The algorithm to estimate fiber diameters works as follows: At the start, each fiber voxel gets assigned the value of the largest sphere fitting into the fiber in this position. Then, it is checked for each fiber voxel if it lays inside of a larger sphere compared to the diameter of the sphere already assigned to the voxel under consideration.

For this comparison, there are two possibilities:

• The center of the smaller sphere is contained within the larger sphere. Then, the diameter value of the larger sphere is assigned to all voxels in the smaller sphere. Voxels at the end of a fiber get like this the same diameter as neighboring parts of the fibers and not a smaller one. See the following figure.
  

• The center of the smaller sphere is not contained within the larger sphere. Then, the overlapping area between both spheres is partitioned between both diameters. Like this, fibers overlapping each other both get the correct diameter. See the following figure.
  

To compute fiber orientations, two algorithms are used:

• Star Length Distribution (SLD):

1. 1. Analysis is done on a per-voxel basis.
   2. For each voxel, it analyzes chord lengths through the voxel for a pre-defined set of directions.

• Principal Component Analysis (PCA):

1. 1. The domain is cut into blocks of a fixed size (currently 32x32x32).
   2. For each block, all connected components are found. Ideally, they correspond to segments of individual fibers.
   3. For each component, it is assumed that the voxels of the component form a point cloud. An ellipsoid which best approximates this point cloud is computed using the PCA method (Principal Component Analysis) and provides the principal orientation.
   4. The principal orientation vectors are averaged to get a per-block orientation.
   5. Per-block orientations are averaged to obtain the desired number of orientation tensors.

In summary, the PCA method cuts the fibers into segments and determines the local average of the orientation of these segments to obtain an orientation tensor.

For both algorithms, GOF files (GeoDict Orientation Files) can be saved to allow for simulation of anisotropic properties (e.g. transverse isotropic) of fibrous structures. In case of transverse isotropy, the material properties in fiber direction differ from properties in the perpendicular direction.

These GOF files can be used e.g. in ElastoDict for computations of mechanical properties in anisotropic materials.

FiberFind can calculate curliness and curvature distributions in models of fibrous structures. Curved fibers occur in materials for numerous industrial sectors: synthetic filter media, cellulose filter media, gas diffusion layers (GDL) in fuel cells, insulation materials, nonwovens for many applications, etc.

The calculated curvature is based on the definition of geometric curvature. At any point on the curve, the curvature is defined as the inverse radius of the osculating circle. The osculating circle has the same tangent vector as the curve at that point and approximates the curve locally.

The curl-index measured by FiberFind is the length of the fiber divided by the distance between its start and its end. For straight fibers, this results in a curl-index of 1.

FiberFind calculates the curvature and curliness based on the centerline and the length of fibers. If no analytic data (.gad) is available, e.g., from a previously run fiber identification, FiberFind identifies the fibers with the classical image processing approach first.

The curvature estimation component is intended to be used with highly porous fibrous media (>90%). Solid volume fractions above 10% can decrease the accuracy of the result.

Since GeoDict 2023, additionally the autocorrelation of the curvature is computed for the fibers identified. This is done by interpolating first the curvature over each fiber and computing the autocorrelation from the interpolated values. The mean of these autocorrelation is then computed over all fibers. The autocorrelation function is cut at the fiber length that is exceeded by only 20% of the fibers.

Since GeoDict 2023, the computation of the torsion of the cross-section area is available for the fibers identified. The torsion is computed for each fiber segment and as mean value for each whole fiber.

Like for the curvature, the autocorrelation of the torsion is computed as well. The torsion is first interpolated over the length of each fiber, and the autocorrelation is computed for each fiber separately. Then, the mean of this autocorrelation is computed over all fibers. At a fiber length that is exceeded by only 20% of the fibers, this autocorrelation function is cut.

So far, the result of the torsion computation is available only in the GeoDict result file under the Report – Map tab.