Annex60.Utilities.Math.Functions.BaseClasses

IEA EBC Annex 60

Annex60.Utilities.Math.Functions.BaseClasses

Package with base classes for Annex60.Utilities.Math.Functions

Information

This package contains base classes that are used to construct the models in Annex60.Utilities.Math.Functions.

Extends from Modelica.Icons.BasesPackage (Icon for packages containing base classes).

Package Content

Name	Description
$Annex60.Utilities.Math.Functions.BaseClasses.der_2_regNonZeroPower$ der_2_regNonZeroPower	Power function, regularized near zero, but nonzero value for x=0
$Annex60.Utilities.Math.Functions.BaseClasses.der_2_smoothTransition$ der_2_smoothTransition	Second order derivative of smoothTransition with respect to x
$Annex60.Utilities.Math.Functions.BaseClasses.der_inverseXRegularized$ der_inverseXRegularized	Derivative of inverseXRegularised function
$Annex60.Utilities.Math.Functions.BaseClasses.der_regNonZeroPower$ der_regNonZeroPower	Power function, regularized near zero, but nonzero value for x=0
$Annex60.Utilities.Math.Functions.BaseClasses.der_smoothTransition$ der_smoothTransition	First order derivative of smoothTransition with respect to x
$Annex60.Utilities.Math.Functions.BaseClasses.der_spliceFunction$ der_spliceFunction	Derivative of splice function
$Annex60.Utilities.Math.Functions.BaseClasses.smoothTransition$ smoothTransition	Twice continuously differentiable transition between the regions

Annex60.Utilities.Math.Functions.BaseClasses.der_2_regNonZeroPower

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

Information

Implementation of the second derivative of the function Annex60.Utilities.Math.Functions.regNonZeroPower.

Inputs

Type	Name	Default	Description
Real	x		Abscissa value
Real	n		Exponent
Real	delta	0.01	Abscissa value where transition occurs
Real	der_x
Real	der_2_x

Outputs

Type	Name	Description
Real	der_2_y	Function value

Modelica definition

function der_2_regNonZeroPower "Power function, regularized near zero, but nonzero value for x=0" input Real x "Abscissa value"; input Real n "Exponent"; input Real delta = 0.01 "Abscissa value where transition occurs"; input Real der_x; input Real der_2_x; output Real der_2_y "Function value"; protected Real a1; Real a3; Real delta2; Real x2; Real y_d "=y(delta)"; Real yP_d "=dy(delta)/dx"; Real yPP_d "=d^2y(delta)/dx^2"; algorithm if abs(x) > delta then der_2_y := n*(abs(x)^(n-1)*sign(x)*der_2_x + (n-1)*abs(x)^(n-2)*der_x^2); else delta2 :=delta*delta; x2 :=x*x; y_d :=delta^n; yP_d :=n*delta^(n - 1); yPP_d :=n*(n - 1)*delta^(n - 2); a1 := -(yP_d/delta - yPP_d)/delta2/8; a3 := (yPP_d - 12 * a1 * delta2)/2; der_2_y := (12*a1*x2+2*a3)*der_x^2 +x * ( 4 * a1 * x2 + 2 * a3)*der_2_x; end if; end der_2_regNonZeroPower;

Annex60.Utilities.Math.Functions.BaseClasses.der_2_smoothTransition

Second order derivative of smoothTransition with respect to x

Information

This function is the 2nd order derivative of Annex60.Utilities.Math.Functions.BaseClasses.smoothTransition.

Implementation

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.

Inputs

Type	Name	Description
Real	x	Abscissa value
Real	delta	Abscissa value below which approximation occurs
Real	deltaInv	Inverse value of delta
Real	a	Polynomial coefficient
Real	b	Polynomial coefficient
Real	c	Polynomial coefficient
Real	d	Polynomial coefficient
Real	e	Polynomial coefficient
Real	f	Polynomial coefficient
Real	x_der	Derivative of x
Real	x_der2	Second order derivative of x

Outputs

Type	Name	Description
Real	y_der2	Second order derivative of function value

Modelica definition

function der_2_smoothTransition "Second order derivative of smoothTransition with respect to x" input Real x "Abscissa value"; input Real delta(min=Modelica.Constants.eps) "Abscissa value below which approximation occurs"; input Real deltaInv "Inverse value of delta"; input Real a "Polynomial coefficient"; input Real b "Polynomial coefficient"; input Real c "Polynomial coefficient"; input Real d "Polynomial coefficient"; input Real e "Polynomial coefficient"; input Real f "Polynomial coefficient"; input Real x_der "Derivative of x"; input Real x_der2 "Second order derivative of x"; output Real y_der2 "Second order derivative of function value"; protected Real aX "Absolute value of x"; Real ex "Intermediate expression"; algorithm aX:= abs(x); ex := 2*c + aX*(6*d + aX*(12*e + aX*20*f)); y_der2 := (b + aX*(2*c + aX*(3*d + aX*(4*e + aX*5*f))))*x_der2 + x_der*x_der*( if x > 0 then ex else -ex); end der_2_smoothTransition;

Annex60.Utilities.Math.Functions.BaseClasses.der_inverseXRegularized

Derivative of inverseXRegularised function

Information

Implementation of the first derivative of the function Annex60.Utilities.Math.Functions.inverseXRegularized.

