Buildings.Utilities.Math.Functions

Package with mathematical functions

Information

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

Package Content

Name Description

$Buildings.Utilities.Math.Functions.bicubic$ bicubic Bicubic function

$Buildings.Utilities.Math.Functions.biquadratic$ biquadratic Biquadratic function

$Buildings.Utilities.Math.Functions.polynomial$ polynomial Polynomial function

$Buildings.Utilities.Math.Functions.powerLinearized$ powerLinearized Power function that is linearized below a user-defined threshold

$Buildings.Utilities.Math.Functions.quadraticLinear$ quadraticLinear Function that is quadratic in first argument and linear in second argument

$Buildings.Utilities.Math.Functions.regNonZeroPower$ regNonZeroPower Power function, regularized near zero, but nonzero value for x=0

$Buildings.Utilities.Math.Functions.smoothExponential$ smoothExponential Once continuously differentiable approximation to exp(-|x|) in interval |x| < delta

$Buildings.Utilities.Math.Functions.smoothHeaviside$ smoothHeaviside Once continuously differentiable approximation to the Heaviside function

$Buildings.Utilities.Math.Functions.smoothMax$ smoothMax Once continuously differentiable approximation to the maximum function

$Buildings.Utilities.Math.Functions.smoothMin$ smoothMin Once continuously differentiable approximation to the minimum function

$Buildings.Utilities.Math.Functions.smoothLimit$ smoothLimit Once continuously differentiable approximation to the limit function

$Buildings.Utilities.Math.Functions.spliceFunction$ spliceFunction

$Buildings.Utilities.Math.Functions.trapezoidalIntegration$ trapezoidalIntegration Integration using the trapezoidal rule

$Buildings.Utilities.Math.Functions.Examples$ Examples Collection of models that illustrate model use and test models

$Buildings.Utilities.Math.Functions.BaseClasses$ BaseClasses Package with base classes for Buildings.Utilities.Math.Functions

$Buildings.Utilities.Math.Functions.inverseXRegularized$ inverseXRegularized Function that approximates 1/x by a twice continuously differentiable function

Name	Description
$Buildings.Utilities.Math.Functions.bicubic$ bicubic	Bicubic function
$Buildings.Utilities.Math.Functions.biquadratic$ biquadratic	Biquadratic function
$Buildings.Utilities.Math.Functions.polynomial$ polynomial	Polynomial function
$Buildings.Utilities.Math.Functions.powerLinearized$ powerLinearized	Power function that is linearized below a user-defined threshold
$Buildings.Utilities.Math.Functions.quadraticLinear$ quadraticLinear	Function that is quadratic in first argument and linear in second argument
$Buildings.Utilities.Math.Functions.regNonZeroPower$ regNonZeroPower	Power function, regularized near zero, but nonzero value for x=0
$Buildings.Utilities.Math.Functions.smoothExponential$ smoothExponential	Once continuously differentiable approximation to exp(-\|x\|) in interval \|x\| < delta
$Buildings.Utilities.Math.Functions.smoothHeaviside$ smoothHeaviside	Once continuously differentiable approximation to the Heaviside function
$Buildings.Utilities.Math.Functions.smoothMax$ smoothMax	Once continuously differentiable approximation to the maximum function
$Buildings.Utilities.Math.Functions.smoothMin$ smoothMin	Once continuously differentiable approximation to the minimum function
$Buildings.Utilities.Math.Functions.smoothLimit$ smoothLimit	Once continuously differentiable approximation to the limit function
$Buildings.Utilities.Math.Functions.spliceFunction$ spliceFunction
$Buildings.Utilities.Math.Functions.trapezoidalIntegration$ trapezoidalIntegration	Integration using the trapezoidal rule
$Buildings.Utilities.Math.Functions.Examples$ Examples	Collection of models that illustrate model use and test models
$Buildings.Utilities.Math.Functions.BaseClasses$ BaseClasses	Package with base classes for Buildings.Utilities.Math.Functions
$Buildings.Utilities.Math.Functions.inverseXRegularized$ inverseXRegularized	Function that approximates 1/x by a twice continuously differentiable function

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

Inputs

Type Name Default Description

Real a[10] Coefficients

Real x1 Independent variable

Real x2 Independent variable

Type	Name	Description
Real	a[10]	Coefficients
Real	x1	Independent variable
Real	x2	Independent variable

