Gradient energy calculation seems wrong

[jijn]

In file applications/allen_cahn_explicit/equations.cc

  ScalarValue f_grad = 0.0;
  for (unsigned int i = 0; i < dim; i++)
    {
      for (unsigned int j = 0; j < dim; j++)
        {
          f_grad += 0.5 * KnV * nx[i] * nx[j];
        }
    }

Note that this is not the dot product. The correct way should be:

  ScalarValue f_grad = 0.0;
  for (unsigned int i = 0; i < dim; i++)
    {
      f_grad += nx[i] * nx[i];
    }
  f_grad *= 0.5 * KnV;

This also affects other applications.

[jijn]

The original code is likely for cases where KnV is a 2nd-rank tensor, like

  ScalarValue f_grad = 0.0;
  for (unsigned int i = 0; i < dim; i++)
    {
      for (unsigned int j = 0; j < dim; j++)
        {
          f_grad += 0.5 * KnV[i][j] * nx[i] * nx[j];
        }
    }

If KnV is an identity tensor, only the diagnal terms are needed.

[david-montiel-t]

The calculation has been has been like this for while but I remember (not entirely sure about this) that somehow the energy was unchanged if we used the correct form. Perhaps an earlier version used a diagonal KnV tensor rather than a constant.

[david-montiel-t]

In any case, we should fix it.