Surface Charge Densities in Uniform Electric Fields

[MrJonaker]

Hi!

I am trying to explore the surface charge distribution on the electrodes of uniform electric fields and I am struggling with Elmer FEM.

The goal is to find the surface charge density as a function of x and y on the electrodes. I want the data in a dataset of x, y, and surface charge.

My current geometry consists of two infinitesimally thin electrodes and a surrounding cylinder from Gmsh saved as .msh. They are converted to Elmers mesh format using ElmerGrid and then I have the .sif file adapted from the GUI tutorial Capacitance of Two Balls. The simulations are ran and using Paraview I can see the potential however ElmerSolver doesn't seem to calculate the surface charges, even if I have "Calculate Surface Charge = True", as "surface charge" doesn't appear in ParaView.

Toggling the other toggle-able values appears to affect what variables are available in the dropdown in ParaView.

Mesh generation:

Code: Select all

gmsh square-mesh-generation.geo

Geometry conversion command:

Code: Select all

ElmerGrid 14 2 squareMesh.msh -autoclean

After moving mesh files to the appropriate folder: ElmerSolver run command:

Code: Select all

ElmerSolver case.sif

Does anyone have any ideas why this might be the case and what could be done to fix it?

[kevinarden]

I think you have a mesh issue, the air needs to be 3D volume elements no surface mesh elements. I would do it this way, then the potential is the potential per surface area, if you want to sum the potential over the surface then there is a method for that.

mymodel.geo

(645 Bytes) Downloaded 92 times

case.sif

(2 KiB) Downloaded 89 times

[MrJonaker]

What do you mean with "the air needs to be 3D volume elements no surface mesh elements"? Because in my .geo file the surrounding air cylinder (I assume that is what you mean) is a volume because it is created with the Cylinder command right?

Or do you think that the electrodes being infinitesimally thin is a problem?

In gmsh my mesh looks, fine in my opinion.

I downloaded your .geo file and used your case.sif however the model still doesn't show surface charge density. I'm not really interested in potential (I mean, it's given in the case.sif file).

Even after I added

Code: Select all

Calulate Surface Charge = True

your case.sif doesn't produce any surface charge that ParaView can show,

[kevinarden]

when I run gmsh on the geo file I only get 2D surface elements no 3D volume elements, I think it is because
Mesh 2;
means mesh 2D so it only creates 2D elements on the surfaces. which looks OK in the viewer, but there are no solid elements.

[kevinarden]

The force on a charge due to an electric field E⃗ is given by F⃗ =qE⃗ , where q is the charge of the particle. The electric field at any point in space, has a given value. This value is called the potential V. Elmer treats variables per area for surface boundaries, both inputs and output. So the potential variable output is the charge per area.

[MrJonaker]

The electric field isn't the same as potential, potential is the integral of electric field strength.

And I'm interested in the surface charge density of the electrodes. Not potential nor electric field strength. Electric field strength per area isn't the same as surface charge density.

In a uniform electric field it doesn't matter where on the electrodes you measure the potential whereas the surface charge density depends on where on the electrode you are looking.

The goal of the simulation is to get a charge density distribution for an infinitely thin electrode, do you think that is possible?

[kevinarden]

You can get these additional outputs from the solver

fields.png (11.96 KiB) Viewed 957 times

case.sif

(2.13 KiB) Downloaded 90 times

[MrJonaker]

In the manual (Elmer Models Manual, p.111) it says there exists a toggle "Calculate Surface Charge Density" but I find it doesn't affect the result at all.

Could it be a deprecated feature?

[kevinarden]

It says the electric flux is the surface charge density


The flux may be defined e.g. by the surface charge density: g = σ.

[kevinarden]

In this case I use the Body ID in the boundary surface and use them as bodies in addition to boundary surfaces. This allows for them to have their own material ID and application of body forces.

case.sif

(2.47 KiB) Downloaded 91 times