This package contains functions for commonly used mathematical operations. The functions are used in the blocks Buildings.Utilities.Math.
Extends from Modelica.Icons.BasesPackage (Icon for packages containing base classes).
Name | Description |
---|---|
![]() | Bicubic function |
![]() | Biquadratic function |
![]() | Interpolate using a cubic Hermite spline with linear extrapolation |
![]() | Polynomial function |
![]() | Power function that is linearized below a user-defined threshold |
![]() | Function that is quadratic in first argument and linear in second argument |
![]() | Power function, regularized near zero, but nonzero value for x=0 |
![]() | Once continuously differentiable approximation to exp(-|x|) in interval |x| < delta |
![]() | Once continuously differentiable approximation to the Heaviside function |
![]() | Once continuously differentiable approximation to the maximum function |
![]() | Once continuously differentiable approximation to the minimum function |
![]() | Once continuously differentiable approximation to the limit function |
![]() | |
![]() | Function to compute the derivatives for cubic hermite spline interpolation |
![]() | Integration using the trapezoidal rule |
![]() | Function that approximates 1/x by a twice continuously differentiable function |
![]() | Returns true if the argument is a monotonic sequence |
![]() | Collection of models that illustrate model use and test models |
![]() | Package with base classes for Buildings.Utilities.Math.Functions |
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
Type | Name | Default | Description |
---|---|---|---|
Real | a[10] | Coefficients | |
Real | x1 | Independent variable | |
Real | x2 | Independent variable |
Type | Name | Description |
---|---|---|
Real | y | Result |
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;
y = a1 + a2 x1 + a3 x12 + a4 x2 + a5 x22 + a6 x1 x2
Type | Name | Default | Description |
---|---|---|---|
Real | a[6] | Coefficients | |
Real | x1 | Independent variable | |
Real | x2 | Independent variable |
Type | Name | Description |
---|---|---|
Real | y | Result |
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;
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.
Type | Name | Default | Description |
---|---|---|---|
Real | x | Abscissa value | |
Real | x1 | Lower abscissa value | |
Real | x2 | Upper abscissa value | |
Real | y1 | Lower ordinate value | |
Real | y2 | Upper ordinate value | |
Real | y1d | Lower gradient | |
Real | y2d | Upper gradient |
Type | Name | Description |
---|---|---|
Real | y | Interpolated ordinate value |
function cubicHermiteLinearExtrapolation "Interpolate using a cubic Hermite spline with linear extrapolation" input Real x "Abscissa value"; input Real x1 "Lower abscissa value"; input Real x2 "Upper abscissa value"; input Real y1 "Lower ordinate value"; input Real y2 "Upper ordinate value"; input Real y1d "Lower gradient"; input Real y2d "Upper gradient"; output Real y "Interpolated ordinate value"; algorithm if (x > x1 and x < x2) then y:=Modelica.Fluid.Utilities.cubicHermite( x=x, x1=x1, x2=x2, y1=y1, y2=y2, y1d=y1d, y2d=y2d); elseif x <= x1 then // linear extrapolation y:=y1 + (x - x1)*y1d; else y:=y2 + (x - x2)*y2d; end if;end cubicHermiteLinearExtrapolation;
y = a1 + a2 x + a3 x2 + ...
Type | Name | Default | Description |
---|---|---|---|
Real | x | Independent variable | |
Real | a[:] | Coefficients |
Type | Name | Description |
---|---|---|
Real | y | Result |
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;
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.
Type | Name | Default | Description |
---|---|---|---|
Real | x | Abscissa value | |
Real | n | Exponent | |
Real | x0 | Abscissa value below which linearization occurs |
Type | Name | Description |
---|---|---|
Real | y | Function value |
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;
y = a1 + a2 x1 + a3 x12 + (a4 + a5 x1 + a6 x12) x2
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 | y | Result |
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;
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.
Type | Name | Default | Description |
---|---|---|---|
Real | x | Abscissa value | |
Real | n | Exponent | |
Real | delta | 0.01 | Abscissa value where transition occurs |
Type | Name | Description |
---|---|---|
Real | y | Function value |
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;
Function to provide a once continuously differentiable approximation to exp(- |x| ) in the interval |x| < δ for some positive δ
Type | Name | Default | Description |
---|---|---|---|
Real | x | Input argument | |
Real | delta | Transition point where approximation occurs |
Type | Name | Description |
---|---|---|
Real | y | Output argument |
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;
Once Lipschitz continuously differentiable approximation to the
Heaviside(.,.)
function.
Type | Name | Default | Description |
---|---|---|---|
Real | x | Argument | |
Real | delta | Parameter used for scaling |
Type | Name | Description |
---|---|---|
Real | y | Result |
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 := Buildings.Utilities.Math.Functions.spliceFunction(1, 0, x, delta);end smoothHeaviside;
Once continuously differentiable approximation to the max(.,.)
function.
Type | Name | Default | Description |
---|---|---|---|
Real | x1 | First argument | |
Real | x2 | Second argument | |
Real | deltaX | Width of transition interval |
Type | Name | Description |
---|---|---|
Real | y | Result |
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;
Once continuously differentiable approximation to the max(.,.)
function.
Type | Name | Default | Description |
---|---|---|---|
Real | x1 | First argument | |
Real | x2 | Second argument | |
Real | deltaX | Width of transition interval |
Type | Name | Description |
---|---|---|
Real | y | Result |
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;
Once continuously differentiable approximation to the limit(.,.)
function.
The output is bounded to be in [l, u].
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 | y | Result |
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;
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.
