cad package for electromagnetic and thermal analysis using...

CAD package for electromagnetic and thermal analysis using finite elements

Mesh skin effect

Flux Table of contents

Mesh skin effect PAGE A

Table of contents

1. Skin effect and mesh: reminders .........................................................................................1 2. An economical mesh (in number of elements) ...................................................................3 3. An easy meshing (few manual operations).........................................................................5 4. The macro: Skin effect mesh (1) ..........................................................................................6 5. The macro : Skin effect mesh (2) .......................................................................................10

Flux Skin effect and mesh: reminders

Mesh skin effect PAGE 1

1. Skin effect and mesh: reminders

Introduction When physical phenomena are concentrated in the skin depth, a good

meshing of the area can be tricky.

This document provides several strategies to mesh skin depth.

Skin effect: reminder

The expression of the skin effect pertains to the phenomena which occur when time varying currents are flowing through a conductor. The AC electric currents are not uniformly distributed in the conductor cross-section for they group themselves in the external layers close to the conductor surface. There is a concentration of current lines towards the outer surface of the conductor.

This happens when the value of either the frequency (f), or the permeability (), or those of the electric conductivity () are high.

Skin depth: definition

The average value of skin depth in which the electric currents flow can be calculated using the following formula:

2 (for a linear, homogenous and isotropic material)

where: is the current pulsation ( f2 )

is the medium permeability

is the conductivity of medium

Skin effect and mesh: reminders Flux

PAGE 2 Mesh skin effect

Mesh of the skin region: rule to observe

The physical quantities of the electric current or the magnetic field are exponentially decreasing in the skin depth. In order to have an accurate evaluation of the physical quantities in the skin depth at the surface of the conductor: it is necessary to have at least two layers of elements (of second order) in the thickness of this region.

A mesh example of the skin depth region is presented in the figure to the right.

!!! Pronounced skin effect !!!

If the skin effect is pronounced, mesh problems appear. Indeed, it becomes too difficult to adhere to the above mesh rule without drastically increase the number of elements. We deal with a pronounced skin effect when the electromagnetic field penetration and the value of the skin depth become too small in relation to the characteristic dimensions of the conductors: L, r , where L is the length or width, and r is the curvature radius of the conductor.

Flux An economical mesh (in number of elements)


2. An economical mesh (in number of elements)

Relevance of rectangular elements (or hexahedron)

To mesh the skin depth it may be judicious to use rectangular type elements (or hexahedron in 3D) in the skin depth zone.

Indeed, with the physical phenomena concentrated in a low thickness depth, it is possible to significantly distort the elements in order that they should get adapted to the physical requirements of the problem.

The rest of the study domain can be meshed by triangular elements.

Choice of mesh generators

The available mesh generators are the following: the mapped mesh generator (2D, 3D) an extruded mesh generator with a mapped base (3D) The rest of the study domain is meshed using the automatic mesh generator.

2D Example A 2D example of mesh on the skin depth is shown in the figure below.

= skin depth

billet

magnetic circuit

inductor

Mapped mesh of skin depth {

Continued on next page

An economical mesh (in number of elements) Flux


3D Example A 3D example of mesh on the skin depth is shown in the figure below.

The most critical volumes (the volumes corresponding to the skin depth of the bar) are meshed using the extrusive mesh generator. The rest of the study domain is meshed using the automatic mesh generator (inside the bar and the surrounding air).

Advantages / drawbacks

Advantages: It is possible to reduce a lot the number of elements in comparison with a triangular mesh (or tetrahedron). This type of mesh is therefore very economical in number of nodes, especially in 3D.

Drawbacks: It is necessary to anticipate the mesh at the moment of the geometric construction. Indeed, you must create points, lines, and faces (and volumes) of the zone relative to the skin depth.

Flux An easy meshing (few manual operations)


3. An easy meshing (few manual operations)

Relevance of triangular elements (or tetrahedron)

In order to mesh the skin depth it can be interesting to use elements of triangular type (or tetrahedron in 3D) in the skin depth zone.

Indeed, by combining the currently available tools in Flux (Aided mesh, and macro Mesh skin effect), the user can rapidly and efficiently mesh any device.

Aided mesh The Aided mesh permits the rapid carrying out of a good quality mesh. The

information for adjustment is supplied by the assembly of geometric entities of the project (points, lines, faces ). The aided mesh administers in a dynamically this information of adjustment over the assembly of the project.

More information on Aided mesh is available in the User guide, Volume 1, chapter 9: Mesh principles

Macro Skin effect mesh

The macro Skin effect mesh permits to add physical constraints to the mesh carried out with Aided mesh. The regions impacted by the skin effect are treated again by the macro Skin effect mesh. Specific adjustment information is then added on the regions in order to carry out a dense mesh in the contour area and very relaxed mesh at the centre.

Advantages / drawbacks

Advantages: It is possible to carry out the mesh of the device in a very simple manner.

Drawbacks: It is possible to considerably increase the number of elements, in comparison to a rectangular mesh (or hexahedron). This type of mesh can therefore be very costly in number of nodes, especially in 3D.

