.TRANSFORM.Media.Interfaces.Solids.PartialAlloy.setSmoothState

Information

This function is used to approximate the equation

      state = if x > 0 then state_a else state_b;
  

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

     state := smooth(1, if x >  x_small then state_a else
                        if x < -x_small then state_b else f(state_a, state_b));
  

This is performed by applying function Media.Common.smoothStep(..) on every element of the thermodynamic state record.

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(state.X) = 1, provided sum(state_a.X) = sum(state_b.X) = 1. This can be shown by evaluating the approximating function in the abs(x) < x_small region (otherwise state.X is either state_a.X or state_b.X):

      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
  

Interface

partial function setSmoothState
  extends Modelica.Icons.Function;
  input Real x "flow variable";
  input ThermodynamicState state_a "Thermodynamic state if x > 0";
  input ThermodynamicState state_b "Thermodynamic state if x < 0";
  input Real x_small(min = 0) "Smooth transition in the region -x_small < x < x_small";
  output ThermodynamicState state "Smooth thermodynamic state for all x (continuous and differentiable)";
end setSmoothState;