Buildings.Utilities.Math.Functions

Package with mathematical functions

Information

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

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

Package Content

Name	Description
average	Average of a vector
besselJ0	Bessel function of the first kind of order 0, J0
besselJ1	Bessel function of the first kind of order 1, J1
besselY0	Bessel function of the second kind of order 0, Y0
besselY1	Bessel function of the second kind of order 1, Y1
bicubic	Bicubic function
binomial	Returns the binomial coefficient
biquadratic	Biquadratic function
booleanReplicator	Replicates Boolean signals
cubicHermiteLinearExtrapolation	Interpolate using a cubic Hermite spline with linear extrapolation
exponentialIntegralE1	Exponential integral, E1
factorial	Returns the value n! as an integer
fallingFactorial	Returns the k-th falling factorial of n
integerReplicator	Replicates integer signals
interpolate	Function for cubic hermite spline interpolation of table data
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
quinticHermite	Quintic Hermite spline function for interpolation between two functions with a continuous second derivative
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
round	Round real number to specified digits
smoothExponential	Once continuously differentiable approximation to exp(-|x|) in interval |x| < delta
smoothHeaviside	Twice continuously differentiable approximation to the Heaviside function
smoothInterpolation	Interpolate using a cubic Hermite spline with linear extrapolation for a vector xSup[], ySup[] and independent variable x
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 Buildings.Utilities.Math.Functions

Buildings.Utilities.Math.Functions.average

Average of a vector

Information

This function outputs the average of the vector.

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

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Modelica definition

Buildings.Utilities.Math.Functions.besselJ0

Bessel function of the first kind of order 0, J0

Information

Evaluates the Bessel function of the first kind of order 0 (J₀), based on the implementations of Press et al. (1986).

References

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes - The Art of Scientific Computing. Cambridge University Press (1986): 988 p.

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

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Modelica definition

Buildings.Utilities.Math.Functions.besselJ1

Bessel function of the first kind of order 1, J1

Information

Evaluates the Bessel function of the first kind of order 1 (J₁), based on the implementations of Press et al. (1986).

References

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes - The Art of Scientific Computing. Cambridge University Press (1986): 988 p.

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

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Modelica definition

Buildings.Utilities.Math.Functions.besselY0

Bessel function of the second kind of order 0, Y0

Information

Evaluates the Bessel function of the second kind of order 0 (Y₀), based on the implementations of Press et al. (1986).

References

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes - The Art of Scientific Computing. Cambridge University Press (1986): 988 p.

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

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Modelica definition

Buildings.Utilities.Math.Functions.besselY1

Bessel function of the second kind of order 1, Y1

Information

Evaluates the Bessel function of the second kind of order 1 (Y₁), based on the implementations of Press et al. (1986).

References

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes - The Art of Scientific Computing. Cambridge University Press (1986): 988 p.

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

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Modelica definition

Buildings.Utilities.Math.Functions.bicubic

Bicubic function

Information

This function 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.Icons.Function (Icon for functions).

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Modelica definition

Buildings.Utilities.Math.Functions.binomial

Returns the binomial coefficient

Information

Function that evaluates the binomial coefficient "n choose k".

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

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Modelica definition

Buildings.Utilities.Math.Functions.biquadratic

Biquadratic function

Information

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

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

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Modelica definition

Buildings.Utilities.Math.Functions.booleanReplicator

Replicates Boolean signals

Information

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

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

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Modelica definition

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

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

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Modelica definition

Buildings.Utilities.Math.Functions.exponentialIntegralE1

Exponential integral, E1

Information

Evaluates the exponential integral (E₁), based on the polynomial and rational approximations of Abramowitz and Stegun (1964).

References

Abramowitz, Milton, and Irene A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards. (1964): 1046 p.

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

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Modelica definition

Buildings.Utilities.Math.Functions.factorial

Returns the value n! as an integer

Information

Function that returns the factorial n! for 0 ≤ n ≤ 12.

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

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Modelica definition

Buildings.Utilities.Math.Functions.fallingFactorial

Returns the k-th falling factorial of n

Information

Function that evaluates the falling factorial "k-permutations of n".

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

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Modelica definition

Buildings.Utilities.Math.Functions.integerReplicator

Replicates integer signals

Information

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

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

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Modelica definition

Buildings.Utilities.Math.Functions.interpolate

Function for cubic hermite spline interpolation of table data

Information

This function returns the value on a cubic hermite spline through the given support points and provided spline derivatives at these points. The spline derivatives at the support points can be computed using Buildings.Utilities.Math.Functions.splineDerivatives. The support points must be monotonically increasing. Outside the provided support points, the function returns a linear extrapolation with the same slope as the cubic spline has at the respective support point.

A similar model is also used in the CONTAM software (Dols and Walton, 2015).

