Fibers may overlap when Allow Fiber Overlap is selected.

Checking Without (Remove) Fiber Overlap eliminates existing overlaps after the generation step.

Overlaps can be removed from:

• Periodic structure models with objects smaller than the domain size. If objects longer than the domain size are present in the model, they might overlap with themselves. This type of overlap will not be removed.
• Non-periodic structure models with arbitrary object sizes.
• The statistical properties might change slightly after removing overlap.

The way in which the overlap removal occurs can be controlled through the Remove Overlap Options accessible via the Edit … button.

In the Remove Overlap Options dialog, the following parameters can be set:

• Iterations: Defines after how many iterations the removal stops.
• Iterations SVP Unchanged: The removal stops if the SVP (Solid Volume Percentage) remained unchanged during this number of iterations.
• Overlap SVP / (%): Defines the final allowed Overlap SVP.
• Allow Object Shifts: Check if shifting the objects should be an allowed operation for overlap removal. Define the Number of Shifts per iteration and the maximal Shift Distance per iteration.
• Allow Object Rotations: Check if rotating the objects should be an allowed operation for removal. Define the maximal Number of Rotations allowed per iteration. Objects will be rotated in all directions, where the maximal rotation angle per iteration is one degree.
• Allow Fiber Deformations: Check if bending fibers should be an allowed operation for overlap removal. Define the Maximal Bending Angle allowed between two segments. Each fiber consists of several segments. The center points of the segment connections are shifted in all directions, while the other shape parameters, as e.g. fiber length and segment length are kept constant. The maximal point shift distance is the same as for the object shift. This option is only applied to fibers, as other objects do not consist of segments.

• Distance Mode: The pull-down menu offers four options:
  • The objects can be Touching (distance zero).
  • Enter a maximal Defined Overlap allowed for the objects.
  • Enter a minimal Defined Isolation distance for the objects.
  • Avoid Contacts means a defined isolation of one voxel volume diagonal.

Important! In a periodic domain, objects which are longer than the domain might overlap with themselves. This type of overlap cannot be removed. In this case, a message appears:

Fibers may touch but not overlap when Prohibit Objects Overlap is selected.

In this case, the overlap of the fibers is prohibited during the creation process. This option might lead to longer run times compared to Remove Objects Overlap. It only works for low porosity. As before, fibers longer than the domain might overlap themselves if a periodic domain is selected.

If a positive value is inserted for Use Isolation Distance, the gaps between fibers have at least this preset distance.

When Use Isolation Distance is set to negative values (e.g. -10 µm), the effect is that the fibers may overlap with maximally 10 µm (showing in a different color which here is blue). Negative values of isolation distance are useful to model synthetic fibers that melt, or natural fibers that deform at touching points, during the generation of the structure. Then, the distance between the centers of the fibers is less than the sum of the two radii.

If checking Create in Current Domain, Prohibit Overlap with Current Structure appears instead of Match SVF Distribution. The fibers in the newly generated structure can intersect with each other but not with those of the already existing structure.

Observe this effect in a structure with one fiber (red) over which a new structure with thinner gray fibers is generated. Whereas the gray fibers overlap with each other, they do not overlap with the red fiber.

For certain applications it is necessary to create statistically distributed inhomogeneities, e.g. to generate a bimodal distribution of objects. The Match Solid Volume Fraction (SVF) Distribution option allows to create such structures with an inhomogeneous solid volume distribution. Objects are shifted and rotated to optimally match a SVF distribution given by a Gaussian random field. This takes place after the generation step. Consider that Periodicity is checked automatically for all directions. This is a prerequisite for the Match Solid Volume Fraction algorithm.

The SVF distribution can be controlled through the Match SVF Distribution dialog accessible via the Edit … button.

In the Match SVF Distribution dialog, the following parameters can be set:

Stopping Criterion: Define the Error Bound to determine when the SVF matching should be stopped. When the normalized relative change of solid volume fraction is smaller than the given error bound, the algorithm is stopped.

Distribution Coarsening Factor: The SVF distribution is matched using a coarser version of the created Gaussian random field. The value must be small enough to resolve the heterogeneity of the structure. A larger value leads to a shorter run-time. If the Distribution Coarsening Factor is set to 1, the Gaussian random field is generated with the same resolution as the grain structure.