Inputs

Type	Name	Default	Description
Real	x		Abscissa value
Real	delta		Abscissa value below which approximation occurs
Real	deltaInv	1/delta	Inverse value of delta
Real	a	-15*deltaInv	Polynomial coefficient
Real	b	119*deltaInv^2	Polynomial coefficient
Real	c	-361*deltaInv^3	Polynomial coefficient
Real	d	534*deltaInv^4	Polynomial coefficient
Real	e	-380*deltaInv^5	Polynomial coefficient
Real	f	104*deltaInv^6	Polynomial coefficient
Real	x_der		Abscissa value

Outputs

Type	Name	Description
Real	y_der	Function value

Modelica definition

function der_inverseXRegularized "Derivative of inverseXRegularised function" input Real x "Abscissa value"; input Real delta(min=Modelica.Constants.eps) "Abscissa value below which approximation occurs"; input Real deltaInv = 1/delta "Inverse value of delta"; input Real a = -15*deltaInv "Polynomial coefficient"; input Real b = 119*deltaInv^2 "Polynomial coefficient"; input Real c = -361*deltaInv^3 "Polynomial coefficient"; input Real d = 534*deltaInv^4 "Polynomial coefficient"; input Real e = -380*deltaInv^5 "Polynomial coefficient"; input Real f = 104*deltaInv^6 "Polynomial coefficient"; input Real x_der "Abscissa value"; output Real y_der "Function value"; algorithm y_der :=if (x > delta or x < -delta) then -x_der/x/x elseif (x < delta/2 and x > -delta/2) then x_der/(delta*delta) else Annex60.Utilities.Math.Functions.BaseClasses.der_smoothTransition( x=x, x_der=x_der, delta=delta, deltaInv=deltaInv, a=a, b=b, c=c, d=d, e=e, f=f); end der_inverseXRegularized;

Annex60.Utilities.Math.Functions.BaseClasses.der_regNonZeroPower

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

Information

Implementation of the first derivative of the function Annex60.Utilities.Math.Functions.regNonZeroPower.

Inputs

Type	Name	Default	Description
Real	x		Abscissa value
Real	n		Exponent
Real	delta	0.01	Abscissa value where transition occurs
Real	der_x

Outputs

Type	Name	Description
Real	der_y	Function value

Modelica definition

function der_regNonZeroPower "Power function, regularized near zero, but nonzero value for x=0" annotation(derivative=der_2_regNonZeroPower); input Real x "Abscissa value"; input Real n "Exponent"; input Real delta = 0.01 "Abscissa value where transition occurs"; input Real der_x; output Real der_y "Function value"; protected Real a1; Real a3; Real delta2; Real x2; Real y_d "=y(delta)"; Real yP_d "=dy(delta)/dx"; Real yPP_d "=d^2y(delta)/dx^2"; algorithm if abs(x) > delta then der_y := sign(x)*n*abs(x)^(n-1)*der_x; else delta2 :=delta*delta; x2 :=x*x; y_d :=delta^n; yP_d :=n*delta^(n - 1); yPP_d :=n*(n - 1)*delta^(n - 2); a1 := -(yP_d/delta - yPP_d)/delta2/8; a3 := (yPP_d - 12 * a1 * delta2)/2; der_y := x * ( 4 * a1 * x * x + 2 * a3) * der_x; end if; end der_regNonZeroPower;

Annex60.Utilities.Math.Functions.BaseClasses.der_smoothTransition

First order derivative of smoothTransition with respect to x

Information

This function is the 1st order derivative of Annex60.Utilities.Math.Functions.BaseClasses.smoothTransition.

Implementation

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. Also, its derivative is provided in Annex60.Utilities.Math.Functions.BaseClasses.der_2__smoothTransition.

