t2)
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), expbased (green) and exp 2 based (blue):
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 fadeIn: sinBurst(1,6,0,0,t):
Sine burst with 2 periods of fadein/out: sinBurst(1,6,2,2,t)
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.
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=rdirection and 2=phidirection.
 x > xcoordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by x.
 y > ycoordinate 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=rdirection, 2=phidirection and 3=zdirection.
 x > xcoordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by x.
 y > ycoordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by y.
 z > zcoordinate of the Cartesian point. For < load > and < dirichletInhom > boundary conditions, it can be literally replaced by z.
sample1D
sampled1D(filename, interpolation, sampleval)
 filename > Name of the file to be read (any path in the filename is taken relative to the location of the .xmlfile). The file must consist of exactly 2 columns, which are separated by any whitespace (space, tab).

interpolation > There are 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

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
We can subscribe 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 xcoordinate 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*(t0.5))+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*(t0.5))+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.