## smoothStepApproximation of a general step, such that the characteristic is continuous and differentiable |

This information is part of the Modelica Standard Library maintained by the Modelica Association.

This function is used to approximate the equation

y =ifx > 0theny1elsey2;

by a smooth characteristic, so that the expression is continuous and differentiable:

y =smooth(1,ifx > x_smalltheny1elseifx < -x_smalltheny2elsef(y1, y2));

In the region -x_small < x < x_small a 2nd order polynomial is used for a smooth transition from y1 to y2.

If **mass fractions** X[:] are approximated with this function then this can be performed
for all **nX** mass fractions, instead of applying it for nX-1 mass fractions and computing
the last one by the mass fraction constraint sum(X)=1. The reason is that the approximating function has the
property that sum(X) = 1, provided sum(X_a) = sum(X_b) = 1
(and y1=X_a[i], y2=X_b[i]).
This can be shown by evaluating the approximating function in the abs(x) < x_small
region (otherwise X is either X_a or X_b):

X[1] = smoothStep(x, X_a[1] , X_b[1] , x_small); X[2] = smoothStep(x, X_a[2] , X_b[2] , x_small); ... X[nX] = smoothStep(x, X_a[nX], X_b[nX], x_small);

or

X[1] = c*(X_a[1] - X_b[1]) + (X_a[1] + X_b[1])/2 X[2] = c*(X_a[2] - X_b[2]) + (X_a[2] + X_b[2])/2; ... X[nX] = c*(X_a[nX] - X_b[nX]) + (X_a[nX] + X_b[nX])/2; c = (x/x_small)*((x/x_small)^2 - 3)/4

Summing all mass fractions together results in

sum(X) = c*(sum(X_a) - sum(X_b)) + (sum(X_a) + sum(X_b))/2 = c*(1 - 1) + (1 + 1)/2 = 1

y = smoothStep(x, y1, y2, x_small)

x |
Type: Real Description: Abscissa value |
---|---|

y1 |
Type: Real Description: Ordinate value for x > 0 |

y2 |
Type: Real Description: Ordinate value for x < 0 |

x_small |
Default Value: 1e-5 Type: Real Description: Approximation of step for -x_small <= x <= x_small; x_small > 0 required |

y |
Type: Real Description: Ordinate value to approximate y = if x > 0 then y1 else y2 |
---|