Tube conductor mode: meshing

Introduction

This section details the four meshing methodologies available for the tube conductors, as well as the parameters that need to be filled out by the user for their implementation.

Uniform meshing

The technique is not implemented in the same manner if the conductor cross-section is rectangular, variously-shaped full or variously-shaped hollow, as described in the table below. Consequently, the configuration required from the user varies from one case to the other.

Cross-section Description
rectangular
  1. the user is invited to fill out the numbers n and m of the PEEC elements according to which the two sides of the cross-section are subdivided
  2. the software creates a regular mesh and in accordance with the n rows and m columns; consequently, all the elements of the mesh have the same size
variously-shaped full
  1. the user is invited to fill out the numbers n and m of the PEEC elements according to which the two sides of the profile are subdivided
  2. the rectangle encompassing the section is subdivided into sub-intervals defined by the points of the profile
  3. each sub-interval is meshed uniformly, observing the maximal size of the elements imposed by the n and m parameters
  4. a technique for the refining on the edges is automatically applied in order to better follow the outline of the geometry

Consequently, the global mesh of the section can become non-conform.
variously-shaped hollow
  1. the user is firstly invited to fill out the maximal width of the PEEC elements according to which the profile perimeter is subdivided
  2. then, the user is invited to identify the number of PEEC elements according to which the thickness of the layer is subdivided; the default value is 3
  3. first the software generates the mesh of the corners of the cross-section: the number of elements in the two directions is determined by the value that the user has filled out at step 2
  4. the profile sides are then meshed in a uniform manner, observing the maximal width chosen by the user at step 1

Geometric meshing

This meshing technique is available only for the rectangular cross-sections and is efficient mainly at high frequencies.

When choosing a geometrical mesh for a section of a tube conductor, the user is invited to fill out:

  • the numbers n and m of the PEEC elements according to which to subdivide the two sides of the profile (width and height);
  • the K (by width) and H (by height) concentrations of these elements. These values, which must be positive, represent the relationship between the sizes of two successive elements.

The Flux PEEC software automatically determines the size of all the elements:

the basic element (a by the width L and b by the height T) is calculated by means of the following formulas:

The calculation of b relies on the same formulas where L, K, n are replaced by T, H and m, respectively.

An example of geometric mesh is shown in the figure below:

It is worth noting that for a concentration value equal to 1, the user has indeed chosen a uniform mesh in the direction considered.

It would be reasonable from the user to make sure that the size of the a and b elements be comparable to the thickness of the skin depth, in order to correctly model the phenomenon of electromagnetic field diffusion. To do that, he must:

  1. choose the number n of the elements he wishes, for example 3, 4, 5, or 6;
  2. calculate the optimal value of K by means of the following formulas:

where δ is the thickness of skin depth that can be estimated by the expression reported in § Choice and definition of the application.

Automatic mesh in function of the frequency

This technique is considered the most efficient, as it combines a very good quality of the results obtained by the minimization of the number of meshes.

The frequency considered by the Flux PEEC software can be the frequency of reference of the project or another one chosen by the user, the algorithm of the mesh is identical.

The technique is not applied in the same manner if the section is rectangular, variously-shaped full or variously-shaped hollow, as described below.

Cross-section Description
rectangular The number of elements in the two directions of the cross-section is imposed by the software; it depends on the skin depth thickness (and implicitly of the frequency) and it is always inferior or equal to five. The details are presented in the table below:
for for
uniform distribution of elements

the size of an element is given by:

 geometric distribution of elements

the concentration factor K is given by:

where:

  • Ent(x) is the entire superior part of x
  • δ is the thickness of skin depth at the considered frequency
  • L is one of the two sizes of the cross-section
variously-shaped full

Depending on the thickness of the skin depth (and implicitly on the frequency), the software proceeds to the positioning of the mesh elements in the following way:

  1. the rectangle encompassing the section is subdivided into sub-intervals defined by the coordinates of the profile points
  2. the meshes across the contour lines of the profile are subdivided so that they are crossed diagonally by these contour lines
  3. a refining technique on the edges is applied in order to better follow the contour lines of the geometry and to take into consideration the skin effect: as a result, the thickness of skin depth is modeled by approximately two meshes
  4. the meshes, that are not included in the thickness of skin depth and whose dimensions enable the merging, are merged in order to reduce the total number of meshes
variously-shaped hollow

The used algorithm is very similar to the one used for the uniform mesh of variously-shaped hollow cross-sections, except that:

  • the maximum width of the PEEC elements into which to subdivide the profile perimeter is not chosen by the user, but is automatically evaluated by the software, and corresponds to the thickness of skin depth calculated at the reference frequency.