For a structure with a domain size of 2048x1024x512 voxels and a voxel length of 1µm a coarsening factor of 32 leads to a random field of 64x32x16 voxels with a voxel length of 32µm.

To resolve the correlation length with a whole number of voxels, the coarsening factor should be chosen as a divisor of the correlation length. For example, if the correlation length is set to 128 voxels and the coarsening factor is set to 32, 128/32 = 4 voxels remain to resolve one feature of the random field.

The correlation length is recommended to be a divisor of the domain size. Otherwise, the runtime increases significantly.

Distribution Mode: For distribution select Gauss or Gradient from the pull-down menu. The following parameters are different for these two options.

• Gauss:
  • Correlation Length: The correlation length is a measure for the inhomogeneity of the material. The parameters for X-, Y- and Z-Direction determine the correlation length of the Gaussian random field, which defines the SVF distribution. It is recommended to select a divisor of the domain size in the corresponding direction. For a constant SVF in one direction choose the corresponding domain size as correlation length.
  • Distribution Standard Deviation: Define the relative standard deviation of the created SVF distribution. Relative means relative to the given object solid volume percentage. A larger value leads to a larger difference between the regions of low and high density. Using the relative standard deviation instead of the absolute standard deviation allows using the same Gaussian field for different solid volume percentages.

• Gradient:
  • Use Relative Position: If checked, the left column values from 0 to 1 correspond to locations in the structure. In the Z-direction, the value 0 is at the origin and the value 1 is at end of the domain. If not checked, the values in the left column correspond to absolute values in the given unit.
  • Distribution Table: The right column assigns relative density values at the locations defined in the left column, e.g. the value 10 means that there are five times more objects at Z = 1 than at Z = 0.1, with a density value of 2. The object density increases and decreases smoothly between the given locations in the Z-direction.
  • Number of Rows: The number of rows in the distribution gradient can be increased or decreased to enter as many pairs of position and density as desired.
  • Load/Save: The density distribution can be saved to and loaded from a .txt file.

Example - Gauss

In the following example curved fibers are generated in a domain of 600x600x100 µm. For the first structure Allow Fiber Overlap is checked and no SVF distribution is matched. The structure of the second picture was generated with the same parameters except for matching the SVF distribution shown below.

A volume field describing the used gaussian random field is saved in the result folder after generating a fiber structure matching an SVF distribution. Select File → Load Volume Field … from the menu bar to load it or choose the Data Visualization tab in the Result Viewer of the result file (*.gdr) and click Load *.gvf. In the Loading volume file dialog click OK.

For the above example, the resulting gaussian random field is shown in 2D view from all three directions.

Viewed from X-direction a high correlation in Z-direction is observed, as the structure size in this direction equals the Correlation Length Z. In Y-direction a more inhomogeneous SVF distribution was generated, as the domain size in Y-direction (600 µm) is 12 times the Correlation Length Y (50 µm).

Viewed from Y-direction a similar observation can be obtained. Still the distribution in Z-direction is very homogeneous, whereas it has low correlation in X-direction.

Viewed from Z-direction, the structure is very inhomogeneous, due to small Correlation Length values for X and Y (50µm) compared to the structure size in these directions (600µm).

Example - Gradient

In the following example observe how the fibers are distributed corresponding to the given density distribution.

The fibers in the first example on the left have the highest density in the center of the Z-axis. Towards both ends of the domain in Z-direction the fibers become fewer. This corresponds to the density values 0 for the relative positions Z=0 and Z=1 and a density of 1 for the relative position Z=0.5.

The second example on the right shows a density distribution in Z-direction with 4 rows. A density of 20 for position Z=0 leads to twice as many fibers near the top as fibers in the lower third with a density value of 10 for position Z=0.7. Towards the center and the bottom, the fibers become fewer for a density value of 0.

Difference Gradient VS Density Distribution

This feature is different from the Density Distribution for Center. The Match SVF Distribution is a post-processing step, and the fibers are moved until they match the given solid volume fraction distribution.

For the center distribution (in the example below shown on the left), the fiber centers are placed according to the given distribution, which is much faster and does not change the orientation of the fibers but leads to a different solid volume fraction distribution and the distribution is not as smooth as with Match SVF Distribution, which is shown on the right in the example below.

The structure’s fibers are required to join each other, forming a large, connected component. This effect is obvious when the structure’s solid volume percentage is low (here 10%), i.e. the fibers do not already form a connected component.