Visualisation

Figure 2.19: A visualisation showing a surface map (A), translucent surface map (B), area mean magnetisation vectors (C), isosurface (D) and streamlines (E)

Visualisation is an important part of scientific computation, both for the analysis of results and their presentation. To visualise computed results, we use MayaVi (Ramachandran, 2001), which makes use of Kitware's VTK (Schroeder et al., 1996, Schroeder et al., 1997) as the middleware for preparing the data prior to being rendered in either POV-Ray or a Pixar Renderman (Pixar, 1989, Pixar, 2000) compliant raytracer.

We can exploit the features of these tools, particularly by adding features commonly found in computational fluid dynamics to further our understanding of the magnetisation patterns resulting from our simulations.

Figure 2.19 shows a typical visualisation. Point A in the upper image shows the surface map of a scalar, in this instance the xy angle of magnetisation. For clarity a wireframe map showing the outline of the finite element mesh is visible. In the lower image, the scalar surface map remains, though it is translucent (point B). The cones indicated by point C represent the mean magnetisation of the small area immediately surrounding the cones; the colour shows a scalar (the $z$ component of the magnetisation) and the direction of the cone reflects the magnetisation vector itself. Where smaller cones are present in a visualisation, these represent an interpolation of the vector where source data is only available around that point rather than at the point itself. This usually takes place at boundaries, arising from a linear interpolation between $\ensuremath{\mathbf{M}}$ and 0.

To highlight points of interest, an isosurface of a scalar (such as that indicated by point D) may be shown. The isosurface in this example is again based on the $z$ component of the magnetisation and attaches a visual representation to the core of a vortex. Finally, point E shows streamlines, which are the result of tracer particles being ``dropped'' into the system. These tracers follow the path of the magnetisation and provide a visual cue for interesting features of the visualisation; here they gradually follow the magnetisation around the surface of the sample, spiralling in until they reach the vortex core.

Figure 2.20: Vortex at the core of a droplet object (see section 5.5) highlighted with streamlines (massless particles, or tracers) injected into the material. At the edges of the sample these follow the magnetisation as if it were a velocity field; at the core these follow the curl of the magnetisation.

Where the volume of the sample is of particular interest, a random point mask can be applied to the visualisation, such as that in figure 2.20. Here streamlines have again been used to add depth to the visualisation and by operating on a derived vector (in this case, the curl of the magnetisation) the bounds of the vortex core are clear.

Python (van Rossum, 2003) and Linux shell scripts (Ramey, 2003) were employed extensively in coordinating the process to take raw simulation results and produce camera-ready images and animations suitable for the analysis of magnetic microstructures.

Schematic drawings are occasionally used to assist understanding physical geometry or aspects of magnetisation. Figure 2.21 shows two schematics of a generic sample with arbitrary shape and a symmetry in the $xy$ plane (point A), here represented by a rhombus. The axes on the left indicate the three-dimensionality of the sample. If the applied field (point B) were initially applied along the $x$ direction, then two possible vortex types emerge. The vortex shown in the left sample has an out-of-plane vortex, where the magnetisation circulates in the $xy$ plane (point C, solid blue arrow) and the core of the vortex (point D, dotted red arrow) points perpendicular to this symmetric plane, i.e. in $z$. The vortex shown in the right sample has an in-plane vortex -- the circulation of the magnetisation (point C, solid blue arrow) is in the $yz$ plane and the core of the vortex (point D, solid red arrow) is aligned with the direction of the initial applied field, i.e. in $x$.

Figure 2.21: Schematic of out-of-plane (left) and in-plane (right) vortices. The dotted red arrow indicates the vortex core direction; the solid blue arrows show the magnetisation circulating around the core