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     Archie Campbell (amc1) Member Username: amc1 Post Number: 7 Registered: 03-2008 |
I am still trying to solve the magnetic problem of wires carrying a given current, which are screened by either superconductivity or the usual skin effect. The equations are similar but the superconductors do not have the time dependence so the equation is the simple diffusion equation div(grad(A))=L*A where A is the vector potential and L a material parameter determining the screening. Both sides should be equal to the current density J. For two wires carrying opposite currents there is no net current so the distant boundary condition on A is A=0. (natural(A)=0 gives similar results). Since the only other condition is an integral one I can see there may be a problem with the uniqueness of the solution. This may explain why putting the integral of the tangential component of B round the wire does not give a solution, it leads to a continuously increasing delta function on the surface (or wherever the condition is imposed). A previous correspondent with this problem says that by doing the integral round the domain the problem disappears, but this cannot be applied to more than one wire and there are many applications in which we need the fields from multiturn coils. An alternative is to define the integral of the current density (i.e. L*A) over the cross section of the wire as the applied current, and at first sight this appears to produce a sensible solution. However there is something wrong with it. The magnetic fields are qualitatively correct but are inconsistent with the currents by a large fairly constant factor, roughly proportional to the applied current in the constraint section. I have therefore plotted the two sides of the equation to be satisfied, div(grad(A)) and the right hand side, L*A which should be the same in a solution (and are in the last program) but are widely different, indeed of different sign. How can this be consistent with an errlim of 1e-4? I attach the program with strong screening (L=200, London1), low screening (L=2,London2) and a fixed uniform current density (resist). The latter two should be the same as the current density in London2 is pretty uniform but the results are not the same. In particular in London2 the vector potential is a maximum at the centre of the wire while for constant J it is a minimum. I suspect this a general problem, not unique to FLEXpde, but any comments would be welcome as there are numerous applications in electrical engineering. Archie Campbell
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| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1215 Registered: 06-2003 |
It appears that there is a conflict between the PDE and the constraint, so that there is no solution that satisfies both. FlexPDE finds a solution by minimizing a functional involving both the nodal equations and the constraint. "Minimizing" may not mean "zeroing", and in cases where it does not, the solution is a compromise between conflicting requirements. So, a solution providing a minimum may not be a very "good" solution at all. In your problem, varying some of the parameters, the constraint is always satisfied, but the PDE is not. Using the constant-current form, the resulting A is negative, which could never satisfy a positive constraint. I infer that there is something amiss, but I have not pursued it any farther than this. | |||||
| nikos adamopoulos (nadam) Member Username: nadam Post Number: 5 Registered: 06-2008 |
With regard to message posted by Archie Campbell, the beloved Prof from Cambridge, I would like to refer to a previous post of mine regarding the time dependent solution of the magnetic field penetration in a superconductor under a transport current and an external magnetic field. The difficulty is how to describe the Ampere’s law and have a unique solution that correctly describes the magnetic field inside the superconductor to mimic the critical state. I have tried to use the integral of the current density in order to describe the magnetic field but it seems the solution is not correct. Anyway, it was a great pleasure to read the comments of Archie Campbell. Nick Adamopoulos |
Small penetration