Modelica.Fluid.Utilities

Information

Package Content

Name	Description
checkBoundary	Check whether boundary definition is correct
regRoot	Anti-symmetric square root approximation with finite derivative in the origin
regRoot_der	Derivative of regRoot
regSquare	Anti-symmetric square approximation with non-zero derivative in the origin
regPow	Anti-symmetric power approximation with non-zero derivative in the origin
regRoot2	Anti-symmetric approximation of square root with discontinuous factor so that the first derivative is finite and continuous
regSquare2	Anti-symmetric approximation of square with discontinuous factor so that the first derivative is non-zero and is continuous
regStep	Approximation of a general step, such that the characteristic is continuous and differentiable
evaluatePoly3_derivativeAtZero	Evaluate polynomial of order 3 that passes the origin with a predefined derivative
regFun3	Co-monotonic and C1 smooth regularization function
cubicHermite	Evaluate a cubic Hermite spline
cubicHermite_withDerivative	Evaluate a cubic Hermite spline, return value and derivative

Modelica.Fluid.Utilities.checkBoundary

Information

Inputs

Name	Description
mediumName
substanceNames[:]	Names of substances
singleState
define_p
X_boundary[:]
modelName

Modelica.Fluid.Utilities.regRoot

Information

This function approximates sqrt(abs(x))*sgn(x), such that the derivative is finite and smooth in x=0.

With the default value of delta=0.01, the difference between sqrt(x) and regRoot(x) is 16% around x=0.01, 0.25% around x=0.1 and 0.0025% around x=1.

Inputs

Name	Description
x
delta	Range of significant deviation from sqrt(abs(x))*sgn(x)

Outputs

Name	Description
y

Modelica.Fluid.Utilities.regRoot_der

Information

Inputs

Name	Description
x
delta	Range of significant deviation from sqrt(x)
dx	Derivative of x

Outputs

Name	Description
dy

Modelica.Fluid.Utilities.regSquare

Information

This function approximates x^2*sgn(x), such that the derivative is non-zero in x=0.

With the default value of delta=0.01, the difference between x^2 and regSquare(x) is 41% around x=0.01, 0.4% around x=0.1 and 0.005% around x=1.

Inputs

Name	Description
x
delta	Range of significant deviation from x^2*sgn(x)

Outputs

Name	Description
y

Modelica.Fluid.Utilities.regPow

Information

This function approximates abs(x)^a*sign(x), such that the derivative is positive, finite and smooth in x=0.

Inputs

Name	Description
x
a
delta	Range of significant deviation from x^a*sgn(x)

Outputs

Name	Description
y

Modelica.Fluid.Utilities.regRoot2

Information

Approximates the function

   y = if x ≥ 0 then sqrt(k1*x) else -sqrt(k2*abs(x)), with k1, k2 ≥ 0

in such a way that within the region -x_small ≤ x ≤ x_small, the function is described by two polynomials of third order (one in the region -x_small .. 0 and one within the region 0 .. x_small) such that

• The derivative at x=0 is finite.
• The overall function is continuous with a continuous first derivative everywhere.
• If parameter use_yd0 = false, the two polynomials are constructed such that the second derivatives at x=0 are identical. If use_yd0 = true, the derivative at x=0 is explicitly provided via the additional argument yd0. If necessary, the derivative yd0 is automatically reduced in order that the polynomials are strict monotonically increasing [Fritsch and Carlson, 1980].

Typical screenshots for two different configurations are shown below. The first one with k1=k2=1:

and the second one with k1=1 and k2=3:

The (smooth) derivative of the function with k1=1, k2=3 is shown in the next figure:

Literature

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Inputs

Name	Description
x	abscissa value
x_small	approximation of function for |x| <= x_small
k1	y = if x>=0 then sqrt(k1*x) else -sqrt(k2*|x|)
k2	y = if x>=0 then sqrt(k1*x) else -sqrt(k2*|x|)
use_yd0	= true, if yd0 shall be used
yd0	Desired derivative at x=0: dy/dx = yd0

