refinement around electrodes - fixes dipole analytical solution disrepancy? 3 spheres comparison

[mdovgialo]

In electrode modelling and signal simulations we want to refine the mesh around electrodes for better accuracy of point sources

For now 2 steps of refinement in 1 cm radius spheres around point sources seem to be the best solution for human sized brains at 0.005 m mesh element sizes

[mdovgialo]

Results for electrode solver:

[mdovgialo]

Simulating point dipole with different refinement levels:

4 spheres:

brain, skull, skin, air
conductivities=[ 0.33, 0.0165, 0.33, 1e-10]
radii=[0.082, 0.086, 0.09, 100.0]

Refinement in a 1cm sphere around dipole position. Each step is one subdivision of elements.

With regrounding at 0.085 to compare decay easier:

Seems like adding more elements around point sources is changing overall offsets and decay in ai!

seems to fix overall incompatability talked about in #50

@danek8317

[mdovgialo]

Experiment with mesh refinement

We refine the mesh near the dipole (radius 1 cm). Each refinement step subdivides the mesh elements which centroids are close to the dipole. At refinement step 4 50% of the mesh elements are in the sphere around the dipole, steps 5 and more --- 90+% of the mesh elements are the elements at the dipole. 2 steps of refinement increase amount of mesh elements from 1641418 to 1668160 by 1.6%.

4 spheres:
brain, skull, skin, air
conductivities=[ 0.33, 0.0165, 0.33, 1e-10]
radii=[0.082, 0.086, 0.09, 100.0]

Comparing with analytical solution.

2 types of meshes: with grounding electrode in the "neck" and without. BC - Dirichlet, outer shell is set to 0 and the grounding electrode also is set to 0. Rest of BC are "Natural":

Meshes in the spheres have element resolution of 5 mm and it decreases in the air when we get further away from the spheres.

No electrode

Other side:

Regrounded at 0.085:

Seems like FEM solver needs to have increased resolution around point dipoles/sources. But increasing it too much can loose accuracy in other regions. In this case of 5 mm mesh, 2 steps of refinement in 1 cm sphere around dipole seems to have the best result...

Neck grounding electrode:

Back side:

Regrounded at 0.085:

Introducing grounding in the neck predictably adds an offset in the potential, and similar to electrode-less solutions, having too high resolution around the dipole changes the rate of decay in the air. Closest to the analytical line in the air above the spheres is again 2 steps of refinement. Interestingly enough, having neck grounding makes the solution be closer to the dipole at low distances, at low refinement steps.

Due to enforced 0 at the neck electrode, the potential below the spheres is "on the wrong side" and decay in the air does not follow the analytical curve at all.

2 steps

Regrounded at 0,085

Due to enforced 0 at the neck electrode, the potential below the spheres is "on the wrong side" and decay in the air does not follow the analytical curve at all.

[mdovgialo]

Example of refinement for electrode leadfield solver:

No refinement:

Refinement:

As we can see we get more accurate representation of a point and faster decay in air

[mdovgialo]

Experiment with disabling dirichet BC

Only Neuman, otherwise setup the same as #52 (comment), 2 steps of refinement.

As we can see without electrode adding or removing far boundary enforced 0 doesn't change the result, most likely because two symmetrical point sources integration constant cancels out. It does change the offset for the solution with electrode. Let's look at the electrode location:

At the electrode location, the solution might be still be disturbed by the enforcement of zero current.

After regrounding at 0.085: