Demagnetization field

The demagnetization (stray) field is the most computationally expensive part of micromagnetics. MagTense computes it from the fully analytical demagnetization tensor of the cell shape; a rectangular prism for the uniform and unstructured-prism grids, and a tetrahedron for the tetrahedral grid. The accuracy of these tensors is documented in Bjørk, R. and d’Aquino, M.: Accuracy of the analytical demagnetization tensor for various geometries, Journal of Magnetism and Magnetic Materials, 587, 171245, 2023.

The tensor is symmetric, so it is stored as the six components \(K_{xx}, K_{xy}, K_{xz}, K_{yy}, K_{yz}, K_{zz}\), each a dense \(n_\mathrm{tot} \times n_\mathrm{tot}\) matrix in single precision. The memory requirement is therefore approximately \(24\,n_\mathrm{tot}^2\) bytes, which sets the problem size the dense tensor handles; the Matlab interface prints an estimate when the problem struct is created, and the Fast Multipole Method lifts the scaling to \(O(N)\) for larger problems. The saturation magnetization is folded into the tensor when it is built, so a spatially varying \(M_s\) costs nothing extra.

Switching the demagnetization off

Setting useDemag (Matlab setUseDemag, Python usedemag) to False removes the demagnetization field from the effective field and skips building the tensor altogether.

Cell-averaged versus point-evaluated tensor

useAvgN (Python useavgn, default on) selects the volume-averaged prism tensor, i.e. the tensor averaged over the receiving cell rather than evaluated at its centre. This is the more accurate choice for a prism mesh and is the default; it uses the averaged prism tile (tile_type = 8).

Some analytical benchmarks are formulated for the tensor evaluated at the cell centre, and for those useAvgN must be switched off. This includes the macrogeometry and shape-correction tests.

For the unstructuredPrisms grid the tensor may additionally be averaged numerically over the receiving cell using N_ave = [nx, ny, nz], which evaluates the tensor on an nx*ny*nz sub-grid inside each receiving prism and averages. The default [1 1 1] means no averaging.

Approximating the tensor

The dense tensor may be sparsified, which trades accuracy for memory and speed. The approximation is selected with setMicroMagDemagApproximation in Matlab and with demag_approx in Python, and the associated cut-off is dem_thres.

Name	Value	Description
`none` (Python `None`)	1	Default. The dense tensor is used as is.
`threshold`	2	Every element with \(\|K\| <\) `dem_thres` is set to zero and the six matrices are converted to MKL sparse matrices.
`threshold_fraction`	4	As above, but `dem_thres` is interpreted as the fraction of tensor elements to discard. The cut-off value is found by bisection. A value \(\geq 1\) zeroes the demagnetization field entirely.

The values 3 (fft_thres) and 5 (fft_threshold_fraction) select earlier Fourier-space variants of the threshold that are superseded by the three options above; the solver says so if they are selected.

Adding noise to the demagnetization field

CV (Python cv) adds a normally distributed relative error to the demagnetization field, with CV being the coefficient of variation, i.e. the ratio of the standard deviation to the mean. The default of zero disables it. The draw is controlled by the same random seed as the thermal field, see Reproducible random numbers.

Storing and reusing the tensor

Because building the tensor can dominate the run time of a short simulation, it can be written to disk and read back:

Matlab	Python	Description
`N_ret`, `N_file_out`	`N_ret`, `N_file_out`	`1` keeps the tensor internal, `2` returns it in memory, and a value `> 2` writes it to the file named by `N_file_out`. The value is the length of that filename.
`N_load`, `N_file_in`	`N_load`, `N_file_in`	The same encoding for loading a previously stored tensor instead of computing it.

The Matlab helpers setReturnNFilename and setLoadNFilename set the filename and the matching length in one call. In Python the filename constructor argument sets all four fields; its default "t" has length one and therefore selects the default of computing the tensor internally.

Parallelism and hardware

useCuda (Matlab setUseCuda, Python cuda) evaluates the tensor-vector product on an NVIDIA GPU at every time step. The Python interface checks for nvidia-smi and runs on the CPU instead, with a note, when no GPU is present. In Matlab, setUseCuda also picks the matching MEX-file.
The Fast Multipole Method replaces the dense tensor by an \(O(N)\) evaluation and is described in Demag field - FMM. It is selected with use_fmm in a build with USE_FMM3D=1.