Unofficial
FEMM
documentation
by Dr Stan Zurek

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===== Symmetry with Dirichlet and Neumann boundary conditions =====

In FEMM:
> If no boundary conditions are explicitly defined, each boundary defaults to a homogeneous Neumann boundary condition. However, a non-derivative boundary condition must be defined somewhere (or the potential must be defined at one reference point in the domain) so that the problem has a unique solution.
> For axisymmetric magnetic problems, A = 0 is enforced on the line r = 0. In this case, a valid solution can be obtained without explicitly defining any boundary conditions, as long as part of the boundary of the problem lies along r = 0. This is not the case for electrostatic problems, however. For electrostatic problems, it is valid to have a solution with a non-zero potential along r = 0.

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Symmetry of the geometry can be exploited by by using the Dirichlet or Neumann conditions:
  * Dirichlet - the condition of A=0 is enforced along the boundary which stops the flux crossing that boundary. It can be thought of as a perfect insulator.
  * Neumann - the condition of dA/dn = 0 which means that the flux lines will be forced to cross this boundary at 90 deg angle. It can be though of as a perfect conductor.

|< 100% 60% 40% >|
^  Comparison of symmetrical models in FEMM, N = number of turns, N/2 = half of turns  ^^
| {{sym_no_symmetry_-_boundary.png?500}} | Fig. 1. \\ Full model, no symmetry, \\ calculated L = 0.117278 H, \\  By(0,0) = 7.02993e-5 T  |
| {{sym_horiz_symmetry_-_boundary.png?500}} | Fig. 2. \\ Horizontal symmetry, \\ calculated L = 0.05875 H (so L * 2 = 0.1175 H), \\ By(0,0) = 7.04286e-5 T \\ \\ Full length of one side of the coil is modelled so full number of turns is used (N). Only "half" of the coil is modelled so some output values (such is inductance or force have to be multiplied by 2).  |
| {{sym_vert_symmetry_-_boundary.png?500}} | Fig. 3. \\ Vertical symmetry, \\ calculated L = 0.0587631 H (so L * 2 = 0.1175262 H), \\ By(0,0) = 7.04353e-5 T \\ \\ Only half length of the coil is modelled so number of turns has to be reduced by half (N/2). Only "half" of the whole coil is modelled so some output values (such is inductance or force have to be multiplied by 2).     |
| {{sym_vert_and_horiz_symmetry_-_boundary.png?500}} | Fig. 4. \\ Horizontal and vertical symmetry, \\ calculated L = 0.0294388 H (so L * 4 = 0.1177552), \\ By(0,0) = 7.05736e-5 T \\ \\ Only half length of the coil is modelled so number of turns has to be reduced by half (N/2). Only quarter of the whole coil is modelled so some output values have to be multiplied by 4.   |
| {{sym_symmetry_comparison.png?500}} | Fig. 5. \\ Visual comparison of all 4 cases. It should be noted that the actual flux density distribution is the same - as can be judged by the coloured map. The difference in the flux line appearance is only apparent, because the limits of flux change and FEMM scales the plotting of the flux lines differently. \\ \\ The flux density at the central point By(0,0) is identical in all 4 cases, with the difference not exceeding 0.4% resulting only from the changes of the mesh.    |

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Download example file: [[http://www.femm.drsz.pl/lib/exe/fetch.php/femm_symmetry.zip|femm_symmetry.zip]]
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Unofficial FEMM documentation by Dr Stan Zurek

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Unofficial
FEMM
documentation
by Dr Stan Zurek
