.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

Interface

function regRoot2
  extends Modelica.Icons.Function;
  input Real x "Abscissa value";
  input Real x_small(min = 0) = 0.01 "Approximation of function for |x| <= x_small";
  input Real k1(min = 0) = 1 "y = if x>=0 then sqrt(k1*x) else -sqrt(k2*|x|)";
  input Real k2(min = 0) = 1 "y = if x>=0 then sqrt(k1*x) else -sqrt(k2*|x|)";
  input Boolean use_yd0 = false "= true, if yd0 shall be used";
  input Real yd0(min = 0) = 1 "Desired derivative at x=0: dy/dx = yd0";
  output Real y "Ordinate value";
end regRoot2;

Revisions

• Sept., 2010 by Martin Otter:
  Improved so that k1=0 and/or k2=0 is also possible.
• Nov., 2005 by Martin Otter:
  Designed and implemented.