The macro: Skin effect mesh (1) Flux


4. The macro: Skin effect mesh (1)

Foreword / reading advice

To correctly understand the macros (definition, utilization, examples, ), you can consult the section : Language PyFlux, command files and macros (Volume 1 / Chapter 6)

Macros for the Skin effect mesh

There are in Flux two macros that permit to carry out the skin effect mesh : one macro 2D for the mesh of the skin depth of the surface regions :

MeshSkinEffectFaceRegions.PFM one macro 3D for the mesh of the skin depth of the volume regions :

MeshSkinEffectVolumeRegions.PFM

These two macros are stored in the following folders : C:\Cedrat\Extensions\Macros\Macros_Flux2D_Mesh C:\Cedrat\Extensions\Macros\Macros_Flux3D_Mesh

Definition The macro Skin effect mesh permits to add a physical constraint

(concentrated mesh on the contour and relaxed at the centre of the regions) to the mesh carried out with Aided mesh.

Approach With the macro applied on the regions, it is necessary to have defined the

physical characteristics of the device before executing it. Here are the stages in the carrying out of a study integrating the skin effect.

Stage Action

1 Define the geometry 2 Define the mesh and mesh the device 3 Define the physical characteristics 4 Charge and execute the macro Skin effect mesh

Flux The macro: Skin effect mesh (1)


Characteristics The macro Skin effect mesh presents the following characteristics : the skin effect mesh is applied on the contour of the chosen regions the internal lines to the regions are ignored the regions can comprise distinct faces The macro Skin effect mesh can be used several times for the same project


The macro: Skin effect mesh (1) Flux


Description The macro Skin effect mesh concentrates several commands in a fixed order.

It receives (input) one or more parameters and executes (output) a series of predefined actions. The parameters of input are given in the table below. The actions performed are presented in the following sections.

Parameter in input Description Entities created

automatically Name permits to identify the entities

created and used by the macro RELAXLINE_NAME RELAXFACE_NAME RELAXVOLUME_NAME (3D) MESHPOINT_NAME

Number of layers definition of number of layers in the thickness of the skin depth (over or equal to 2)

Parameter NBRLAYERS_NAME

Skin depth thickness definition of the value of the skin depth thickness (in meter)

Parameter SKIN_DEPTH_NAME

Surface regions (2D) Volume regions (3D)

regions where it is desired to carry out the skin effect mesh

Associated color permits to identify the skin effect mesh generated across the entities of mesh created

Excluded lines (2D) Excluded faces(3D)

lines (in 2D) or faces (in 3D) excluded from contour

Note : The lines or faces excluded are the lines or faces of the contour over which the skin effect mesh will not be applied. This entry is particularly useful in 3D in order to simplify the mesh wherever this is possible (see example 3D below).

Actions of the macro (2D)

The main actions carried out by the macro 2D are described in the table below :

Phase Description

1 Creation of a medium relaxation face entity (RELAXFACE_NAME) 2 Assignment of the entity on the faces 3 Creation of an inactive relaxation line entity (RELAXLINE_NAME) 4 Assignment of the entity on the contour lines 5 Creation of a mesh point entity equal to the skin depth thickness

divided by the number of layers (MESHPOINT_NAME) 6 Assignment of the entity on the contour points


Flux The macro: Skin effect mesh (1)


Actions of the macro (3D)

The main actions carried out by the macro 3D are described in the table below :

Phase Description

1 Creation of a medium relaxation volume entity (RELAXVOLUME_NAME)

2 Assignment of the entity on the volumes 3 Creation of an inactive relaxation face entity (RELAXFACE_NAME) 4 Assignment of the entity on the contour of the faces 5 Creation of an inactive relaxation line entity (RELAXLINE_NAME) 6 Assignment of the entity on the contour lines 7 Creation of a mesh point entity equal to the skin depth thickness

divided by the number of layers (MESHPOINT_NAME) 8 Assignment of the entity on the contour points

The macro : Skin effect mesh (2) Flux


5. The macro : Skin effect mesh (2)

Example 2D An example 2D of skin effect mesh is represented in the figure below.


Flux The macro : Skin effect mesh (2)


Example 3D An example 3D of skin effect mesh is represented in the figure below.

In this case, the extremity faces of the bar have been un-selected from the skin effect mesh (due to the optional entry of the macro interface). This permits to avoid useless over meshing.

Advice and tips on use

During the stage of definition of mesh, it is not advisable to add mesh information on the zone where there is the skin effect at the moment when the skin effect macro is being executed, as they can be suppressed. Only the information referring to the mesh shape (mesh generators) are preserved by the macro.

If after having applied the macro, it is wished to change the thickness of the skin depth or the number of layers, it is advisable to do this by modifying within the geometric parameters created by the macro (SKIN_DEPTH_NAME & NBRLAYERS_NAME)

In 3D, it is sometimes difficult to select in the optional entry the faces that should not be into consideration for the skin effect mesh, as they are not all visible. One tip is to select them beforehand and regroup them by one characteristic (the color for example). In the interface of the macro it is then possible to choose the faces by this characteristic.

The macro : Skin effect mesh (2) Flux


Warnings It is not advisable to use the optional entry permitting to exclude the lines (in 2D) / faces (in 3D) if it is not the first application of the macro on a given region. To avoid this problem, it is possible to delete and replace the mesh information manually by editing geometric entities to ignore: If the aided mesh is activated, it is enough to replace the relaxation of the

lines in 2D, the faces and lines (at the intersection of two faces to be ignored) in 3D by the aided relaxation (AIDED_RELAXLINE / AIDED_RELAXFACE).

If the aided mesh is deactivated, the user can create a local relaxation and assign it to the entities concerned.

Once the macro has been executed, it is advisable not to reopen the aided mesh box as this can automatically place aided mesh information by clicking on Apply or Ok .

1. Skin effect and mesh: reminders2. An economical mesh (in number of elements)3. An easy meshing (few manual operations)4. The macro: Skin effect mesh (1)5. The macro: Skin effect mesh (2)

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