IEA EBC Annex 60

Annex60.Utilities.Math

Library with functions such as for smoothing

Information

This package contains blocks and functions for commonly used mathematical operations. The classes in this package augment the classes Modelica.Blocks.

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

Package Content

Name	Description
Average	Average of a vector
Bicubic	Bicubic function
Biquadratic	Biquadratic function
BooleanReplicator	Boolean signal replicator
IntegerReplicator	Integer signal replicator
IntegratorWithReset	Output the integral of the input signal
InverseXRegularized	Function that approximates 1/x by a twice continuously differentiable function
Max	Maximum element of a vector
Min	Minimum element of a vector
Polynominal	Polynominal function
PowerLinearized	Power function that is linearized below a user-defined threshold
QuadraticLinear	Function that is quadratic in first argument and linear in second argument
RegNonZeroPower	Power function, regularized near zero, but nonzero value for x=0
SmoothExponential	Once continuously differentiable approximation to exp(-|x|) in interval |x| < delta
SmoothHeaviside	Once continuously differentiable approximation to the Heaviside function
SmoothLimit	Once continuously differentiable approximation to the limit function
SmoothMax	Once continuously differentiable approximation to the maximum function
SmoothMin	Once continuously differentiable approximation to the minimum function
Splice	Block for splice function opertation
TrapezoidalIntegration	Integration using the trapezoidal rule
Functions	Package with mathematical functions
Examples	Collection of models that illustrate model use and test models

Annex60.Utilities.Math.Average

Average of a vector

Information

This block outputs the average of the vector.

Extends from Modelica.Blocks.Interfaces.MISO (Multiple Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.Bicubic

Bicubic function

Information

This block computes

y = a₁ + a₂ x₁ + a₃ x₁² + a₄ x₂ + a₅ x₂² + a₆ x₁ x₂ + a₇ x₁^3 + a₈ x₂^3 + a₉ x₁² x₂ + a₁0 x₁ x₂²

Extends from Modelica.Blocks.Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Connectors

Modelica definition

Annex60.Utilities.Math.Biquadratic

Biquadratic function

Information

This block computes

y = a₁ + a₂ x₁ + a₃ x₁² + a₄ x₂ + a₅ x₂² + a₆ x₁ x₂

Extends from Modelica.Blocks.Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.BooleanReplicator

Boolean signal replicator

Information

This block replicates the boolean input signal to an array of nout identical output signals.

Extends from Modelica.Blocks.Icons.BooleanBlock (Basic graphical layout of Boolean block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.IntegerReplicator

Integer signal replicator

Information

This block replicates the integer input signal to an array of nout identical output signals.

Extends from Modelica.Blocks.Icons.IntegerBlock (Basic graphical layout of Integer block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.IntegratorWithReset

Output the integral of the input signal

Information

This model is similar to Modelica.Blocks.Continuous.Integrator except that it optionally allows to reset the output y of the integrator.

The output of the integrator can be reset as follows:

• If reset = Annex60.Types.Reset.Disabled, which is the default, then the integrator is never reset.
• If reset = Annex60.Types.Reset.Parameter, then a boolean input signal trigger is enabled. Whenever the value of this input changes from false to true, the integrator is reset by setting y to the value of the parameter y_reset.
• If reset = Annex60.Types.Reset.Input, then a boolean input signal trigger is enabled. Whenever the value of this input changes from false to true, the integrator is reset by setting y to the value of the input signal y_reset_in.

See Annex60.Utilities.Math.Examples.IntegratorWithReset for an example.

Implementation

To adjust the icon layer, the code of Modelica.Blocks.Continuous.Integrator has been copied into this model rather than extended.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.InverseXRegularized

Function that approximates 1/x by a twice continuously differentiable function

Information

Function that approximates y=1 ⁄ x inside the interval -δ ≤ x ≤ δ. The approximation is twice continuously differentiable with a bounded derivative on the whole real line.