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function bicubic "Bicubic function"
 input Real a[10] "Coefficients";
 input Real x1 "Independent variable";
 input Real x2 "Independent variable";
 output Real y "Result";
protected 
 Real x1Sq "= x1^2";
 Real x2Sq "= x2^2";
algorithm 
  x1Sq :=x1*x1;
  x2Sq :=x2*x2;
  y := a[1] + a[2] * x1 + a[3] * x1^2
            + a[4] * x2 + a[5] * x2^2
            + a[6] * x1 * x2
            + a[7] * x1Sq * x1
            + a[8] * x2Sq * x2
            + a[9] * x1Sq * x2
            + a[10] * x1 * x2Sq;

end bicubic;

Buildings.Utilities.Math.Functions.biquadratic

Biquadratic function

Information

This function computes

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

Inputs

Type Name Default Description

Real a[6] Coefficients

Real x1 Independent variable

Real x2 Independent variable

Type	Name	Description
Real	a[6]	Coefficients
Real	x1	Independent variable
Real	x2	Independent variable

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function biquadratic "Biquadratic function"
 input Real a[6] "Coefficients";
 input Real x1 "Independent variable";
 input Real x2 "Independent variable";
 output Real y "Result";
algorithm 
  y :=a[1] + x1*(a[2] + a[3]*x1) + x2*(a[4]+ a[5]*x2) + a[6]*x1*x2;

end biquadratic;

Buildings.Utilities.Math.Functions.polynomial

Polynomial function

Information

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

y = a₁ + a₂ x + a₃ x² + ...

Inputs

Type Name Default Description

Real x Independent variable

Real a[:] Coefficients

Type	Name	Default	Description
Real	x		Independent variable
Real	a[:]		Coefficients

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function polynomial "Polynomial function"
  annotation(derivative=Buildings.Utilities.Math.Functions.BaseClasses.der_polynomial);
 input Real x "Independent variable";
 input Real a[:] "Coefficients";
 output Real y "Result";
protected 
 parameter Integer n = size(a, 1)-1;
 Real xp[n+1] "Powers of x";
algorithm 
  xp[1] :=1;
  for i in 1:n loop
     xp[i+1] :=xp[i]*x;
  end for;
  y :=a*xp;
end polynomial;

Buildings.Utilities.Math.Functions.powerLinearized

Power function that is linearized below a user-defined threshold

Information

Function that approximates y=xⁿ where 0 < n so that

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.

Inputs

Type Name Default Description

Real x Abscissa value

Real n Exponent

Real x0 Abscissa value below which linearization occurs

Type	Name	Description
Real	x	Abscissa value
Real	n	Exponent
Real	x0	Abscissa value below which linearization occurs

Outputs

Type Name Description

Real y Function value

Type	Name	Description
Real	y	Function value

Modelica definition

function powerLinearized 
  "Power function that is linearized below a user-defined threshold"

 input Real x "Abscissa value";
 input Real n "Exponent";
 input Real x0 "Abscissa value below which linearization occurs";
 output Real y "Function value";
algorithm 
  if x > x0 then
   y := x^n;
  else
   y := x0^n * (1-n) + n * x0^(n-1) * x;
  end if;
end powerLinearized;

Buildings.Utilities.Math.Functions.quadraticLinear

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

Information

This function computes

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

Inputs

Type Name Default Description

Real a[6] Coefficients

Real x1 Independent variable for quadratic part

Real x2 Independent variable for linear part

Type	Name	Description
Real	a[6]	Coefficients
Real	x1	Independent variable for quadratic part
Real	x2	Independent variable for linear part

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function quadraticLinear 
  "Function that is quadratic in first argument and linear in second argument"
 input Real a[6] "Coefficients";
 input Real x1 "Independent variable for quadratic part";
 input Real x2 "Independent variable for linear part";
 output Real y "Result";
protected 
 Real x1Sq;
algorithm 
  x1Sq :=x1*x1;
  y :=a[1] + a[2]*x1 + a[3]*x1Sq + (a[4] + a[5]*x1 + a[6]*x1Sq)*x2;

end quadraticLinear;

Buildings.Utilities.Math.Functions.regNonZeroPower

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

Information

Function that approximates y=|x|ⁿ where n > 0 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.

