Collection of models that illustrate model use and test models
Information
This package contains examples for the use of models that can be found in
Buildings.Utilities.Math.Functions.
Extends from Modelica.Icons.ExamplesPackage (Icon for packages containing runnable examples).
Package Content
Test problem for cubic hermite splines
Information
This example demonstrates the use of the function for cubic hermite interpolation
and linear extrapolation.
The example use interpolation with two different settings: One settings
produces a monotone cubic hermite, whereas the other setting
does not enforce monotonicity.
The resulting plot should look as shown below, where for better visibility, the support points have been marked with black dots.
Notice that the red curve is monotone increasing.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Parameters
| Type | Name | Default | Description |
|---|
| Real | xd[:] | {-1,1,5,6} | Support points |
| Real | yd[size(xd, 1)] | {-1,1,2,10} | Support points |
| Real | d[size(xd, 1)] | | Derivatives at the support points |
| Real | dMonotone[size(xd, 1)] | | Derivatives at the support points |
| Boolean | ensureMonotonicity | true | |
Modelica definition
model CubicHermite
"Test problem for cubic hermite splines"
extends Modelica.Icons.Example;
parameter Real[:] xd={-1,1,5,6}
"Support points";
parameter Real[
size(xd, 1)] yd={-1,1,2,10}
"Support points";
parameter Real[
size(xd, 1)] d(
each fixed=false)
"Derivatives at the support points";
parameter Real[
size(xd, 1)] dMonotone(
each fixed=false)
"Derivatives at the support points";
parameter Boolean ensureMonotonicity=true;
Real x
"Independent variable";
Real y
"Dependent variable without monotone interpolation";
Real yMonotone
"Dependent variable with monotone interpolation";
Integer i
"Integer to select data interval";
initial algorithm
// Get the derivative values at the support points
d :=
Buildings.Utilities.Math.Functions.splineDerivatives(
x=xd,
y=yd,
ensureMonotonicity=false);
dMonotone :=
Buildings.Utilities.Math.Functions.splineDerivatives(x=xd, y=yd,
ensureMonotonicity=true);
algorithm
x := xd[1] + time*1.2*(xd[
size(xd, 1)] - xd[1]) - 0.5;
// i is a counter that is used to pick the derivative of d or dMonotonic
// that correspond to the interval that contains x
i := 1;
for j
in 1:
size(xd, 1) - 1
loop
if x > xd[j]
then
i := j;
end if;
end for;
// Extrapolate or interpolate the data
y :=
Buildings.Utilities.Math.Functions.cubicHermiteLinearExtrapolation(
x=x,
x1=xd[i],
x2=xd[i + 1],
y1=yd[i],
y2=yd[i + 1],
y1d=d[i],
y2d=d[i + 1]);
yMonotone :=
Buildings.Utilities.Math.Functions.cubicHermiteLinearExtrapolation(
x=x,
x1=xd[i],
x2=xd[i + 1],
y1=yd[i],
y2=yd[i + 1],
y1d=dMonotone[i],
y2d=dMonotone[i + 1]);
end CubicHermite;
Test problem for function that replaces 1/x around the origin by a twice continuously differentiable function
Information
This example tests the implementation of
Buildings.Utilities.Math.Functions.inverseXRegularized.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Parameters
| Type | Name | Default | Description |
|---|
| Real | delta | 0.5 | Small value for approximation |
Modelica definition
model InverseXRegularized
"Test problem for function that replaces 1/x around the origin by a twice continuously differentiable function"
extends Modelica.Icons.Example;
Real x
"Independent variable";
parameter Real delta = 0.5
"Small value for approximation";
Real y
"Function value";
Real xInv
"Function value";
equation
x=2*time-1;
xInv =
if (
abs(x) > 0.1)
then 1 / x
else 0;
y =
Buildings.Utilities.Math.Functions.inverseXRegularized(x=x, delta=delta);
end InverseXRegularized;
Tests the correct implementation of the function isMonotonic
Information
This example tests the correct implementation of the function
Buildings.Utilities.Math.Functions.isMonotonic.
