As you are thinking about a tree-cotree gauge I assume you are using edge elements. There was a long discussion on this issue in the past.

See, for instance,

1) Biro, O.; Preis, K.; Richter, K. R.: On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic fields. IEEE transactions on magnetics 32 (1996) , S. 651-654.

2) A discrete sequence associated with mixed finite elements and its gauge

condition for vector potentials, P. Dular, A. Nicolet, A. Genon, W.

Legros, IEEE Transactions on Magnetics, Vol.31, No.3, (1995), pp. 1356-1359

My experience is that it is better to let your system ungauged and use an iterative solver (like a preconditioned conjugate gradient ), which finds one of the solutions then trying to use a tree-cotree gauge for this kind of formulation.

Regards,

Renato.

Thanks very much for the useful comment.

Another question is that is that is there any difference for magnetostatic and eddy-current problems when applying the tree-cotree gauge? Suppose the magnetostatic case with natural boudanry condition, let Nn denote the number of nodes in the mesh, then the rank of the resultant curl-curl linear system is Ne-(Nn-1), where Ne is the number of edges in the mesh.

Then the question is that if the edge element for A, nodal element for phi and Backward Euler scheme is used to perform time-stepping, then what is the rank of coefficient matrix of the resultant linear system? For example, if there are Nn nodes in the mesh, Ne edges in the mesh, Nn_c nodes in the conducting object and Ne_c edges in the conducting object.

I thinks mayne the above problem is closely related to the following quesion,

sigma*dA_dt + rot(nu*rot(A)) + sigma*grad(phi) = Js ------ (1)

div(-sigma*dA_dt-sigma*grad(phi)) = 0 ------ (2)

In the conducting region, is the eddy-current equation (1)(2) uniquely solvable? Some papers say YES. However, if we make div(Js)=0, then take the divergence of equation (1), we arrive at equation (2). Hence (2) is dependent on (1), how the solution be unique then?

As for the ungauged method, if the ICCG linear solver is used for transient problem, since the sigular system has infinite solutions and the ICCG only arrives at some specific solution, how can I know that each time-step, the converged linear solution is the same "specific solution". I mean that for static problem, the ICCG is no problem. For time-stepping FEM, the answer is still YES?

I have used another gauge method, that is to modify the conductivity to be small positive constant in non-conducting region and thus the A-phi formulation possess unique solution then. However I am not sure which gauge condition is the most popular method used by people. Besides, the gauge condition for higher-order edge element is not common to see, so I think the gauge method is difficult to use.

Thanks very much for your attention!

Regards,

Yanpu

Dear Yanpu,

Sorry for not having enough time to discuss each point of your interesting questions (too busy these days), but I suggest that you search for Oszkár Bíró papers on the 3D eddy currents problem and its ungauged solution. I suggest that you begin with:

Biro, O. , Edge element formulations of eddy current problems, Comput. Methods Appl. Mech, Engrg, 169 (1999) 391-405

He shows that the ungauged system is singular, but can be solved by iterative methods if the right-hand side is consistent. He also shows how you can enforce its consistency.

Then, search for Oszkár papers on the same problem in time-domain. I think all the answers for your questions are there. But if they are not, please write again.

Your trick of using A-phi for all the problem with very small conductivity on the air is not good in my opinion, because you add one more unknown for each node on the air region.

As regard the most popular formulation for this problem, I think it is the ungauged magnetic (or electric) vector potential edge elements formulation solved with iterative solvers.

Regards,

Renato.

Dear Prof. Renato,

Thanks very much for the valuable information. The paper of Biro (1999) is very classic and I learned a lot from it.

Since the intel PARDISO direct solver is fast and powerful (suppose the computer memory is sufficient), I think it is very necessary to use the gauged version of edge element method.

As for as the "small positive conductivity gauge" method mentioned, there has been some work.

