Hi everybody
Just got acquainted with Elmer yesterday. I was wondering if the Poisson equation solver can handle the well known boundary condition
epsilon1 * E1|n - epsilon2 * E2|n = sigma
at an interface between two different dielectrics and with sigma being a given arbitrary surface charge distribution.
It is possible also that the mesh for dielectric 1 might not be conformal with that of dielectric 2 at that interface.
In the event that Elmer is capable of dealing with this situation, I hope it does not involve iterating between successive Poisson equation solving on the above two meshes.
Regards to all
boundary condition at a dielectric interface
Re: boundary condition at a dielectric interface
Hello again
OK, first I'm going to show that for a single uniform Cartesian 2D mesh covering two dielectric regions, a very simple program can be coded (for example in C++ or Matlab). I will write the Poisson equations using FDM for the sake of argument but of course the problem could have been written in terms of FVM or FEM.
The cell area is delX * delY. The dielectric constants are eps1, eps2 respectively. I let region 1 be air with a charge volume density rho and region 2 be a solid empty of charge except for the surface charge density sigma at the dielectric interface.
For the top air domain,we have with j > 0
( eps1 * (V(i,j+1) - V(i,j))/delY - eps1 * (V(i,j) - V(i,j-1))/delY )/delY +
( eps1 * (V(i+1,j) - V(i,j))/delX - eps1 * (V(i,j) - V(i-1,j))/delX )/delX
= rho(i,j)
At the dielectric interface, the boundary condition is
eps1 * (V(i,1) - V(i,0))/delY + eps2 * (V(i,0) - V(i,-1))/delY = sigma(i)
For the bottom solid non conducting domain we have with j < 0
( eps2 * (V(i,j+1) - V(i,j))/delY - eps2 * (V(i,j) - V(i,j-1))/delY )/delY +
( eps2 * (V(i+1,j) - V(i,j))/delX - eps2 * (V(i,j) - V(i-1,j))/delX )/delX
= 0
Altogether, we have a system of linear equations Ax = b where the unknowns are V(i,j).
So I want to do the above but generalized to two arbitrary 3D domain shapes and two unstructured meshes perhaps non conformal on the dielectric interface.
Maybe the answer is as easy as declaring 2 bodies with their respective dielectric constants, generating a mesh for each and calling the solver for the Poisson equation. Perhaps the volume density rho(i,j) would be defined in a UDF. But how would you define the surface charge density sigma(i) at the dielectric interface ?
Any reply welcome
OK, first I'm going to show that for a single uniform Cartesian 2D mesh covering two dielectric regions, a very simple program can be coded (for example in C++ or Matlab). I will write the Poisson equations using FDM for the sake of argument but of course the problem could have been written in terms of FVM or FEM.
The cell area is delX * delY. The dielectric constants are eps1, eps2 respectively. I let region 1 be air with a charge volume density rho and region 2 be a solid empty of charge except for the surface charge density sigma at the dielectric interface.
For the top air domain,we have with j > 0
( eps1 * (V(i,j+1) - V(i,j))/delY - eps1 * (V(i,j) - V(i,j-1))/delY )/delY +
( eps1 * (V(i+1,j) - V(i,j))/delX - eps1 * (V(i,j) - V(i-1,j))/delX )/delX
= rho(i,j)
At the dielectric interface, the boundary condition is
eps1 * (V(i,1) - V(i,0))/delY + eps2 * (V(i,0) - V(i,-1))/delY = sigma(i)
For the bottom solid non conducting domain we have with j < 0
( eps2 * (V(i,j+1) - V(i,j))/delY - eps2 * (V(i,j) - V(i,j-1))/delY )/delY +
( eps2 * (V(i+1,j) - V(i,j))/delX - eps2 * (V(i,j) - V(i-1,j))/delX )/delX
= 0
Altogether, we have a system of linear equations Ax = b where the unknowns are V(i,j).
So I want to do the above but generalized to two arbitrary 3D domain shapes and two unstructured meshes perhaps non conformal on the dielectric interface.
Maybe the answer is as easy as declaring 2 bodies with their respective dielectric constants, generating a mesh for each and calling the solver for the Poisson equation. Perhaps the volume density rho(i,j) would be defined in a UDF. But how would you define the surface charge density sigma(i) at the dielectric interface ?
Any reply welcome
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raback
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Re: boundary condition at a dielectric interface
Hi
The internal BC should work ok for any Poisson type of equation of Elmer. It does not matter whether the Neumann condition is on internal or external boundary.
If the boundary is not conforming, you should add the flux on either side and decleare mortar BCs between the the two boundaries. There might be some tinkering to find the correct mortar settings but I probably there would be no problem there either.
Both approaches are of course fully implicit.
-Peter
The internal BC should work ok for any Poisson type of equation of Elmer. It does not matter whether the Neumann condition is on internal or external boundary.
If the boundary is not conforming, you should add the flux on either side and decleare mortar BCs between the the two boundaries. There might be some tinkering to find the correct mortar settings but I probably there would be no problem there either.
Both approaches are of course fully implicit.
-Peter
Re: boundary condition at a dielectric interface
I would have saved myself the trouble of posting this question and viewers from replying if only I had read Advanced Electromagnetic Computation 2nd edition by Kalluri a day in advance since it is proven in his Appendix 16C that Neumann BCs are implicity enforced in FEM's minimizing of the functional. Nevertheless, I am very grateful for your reply. There only remains for me to become familiar with how to code a given surface charge density on a dielectric. By definition, perfect conductors have zero skin depth and therefore only a surface charge. Therefore, since problems involving perfect conductors with a floating voltage (so not a Dirichlet BC) must be fairly common, I would think that surface charges can be handled by Elmer. I certainly hope so because my air region can be ionized and deposit ions on the dielectric surface.
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raback
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Re: boundary condition at a dielectric interface
For floating potential see, viewtopic.php?f=3&t=6715&p=21080&hilit=floating#p21071
Re: boundary condition at a dielectric interface
The link was enlightening. Well I'm glad Elmer seems to be able to take care of a surface charge density that varies with location on an arbitrary surface. I am now going to study the tutorials before writing my own program and before starting threads on Elmer forums.
Yours truly
Yours truly