WOLFRAM SYSTEM MODELER
Co-monotonic and C1 smooth regularization function
This information is part of the Modelica Standard Library maintained by the Modelica Association.
Approximates a function in a region between
In this region, a continuation is constructed from the given points
(x1, y1) and the respective
derivatives. For this purpose, a single polynomial of third order or two
cubic polynomials with a linear section in between are used [Gasparo
and Morandi, 1991]. This algorithm was extended with two additional
conditions to avoid saddle points with zero/infinite derivative that lead to
integrator step size reduction to zero.
This function was developed for pressure loss correlations properly
addressing the static head on top of the established requirements
for monotonicity and smoothness. In this case, the present function
allows to implement the exact solution in the limit of
x1-x0 -> 0 or
y1-y0 -> 0.
Typical screenshots for two different configurations
are shown below. The first one illustrates five different settings of
The second graph shows the continuous derivative of this regularization function:
Description: Abscissa value
Description: Lower abscissa value
Description: Upper abscissa value
Description: Ordinate value at lower abscissa value
Description: Ordinate value at upper abscissa value
Description: Derivative at lower abscissa value
Description: Derivative at upper abscissa value
Description: Ordinate value
Description: Slope of linear section between two cubic polynomials or dummy linear section slope if single cubic is used
(x1,y1)on horizontal line, then return value
cwas undefined. This was corrected. Furthermore, an additional term was included for the computation of
yin this case to assist automatic differentiation.