For example, in the thesis of Benjamin Bihler titled "Nonlinear Multigrid Methods for

the Numerical Solution of TEAM Workshop Problem 20" in 2005, the magnetostatic problem whose solution is not unique

curl(nu*curl(A)) = Js ------ (1)

is replaced by a perturbed one with small positive beta:

curl(nu*curl(A))+beta*A = Js ------(2).

Since the resultant matrix is positive definite now (stiffness matrix semi-positive definite, the mass matrix positive definite), so the solution of (2) is unique and the tree-cotree gauge is not needed any more. If we directly solve (1), to ensure the uniqueness of the solution, the tree-cotree gauge can be applied which however will make the matrix system bad-conditioned, since the optimal tree is not easy to get.

For transient eddy-current problems, the paper "Algebraic MultiGrid Methods for Nodal and Edge based Discretizations of Maxwell's Equations" which can be found by Google and the paper "Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems (2004)" by F. Bachinger, U. Langer and J. Sch¨oberl both used "the small positive conductivity gauge" for the A*-formulation. I think maybe the small positve conductivity is to make sure that the solution in the non-conducting region unique.

In the A-phi formulation, I use the same method without introducing phi in the non-conducting region. In practice it is feasible however I don't know the error or the accuracy of the "proposed method".

Regards,

Yanpu.

Dear Yanpu,

Yes, if you want to use a direct solver, then you should gauge your problem. However, even the new fast solvers have serious memory limitations and I don´t think they are, in general, a good choice for real sized electromagnetic problems.

The 3 references you cite only use the magnetic vector potential as unknown, and then they don´t have the problem of adding a new unknown to the non-conducting region.

As you are working with A-phi, I thought you were adding phi to the non-conducting region when you were using the small conductivity trick. Then I agree: you should test your trick solving some problems with known solutions, in order to evaluate the error you are introducing. If it works well, it would be nice to have a theoretical explanation on why phi could be removed.

Regards,

Renato.

Thanks for the nice comments.

The direct solver demands huge memory for practical problems although it is very fast.

I will try some examples with known artificial solutions and check the validity of the gauging method with small sigma.

It would be nice if you could later report your achievements in a paper. I am very interested in your findings.

Regards,

Renato.

Dear Yanpu:

I have the same question, and it has puzzled me for a long time.

Is "A"(magnetic vector potential) unique in conducting region when "A, phi" formulation is used? According to some papers, it is non-unique, and tree-cotree gauge can be still imposed in conducting region. And also ANSYS-emag does it, and also my procedure. At least, imposing tree-cotree gauge in conducting region with edge based elements won't bring inaccuracy in my experience, if a direct solver is used. On the other hand, as Prof. Renato said, ungauged equations can be handled by an iterative solvers, such as cg/iccg, usually more efficient than the direct solver with tree-cotree gauge.

However, I haven't find clear theoretical evidence of the uniqueness or non-uniqueness of the solution in conducting region with edge based element in papers. I guess that  the solution of "A" and "phi" is not unique, explained in the following way:

Suppose A1 and phi1 are solutions of the governing equations you gave. Then another solution A2 = A1 + grad lambda and phi2 = phi -  d(lambada)/dt can be acquired, in which lambada is an arbitrary scalar potential field satisfying d(lambada)/dt = 0 on boundaries of conducting region . 

Do you agree?

Best regards,

Sincerely,

Peng Jiang

Dear Peng,

I think that using A-phi formulation, the solution of A and phi is not unique. Because we doesn't have defined any information about the divergence of A. Only the curl of A is defined in conductors where both A and phi exist. Thus the vector variable A can't be uniquely defined. 

The iterative solver can be used to solve the ungauged A-phi formulation, because the divergence of A is implicitly defined in the iterative process (the div of A (in weak sense) equals the div of initial guess of A when starting the iterative process), so we can get a converged solution even we doesn't specify the div of A explicitly.

Do you have some references of applying tree-gauge of A in all the domain (including air and conductors?)

Thanks.

Interesting..