See the package Examples for the graph.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.Max

Maximum element of a vector

Information

Outputs the maximum of the vector.

Extends from Modelica.Blocks.Interfaces.MISO (Multiple Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.Min

Minimum element of a vector

Information

Outputs the minimum of the vector.

Extends from Modelica.Blocks.Interfaces.MISO (Multiple Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.Polynominal

Polynominal function

Information

This block computes a polynomial of arbitrary order. The polynomial has the form

y = a1 + a2 x + a3 x2 + ...

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.PowerLinearized

Power function that is linearized below a user-defined threshold

Information

• the function is defined and monotone increasing for all x.
• dy/dx is bounded and continuous everywhere (for n < 1).

For x < x₀, this function replaces y=xⁿ by a linear function that is continuously differentiable everywhere.

A typical use of this function is to replace T = T4^(1/4) in a radiation balance to ensure that the function is defined everywhere. This can help solving the initialization problem when a solver may be far from a solution and hence T4 < 0.

See the package Examples for the graph.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.QuadraticLinear

Function that is quadratic in first argument and linear in second argument

Information

Block for function quadraticLinear, which computes

y = a1 + a2 x1 + a3 x12 + (a4 + a5 x1 + a6 x12) x2

Extends from Modelica.Blocks.Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.RegNonZeroPower

Power function, regularized near zero, but nonzero value for x=0

Information

• y(0) is not equal to zero.
• dy/dx is bounded and continuous everywhere.

This function replaces y=|x|ⁿ in the interval -δ...+δ by a 4-th order polynomial that has the same function value and the first and second derivative at x=± δ.

A typical use of this function is to replace the function for the convective heat transfer coefficient for forced or free convection that is of the form h=c |dT|ⁿ for some constant c and exponent 0 ≤ n ≤ 1. By using this function, the original function that has an infinite derivative near zero and that takes on zero at the origin is replaced by a function with a bounded derivative and a non-zero value at the origin. Physically, the region -δ...+δ may be interpreted as the region where heat conduction dominates convection in the boundary layer.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.SmoothExponential

Once continuously differentiable approximation to exp(-|x|) in interval |x| < delta

Information

Function to provide a once continuously differentiable approximation to exp(- |x| ) in the interval |x| for some positive δ

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.SmoothHeaviside

Once continuously differentiable approximation to the Heaviside function

Information

Once Lipschitz continuously differentiable approximation to the Heaviside(.,.) function. See Example Annex60.Utilities.Math.Examples.SmoothHeaviside.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.SmoothLimit

Once continuously differentiable approximation to the limit function

Information

Once continuously differentiable approximation to the limit(.,.) function. The output is bounded to be in [lower, upper].

Note that the limit need not be respected, such as illustrated in Annex60.Utilities.Math.Examples.SmoothMin.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.SmoothMax

Once continuously differentiable approximation to the maximum function

Information

Once continuously differentiable approximation to the max(.,.) function.

Note that the maximum need not be respected, such as illustrated in Annex60.Utilities.Math.Examples.SmoothMin.

Extends from Modelica.Blocks.Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.SmoothMin

Once continuously differentiable approximation to the minimum function

Information

Once continuously differentiable approximation to the min(.,.) function.

Note that the minimum need not be respected, such as illustrated in Annex60.Utilities.Math.Examples.SmoothMin.

Extends from Modelica.Blocks.Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.Splice

Block for splice function opertation

Information

This block implements Annex60.Utilities.Math.Functions.spliceFunction, which provides a continuously differentiable transition between two arguments.

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica definition

Annex60.Utilities.Math.TrapezoidalIntegration

Integration using the trapezoidal rule

Information

This function computes a definite integral using the trapezoidal rule.

Extends from Modelica.Blocks.Interfaces.MISO (Multiple Input Single Output continuous control block).

Parameters

Connectors

Modelica definition