f3d 920 generic tutorial

7/30/2019 F3D 920 Generic Tutorial

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CAD Package for Electromagnetic and Thermal

Analysis using Finite Elements

FLUX3D ApplicationGeneric tutorial

of

geometry and mesh

Copyright February 2006


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FLUX is a registered trademark.

FLUX software : COPYRIGHT CEDRAT/INPG/CNRS/EDF

FLUX tutorials : COPYRIGHT CEDRAT

FLUX's Quality Assessment

(Electricit de France, registered number AQMIL013)

This tutorial was edited on 6 February 2006

Ref.: K305-920-EN-02/06

CEDRAT

15 Chemin de Malacher - Inovalle

38246 Meylan Cedex

FRANCE

Phone: +33 (0)4 76 90 50 45

Fax: +33 (0)4 56 38 08 30

E-mail: [email protected]

Web: http://www.cedrat.com


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How to get the most from this document

Introduction To help you use this tutorial more efficiently, it has:

adopted special formats (typographic conventions) for the most common

types of information

followed some rules to separate types of information: definition of new

concepts, generalities about specific features or logical sequence of

commands, etc.

Contents This section contains the following topics:

Topic See Page

Information division, reading advice

FLUX files


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Information division, reading advice

Different types

of informationYou will find in this document the following different types of information:

definitions of new concepts used by FLUX and general information about

specific features actions you must perform to construct the model.

Organization

informationThe organization of the chapters is the following.

all topics beginning with a verb (create, add, assign, ) contain

information about actions you must complete

all topics beginning with the word about contain definitions or general

information about specific features.

Reading advice If you are a beginner with FLUX, it is recommended that you read and work

through the complete text of the chapters.

If you are an experienced user of FLUX, you may be able to enter the

problem information quickly without having to read the about paragraphs.


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FLUX files

FLUX files

locationFLUX files corresponding to the problem studied in this tutorial are included

in the CD-ROM:

PROBE_3D.FLU WHEEL_BASE_3D.FLU

SENSOR_3D.FLU

If you install FLUX with the documentation and the examples, files are

placed in the folder:

C:\CEDRAT (or your installation folder)

\Flux_XXX_Doc_examples\Examples\Tutorials\F3D_Tutorial_Geometry

&mesh

Use of FLUXfiles

The FLUX files, included in the CD-ROM, are ready to be used.

You can refer to these files in case of difficulties completing this tutorial, or

to directly adapt this tutorial to your needs, without going through all the

steps to construct the model.


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FLUX9.20 TABLE OF CONTENTS

TABLE OF CONTENTS

PART A: GENERAL INFORMATION..........................................................1

1. Overview.................................................................................................................................3

1.1. Introduction.......................................................................................................................4

1.2.The studied device: a variable reluctance speed sensor .................................................5

1.3.The device description in FLUX: which strategy? ............................................................6

2. Get started with FLUX ...........................................................................................................9

2.1.Starting FLUX.................................................................................................................11

2.1.1. Start the FLUX Supervisor................................................................................................12

2.1.2. About the FLUX Supervisor ..............................................................................................132.2.Starting Preflux...............................................................................................................15

2.2.1. Open Preflux.....................................................................................................................16

PART B: DESCRIPTION OF THE PROBE ...............................................17

3. Geometry description of the probe object ........................................................................19

3.1.Create a FLUX project for the probe ..............................................................................21

3.1.1. Create a new project for the probe...................................................................................223.1.2. About the Preflux window .................................................................................................233.1.3. About the Help menu / Users guide.................................................................................243.1.4.

About the geometry context..............................................................................................26

3.1.5. Name the project...............................................................................................................27

3.2.Strategy and tools for geometry description of the probe...............................................29

3.2.1. Available geometric tools and analysis before geometry description...............................303.2.2. Main phases for geometry description of the probe .........................................................32

3.3.Creation of geometric tools ............................................................................................33

3.3.1. About creation of an entity ................................................................................................343.3.2. About geometric parameters ............................................................................................363.3.3. Create the geometric parameters .....................................................................................373.3.4. About the Tools menu / toolbar.........................................................................................393.3.5. About selection of graphic entities....................................................................................403.3.6. About modification and deletion of an entity.....................................................................42

3.3.7. About graphic view............................................................................................................453.3.8. Change the background color...........................................................................................473.3.9. About coordinate systems ................................................................................................483.3.10.Create the coordinate systems.........................................................................................50