Inputs

Type Name Default Description

Real x Abscissa value

Real n Exponent

Real delta 0.01 Abscissa value where transition occurs

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

Outputs

Type Name Description

Real y Function value

Type	Name	Description
Real	y	Function value

Modelica definition

function regNonZeroPower 
  "Power function, regularized near zero, but nonzero value for x=0"
  annotation(derivative=BaseClasses.der_regNonZeroPower);

 input Real x "Abscissa value";
 input Real n "Exponent";
 input Real delta = 0.01 "Abscissa value where transition occurs";
 output Real y "Function value";
protected 
  Real a1;
  Real a3;
  Real a5;
  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
   y := abs(x)^n;
  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;
   a5 := (y_d - delta2 * (a3 + delta2 * a1));
   y := a5 + x2 * (a3 + x2 * a1);
   assert(a5>0, "Delta is too small for this exponent.");
  end if;
end regNonZeroPower;

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

Inputs

Type Name Default Description

Real x Input argument

Real delta Transition point where approximation occurs

Type	Name	Default	Description
Real	x		Input argument
Real	delta		Transition point where approximation occurs

Outputs

Type Name Description

Real y Output argument

Type	Name	Description
Real	y	Output argument

Modelica definition

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

  input Real x "Input argument";
  input Real delta "Transition point where approximation occurs";
  output Real y "Output argument";
protected 
  Real absX "Absolute value of x";
  Real a2 "Coefficient for approximating function";
  Real a3 "Coefficient for approximating function";
  Real e;
  Real d2;
  Real x2;
algorithm 
  absX :=abs(x);
  if absX > delta then
    y :=  exp(-absX);
  else
    d2 := delta*delta;
    e  := exp(-delta);
    a2 := (delta*e-4*(1-e))/2/d2;
    a3 := (e-1-a2*d2)/d2/d2;
    x2 := x*x;
    y  := 1+x2*(a2+x2*a3);
  end if;
end smoothExponential;

Buildings.Utilities.Math.Functions.smoothHeaviside

Once continuously differentiable approximation to the Heaviside function

Information

Once Lipschitz continuously differentiable approximation to the Heaviside(.,.) function.

Inputs

Type Name Default Description

Real x Argument

Real delta Parameter used for scaling

Type	Name	Default	Description
Real	x		Argument
Real	delta		Parameter used for scaling

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function smoothHeaviside 
  "Once continuously differentiable approximation to the Heaviside function"
  input Real x "Argument";
  input Real delta "Parameter used for scaling";
  output Real y "Result";
algorithm 
 y := spliceFunction(1, 0, x, delta);
end smoothHeaviside;

Buildings.Utilities.Math.Functions.smoothMax

Once continuously differentiable approximation to the maximum function

Information

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

Inputs

Type Name Default Description

Real x1 First argument

Real x2 Second argument

Real deltaX Width of transition interval

Type	Name	Description
Real	x1	First argument
Real	x2	Second argument
Real	deltaX	Width of transition interval

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function smoothMax 
  "Once continuously differentiable approximation to the maximum function"
  input Real x1 "First argument";
  input Real x2 "Second argument";
  input Real deltaX "Width of transition interval";
  output Real y "Result";
algorithm 
  y := Buildings.Utilities.Math.Functions.spliceFunction(
         pos=x1, neg=x2, x=x1-x2, deltax=deltaX);
end smoothMax;

Buildings.Utilities.Math.Functions.smoothMin

Once continuously differentiable approximation to the minimum function

Information

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

Inputs

Type Name Default Description

Real x1 First argument

Real x2 Second argument

Real deltaX Width of transition interval

Type	Name	Description
Real	x1	First argument
Real	x2	Second argument
Real	deltaX	Width of transition interval

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function smoothMin 
  "Once continuously differentiable approximation to the minimum function"
  input Real x1 "First argument";
  input Real x2 "Second argument";
  input Real deltaX "Width of transition interval";
  output Real y "Result";
algorithm 
  y := Buildings.Utilities.Math.Functions.spliceFunction(
       pos=x1, neg=x2, x=x2-x1, deltax=deltaX);
end smoothMin;

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 [0, 1].

