FEM-BEM hybrid method to solve the demagnetisation field

From Maxwell's equations, the main PDE we get to solve for the demagnetisation field in the absence of an electric current is:

$$ \Delta u = \nabla \cdot \vec{M}, $$

where $u$ magnetosatic potential, and $\vec{M}$ is the magnetisation. This equation holds over a domain $\Omega$ where magnetisation is finite. The Poisson equation has an open boundary condition as $u$ decays to zero at infinity. Further, one can obtain the demagnetisation field from the magnetistatic potential as:

$$ \vec{H}_{dmg} = - \nabla{u}. $$

This can be solved using standard FEM formulation, however, one needs to mesh region outside $\Omega$ to apply the Dirichlet boundary condition $\lim_{x\to\infty} u(x) = 0$. To obtain $u$ over $\Omega$ without meshing region with no magnetisation, a hybrid FEM-BEM technique is used as presented by Fredkin & Koehler. The method essentially divides $u$ in two parts, $u_1$ and $u_2$, such that over the domain $\Omega$:

$$ \begin{align} \Delta{u_1} &= \nabla \cdot \vec{M}; \ \frac{\partial{u_1}}{\partial{n}} = \vec{M} \cdot \hat{n} \text{ on d} \Omega\\ u_2(x') &= \oint_{\text{d}\Omega} u_1(x) \cdot \frac{\partial{G(x, x')}}{\partial{n}} \ \text{d}S + \big( \frac{\Psi(x')}{4\pi} - 1 \big) u_1(x'); \ x' \in \text{d}\Omega \\ \Delta{u_2} &= 0; \ \text{Dirichlet BC from above.} \\ u &= u_1 + u_2. \end{align} $$

The second step is obtained through BEM formulation. The goal here is to go over each step and solve for $u_1$ and $u_2$, and thus, ultimately $u$.

Running the notebooks

1. Install pixi: https://pixi.prefix.dev/latest/installation/
2. To launch notebook for mesh calculation, run pixi run mesh-calculation in the terminal.
3. To launch notebook for demagnetisation field calculation, run pixi run demag-calculation in the terminal.