Galerkin FEM for elliptic PDEs
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  1. \chapter{Examples and Verification}
  2. \section{Coulombic Magnetic Potential}
  3. In the absence of external current densities, we may write $\nabla \times \mathbf{H} = 0$.
  4. This allows the $H$ field to be represented using a scalar potential term
  5. \begin{align}
  6. \mathbf{H} = \nabla \phi_M.
  7. \end{align}
  8. Static magnetics problems (in linear media) can be solved using a scalar potential term obeying the
  9. following relationship \cite[e.g.,][]{Jackson1998}
  10. \begin{align} \label{eq:permMagnet}
  11. %-\nabla \cdot \mu(\mathbf{r}) \nabla \phi_M(\mathbf{r}) = \mu(\mathbf{r}) M_0(\mathbf{r}).
  12. -\LAPL{\phi_M} = \rho_M(\mathbf{r}).
  13. \end{align}
  14. Where $\phi_M$ represents the Coulomb magnetic potential (appropriate for static magnetics problems)
  15. and $\rho_M$ is the effective magnetic charge density (=$\nabla \cdot \mathbf{M}$).
  16. This formulation is particularity useful in calculating the fields of permanent magnets, where the
  17. magnetisation of the magnet is given by $\mathbf{M}(\mathbf{r})$.
  18. The left hand side of (\ref{eq:permMagnet}) can easily be solved using {\bf{FEM4EllipticPDE}}, however
  19. the right hand side requires some special care.
  20. \subsection{Calculation of Charge Density}
  21. For uniformly magnetised media $\rho_M$ vanishes everywhere except at the boundaries of the magnet.
  22. In truth, $\mathbf{M}$ is well behaved and there are no truly uniformly magnetized magnets, but the
  23. true nature of $\mathbf{M}$ is rarely known, and the assumption of uniform magnetisation becomes necessary.
  24. Practical permanent magnets are \emph{effectively} uniformly magnetised, and this assumption is completely
  25. reasonable, and even necessary.
  26. We can therefore apply the divergence theorem to to the boundaries to calculate the surface charge density
  27. $\sigma_M = \hat{\mathbf{n}} \cdot \mathbf{M}$ which exists on the 2D surface of the media and has zero
  28. Riemann measure, formally necessitating a Lebesgue integral.
  29. We may then let $\rho_M \rightarrow \sigma_M$ in
  30. (\ref{eq:permMagnet}).
  31. The FEM variational formulation for an particular element in the load vector $g$ that spans
  32. the boundary of the media at point $\mathbf{r}_0$ can be written
  33. \begin{align}
  34. [g]_i & = \left< \rho_M, \phi_i \right> = \int \rho_M \phi_i \\
  35. %//& = \int_L \delta\{\hat{\mathbf{r}_0}\} \sigma_M \phi_i \\
  36. & = \int_L \sigma_M \phi_i \\
  37. & = \sigma_M \phi_i(\mathbf{r}_0).
  38. \end{align}
  39. \subsection{Uniformly Charged Sphere}
  40. The scalar potential of a uniformly charged sphere is given \cite[p. 198, ][]{Jackson1998}
  41. \begin{align}
  42. \phi_M(\rho, \theta) = \frac{1}{3} M_0 a^2 \frac{r_<}{r_>^2} \cos(\theta).
  43. \end{align}
  44. Where the notation $r_<, r_>$ represents the smaller or larger of the distance $\rho$ and sphere radius $a$.
  45. Inside the sphere
  46. \begin{align}
  47. %\phi_M &= \frac{1}{3} M_0 \rho \cos \theta & \text{inside the sphere} \\
  48. \phi_M &= \frac{1}{3} M_0 z & \text{inside the sphere} \\
  49. \phi_M &= \frac{1}{3} M_0 a^3 \frac{\cos \theta}{\rho^2} & \text{outside the sphere}
  50. \end{align}