Trying to understand _disperse_step

[RomanLF]

Hello Trever,

In rbf/pde /nodes.py at _disperse_step function you compute a proportionality constant based on electrorepulsion law with the support of $\rho$ density function. You wrote :

 all_nodes = np.vstack((nodes, fixed_nodes))
    # find index and distance to nearest nodes
    dist, idx = KDTree(all_nodes).query(nodes, neighbors + 1)
    # dont consider a node to be one of its own nearest neighbors
    dist, idx = dist[:, 1:], idx[:, 1:]
    # compute the force proportionality constant between each node
    # based on their charges
    c = 1.0/(rho(all_nodes)[idx, None]*rho(nodes)[:, None, None])
    # calculate forces on each node resulting from the neighboring nodes. This
    # will result in a division by zero warning if there are duplicate nodes.
    # Do not suppress the warning because it is a real problem.
    forces = c*(nodes[:, None, :] - all_nodes[idx, :])/dist[:, :, None]**3
    # sum up all the forces for each node to get the direction that the nodes
    # should move.
    direction = np.sum(forces, axis=1)

• $\mathbf{X}$ the set of all nodes of dimension $(N,d)$,
• $\mathbf{S}$ the set neighbour nodes dimension $(m,d)$, Basically if no fix nodes are specified you have : $\mathbf{X}$ the set of all nodes of dimension $(N,d)$,
• $\mathbf{S}$ the set neighbour nodes dimension $(m,d)$, Basically if no fix nodes are specified you have :

$$ c(\mathbf{X, S}) = 1 / (\rho(\mathbf{X}) \rho(\mathbf{S})) $$

And compute forces with,

$$ \mathbf{F}_{\mathbf{X}~on~ \mathbf{S}} = \frac{1}{\rho(\mathbf{X}) \rho(\mathbf{S})} \frac{\mathbf{S} - \mathbf{X}}{| \mathbf{S} - \mathbf{X}|^3} $$

Where for Cloumb's law force exerced by charged by a particule 1 on particule 2 is,

$$ \vec{F}_{1 / 2}= {q_1 q_2} \frac{\vec{r}_2-\vec{r}_1}{\left|\vec{r}_2-\vec{r}_1\right|^3} $$

Does it mean that you assume that node density is $\rho$ is equivalent to $1/q$ in term of charge density ?

Best regards, Roman

[treverhines]

I am using the inverse of the desired node density for the charge. However, if you run _disperse_step enough times, it wont actually arrive at that desired node density as a steady state. So this function is really just used to refine nodes that are already placed close to the desired node density. By setting the charge to the inverse of the node density, I am just trying to keep _disperse_step from significantly changing the node density.

[RomanLF]

Ok thank you very much for your explanation.