IEA EBC Annex 60

Annex60.Utilities.Math.Functions

Package with mathematical functions

Information

This package contains functions for commonly used mathematical operations. The functions are used in the blocks Annex60.Utilities.Math.

Extends from Modelica.Icons.VariantsPackage (Icon for package containing variants).

Package Content

Name	Description
average	Average of a vector
bicubic	Bicubic function
biquadratic	Biquadratic function
booleanReplicator	Replicates Boolean signals
cubicHermiteLinearExtrapolation	Interpolate using a cubic Hermite spline with linear extrapolation
integerReplicator	Replicates integer signals
inverseXRegularized	Function that approximates 1/x by a twice continuously differentiable function
isMonotonic	Returns true if the argument is a monotonic sequence
polynomial	Polynomial 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
regStep	Approximation of a general step, such that the approximation is continuous and differentiable
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
spliceFunction
splineDerivatives	Function to compute the derivatives for cubic hermite spline interpolation
trapezoidalIntegration	Integration using the trapezoidal rule
Examples	Collection of models that illustrate model use and test models
BaseClasses	Package with base classes for Annex60.Utilities.Math.Functions

Annex60.Utilities.Math.Functions.average

Average of a vector

Information

This function outputs the average of the vector.

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Modelica definition

Annex60.Utilities.Math.Functions.bicubic

Bicubic function

Information

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

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Modelica definition

Annex60.Utilities.Math.Functions.biquadratic

Biquadratic function

Information

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

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Modelica definition

Annex60.Utilities.Math.Functions.booleanReplicator

Replicates Boolean signals

Information

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

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Modelica definition

Annex60.Utilities.Math.Functions.cubicHermiteLinearExtrapolation

Interpolate using a cubic Hermite spline with linear extrapolation

Information

For x₁ < x < x₂, this function interpolates using cubic hermite spline. For x outside this interval, the function linearly extrapolates.

For how to use this function, see Annex60.Utilities.Math.Functions.Examples.CubicHermite.

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Modelica definition

Annex60.Utilities.Math.Functions.integerReplicator

Replicates integer signals

Information

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

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Outputs

Modelica definition

Annex60.Utilities.Math.Functions.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 plot of Annex60.Utilities.Math.Functions.Examples.InverseXRegularized for the graph.

For efficiency, the polynomial coefficients a, b, c, d, e, f and the inverse of the smoothing parameter deltaInv are exposed as arguments to this function. Typically, these coefficients only depend on parameters and hence can be computed once. They must be equal to their default values, otherwise the function is not twice continuously differentiable. By exposing these coefficients as function arguments, models that call this function can compute them as parameters, and assign these parameter values in the function call. This avoids that the coefficients are evaluated for each time step, as they would otherwise be if they were to be computed inside the body of the function. However, assigning the values is optional as otherwise, at the expense of efficiency, the values will be computed each time the function is invoked. See Annex60.Utilities.Math.Functions.Examples.InverseXRegularized for how to efficiently call this function.

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Modelica definition

Annex60.Utilities.Math.Functions.isMonotonic

Returns true if the argument is a monotonic sequence

Information

This function returns true if its argument is monotonic increasing or decreasing, and false otherwise. If strict=true, then strict monotonicity is tested, otherwise weak monotonicity is tested.

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Modelica definition

Annex60.Utilities.Math.Functions.polynomial

Polynomial function

Information

y = a₁ + a₂ x + a₃ x² + ...

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Modelica definition

Annex60.Utilities.Math.Functions.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.

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Modelica definition

Annex60.Utilities.Math.Functions.quadraticLinear

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

Information

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

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Modelica definition

Annex60.Utilities.Math.Functions.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.

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Modelica definition

Annex60.Utilities.Math.Functions.regStep

Approximation of a general step, such that the approximation is continuous and differentiable

Information

This function is used to approximate the equation

    y = if x > 0 then y1 else y2;

by a smooth characteristic, so that the expression is continuous and differentiable:

   y = smooth(1, if x >  x_small then y1 else
                 if x < -x_small then y2 else f(y1, y2));

In the region -x_small < x < x_small a 2nd order polynomial is used for a smooth transition from y1 to y2.

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

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Modelica definition

Annex60.Utilities.Math.Functions.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| < δ for some positive δ

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Modelica definition

Annex60.Utilities.Math.Functions.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.

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Modelica definition

Annex60.Utilities.Math.Functions.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 [l, u].

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

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Modelica definition

Annex60.Utilities.Math.Functions.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.

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Modelica definition

Annex60.Utilities.Math.Functions.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.

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Modelica definition

Annex60.Utilities.Math.Functions.spliceFunction

Information

Function to provide a once continuously differentiable transition between to arguments.

The function is adapted from Modelica.Media.Air.MoistAir.Utilities.spliceFunction and provided here for easier accessability to model developers.

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Modelica definition

Annex60.Utilities.Math.Functions.splineDerivatives

Function to compute the derivatives for cubic hermite spline interpolation

Information

This function computes the derivatives at the support points x_i that can be used as input for evaluating a cubic hermite spline.

If ensureMonotonicity=true, then the support points y_i need to be monotone increasing (or increasing), and the computed derivatives d_i are such that the cubic hermite is monotone increasing (or decreasing). The algorithm to ensure monotonicity is based on the method described in Fritsch and Carlson (1980) for ρ = ρ₂.

This function is typically used with Annex60.Utilities.Math.Functions.cubicHermiteLinearExtrapolation which is used to evaluate the cubic spline. Because in many applications, the shape of the spline depends on parameters, this function has been implemented in such a way that all derivatives can be computed at once and then stored for use during the time stepping, in which the above function may be called.

References

F.N. Fritsch and R.E. Carlson, Monotone piecewise cubic interpolation. SIAM J. Numer. Anal., 17 (1980), pp. 238-246.

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Modelica definition

Annex60.Utilities.Math.Functions.trapezoidalIntegration

Integration using the trapezoidal rule

Information

This function computes a definite integral using the trapezoidal rule.

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Modelica definition