Modelica.Math.Nonlinear.Examples

Information

Package Content

Name	Description
quadratureLobatto1	Integrate integral with fixed inputs
quadratureLobatto2	Integrate integral with user dependent inputs
solveNonlinearEquations1	Solve nonlinear equations with fixed inputs
solveNonlinearEquations2	Solve nonlinear equations with user dependent inputs
quadratureLobatto3	Integrate function in a model
UtilityFunctions	Utility functions that are used as function arguments to the examples

Modelica.Math.Nonlinear.Examples.quadratureLobatto1

Information

This examples integrates the following integrands with function quadratureLobatto and compares the result with an analytical solution. The examples also demonstrate how additional input arguments to the integrand function can be passed as additional arguments. The following integrals are computed:

• integral(sin(x)*dx) from x=0 to x=1
• integral(sin(5*x)*dx) from x=0 to x=13
• elliptic integral from x=0 to pi/2

Inputs

Type	Name	Default	Description
Real	tolerance	1e-5	Error tolerance of integral values

Modelica definition

function quadratureLobatto1 "Integrate integral with fixed inputs"
  input Real tolerance=1e-5 "Error tolerance of integral values";

  import Modelica.Utilities.Streams.print;

protected 
  Real I_numerical[3] "Numerical integral values";
  Real I_analytical[size(I_numerical, 1)] "Analytical integral values";
  Real I_err[size(I_numerical, 1)] 
    "Absolute errors between numerical and analytical integral values";

algorithm 
  I_numerical[1] := Modelica.Math.Nonlinear.quadratureLobatto(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun4(),
      0,
      1,
      tolerance);
  I_analytical[1] := -cos(1) + cos(0);

  I_numerical[2] := Modelica.Math.Nonlinear.quadratureLobatto(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun5(w=5),
      0,
      13,
      tolerance);
  I_analytical[2] := -cos(5*13)/5 + cos(5*0)/5;

  I_numerical[3] := Modelica.Math.Nonlinear.quadratureLobatto(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun6(k=1/
      sqrt(2)),
      0,
      Modelica.Constants.pi/2,
      tolerance);
  I_analytical[3] := 1.8540746773013719184338503;

  I_err := abs(I_numerical - I_analytical);

  print("\n... Results of Modelica.Math.Nonlinear.Examples.quadratureLobatto1:");
  print("Function 1 ( integral(sin(x)*dx) from x=0 to x=1): ");
  print("Analytical integral value = " + String(I_analytical[1], format=
    "2.16f"));
  print("Numerical integral value  = " + String(I_numerical[1], format=
    "2.16f"));
  print("Absolute difference       = " + String(I_err[1], format="2.0e"));

  print("");
  print("Function 2 (integral(sin(5*x)*dx) from x=0 to x=13): ");
  print("Analytical integral value = " + String(I_analytical[2], format=
    "2.16f"));
  print("Numerical integral value  = " + String(I_numerical[2], format=
    "2.16f"));
  print("Absolute difference       = " + String(I_err[2], format="2.0e"));

  print("");
  print("Function 3 (Elliptic integral from x=0 to pi/2): ");
  print("Analytical integral value = " + String(I_analytical[3], format=
    "2.16f"));
  print("Numerical integral value  = " + String(I_numerical[3], format=
    "2.16f"));
  print("Absolute difference       = " + String(I_err[3], format="2.0e"));

end quadratureLobatto1;

Modelica.Math.Nonlinear.Examples.quadratureLobatto2

Information

This examples solves the following integrands with function quadratureLobatto. The user can set the parameters, like "w" or "k", and can experiment with different integration intervals. The following integrals are computed:

• integral(sin(x)*dx)
• integral(sin(w*x)*dx)
• elliptic integral

Inputs

Type	Name	Default	Description
General
Real	Tolerance	1e-5	Error tolerance of integral value
Sine
Real	a1	0	Lower limit
Real	b1	1	Upper limit
Sine w
Real	a2	0	Lower limit
Real	b2	13	Upper limit
Real	w	5	Angular velocity
Elliptic integral
Real	a3	0	Lower limit
Real	b3	Modelica.Constants.pi/2	Upper limit
Real	k	1/sqrt(2)	Modul