GEOMETRY AND MESH TUTORIAL PAGE A


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TABLE OF CONTENTS FLUX9.20

3.4.Creation of points and lines of the probe base .............................................................. 53

3.4.1. Change to the plane view .................................................................................................543.4.2. About points ......................................................................................................................553.4.3. Create points for the probe base ......................................................................................563.4.4. About display of entities in the graphic scene...................................................................583.4.5. Display point numbers ......................................................................................................593.4.6. About lines ........................................................................................................................60

3.4.7. Create lines for the probe base.........................................................................................613.5.Building faces and volumes for the probe...................................................................... 63

3.5.1. About automatic construction............................................................................................643.5.2. Build faces of the probe base ...........................................................................................653.5.3. Change to the standard view ............................................................................................663.5.4. About transformations.......................................................................................................673.5.5. Create the geometric transformation ................................................................................693.5.6. About propagation and extrusion......................................................................................713.5.7. About selection by criterion...............................................................................................733.5.8. Extrude faces ....................................................................................................................743.5.9. Complete the construction by automatic building .............................................................77

3.6.Check and correct the geometry.................................................................................... 79

3.6.1. About intersections of entities ...........................................................................................803.6.2. Check the geometry..........................................................................................................813.6.3. Modify the visibility of faces to visualize the geometric defects........................................823.6.4. Change the nature of the faces.........................................................................................83

4. Mesh generation of the probe object.................................................................................85

4.1.Strategy and tools for mesh generation of the probe..................................................... 87

4.1.1. Available meshing tools and analysis before mesh generation........................................884.1.2. Main phases for mesh generation of the probe ................................................................89

4.2.Creation and assignment of mesh points....................................................................... 91

4.2.1. Change to the mesh context .............................................................................................92

4.2.2. About the mesh context ....................................................................................................934.2.3. About meshing tools .........................................................................................................944.2.4. Create the mesh points.....................................................................................................964.2.5. Modify the visibility of faces before the assignment of mesh entities ...............................984.2.6. Assign the mesh points to points ......................................................................................99

4.3.Creation and assignment of mesh lines....................................................................... 101

4.3.1. Create the mesh line...................................................................................................... 1024.3.2. Modify the visibility of faces and change the view......................................................... 1034.3.3. Assign the mesh line to lines ......................................................................................... 104

4.4.Meshing the probe ....................................................................................................... 107

4.4.1. Change to the standard view......................................................................................... 1084.4.2. Mesh lines, faces and volumes...................................................................................... 1094.4.3. Delete the mesh............................................................................................................. 1114.4.4. Save and close the project............................................................................................. 112

PAGE B GEOMETRY AND MESH TUTORIAL


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FLUX9.20 TABLE OF CONTENTS

PART C: DESCRIPTION OF THE WHEEL BASE..................................113

5. Geometric description of the wheel base object............................................................115

5.1.Create a FLUX project for the wheel base ...................................................................117

5.1.1. Create and name a new project for the wheel base.......................................................118

5.2.Strategy and tools for geometry description of the wheel base object .........................1195.2.1. Available geometric tools and analysis before geometry description.............................1205.2.2. Main phases for geometric description of the wheel base..............................................122

5.3.Creation of geometric tools ..........................................................................................123

5.3.1. Create the geometric parameters ...................................................................................1245.3.2. Create the coordinate system.........................................................................................126

5.4.Creation of points and lines for the wheel base ...........................................................129

5.4.1. Create the points for the wheel base..............................................................................1305.4.2. Create the lines for the wheel base ................................................................................132

5.5.Building the face for the wheel base ............................................................................135

5.5.1. Build the face ..................................................................................................................136

5.6.Creation of transformations..........................................................................................137

5.6.1. Create transformations ...................................................................................................138

6. Mesh generation of the wheel base object......................................................................141

6.1.Strategy and tools for mesh generation of the wheel base ..........................................143

6.1.1. Available meshing tools and analysis before mesh generation......................................1446.1.2. Main phases for mesh generation of the wheel..............................................................145

6.2.Creation and assignment of mesh points .....................................................................147

6.2.1. Change to the mesh context...........................................................................................1486.2.2. Create mesh points.........................................................................................................1496.2.3. Assign mesh points to points ..........................................................................................151

6.3.Meshing the wheel base...............................................................................................153