Inputs

Type Name Default Description

Real x Variable

Real l Low limit

Real u Upper limit

Real deltaX Width of transition interval

Type	Name	Description
Real	x	Variable
Real	l	Low limit
Real	u	Upper limit
Real	deltaX	Width of transition interval

Outputs

Type Name Description

Real y Result

Type	Name	Description
Real	y	Result

Modelica definition

function smoothLimit 
  "Once continuously differentiable approximation to the limit function"
  input Real x "Variable";
  input Real l "Low limit";
  input Real u "Upper limit";
  input Real deltaX "Width of transition interval";
  output Real y "Result";

protected 
  Real cor;
algorithm 
  cor :=deltaX/10;
  y := Buildings.Utilities.Math.Functions.smoothMax(x,l+deltaX,cor);
  y := Buildings.Utilities.Math.Functions.smoothMin(y,u-deltaX,cor);
end smoothLimit;

Buildings.Utilities.Math.Functions.spliceFunction

Information

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

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

Inputs

Type Name Default Description

Real pos Argument of x > 0

Real neg Argument of x < 0

Real x Independent value

Real deltax Half width of transition interval

Type	Name	Description
Real	pos	Argument of x > 0
Real	neg	Argument of x < 0
Real	x	Independent value
Real	deltax	Half width of transition interval

Outputs

Type Name Description

Real out Smoothed value

Type	Name	Description
Real	out	Smoothed value

Modelica definition

function spliceFunction
  annotation(derivative=BaseClasses.der_spliceFunction);
    input Real pos "Argument of x > 0";
    input Real neg "Argument of x < 0";
    input Real x "Independent value";
    input Real deltax "Half width of transition interval";
    output Real out "Smoothed value";
protected 
    Real scaledX;
    Real scaledX1;
    Real y;
algorithm 
    scaledX1 := x/deltax;
    scaledX := scaledX1*Modelica.Math.asin(1);
    if scaledX1 <= -0.999999999 then
      y := 0;
    elseif scaledX1 >= 0.999999999 then
      y := 1;
    else
      y := (Modelica.Math.tanh(Modelica.Math.tan(scaledX)) + 1)/2;
    end if;
    out := pos*y + (1 - y)*neg;
end spliceFunction;

Buildings.Utilities.Math.Functions.trapezoidalIntegration

Integration using the trapezoidal rule

Information

This function computes a definite integral using the trapezoidal rule.

Inputs

Type Name Default Description

Integer N Number of integrand points

Real f[:] Integrands

Real deltaX Width of interval for Trapezoidal integration

Type	Name	Description
Integer	N	Number of integrand points
Real	f[:]	Integrands
Real	deltaX	Width of interval for Trapezoidal integration

Outputs

Type Name Description

Real result Result

Type	Name	Description
Real	result	Result

Modelica definition

function trapezoidalIntegration 
  "Integration using the trapezoidal rule"
  input Integer N "Number of integrand points";
  input Real[:] f "Integrands";
  input Real deltaX "Width of interval for Trapezoidal integration";
  output Real result "Result";
algorithm 
  assert(N >= 2, "N must be no less than 2.");
  result := 0;
  for i in 1:N loop
    result := result + f[i];
  end for;

  result := 2*result;
  result := result - f[1] - f[N];
  result := result*deltaX/2;
end trapezoidalIntegration;

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 package Examples for the graph.

Inputs

Type Name Default Description

Real x Abscissa value

Real delta Abscissa value below which approximation occurs

Type	Name	Default	Description
Real	x		Abscissa value
Real	delta		Abscissa value below which approximation occurs

Outputs

Type Name Description

Real y Function value

Type	Name	Description
Real	y	Function value

Modelica definition

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

 input Real x "Abscissa value";
 input Real delta(min=0) "Abscissa value below which approximation occurs";
 output Real y "Function value";
protected 
 Real delta2 "Delta^2";
 Real x2_d2 "=x^2/delta^2";
algorithm 
  delta2 :=delta*delta;
  x2_d2  := x*x/delta2;
  y :=smooth(2, if (abs(x) > delta) then 1/x else 
    x/delta2 + x*abs(x/delta2/delta*(2 - x2_d2*(3 - x2_d2))));
end inverseXRegularized;

Automatically generated Fri May 06 14:14:58 2011.