Modelica definition

function quadratureLobatto2 
  "Integrate integral with user dependent inputs"
  input Real Tolerance=1e-5 "Error tolerance of integral value";
  input Real a1=0 "Lower limit";
  input Real b1=1 "Upper limit";

  input Real a2=0 "Lower limit";
  input Real b2=13 "Upper limit";
  input Real w=5 "Angular velocity";

  input Real a3=0 "Lower limit";
  input Real b3=Modelica.Constants.pi/2 "Upper limit";
  input Real k=1/sqrt(2) "Modul";

  import Modelica.Utilities.Streams.print;

protected 
  Real I[3] "Numerical integral values";

algorithm 
  I[1] := Modelica.Math.Nonlinear.quadratureLobatto(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun4(),
      a1,
      b1,
      Tolerance);

  I[2] := Modelica.Math.Nonlinear.quadratureLobatto(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun5(w=w),
      a2,
      b2,
      Tolerance);

  I[3] := Modelica.Math.Nonlinear.quadratureLobatto(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun6(k=k),
      a3,
      b3,
      Tolerance);

  print("\n... Results of Modelica.Math.Nonlinear.Examples.quadratureLobatto2:");
  print("Function 1 (integral(sin(x)*dx)): ");
  print("Numerical integral value  = " + String(I[1], format="2.16f"));

  print("");
  print("Function 2 (integral(sin(w*x)*dx)): ");
  print("Numerical integral value  = " + String(I[2], format="2.16f"));

  print("");
  print("Function 3 (Elliptic integral): ");
  print("Numerical integral value  = " + String(I[3], format="2.16f"));

end quadratureLobatto2;

Modelica.Math.Nonlinear.Examples.solveNonlinearEquations1

Information

This examples solves the following nonlinear equations with function solveOneNonlinearEquation and compares the result with the available analytical solution. The examples also demonstrate how additional input arguments to the nonlinear equation function can be passes as additional arguments. The following nonlinear equations are solved:

• 0 = u^2 - 1
• 0 = 3*u - sin(3*u) - 1
• 0 = 5 + log(u) - u

Inputs

Type	Name	Default	Description
Real	tolerance	100*Modelica.Constants.eps	Relative tolerance of solution u

Modelica definition

function solveNonlinearEquations1 
  "Solve nonlinear equations with fixed inputs"
  input Real tolerance=100*Modelica.Constants.eps 
    "Relative tolerance of solution u";

protected 
  Real u_numerical[3];
  Real u_analytical[3];
  Real u_err[3];

  import Modelica.Utilities.Streams.print;

algorithm 
  u_numerical[1] := Modelica.Math.Nonlinear.solveOneNonlinearEquation(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun1(),
      -0.5,
      10,
      tolerance);
  u_analytical[1] := 1.0;

  u_numerical[2] := Modelica.Math.Nonlinear.solveOneNonlinearEquation(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun2(w=3),
      0,
      5,
      tolerance);
  u_analytical[2] := 0.6448544035840080891877538;

  u_numerical[3] := Modelica.Math.Nonlinear.solveOneNonlinearEquation(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun3(p={5,1},
      m=1),
      1,
      8,
      tolerance);
  u_analytical[3] := 6.9368474072202187221643182;

  u_err := abs(u_numerical - u_analytical);

  print("\n... Results of Modelica.Math.Nonlinear.Examples.solveNonlinearEquations1:");
  print("Solve 3 nonlinear equations with relative tolerance = " + String(tolerance) +"\n");
  print("Function 1 (u^2 - 1 = 0): ");
  print("Analytical zero     = " + String(u_analytical[1], format="2.16f"));
  print("Numerical zero      = " + String(u_numerical[1], format="2.16f"));
  print("Absolute difference = " + String(u_err[1], format="2.0e"));

  print("");
  print("Function 2 (3*u - sin(3*u) - 1 = 0): ");
  print("Analytical zero     = " + String(u_analytical[2], format="2.16f"));
  print("Numerical zero      = " + String(u_numerical[2], format="2.16f"));
  print("Absolute difference = " + String(u_err[2], format="2.0e"));

  print("");
  print("Function 3 (5 + log(u) - u = 0): ");
  print("Analytical zero     = " + String(u_analytical[3], format="2.16f"));
  print("Numerical zero      = " + String(u_numerical[3], format="2.16f"));
  print("Absolute difference = " + String(u_err[3], format="2.0e"));

end solveNonlinearEquations1;