If the function is implemented incorrect, the example will stop
with an error.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model IsMonotonic
"Tests the correct implementation of the function isMonotonic"
extends Modelica.Icons.Example;
Real x_incStrict[3] = {0, 1, 2}
"strictly increasing";
Real x_notMon[3] = {0, 3, 2}
"not monotonic";
Real x_incWeak[4] = {0, 1, 1, 2}
"weakly increasing";
Real x_notWeak[4] = {0, 3, 3, 2}
"not weakly monotonic";
Real x_decStrict[3] = {2.5, 2, 0.1}
"strictly decreasing";
Real x_decWeak[4] = {3, 1, 1, 0.5}
"weakly decreasing";
equation
// Tests with weak monotonicity
//strictly increasing
assert(
Buildings.Utilities.Math.Functions.isMonotonic(x_incStrict, strict=false),
"Error. Function should have returned true.");
//not monotonic
assert(false ==
Buildings.Utilities.Math.Functions.isMonotonic(x_notMon, strict=false),
"Error. Function should have returned true.");
//weakly increasing
assert(
Buildings.Utilities.Math.Functions.isMonotonic(x_incWeak, strict=false),
"Error. Function should have returned true.");
//not weakly monotonic
assert(false ==
Buildings.Utilities.Math.Functions.isMonotonic(x_notWeak, strict=false),
"Error. Function should have returned true.");
//strictly decreasing
assert(
Buildings.Utilities.Math.Functions.isMonotonic({2.5, 2, 0.1}, strict=false),
"Error. Function should have returned true.");
//weakly decreasing
assert(
Buildings.Utilities.Math.Functions.isMonotonic({3, 1, 1, 0.5}, strict=false),
"Error. Function should have returned true.");
// Tests with strict monotonicity
//strictly increasing
assert(
Buildings.Utilities.Math.Functions.isMonotonic(x_incStrict, strict=true),
"Error. Function should have returned true.");
//not monotonic
assert(false ==
Buildings.Utilities.Math.Functions.isMonotonic(x_notMon, strict=true),
"Error. Function should have returned true.");
//weakly increasing
assert(false ==
Buildings.Utilities.Math.Functions.isMonotonic(x_incWeak, strict=true),
"Error. Function should have returned true.");
//not weakly monotonic
assert(false ==
Buildings.Utilities.Math.Functions.isMonotonic(x_notWeak, strict=true),
"Error. Function should have returned true.");
//strictly decreasing
assert(
Buildings.Utilities.Math.Functions.isMonotonic(x_decStrict, strict=true),
"Error. Function should have returned true.");
//weakly decreasing
assert(false ==
Buildings.Utilities.Math.Functions.isMonotonic(x_decWeak, strict=true),
"Error. Function should have returned true.");
end IsMonotonic;
Information
This example checks whether the function derivative
is implemented correctly. If the derivative implementation
is incorrect, the model will stop with an assert statement.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model PolynomialDerivativeCheck
extends Modelica.Icons.Example;
Real x;
Real y;
initial equation
y=x;
equation
x=
Buildings.Utilities.Math.Functions.polynomial(x=time-2, a={2, 4, -4, 5});
der(y)=
der(x);
// Trigger an error if the derivative implementation is incorrect.
assert(
abs(x-y) < 1E-2, "Model has an error.");
end PolynomialDerivativeCheck;
Test problem for function that linearizes y=x^n below some threshold
Information
This example tests the implementation of
Buildings.Utilities.Math.Functions.powerLinearized.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model PowerLinearized
"Test problem for function that linearizes y=x^n below some threshold"
extends Modelica.Icons.Example;
Real T4(start=300^4)
"Temperature raised to 4-th power";
Real T
"Temperature";
Real TExact
"Temperature";
equation
T = (1+500*time);
T =
Buildings.Utilities.Math.Functions.powerLinearized(x=T4, x0=243.15^4, n=0.25);
TExact =
abs(T4)^(1/4);
end PowerLinearized;
Information
This example tests the implementation of
Buildings.Utilities.Math.Functions.regNonZeroPower.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model RegNonZeroPower
extends Modelica.Icons.Example;
Real y
"Function value";
equation
y=
Buildings.Utilities.Math.Functions.regNonZeroPower(
time, 0.3, 0.5);
end RegNonZeroPower;
Information
This example checks whether the function derivative
is implemented correctly. If the derivative implementation
