Buildings.Utilities.Math.Functions

Package with mathematical functions

Information

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

Package Content

NameDescription
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

TypeNameDefaultDescription
Reala[10] Coefficients
Realx1 Independent variable
Realx2 Independent variable

Outputs

TypeNameDescription
RealyResult

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

TypeNameDefaultDescription
Reala[6] Coefficients
Realx1 Independent variable
Realx2 Independent variable

Outputs

TypeNameDescription
RealyResult

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

TypeNameDefaultDescription
Realx Independent variable
Reala[:] Coefficients

Outputs

TypeNameDescription
RealyResult

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

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

TypeNameDefaultDescription
Realx Abscissa value
Realn Exponent
Realx0 Abscissa value below which linearization occurs

Outputs

TypeNameDescription
RealyFunction 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

TypeNameDefaultDescription
Reala[6] Coefficients
Realx1 Independent variable for quadratic part
Realx2 Independent variable for linear part

Outputs

TypeNameDescription
RealyResult

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

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

TypeNameDefaultDescription
Realx Abscissa value
Realn Exponent
Realdelta0.01Abscissa value where transition occurs

Outputs

TypeNameDescription
RealyFunction 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

TypeNameDefaultDescription
Realx Input argument
Realdelta Transition point where approximation occurs

Outputs

TypeNameDescription
RealyOutput 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

TypeNameDefaultDescription
Realx Argument
Realdelta Parameter used for scaling

Outputs

TypeNameDescription
RealyResult

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

TypeNameDefaultDescription
Realx1 First argument
Realx2 Second argument
RealdeltaX Width of transition interval

Outputs

TypeNameDescription
RealyResult

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

TypeNameDefaultDescription
Realx1 First argument
Realx2 Second argument
RealdeltaX Width of transition interval

Outputs

TypeNameDescription
RealyResult

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

TypeNameDefaultDescription
Realx Variable
Reall Low limit
Realu Upper limit
RealdeltaX Width of transition interval

Outputs

TypeNameDescription
RealyResult

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

TypeNameDefaultDescription
Realpos Argument of x > 0
Realneg Argument of x < 0
Realx Independent value
Realdeltax Half width of transition interval

Outputs

TypeNameDescription
RealoutSmoothed 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

TypeNameDefaultDescription
IntegerN Number of integrand points
Realf[:] Integrands
RealdeltaX Width of interval for Trapezoidal integration

Outputs

TypeNameDescription
RealresultResult

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

TypeNameDefaultDescription
Realx Abscissa value
Realdelta Abscissa value below which approximation occurs

Outputs

TypeNameDescription
RealyFunction 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.