Modelica.Math.Nonlinear.Examples.solveNonlinearEquations2

Information

This examples solves the following nonlinear equations with function solveOneNonlinearEquation. The user can set the parameters, like "w" or "m", and can experiment with different start intervals. The following nonlinear equations are solved:

• 0 = u^2 - 1
• 0 = 3*u - sin(w*u) - 1
• 0 = p[1] + log(p[2]*u) - m*u

Inputs

Type	Name	Default	Description
General
Real	tolerance	100*Modelica.Constants.eps	Relative tolerance of solution u
u^2-1
Real	u_min1	-0.5	Lower limit
Real	u_max1	10	Upper limit
3*u - sin(w*u) - 1
Real	u_min2	0	Lower limit
Real	u_max2	5	Upper limit
Real	w	3	Angular velocity
p[1] + log(p[2]*u) - m*u
Real	u_min3	1	Lower limit
Real	u_max3	8	Upper limit
Real	p[2]	{5,1}	Parameter vector
Real	m	1	Parameter

Modelica definition

function solveNonlinearEquations2 
  "Solve nonlinear equations with user dependent inputs"
    input Real tolerance=100*Modelica.Constants.eps 
    "Relative tolerance of solution u";
  input Real u_min1=-0.5 "Lower limit";
  input Real u_max1=10 "Upper limit";
  input Real u_min2=0 "Lower limit";
  input Real u_max2=5 "Upper limit";
  input Real w=3 "Angular velocity ";
  input Real u_min3=1 "Lower limit";
  input Real u_max3=8 "Upper limit";
  input Real p[2]={5,1} "Parameter vector";
  input Real m=1 "Parameter";

protected 
  Real u[3];

  import Modelica.Utilities.Streams.print;

algorithm 
  u[1] := Modelica.Math.Nonlinear.solveOneNonlinearEquation(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun1(),
      u_min1,
      u_max1,
      tolerance);

  u[2] := Modelica.Math.Nonlinear.solveOneNonlinearEquation(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun2(w=w),
      u_min2,
      u_max2,
      tolerance);

  u[3] := Modelica.Math.Nonlinear.solveOneNonlinearEquation(
      function Modelica.Math.Nonlinear.Examples.UtilityFunctions.fun3(p=p, m=
      m),
      u_min3,
      u_max3,
      tolerance);

  print("\n... Results of Modelica.Math.Nonlinear.Examples.solveNonlinearEquations2:");
  print("Solve 3 nonlinear equations with relative tolerance = " + String(tolerance) +"\n");

  print("Function 1 (u^2 - 1): ");
  print("Numerical zero = " + String(u[1], format="2.16f"));

  print("");
  print("Function 2 (3*u - sin(w*u) - 1): ");
  print("Numerical zero = " + String(u[2], format="2.16f"));

  print("");
  print("Function 3 (p[1] + log(p[2]*u) - m*u): ");
  print("Numerical zero = " + String(u[3], format="2.16f"));

end solveNonlinearEquations2;

Modelica.Math.Nonlinear.Examples.quadratureLobatto3

Information

This example demonstrates how to utilize a function as input argument to a function in a model.

Extends from Modelica.Icons.Example (Icon for runnable examples).

Parameters

Type	Name	Default	Description
Real	A	1	Amplitude of integrand of s
Real	ws	2	Angular frequency of integrand of s
Real	wq	3	Angular frequency of q

Modelica definition

model quadratureLobatto3 "Integrate function in a model"
  extends Modelica.Icons.Example;
  parameter Real A=1 "Amplitude of integrand of s";
  parameter Real ws=2 "Angular frequency of integrand of s";
  parameter Real wq=3 "Angular frequency of q";
  Real q(start=1, fixed=true);
  Real qd(start=0, fixed=true);
  Real x;
  final parameter Real s = Modelica.Math.Nonlinear.quadratureLobatto(
                              function UtilityFunctions.fun7(A=A, w=ws),
                              0,1);
equation 
  qd = der(q);
  der(qd) + wq*q = 0;
  x = s*q;
end quadratureLobatto3;