is not correct, the model will stop with an assert statement.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Parameters
| Type | Name | Default | Description |
|---|
| Real | n | 0.33 | Exponent |
| Real | delta | 0.1 | Abscissa value where transition occurs |
Modelica definition
model RegNonZeroPowerDerivativeCheck
extends Modelica.Icons.Example;
parameter Real n=0.33
"Exponent";
parameter Real delta = 0.1
"Abscissa value where transition occurs";
Real x;
Real y;
initial equation
y=x;
equation
x=
Buildings.Utilities.Math.Functions.regNonZeroPower(
time,n, delta);
der(y)=
der(x);
assert(
abs(x-y) < 1E-2, "Model has an error");
end RegNonZeroPowerDerivativeCheck;
Information
This example checks whether the function derivative
is implemented correctly. If the derivative implementation
is not correct, the model will stop with an assert statement.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Parameters
| Type | Name | Default | Description |
|---|
| Real | n | 0.33 | Exponent |
| Real | delta | 0.1 | Abscissa value where transition occurs |
Modelica definition
model RegNonZeroPowerDerivative_2_Check
extends Modelica.Icons.Example;
parameter Real n=0.33
"Exponent";
parameter Real delta = 0.1
"Abscissa value where transition occurs";
Real x;
Real y;
initial equation
y=x;
equation
x=
Buildings.Utilities.Math.Functions.BaseClasses.der_regNonZeroPower(
time,n, delta, time);
der(y)=
der(x);
assert(
abs(x-y) < 1E-2, "Model has an error");
end RegNonZeroPowerDerivative_2_Check;
Information
This example checks whether the function derivative
is implemented correctly. If the derivative implementation
is not correct, the model will stop with an assert statement.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model SmoothExponentialDerivativeCheck
extends Modelica.Icons.Example;
Real x;
Real y;
Real ex
"exact function value";
initial equation
y=x;
equation
x=
Buildings.Utilities.Math.Functions.smoothExponential(
x=time-2, delta=0.5);
der(y)=
der(x);
assert(
abs(x-y) < 1E-2, "Model has an error");
ex=
exp(-
abs(time-2));
end SmoothExponentialDerivativeCheck;
Information
This example checks whether the function derivative
is implemented correctly. If the derivative implementation
is not correct, the model will stop with an assert statement.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model SpliceFunction
extends Modelica.Icons.Example;
Real y
"Function value";
equation
y=
Buildings.Utilities.Math.Functions.spliceFunction(
pos=10, neg=-10, x=time-0.4, deltax=0.2);
end SpliceFunction;
Information
This example checks whether the function derivative
is implemented correctly. If the derivative implementation
is not correct, the model will stop with an assert statement.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model SpliceFunctionDerivativeCheck
extends Modelica.Icons.Example;
Real x;
Real y;
initial equation
y=x;
equation
x=
Buildings.Utilities.Math.Functions.spliceFunction(
10, -10, time+0.1, 0.2);
der(y)=
der(x);
assert(
abs(x-y) < 1E-2, "Model has an error");
end SpliceFunctionDerivativeCheck;
Tests the correct implementation of the function trapezoidalIntegration
Information
Tests the correct implementation of function
Buildings.Utilities.Math.Functions.trapezoidalIntegration.
Integrands y1[7]={72, 70, 64, 54, 40, 22, 0} are the function values of y = -2*x^2-72 for x = {0,1,2,3,4,5,6}. The trapezoidal integration over the 7 integrand points should give a result of 286.
Extends from Modelica.Icons.Example (Icon for runnable examples).
Modelica definition
model TrapezoidalIntegration
"Tests the correct implementation of the function trapezoidalIntegration"
extends Modelica.Icons.Example;
Real y1[7] = {72, 70, 64, 54, 40, 22, 0};
//function values of y = -2*x^2-72 for x={0,1,2,3,4,5,6}
Real y
"Integration result";
//Real y2[7] = {0.3333, 1.0, 3.0, 9.9, 27.0, 81.0, 243.0}; // //function values of y = 3^(3x-1) for x=0:0.3333:2
algorithm
y :=
Buildings.Utilities.Math.Functions.trapezoidalIntegration(N=7, f=y1, deltaX=1);
assert(y - 286.0 < 1E-4,
"Error. Function should have returned 286.");
end TrapezoidalIntegration;
Automatically generated Mon May 4 10:27:20 2015.
CubicHermite
Buildings.Utilities.Math.Functions.Examples.CubicHermite