Demagnetizing field using scalar potential method

Question

I want to calculate the stray magnetic field from a ferromagnet using the scalar potential method (1). The problem consists of a ferromagnetic cuboid divided into small cuboidal cells in which the magnetization $\mathbf{M}$ is assumed to be constant. To do this, I have to evaluate a convolution integral, discretized to a 3D - convolution sum (assuming all magnetization vectors point along $z$-axis)

$$\phi(\mathbf{r}_i) = \sum_j S_z(\mathbf{r}_i - \mathbf{r}_j)M_z(\mathbf{r}_j)$$

where $\phi$ is the scalar potential and $\mathbf{r}_i$ is the position vector a cell and $S_z$ is a known function. The solution is via FFT

$$\phi = \mathrm{F}^{-1}\left(\mathrm{F(S_z)} \, \mathrm{F}(M_z) \right)\, .$$

I am struggling to understand, especially the matrix representation of $S_z$, and how to proceed to perform the convolution with FFT.

References

1. Abert, Claas, et al. "A fast finite-difference method for micromagnetics using the magnetic scalar potential." IEEE Transactions on Magnetics 48.3 (2011): 1105-1109.

Answer 1

I think I see the confusion.

Equation 14 of the cited publication indicates that $S_z$ is a scalar field and not a matrix. The i,j, and k components that appear in $S_z$ definition at equation 17 are arguments that are passed to $F(x,y,z)$ defined in equation 16. That is, $F(x,y,z)$ returns a scalar.

To perform a 3D Fourier Transform on $S_z$ in python, see

https://docs.scipy.org/doc/numpy/reference/generated/numpy.fft.fftn.html

But a better approach would be to use the scipy fft convolve on the 3D grid containing the numerical approximation for $S_z$ and $M_z$. See

https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.fftconvolve.html.

Doing so would handle a lot of the tedious complexity of padding, centering, etc... associated with manual FFT convolution. The final result from this method is the final result you're looking for.