Description: "Collection of models that illustrate model use and test models"

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## Buildings.Utilities.Math.Functions.Examples

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

Name | Description
--- | ---
CubicHermite | Test problem for cubic hermite splines
InverseXRegularized | Test problem for function that replaces 1/x around the origin by a twice continuously differentiable function
IsMonotonic | Tests the correct implementation of the function isMonotonic
PolynomialDerivativeCheck |
PowerLinearized | Test problem for function that linearizes y=x^n below some threshold
RegNonZeroPower |
RegNonZeroPowerDerivativeCheck |
RegNonZeroPowerDerivative_2_Check |
SmoothExponentialDerivativeCheck |
SpliceFunction |
SpliceFunctionDerivativeCheck |
TrapezoidalIntegration | Tests the correct implementation of the function trapezoidalIntegration

## Buildings.Utilities.Math.Functions.Examples.CubicHermite

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;

## Buildings.Utilities.Math.Functions.Examples.InverseXRegularized

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;

## Buildings.Utilities.Math.Functions.Examples.IsMonotonic

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;

## Buildings.Utilities.Math.Functions.Examples.PolynomialDerivativeCheck

### 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;

## Buildings.Utilities.Math.Functions.Examples.PowerLinearized

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;

## Buildings.Utilities.Math.Functions.Examples.RegNonZeroPower

### 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;

## Buildings.Utilities.Math.Functions.Examples.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 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;

## Buildings.Utilities.Math.Functions.Examples.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).

### 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;

## Buildings.Utilities.Math.Functions.Examples.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 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;

## Buildings.Utilities.Math.Functions.Examples.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 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;

## Buildings.Utilities.Math.Functions.Examples.SpliceFunctionDerivativeCheck

### 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;

## Buildings.Utilities.Math.Functions.Examples.TrapezoidalIntegration

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;