r-refinement
 

Dear Friends,

I am currently looking at r-refinement, wherein the nodes are moved according to the solution strengths. Currently, I am looking at a modified laplacian strategy employed by Marchant and Weatherill. In essence they make use of the following expression.

r_o^(n+1) = r_o^(n) + delta

delta = sum(w_i*(r_i - r_o))/sum(w_i)

where w_i is the weights computed as w_i = k1 + k2 * dabs((phi_i - phi_o)/(phi_i + phi_o)).

Here, phi stands for the solution variable, r the position vector and the summation is over all neighbours i of the point o. k1 and k2 are constants for noise damping and gradient amplification. Also, a slight modification of the aforementioned strategy is employed for boundary nodes. I have two questions.

1. What are the commonly used values of k1 and k2, and what is the basic idea behind selecting those ?

2. On the boundaries, how is the procedure modified, so as to also take care of the fact that it always lies on the boundary itself.

I will be grateful if anyone with experience on these issues is able to help me.

Thanks and Regards,

Ganesh

Re: r-refinement
 

In constant or nearly-constant regions, you would expect the grid to be isotropic. The node movement should reduce to an averaging formula, all weights should be equal. But there will be usually noise in the solution, may be due to limiters. Adding a k1 term which is larger than the magnitude of the second term due to noise would give nearly constant weights.

k2 seems to be just a scaling term for the gradients. I guess one needs to do some trial and error to determine good values for these.