Outputs

Name	Description
y	ordinate value

Modelica.Fluid.Utilities.regSquare2

Information

Approximates the function

   y = if x ≥ 0 then k1*x*x else -k2*x*x, with k1, k2 > 0

in such a way that within the region -x_small ≤ x ≤ x_small, the function is described by two polynomials of third order (one in the region -x_small .. 0 and one within the region 0 .. x_small) such that

• The derivative at x=0 is non-zero (in order that the inverse of the function does not have an infinite derivative).
• The overall function is continuous with a continuous first derivative everywhere.
• If parameter use_yd0 = false, the two polynomials are constructed such that the second derivatives at x=0 are identical. If use_yd0 = true, the derivative at x=0 is explicitly provided via the additional argument yd0. If necessary, the derivative yd0 is automatically reduced in order that the polynomials are strict monotonically increasing [Fritsch and Carlson, 1980].

A typical screenshot for k1=1, k2=3 is shown in the next figure:

The (smooth, non-zero) derivative of the function with k1=1, k2=3 is shown in the next figure:

Literature

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Inputs

Name	Description
x	abscissa value
x_small	approximation of function for |x| <= x_small
k1	y = (if x>=0 then k1 else k2)*x*|x|
k2	y = (if x>=0 then k1 else k2)*x*|x|
use_yd0	= true, if yd0 shall be used
yd0	Desired derivative at x=0: dy/dx = yd0

Outputs

Name	Description
y	ordinate value

Modelica.Fluid.Utilities.regStep

Information

This function is used to approximate the equation

    y = if x > 0 then y1 else y2;

by a smooth characteristic, so that the expression is continuous and differentiable:

   y = smooth(1, if x >  x_small then y1 else
                 if x < -x_small then y2 else f(y1, y2));

In the region -x_small < x < x_small a 2nd order polynomial is used for a smooth transition from y1 to y2.

Inputs

Name	Description
x	Abscissa value
y1	Ordinate value for x > 0
y2	Ordinate value for x < 0
x_small	Approximation of step for -x_small <= x <= x_small; x_small >= 0 required

Outputs

Name	Description
y	Ordinate value to approximate y = if x > 0 then y1 else y2

Modelica.Fluid.Utilities.evaluatePoly3_derivativeAtZero

Information

Inputs

Name	Description
x	Value for which polynomial shall be evaluated
x1	Abscissa value
y1	y1=f(x1)
y1d	First derivative at y1
y0d	First derivative at f(x=0)

Outputs

Name	Description
y

Modelica.Fluid.Utilities.regFun3

Information

Approximates a function in a region between x0 and x1 such that

• The overall function is continuous with a continuous first derivative everywhere.
• The function is co-monotone with the given data points.

In this region, a continuation is constructed from the given points (x0, y0), (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 xi and yid:

The second graph shows the continuous derivative of this regularization function:

Literature

Gasparo M. G. and Morandi R. (1991):

Piecewise cubic monotone interpolation with assigned slopes. Computing, Vol. 46, Issue 4, December 1991, pp. 355 - 365.

Inputs

Name	Description
x	Abscissa value
x0	Lower abscissa value
x1	Upper abscissa value
y0	Ordinate value at lower abscissa value
y1	Ordinate value at upper abscissa value
y0d	Derivative at lower abscissa value
y1d	Derivative at upper abscissa value

Outputs

Name	Description
y	Ordinate value
c	Slope of linear section between two cubic polynomials or dummy linear section slope if single cubic is used

Modelica.Fluid.Utilities.cubicHermite

Information

Inputs

Name	Description
x	Abscissa value
x1	Lower abscissa value
x2	Upper abscissa value
y1	Lower ordinate value
y2	Upper ordinate value
y1d	Lower gradient
y2d	Upper gradient

Outputs

Name	Description
y	Interpolated ordinate value

Modelica.Fluid.Utilities.cubicHermite_withDerivative

Information

Inputs

Name	Description
x	Abscissa value
x1	Lower abscissa value
x2	Upper abscissa value
y1	Lower ordinate value
y2	Upper ordinate value
y1d	Lower gradient
y2d	Upper gradient

Outputs

Name	Description
y	Interpolated ordinate value
dy_dx	Derivative dy/dx at abscissa value x