Internal:Adaptive CDT3D - Smoothing

NASA/refine

Pseudocode

We solve the nonsmooth optimization problem... ...Using this information, we can predict the points at which the active set A will change by computing the intersection of each linear approximation with the pro jection of the current active function in the search direction. The distance to the nearest intersection point from the current location gives the initial step length, a. The initial step is accepted if the actual improvement exceeds 90 percent of the estimated improvement or the subsequent step results in a smaller function improvement. Otherwise is halved recursively until a step is accepted or falls below some minimum step length tolerance.

1. Evaluate the degree of the node and collect adajcent cells
2. For each attached cell evaluate quality and gradient of quality as well as min quality of set.
3. activate tets that (quals[i]-quals[worst]) < 1e-12
4. If number of active elements >= 4 exit
   4.1 else if number of active elements ==1 set dir=grad[worst]
   4.2 else use Charalambous and Conn approach  to find best dir
5. set m0 = dot(dir,grad[worst])
6. find stepsize alpha:
//for our initial step size, we use the distance
//to the next intersection with another quality funtion.
//this is the point at which the other quality function
///becomes the worst. we use a single-term taylor expansion
///to approximate all of the quality functions as lines,
//so we will have to do a line search to find our ultimate
//step size. 
7. compute the approximation of when this funnction will become the objective /
8. for each non active incident : alpha = (quals[worst] - quals[i]) / (m1 - m0); min(min_alpha,alpha)
9. if min_alpha didn’t change min_alpha = (1.0 - quals[0]) / ref_math_dot(dir, &(grads[3 * 0]))
10. for (i = 1; i < degree; i++) min_alpha = MIN(min_alpha, (1.0 - quals[i]) / ref_math_dot(dir, &(grads[3 * i])));
11. line serach as descriped above :
  11.1 find smallest dot product between dir and grad of active elements (m0 in refine)
  11.2 Do at most 8 loops of linear walk to optimize quality of cavity

Code  : https://github.com/nasa/refine/blob/master/src/ref_smooth.c#L1258