Modelica.Media.Common.OneNonLinearEquation

Determine solution of a non-linear algebraic equation in one unknown without derivatives in a reliable and efficient way

Information

This function should currently only be used in Modelica.Media, since it might be replaced in the future by another strategy, where the tool is responsible for the solution of the non-linear equation.

This library determines the solution of one non-linear algebraic equation "y=f(x)" in one unknown "x" in a reliable way. As input, the desired value y of the non-linear function has to be given, as well as an interval x_min, x_max that contains the solution, i.e., "f(x_min) - y" and "f(x_max) - y" must have a different sign. If possible, a smaller interval is computed by inverse quadratic interpolation (interpolating with a quadratic polynomial through the last 3 points and computing the zero). If this fails, bisection is used, which always reduces the interval by a factor of 2. The inverse quadratic interpolation method has superlinear convergence. This is roughly the same convergence rate as a globally convergent Newton method, but without the need to compute derivatives of the non-linear function. The solver function is a direct mapping of the Algol 60 procedure "zero" to Modelica, from:

Brent R.P.:
Algorithms for Minimization without derivatives. Prentice Hall, 1973, pp. 58-59.

Due to current limitations of the Modelica language (not possible to pass a function reference to a function), the construction to use this solver on a user-defined function is a bit complicated (this method is from Hans Olsson, Dassault Systèmes AB). A user has to provide a package in the following way:

  package MyNonLinearSolver
    extends OneNonLinearEquation;

    redeclare record extends Data
      // Define data to be passed to user function
      ...
    end Data;

    redeclare function extends f_nonlinear
    algorithm
       // Compute the non-linear equation: y = f(x, Data)
    end f_nonlinear;

    // Dummy definition that has to be present for current Dymola
    redeclare function extends solve
    end solve;
  end MyNonLinearSolver;

  x_zero = MyNonLinearSolver.solve(y_zero, x_min, x_max, data=data);

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name Description
Modelica.Media.Common.OneNonLinearEquation.f_nonlinear_Data f_nonlinear_Data Data specific for function f_nonlinear
Modelica.Media.Common.OneNonLinearEquation.f_nonlinear f_nonlinear Nonlinear algebraic equation in one unknown: y = f_nonlinear(x,p,X)
Modelica.Media.Common.OneNonLinearEquation.solve solve Solve f_nonlinear(x_zero)=y_zero; f_nonlinear(x_min) - y_zero and f_nonlinear(x_max)-y_zero must have different sign

Modelica.Media.Common.OneNonLinearEquation.f_nonlinear_Data Modelica.Media.Common.OneNonLinearEquation.f_nonlinear_Data

Data specific for function f_nonlinear

Information

Extends from Modelica.Icons.Record (Icon for records).

Modelica.Media.Common.OneNonLinearEquation.f_nonlinear Modelica.Media.Common.OneNonLinearEquation.f_nonlinear

Nonlinear algebraic equation in one unknown: y = f_nonlinear(x,p,X)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

NameDescription
xIndependent variable of function
pDisregarded variables (here always used for pressure)
X[:]Disregarded variables (her always used for composition)
f_nonlinear_dataAdditional data for the function

Outputs

NameDescription
y= f_nonlinear(x)

Modelica.Media.Common.OneNonLinearEquation.solve Modelica.Media.Common.OneNonLinearEquation.solve

Solve f_nonlinear(x_zero)=y_zero; f_nonlinear(x_min) - y_zero and f_nonlinear(x_max)-y_zero must have different sign

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

NameDescription
y_zeroDetermine x_zero, such that f_nonlinear(x_zero) = y_zero
x_minMinimum value of x
x_maxMaximum value of x
pressureDisregarded variables (here always used for pressure)
X[:]Disregarded variables (here always used for composition)
f_nonlinear_dataAdditional data for function f_nonlinear
x_tolRelative tolerance of the result

Outputs

NameDescription
x_zerof_nonlinear(x_zero) = y_zero
Automatically generated Thu Dec 19 17:20:15 2019.