This function is used in Buildings.Airflow.Multizone.Table_m_flow and Buildings.Airflow.Multizone.Table_V_flow

References

• W. S. Dols and B. J. Polidoro,2015. CONTAM User Guide and Program Documentation Version 3.2, National Institute of Standards and Technology, NIST TN 1887, Sep. 2015. doi: 10.6028/NIST.TN.1887.

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

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Modelica definition

Buildings.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 Buildings.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 Buildings.Utilities.Math.Functions.Examples.InverseXRegularized for how to efficiently call this function.

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

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Modelica definition

Buildings.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.

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

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Modelica definition

Buildings.Utilities.Math.Functions.polynomial

Polynomial function

Information

y = a₁ + a₂ x + a₃ x² + ...

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

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Modelica definition

Buildings.Utilities.Math.Functions.powerLinearized

Power function that is linearized below a user-defined threshold

Information

• the function is defined and monotonically 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.Icons.Function (Icon for functions).

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Modelica definition

Buildings.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

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

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Modelica definition

Buildings.Utilities.Math.Functions.quinticHermite

Quintic Hermite spline function for interpolation between two functions with a continuous second derivative

Information

Returns the result y of a quintic Hermite spline, which is a C² continuous interpolation between two functions. The abscissa value x has to be between x1 and x2. Variables y1, y1d, y1dd are the ordinate, ordinate derivative and ordinate second derivative of the function at x1. Variables y2, y2d, y2dd are respectively the ordinate, ordinate derivative and ordinate second derivative of the function at x2.

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

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Modelica definition

Buildings.Utilities.Math.Functions.regNonZeroPower

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

Information

Function that approximates y=|x|ⁿ where 0 < n < 2 so that

• 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.

See the package Examples for the graph.

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

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Modelica definition

Buildings.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

Buildings.Utilities.Math.Functions.round

Round real number to specified digits

Information

Function that outputs the input after rounding it to n digits.

For example,

• set n = 0 to round to the nearest integer,
• set n = 1 to round to the next decimal point, and
• set n = -1 to round to the next multiple of ten.

Hence, the function outputs

    y = floor(x*(10^n) + 0.5)/(10^n)  for  x > 0,
    y = ceil( x*(10^n) - 0.5)/(10^n)  for  x < 0.

To use this function as a block, use Buildings.Controls.OBC.CDL.Reals.Round.

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

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Modelica definition

Buildings.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

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

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Modelica definition

Buildings.Utilities.Math.Functions.smoothHeaviside

Twice continuously differentiable approximation to the Heaviside function

Information

Twice Lipschitz continuously differentiable approximation to the Heaviside(.,.) function.
Function is derived from a quintic polynomial going through (0,0) and (1,1), with zero first and second order derivatives at those points.
See Example Buildings.Utilities.Math.Examples.SmoothHeaviside.

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

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Modelica definition

Buildings.Utilities.Math.Functions.smoothInterpolation

Interpolate using a cubic Hermite spline with linear extrapolation for a vector xSup[], ySup[] and independent variable x

Information

For xSup₁ ≤ x ≤ xSup_n, where n is the size of the support points xSup, which must be strictly monotonically increasing, this function interpolates using cubic hermite spline. For x outside this interval, the function linearly extrapolates.

If n=2, linear interpolation is used an if n=1, the function value y¹ is returned.

Note that if xSup and ySup only depend on parameters or constants, and therefore will not change during the simulation, it is more efficient to first call Buildings.Utilities.Math.Functions.splineDerivatives to find the derivatives, and then call Buildings.Utilities.Math.Functions.interpolate to perform the interpolation. This way the derivatives only need to be computed once upon initialisation, not at each step during the simulation. See the example implemented in Buildings.Utilities.Math.Functions.Examples.Interpolate.

In contrast to the function Modelica.Math.Vectors.interpolate which provides linear interpolation, this function does not trigger events.

For how to use this function, see Buildings.Utilities.Math.Functions.Examples.SmoothInterpolation.

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

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Modelica definition

Buildings.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 Buildings.Utilities.Math.Examples.SmoothMin.

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

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Modelica definition

Buildings.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 Buildings.Utilities.Math.Examples.SmoothMin.

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

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Modelica definition

Buildings.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 Buildings.Utilities.Math.Examples.SmoothMin.

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

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Modelica definition

Buildings.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.

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

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Modelica definition

Buildings.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 decreasing), 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 Buildings.Utilities.Math.Functions.interpolate which is used to evaluate the cubic spline. Because in many applications, the shape of the spline depends on parameters which will no longer change once the initialisation is complete, this function computes and returns the derivatives so that they can be stored by the calling model to avoid repetitive computations.

References

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

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

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Modelica definition

Buildings.Utilities.Math.Functions.trapezoidalIntegration

Integration using the trapezoidal rule

Information

This function computes a definite integral using the trapezoidal rule.

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

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Modelica definition