Grids and meshes

The micromagnetic model supports three grid types. The grid type is selected with problem.setMicroMagGridType(...) in Matlab and with the grid_type argument in Python, and it decides which of the geometry parameters are read.

Grid type	Value	Required geometry input
uniform	1	grid_n (res in Python) and grid_L. The grid is generated internally.
tetrahedron	2	grid_pts, grid_nod, grid_ele and grid_nnod.
unstructuredPrisms	3	grid_pts and grid_abc.

Uniform grid

This is the default grid type, and by far the simplest. The domain is a rectangular box of size grid_L \(= (L_x, L_y, L_z)\) divided into grid_n \(= (n_x, n_y, n_z)\) identical rectangular prism cells, so that \(\Delta x = L_x/n_x\) and similarly for \(y\) and \(z\). The box is centred on the origin, i.e. it spans \([-L_x/2, L_x/2]\) and so on, and the cell centres are placed at the centre of each cell. A direction with a single cell is placed at zero and given the full length of the domain as its cell size.

The cells are ordered with \(x\) running fastest and \(z\) slowest,

\[\mathrm{ind} = i + (j-1)\,n_x + (k-1)\,n_x n_y .\]

The same ordering applies to every array of single-cell properties (Ms, K0, u_ea, m0, temperature, …) and to the returned magnetization.

For a uniform grid the exchange operator is built as a finite-difference Laplacian, see Exchange interaction.

Unstructured prisms

The unstructuredPrisms grid is a mesh of rectangular prisms of varying size. The prisms are all aligned in the sense their faces are perpendicular to the global xyz-axes, so every face is either parallel or perpendicular to every other. The mesh is specified through

• grid_pts - an (ntot,3) array with the centre of every prism;

• grid_abc - an (ntot,3) array with the full side lengths \((a,b,c)\) of every prism.

The problem resolution must be set to [ntot,1,1], since the number of cells is what the solver uses, not a rectangular subdivision:

resolution = [length(mesh.pos_out) 1 1];
problem = DefaultMicroMagProblem( resolution(1), resolution(2), resolution(3) );
problem = problem.setMicroMagGridType('unstructuredPrisms');
problem.grid_pts = mesh.pos_out;
problem.grid_abc = mesh.dims_out;

problem = MicromagProblem(
    res=(len(grid_pts), 1, 1),
    grid_type="unstructuredPrisms",
    grid_pts=grid_pts,
    grid_abc=grid_abc,
)

For this grid type the mesh is analysed by CartesianUnstructuredMeshAnalysis, which finds the faces, face normals, face areas, cell volumes and the interpolation stencils that the exchange operator is built from. The result is returned to Matlab as the GridInfo struct. The mesh analysis is also where periodic exchange boundaries are resolved, see Periodic exchange boundaries.

An unstructured mesh may be plotted from Matlab with cartesianUnstructuredMeshPlot.

Tetrahedral grid

A tetrahedral mesh is specified through the node coordinates grid_nod, the number of nodes grid_nnod, the element definitions grid_ele (the four node indices of each element) and the element centres grid_pts. As for the unstructured prisms, the resolution is [ntot,1,1].

Giving the mesh is all that is required. MagTense analyses it and builds the exchange operator itself, exactly as it does for a grid of unstructured prisms given its centres and sizes. In MATLAB the whole grid is set up by

problem = problem.setMicroMagGridTetrahedron(model.Mesh.Nodes, model.Mesh.Elements);

which fills in grid_nod, grid_ele, grid_nnod and the element centres grid_pts, and in Python by passing grid_nod and grid_ele to MicromagProblem, where grid_nnod and grid_pts likewise default to the mesh. A quadratic mesh is accepted and analysed as its linear counterpart.

The mesh analysis is in TetrahedralMeshAnalysis.f90. It works from the connectivity rather than from the geometry: two tetrahedra are neighbours when they share three nodes, and the face they share is the one opposite the node that they do not.

Periodic exchange boundaries are requested through exchPBC. They are linked either by identifying the nodes on the two boundary planes, when the mesh is periodic along that direction, or by mortar coupling the boundary faces when it is not, with the choice made per direction and a warning printed when the fallback is used. See Periodic boundaries on a tetrahedral mesh for what each mechanism does and what the fallback costs.

The demagnetization tensor is computed from the analytical tetrahedron tensor. Note the following limitations of this grid type:

• Averaging of the demagnetization tensor over the receiving cell (N_ave) is not supported and a warning is printed if it is requested.

• The shape correction is disabled, because a tetrahedral mesh carries no per-element size array from which the occupied volume fraction can be found.

Passing an externally computed exchange matrix

For any grid type the exchange operator may be computed outside MagTense and passed in directly, which bypasses the mesh analysis entirely. This is described in Supplying the exchange matrix directly.