The Anisotropy Parameters, Anisotropy 1 () and Anisotropy 2 (), allow to define the orientation of the objects. Anisotropy Parameters and the Orientation Tensor are equivalent representations for the orientation, and GeoDict can compute one from the other automatically. The objects are created with random orientations, following a distribution defined by the given parameters.

The Algorithm

For each object, a random direction is computed

which follows an isotropic distribution. Afterwards, the anisotropy parameters are used to influence these orientations. Before normalization, the resulting vector for the object orientation is:

With normalization follows:

Before normalization, Anisotropy 1 controls the first two coordinates, while Anisotropy 2 only affects the second coordinate and none of them changes the third. After normalization, the anisotropy parameters both have impact on all three coordinates, where the influence on the object direction differs. Whereas high values for Anisotropy 1 lead to objects parallel to the XY-plane, high values for Anisotropy 2 result in objects oriented in Y-direction. Objects oriented in Z-direction are obtained by low values for Anisotropy 1, and low values for Anisotropy 2 result in objects parallel to the XZ-plane.

In the following, five main cases for the anisotropy parameters are described:

1. Anisotropy 1 = Anisotropy 2 = 1 (Isotropic material)

The object orientation is isotropic. All directions have the same probability, as the direction obtained from the random algorithm is not changed:

2. Anisotropy 1 < 1 and Anisotropy 2 = 1

Values smaller than 1 for Anisotropy 1 lead to orientations in Z-direction, so that the objects are parallel to the YZ-plane and the XZ-plane when viewed from the X- or Y-direction. The smaller the value of Anisotropy 1, the more anisotropic the objects have.

For example, with the values 0.01 for Anisotropy 1 and 1 for Anisotropy 2 it follows:

Thus, after normalization the direction vectors of the objects approximate the unit vector:

That results in objects oriented in Z-direction.

3. Anisotropy 1 > 1 and Anisotropy 2 =1

The objects are isotropic in the XY-plane (Z-slice). The higher the value of Anisotropy 1, the stronger is the anisotropy.

For example, with the values 100 for Anisotropy 1 and 1 for Anisotropy 2 it follows:

Thus, after normalization the direction vectors of the objects approximate the vector:

That results in objects oriented in the XY-plane.

4. Anisotropy 1 = 1 and Anisotropy 2 < 1

Small values for Anisotropy 2 lead to objects oriented parallel to the XZ-plane. Observe that the objects are isotropic in the XZ-plane (Y-slice).

For example, with the values 1 for Anisotropy 1 and 0.01 for Anisotropy 2 it follows:

Thus, after normalization the direction vectors of the objects approximate the vector:

That results in objects oriented in the XZ-plane.

5. Anisotropy 1 = 1 and Anisotropy 2 > 1

The higher the values for Anisotropy 2, the more the objects tend to be oriented along the Y-axis. Observe that the objects cut the XZ-plane (Y-slice), while they run parallel to the YZ-plane and the XY-plane when viewing them from the X- or the Z-direction.

For example, with the values 1 for Anisotropy 1 and 100 for Anisotropy 2, it follows:

Thus, after normalization the direction vectors of the objects approximate the unit vector:

That results in objects oriented in Y-direction.

Different anisotropy values for different object types

As mentioned before, the orientation can be separately set for each object type so that differently oriented objects may coexist within the same structure. In the following structure, observe that Anisotropy 1 and Anisotropy 2 are set to 0.01 and 1 for the short circular (red) fibers. The short circular (gray) fibers have Anisotropy 1 and Anisotropy 2 of 1 and 100, respectively.

The red fibers tend to be oriented along the Z-axis, and the fibers run parallel when viewed from the X- or Y-direction.

The gray fibers tend to be oriented in the Y-axis and cut the XZ-plane, while they run parallel to the YZ-plane and the XY-plane, as observed from the X- or the Z-direction.

The Euler rotation angles are applied to the objects during the generation after applying the anisotropy parameters following the order Phi → Theta → Psi.

Alternatively, when selecting Orientation Tensor from the Direction Mode pull-down menu, the values for the orientation tensor for the anisotropic objects can be entered.

The Orientation Tensor provides an equivalent alternative to the combination of the Anisotropy Parameters and the Euler Angles.

Clicking the Normalize + Calculate Anisotropy Parameters button enters the adjusted values in the orientation tensor and the Anisotropy 1 and Anisotropy 2 boxes.

In detail, let

be the unit vector describing the direction of the k-th object and n the number of objects. Then the orientation tensor T is the sum over the dyadic products of the dk from all n objects, divided by n:

The diagonal elements define the orientation strength for the corresponding directions and sum up to 1. Thus, if for example t11=1 (and t22=t33=0), all objects are oriented in the X-direction. For t11=0 all objects are oriented normally to the X-direction and same values for all diagonal elements (t11=t22=t33=1/3) result in a uniform distribution for the object orientation.

Observe the effect of changing the default values of the isotropic Orientation Tensor to obtain anisotropic structures oriented in the X-, Y-, or Z-direction.

The extra-diagonal values define the statistical variation of the orientations given by the diagonal values. For example, if t11 has a high value near 1, and t12 has a relatively high value near 0.1, less fibers are oriented straight in X-direction than it is the case for a lower value for t12.

Clicking the Normalize + Calculate Anisotropy Parameters button, the Orientation Tensor is normalized, and the Anisotropy values and the Euler Angles are calculated and entered in the boxes above.

The angle between the object cross section axis and XY-plane can be entered in the Profile Angle to XY-Plane dialog.

The object Profile Angle to XY-Plane, can be set to be a Constant value, or to follow a distribution (Uniformly in interval, Gaussian, or Probability Distribution).

Observe the effect of varying the Profile Angle To XY-Plane in a structure made of Short Elliptical fibers strongly oriented in the Y-axis (Anisotropy 1 = 1, Anisotropy 2 = 1000). Other parameters are unchanged.

In the following example, a Probability Distribution is defined: 50% of the elliptical fibers are at an angle of 45º with the XY-plane, 30% lie on the plane (0º angle), and 20% are at a 25º angle.

In the left column the Count Probability can be entered. A value of 1 means all objects will have the given direction on the right. For example, a Count Probability of 1 for the direction vector 0, 0, 1 means that all objects follow the Z-direction, and 1, 0, 0 means that all objects follow the X-direction.

If more rows are given via Number of Rows, every given direction will be applied to the generated objects with the corresponding count probability. In the following example observe how 80% of the short circular fibers are oriented in X-direction while the other 20% are oriented along the Z-axis.  

Every direction in the 3-dimensional space is possible, as seen in these examples:

Clicking the Normalize button ensures the Count Probability values sum up to one.

The Given Directions distribution settings can be saved as text file and loaded by clicking Save and Load, respectively.

For example, enter 0°, 30°, 45°, or 90°, to obtain the following orientations in the XY-Plane:

Observe how the orientation varies in the following example with angles around direction of 0°, 5°, and 10° together with a Main Direction Vector of (1, 0, 0).