functionsmoothStep

Information

This function is used to approximate the equation  


    y = if x > 0 then y1 else y2;  


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


   y = smooth(1, if x >  x_small then y1 else  
                 if x < -x_small then y2 else f(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  

## Copyright © EDF 2002 - 2025

## ThermoSysPro Version 4.2

Inputs

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Revisions

April 29, 2008  
    by Martin Otter:  
    Designed and implemented.  
August 12, 2008  
    by Michael Sielemann:  
    Minor modification to cover the limit case x_small -> 0 without division by zero.