Inputs

Type	Name	Description
Real	x	Abscissa value
Real	delta	Abscissa value below which approximation occurs
Real	deltaInv	Inverse value of delta
Real	a	Polynomial coefficient
Real	b	Polynomial coefficient
Real	c	Polynomial coefficient
Real	d	Polynomial coefficient
Real	e	Polynomial coefficient
Real	f	Polynomial coefficient
Real	x_der	Derivative of x

Outputs

Type	Name	Description
Real	y_der	Derivative of function value

Modelica definition

function der_smoothTransition "First order derivative of smoothTransition with respect to x" annotation(derivative=Annex60.Utilities.Math.Functions.BaseClasses.der_2_smoothTransition); input Real x "Abscissa value"; input Real delta(min=Modelica.Constants.eps) "Abscissa value below which approximation occurs"; input Real deltaInv "Inverse value of delta"; input Real a "Polynomial coefficient"; input Real b "Polynomial coefficient"; input Real c "Polynomial coefficient"; input Real d "Polynomial coefficient"; input Real e "Polynomial coefficient"; input Real f "Polynomial coefficient"; input Real x_der "Derivative of x"; output Real y_der "Derivative of function value"; protected Real aX "Absolute value of x"; algorithm aX:= abs(x); y_der := (b + aX*(2*c + aX*(3*d + aX*(4*e + aX*5*f))))*x_der; end der_smoothTransition;

Annex60.Utilities.Math.Functions.BaseClasses.der_spliceFunction

Derivative of splice function

Information

Implementation of the first derivative of the function Annex60.Utilities.Math.Functions.spliceFunction.

Inputs

Type	Name	Default
Real	pos
Real	neg
Real	x
Real	deltax	1
Real	dpos
Real	dneg
Real	dx
Real	ddeltax	0

Outputs

Type	Name	Description
Real	out

Modelica definition

function der_spliceFunction "Derivative of splice function" input Real pos; input Real neg; input Real x; input Real deltax=1; input Real dpos; input Real dneg; input Real dx; input Real ddeltax=0; output Real out; protected Real scaledX; Real scaledX1; Real dscaledX1; Real y; constant Real asin1 = Modelica.Math.asin(1); algorithm scaledX1 := x/deltax; if scaledX1 <= -0.99999999999 then out := dneg; elseif scaledX1 >= 0.9999999999 then out := dpos; else scaledX := scaledX1*asin1; dscaledX1 := (dx - scaledX1*ddeltax)/deltax; y := (Modelica.Math.tanh(Modelica.Math.tan(scaledX)) + 1)/2; out := dpos*y + (1 - y)*dneg; out := out + (pos - neg)*dscaledX1*asin1/2/( Modelica.Math.cosh(Modelica.Math.tan(scaledX))*Modelica.Math.cos( scaledX))^2; end if; end der_spliceFunction;

Annex60.Utilities.Math.Functions.BaseClasses.smoothTransition

Twice continuously differentiable transition between the regions

Information

This function is used by Annex60.Utilities.Math.Functions.inverseXRegularized to provide a twice continuously differentiable transition between the different regions. The code has been implemented in a function as this allows to implement the function Annex60.Utilities.Math.Functions.inverseXRegularized in such a way that Dymola inlines it. However, this function will not be inlined as its body is too large.

Implementation

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. Also, derivatives are provided in Annex60.Utilities.Math.Functions.BaseClasses.der_smoothTransition and in Annex60.Utilities.Math.Functions.BaseClasses.der_2__smoothTransition.

Inputs

Type	Name	Default	Description
Real	x		Abscissa value
Real	delta		Abscissa value below which approximation occurs
Real	deltaInv	1/delta	Inverse value of delta
Real	a	-15*deltaInv	Polynomial coefficient
Real	b	119*deltaInv^2	Polynomial coefficient
Real	c	-361*deltaInv^3	Polynomial coefficient
Real	d	534*deltaInv^4	Polynomial coefficient
Real	e	-380*deltaInv^5	Polynomial coefficient
Real	f	104*deltaInv^6	Polynomial coefficient

Outputs

Type	Name	Description
Real	y	Function value

Modelica definition

function smoothTransition "Twice continuously differentiable transition between the regions" annotation(derivative=Annex60.Utilities.Math.Functions.BaseClasses.der_smoothTransition); // The function that transitions between the regions is implemented // using its own function. This allows Dymola 2016 to inline the function // inverseXRegularized. input Real x "Abscissa value"; input Real delta(min=Modelica.Constants.eps) "Abscissa value below which approximation occurs"; input Real deltaInv = 1/delta "Inverse value of delta"; input Real a = -15*deltaInv "Polynomial coefficient"; input Real b = 119*deltaInv^2 "Polynomial coefficient"; input Real c = -361*deltaInv^3 "Polynomial coefficient"; input Real d = 534*deltaInv^4 "Polynomial coefficient"; input Real e = -380*deltaInv^5 "Polynomial coefficient"; input Real f = 104*deltaInv^6 "Polynomial coefficient"; output Real y "Function value"; protected Real aX "Absolute value of x"; algorithm aX:= abs(x); y := a + aX*(b + aX*(c + aX*(d + aX*(e + aX*f)))); if x < 0 then y := -y; end if; end smoothTransition;

http://iea-annex60.org