## Electric field calculation and charge transport into a dielectric

Numerical methods and mathematical models of Elmer
jeremie
Posts: 12
Joined: 11 Jul 2019, 11:31
Antispam: Yes

### Electric field calculation and charge transport into a dielectric

Hello ELMER-users,

I am trying to simulate the electric field produced by a charge localized deposition into a dielectric.

The charge deposition is constant in time and is a function of space. The charge conduction must be calculated to allow charges to flow down to a ground element. I used the following simple geometry: a dielectric cube with a grounded boundary.

Typically, the Gauss’s law, the Ohm’s law and the continuity equation are combined to give the following equation to solve (with a source term):
• ∇.E = ρ/ε
• J = σE
• ∂ρ/∂t + ∇.J = ρ_s
=> d/dt ( ε ∇grad(ϕ) ) + ∇( σ grad(ϕ) ) = -ρ_s

I used the Static Current Conduction model (StatCurrentSolveVec) to calculate the potential as a function of time. I used a user function to inject charges in only one element of the volume. This user function is called into the body force section of the .sif. Then I used the FluxSolver to calculate the electric field from the potential.

The potential and the electric field are well calculated for each step time but they are constant. I think the problem comes from the body force: the charges do not build up and do not flow.

Does the StatCurrentSolveVec allows the transport of charges?
How to set the body force to inject charge into an element?

Thanks

Jeremie
raback
Posts: 4173
Joined: 22 Aug 2009, 11:57
Antispam: Yes
Location: Espoo, Finland
Contact:

### Re: Electric field calculation and charge transport into a dielectric

Hi Jeremie,

Maybe you could offer your sif for inspection. For example, you should be running in transient but it difficult to check that without the sif.

-Peter
jeremie
Posts: 12
Joined: 11 Jul 2019, 11:31
Antispam: Yes

### Re: Electric field calculation and charge transport into a dielectric

Hi Peter,

Indeed, I set in the .sif: "Simulation Type = Transient".
The .sif file is attached here:
case.sif

I used the StatCurrentSolveVec and I activated the calculation of the electric field and the volume current.
I used a user defined function (UDF) in the Body force section of the .sif file to inject charges at a specific node, corresponding to the middle of a cube face.
See here:
source_term.f90
I tried several test cases with different parameters (BC, conductivity, timesteps) but it seems that there is no charge relaxation.
I expected that the charges build up and the potential and electric field grow up but they are constant in time, there is no variations.
Also, if I put a time condition in the UDF to inject charges while t < t1 for example, the potential immediately goes back to zero at t1.

Jeremie
raback
Posts: 4173
Joined: 22 Aug 2009, 11:57
Antispam: Yes
Location: Espoo, Finland
Contact:

### Re: Electric field calculation and charge transport into a dielectric

Hi Jeremie,

This should give something. Note however that you integrate over the point source, and that only in the one element. So it will be multplied by some fraction of the element volume.

You can for any field give also a load directly as r.h.s. term so that here the unit would be directly in terms of charge, perhaps a true point load you're looking for. E.g.

Body Force 1

Other than that your sif looks god for the quick inspection.

-Peter
jeremie
Posts: 12
Joined: 11 Jul 2019, 11:31
Antispam: Yes

### Re: Electric field calculation and charge transport into a dielectric

Hello Peter,

Thank you for your support. I have another issue by using a non-uniform conductivity.

I used the StatCurrentSolveVec solver to calculate the electric field and the volume current in a dielectric with a non-uniform conductivity.

The geometry is a parallelepiped of 1 x 1 x 30 mm (X x Y x Z). The conductivity of the dielectric is non-uniform along the Z-axis therefore I used a UDF to give the conductivity in function of space.

However, when a non-uniform conductivity is used, there are some oscillations for the volume current (the red curve in attachment). When I set the conductivity as a constant there are no oscillations (the black curve in attachment). These oscillations are reduced when the mesh is refined but they still exist.

The "Variable Time" in the Material section of the .sif is just used as a dummy variable and is not used in the UDF.

Does the StatCurrentSolveVec solver work with a non-uniform conductivity?

Thank you

Jeremie
Attachments
Volume_current.PNG
case.sif
jeremie
Posts: 12
Joined: 11 Jul 2019, 11:31
Antispam: Yes

### Re: Electric field calculation and charge transport into a dielectric

Hi Peter,

I would like to compute the diffusion of electric charges in a dielectric in function of time.
By using a constant source term with the StatCurrentSolveVec, I don't see the diffusion of the electric charges.
I tried many test cases: variation of the timestep, mesh size, conductivity, geometry, BC, ... But there is any temporal variation. Any suggestions?

Moreover, in the Body Force section, I use a UDF to set the Current Source. For a test case, I set f = 1.0 C.m-3.s-1 for one node and f = 0.0 C.m-3.s-1 for the other, and I use several mesh sizes. However when the mesh size is increased, the potential is increased. I guess ELMER interpolates the space charge density between nodes, so in each case the total space charge is different, depending on the mesh size. I think the source term (f in C.m-3.s-1) must be normalized. I am a beginner in FEM, so what do you mean by "multplied by some fraction of the element volume" ?
raback wrote: 11 Aug 2019, 01:44 ...
This should give something. Note however that you integrate over the point source, and that only in the one element. So it will be multplied by some fraction of the element volume.
...
Thank you

Jeremie
raback
Posts: 4173
Joined: 22 Aug 2009, 11:57
Antispam: Yes
Location: Espoo, Finland
Contact:

### Re: Electric field calculation and charge transport into a dielectric

Hi Jeremie

The conductivity can be a function of things. For example, function of x-coordinate. You can use MATC, LUA, UDF or table to generate profiles.

If you want discontinuous conductivities then define different materials and given them different conductivities such that the same node will gen different value for conductivity when the elements is in different material.

Generally specifying super-high conductivity in one node is not perhaps the best idea. The FEM does not really use the nodal values. It uses the values at integration points. And the way to obtain the value at integration point is using the basis functions to evaluate the value therein. For linear basis the effect will thus be that you have linearly altering profile that you integrate over.

What kind of conductivity are you looking in the end?

-Peter
jeremie
Posts: 12
Joined: 11 Jul 2019, 11:31
Antispam: Yes

### Re: Electric field calculation and charge transport into a dielectric

Hi,

When a body force assigns a source term to one node of the mesh, does the source term value is weighted by the element volume?
(Assuming that Constant Weights = False).

Thanks

Jeremie
raback
Posts: 4173
Joined: 22 Aug 2009, 11:57
Antispam: Yes
Location: Espoo, Finland
Contact:

### Re: Electric field calculation and charge transport into a dielectric

Hi

If you add source term with "Potential Load" then that is a nodal quantity and it not integrated over. This is a generic matrix level operation. You would use this type of source usually generated as a reaction force by some other equation, or sometimes you really introduce point sources.

If you add source term with "Current Source" (or something physics specific) this is integrated over (gets the weighted by the representative volume of the node). This is what you usually have in PDEs.

-Peter
jeremie
Posts: 12
Joined: 11 Jul 2019, 11:31
Antispam: Yes

### Re: Electric field calculation and charge transport into a dielectric

OK, yes I added a source term by using "Current source".