II. Faces: Each face is tested separately for refinement or derefinement. For the tetrahedral face defined by nodes 1, 2, and 3, we have
 

  where A is the area of the face.

     Criteria:

1) Junction: Refine if any of the faces field values straddle c.

          Tag for refinement if

     f(1) > c and f(2), f(3) < c
or
      f(2) > c and f(1), f(3) < c

or
      f(3) > c and f(1), f(2) < c

or
all of the above with > and < interchanged.
example: For c = 0, refine if f changes sign between any of the three nodes.

 2) Constant: Refine if any of the face field values exceed c.

      Tag for refinement if
 

     f(1) > 2 or f(2) > c or f(3) > c.

 
3)Maxsize:  Refine if the face area exceeds c.

    Tag if A > c.

4) Aspect Ratio: Refine if the faces aspect ratio is less than c.

The aspect ratio (AR) is defined as the ratio of the radius of the inscribed circle of the  triangular face to the radius of the circumscribed circle. We renormalize this ratio of multiplying by 2 so that the ratio equals one for an equilateral triangle.
 

AR = 2 RIN/ROUT  where

RIN = radius of inscribed circle
ROUT = radius of circumscribed circle
AR is never greater than one.

Tag if AR < c.  Generally the smaller AR is the more elongated the face is.

5) Lambda Refine: Refine if lamda/dx  <c.  Where dx  is taken to be the radius of the circumscribed circle  RIN of the triangular face
 

lambda = f(xcen)/grad f
f(xcen) = (f(1) + f(2) +f(3))/3 where
 

xcen is the centroid of  the triangular face,  and we have assumed a linear interpolation of f.
grad f is evaluated on the face by a suitable approximation involving a linear interpolation of f and  the line integral around the edge of the face.