The default bunching law. The two initial heights and ratios define parabolas in a coordinate system where the number of node points is the X-axis and the cumulative distance along the edge is the Y-axis. The parabolas are truncated where their tangent lines are identical; the spacing is linear between these points. If there are not enough nodal points to form this linear segment, a hyperbolic law is used and the ratios are ignored.

The nodes along the edge are uniformly distributed.

The spacing at each end are used to define a hyperbolic distribution of the nodes along the edge. You can set Spacing 1 and Spacing 2, and the growth ratios are determined internally.

The Hyperbolic Tangential Bunching law is described by the following equations:

The parameter limitations are:

The spacing of the node intervals is calculated according to a Poisson distribution. Requested values of Spacing 1 and Spacing 2 are used. Requested values of Ratio 1 and Ratio 2 are ignored.

The mapping function is obtained by solving the following differential equation: ,

with the following boundary conditions:

,

where Sp1 = Spacing 1 and Sp2 = Spacing 2.

The function P is required to satisfy the Neumann boundary condition. It is computed by an iterative optimization process loop. Some parameter limitations are:

0.0 < Sp1 < 1.0

0.0 < Sp2 < 1.0 - Sp2

500 < Number of iterations < 9999

The spacing of the node intervals is calculated according to the curvature of the function defining the distribution.

Spacing 1 is used to set the first distance from the starting end of the edge, with the remaining nodes are spaced with a constant growth ratio. Only Spacing 1 is specified.

The Geometric bunching laws are described by the following equation:,

where is the distance from the starting end to node i, R is the ratio, and N is the total number of nodes. The ratio R is limited by 0.25 > R > 4.0.

The same as Geometric 1, except that Spacing 2 is used to define the distribution starting from the terminating end of the edge.

The Exponential 1 bunching law is described by the following equation:

where is the distance from the starting end to node i, Sp1 is Spacing 1, N is the total number of nodes, and R is the ratio, where: .

The same as Exponential 1, except that the Spacing 2 and Ratio 2 parameters are used and the distribution starting point is the terminating end of the edge.

The spacing of the node intervals is calculated according to the law specified for Exponential 1 and 2. However, Spacing 1, Ratio 1, Spacing 2 , and Ratio 2 are used to define the distribution. The Spacing 1 and Ratio 1 parameters define the distribution from the beginning of the edge to the midpoint of the edge, and Spacing 2 and Ratio 2 define the distribution from the terminating end of the edge to the midpoint of the edge.

This bunching law is described by the following equation:

The parameters a are computed according to the vertex constraints. If a ratio equals 0 at a vertex, the spacing constraint at this vertex only is taken into account and the ratio constraint with the neighbor spacing is left free by decreasing the polynomial order in the mathematical function.

The spacing of the node intervals is calculated using a linear function.