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 | out | Smoothed value |
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 scaledX1; Real y; constant Real asin1 = Modelica.Math.asin(1); algorithm scaledX1 := x/deltax; if scaledX1 <= -0.999999999 then out := neg; elseif scaledX1 >= 0.999999999 then out := pos; else y := (Modelica.Math.tanh(Modelica.Math.tan(scaledX1*asin1)) + 1)/2; out := pos*y + (1 - y)*neg; end if;end spliceFunction;
This function computes the derivatives at the support points xi
that can be used as input for evaluating a cubic hermite spline.
If ensureMonotonicity=true
, then the support points yi
need to be monotone increasing (or increasing), and the computed derivatives
di 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.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.
F.N. Fritsch and R.E. Carlson, Monotone piecewise cubic interpolation. SIAM J. Numer. Anal., 17 (1980), pp. 238?246.
Type | Name | Default | Description |
---|---|---|---|
Real | x[:] | Support point, strict monotone increasing | |
Real | y[size(x, 1)] | Function values at x | |
Boolean | ensureMonotonicity | isMonotonic(y, strict=false) | Set to true to ensure monotonicity of the cubic hermite |
Type | Name | Description |
---|---|---|
Real | d[size(x, 1)] | Derivative at the support points |
function splineDerivatives "Function to compute the derivatives for cubic hermite spline interpolation" input Real x[:] "Support point, strict monotone increasing"; input Real y[size(x, 1)] "Function values at x"; input Boolean ensureMonotonicity=isMonotonic(y, strict=false) "Set to true to ensure monotonicity of the cubic hermite"; output Real d[size(x, 1)] "Derivative at the support points"; protected Integer n=size(x, 1) "Number of data points"; Real delta[n - 1] "Slope of secant line between data points"; Real alpha "Coefficient to ensure monotonicity"; Real beta "Coefficient to ensure monotonicity"; Real tau "Coefficient to ensure monotonicity"; algorithm if (n>1) then assert(x[1] < x[n], "x must be strictly increasing. Received x[1] = " + String(x[1]) + " x[" + String(n) + "] = " + String(x[n])); // Check data assert(isMonotonic(x, strict=true), "x-values must be strictly monontone increasing or decreasing."); if ensureMonotonicity then assert(isMonotonic(y, strict=false), "If ensureMonotonicity=true, y-values must be monontone increasing or decreasing."); end if; end if; // Compute derivatives at the support points if n == 1 then // only one data point d[1] :=0; elseif n == 2 then // linear function d[1] := (y[2] - y[1])/(x[2] - x[1]); d[2] := d[1]; else // Slopes of the secant lines between i and i+1 for i in 1:n - 1 loop delta[i] := (y[i + 1] - y[i])/(x[i + 1] - x[i]); end for; // Initial values for tangents at the support points. // End points use one-sided derivatives d[1] := delta[1]; d[n] := delta[n - 1]; for i in 2:n - 1 loop d[i] := (delta[i - 1] + delta[i])/2; end for; end if; // Ensure monotonicity if n > 2 and ensureMonotonicity then for i in 1:n - 1 loop if (abs(delta[i]) < Modelica.Constants.small) then d[i] := 0; d[i + 1] := 0; else alpha := d[i]/delta[i]; beta := d[i + 1]/delta[i]; // Constrain derivative to ensure monotonicity in this interval if (alpha^2 + beta^2) > 9 then tau := 3/(alpha^2 + beta^2)^(1/2); d[i] := delta[i]*alpha*tau; d[i + 1] := delta[i]*beta*tau; end if; end if; end for; end if;end splineDerivatives;
This function computes a definite integral using the trapezoidal rule.
Type | Name | Default | Description |
---|---|---|---|
Integer | N | Number of integrand points | |
Real | f[:] | Integrands | |
Real | deltaX | Width of interval for Trapezoidal integration |
Type | Name | Description |
---|---|---|
Real | result | Result |
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;
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.
Type | Name | Default | Description |
---|---|---|---|
Real | x | Abscissa value | |
Real | delta | Abscissa value below which approximation occurs |
Type | Name | Description |
---|---|---|
Real | y | Function value |
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 if (abs(x) > delta) then y := 1/x; else delta2 :=delta*delta; x2_d2 := x*x/delta2; y := x/delta2 + x*abs(x/delta2/delta*(2 - x2_d2*(3 - x2_d2))); end if;end inverseXRegularized;
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.
Type | Name | Default | Description |
---|---|---|---|
Real | x[:] | Sequence to be tested | |
Boolean | strict | false | Set to true to test for strict monotonicity |
Type | Name | Description |
---|---|---|
Boolean | monotonic | True if x is monotonic increasing or decreasing |
function isMonotonic "Returns true if the argument is a monotonic sequence" input Real x[:] "Sequence to be tested"; input Boolean strict=false "Set to true to test for strict monotonicity"; output Boolean monotonic "True if x is monotonic increasing or decreasing"; protected Integer n=size(x, 1) "Number of data points"; algorithm if n == 1 then monotonic := true; else monotonic := true; if strict then if (x[1] >= x[n]) then for i in 1:n - 1 loop if (not x[i] > x[i + 1]) then monotonic := false; end if; end for; else for i in 1:n - 1 loop if (not x[i] < x[i + 1]) then monotonic := false; end if; end for; end if; else // not strict if (x[1] >= x[n]) then for i in 1:n - 1 loop if (not x[i] >= x[i + 1]) then monotonic := false; end if; end for; else for i in 1:n - 1 loop if (not x[i] <= x[i + 1]) then monotonic := false; end if; end for; end if; end if; // strict end if;end isMonotonic;