6.3.1. Mesh lines and faces ......................................................................................................1546.3.2. Delete the mesh..............................................................................................................1566.3.3. Save and close the project .............................................................................................157

GEOMETRY AND MESH TUTORIAL PAGE C


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TABLE OF CONTENTS FLUX9.20

PART D: DESCRIPTION OF THE SENSOR........................................... 159

7. Geometry description of the sensor................................................................................161

7.1.Create a FLUX project for the sensor .......................................................................... 163

7.1.1. Create and name a new project for the sensor.............................................................. 164

7.2.Strategy and tools for geometric description of the sensor.......................................... 1657.2.1. Available geometric tools and analysis before geometry description............................ 1667.2.2. Main phases for geometric description .......................................................................... 167

7.3. Importation of the wheel base object and building the whole wheel ............................ 169

7.3.1. Import the wheel base object ......................................................................................... 1707.3.2. Geometry building process of the wheel........................................................................ 1717.3.3. Propagate the face (tooth) ............................................................................................. 1727.3.4. Extrude the line.............................................................................................................. 1757.3.5. Create an arc ................................................................................................................. 1777.3.6. Propagate the arc .......................................................................................................... 1797.3.7. Build faces...................................................................................................................... 1817.3.8. Extrude the wheel base ................................................................................................. 182

7.4. Importation of the probe objects and positioning of the wheel and probes.................. 185

7.4.1. Modify the coordinate system ........................................................................................ 1867.4.2. Import the first probe object ........................................................................................... 1887.4.3. Modify the parameters ................................................................................................... 1907.4.4. Import the second probe object ..................................................................................... 191

7.5.Completing the domain................................................................................................ 193

7.5.1. About an infinite box ...................................................................................................... 1947.5.2. Add an infinite box ......................................................................................................... 1967.5.3. Build faces...................................................................................................................... 1977.5.4. Check the geometry....................................................................................................... 1987.5.5. Build volumes................................................................................................................. 1997.5.6. Modify the visibility of faces ........................................................................................... 200

8. Mesh generation of the sensor ........................................................................................201

8.1.Strategy and tools for mesh generation of the sensor ................................................. 203

8.1.1. Available meshing tools and analysis before mesh generation..................................... 2048.1.2. Main phases for mesh description................................................................................. 205

8.2.Modification and assignment of mesh points ............................................................... 207

8.2.1. Change to the mesh context .......................................................................................... 2088.2.2. Modify the mesh points.................................................................................................. 2098.2.3. Change the display and the view................................................................................... 2108.2.4. Assign mesh points to points ......................................................................................... 211

8.3.Meshing the sensor...................................................................................................... 213

8.3.1. Mesh lines, mesh faces, mesh volumes and generate second order elements............ 2148.3.2. Save the project and close the Preflux window ............................................................. 217

9. Annex .................................................................................................................................219

9.1.Use of command files .................................................................................................. 221

9.1.1. About command files and the Python language............................................................ 2229.1.2. Execute command file.................................................................................................... 223

PAGE D GEOMETRY AND MESH TUTORIAL


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FLUX9.20 Part A: General informationOverview

PART A: GENERAL INFORMATION

Introduction This part A contains the presentation of the studied device and the FLUX

software.

Contents This part contains the following topics:

Topic See Page

Overview 3

Get started with FLUX 9

GEOMETRY AND MESH TUTORIAL PAGE 1


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Part A: General information FLUX9.20Overview

PAGE 2 GEOMETRY AND MESH TUTORIAL


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1. Overview

Introduction This chapter presents the studied device (a variable reluctance speed sensor)

and the strategy of the device description in FLUX.

Contents This chapter contains the following topics:

Topic See Page

Introduction 4

The studied device: a variable reluctance speed sensor 5

The device description in FLUX: which strategy? 6



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1.1. Introduction

Introduction FLUX is finite elements software for electromagnetic simulation. FLUX

handles the design and analysis of any electromagnetic device.

To perform a study with FLUX, you build a finite elements project. This

process is broken into 5 phases:

geometry description

mesh generation

description of the physical properties

solving process

analysis of the results

Only the first two phases are presented in this document.

Objective The objective of this document is discovery and mastering various

functionalities in the software through the example of a simple device.

The device, which will be used as example, is a variable reluctance speed

sensor described in the following paragraph.

The studied functionalities* of the software are those related to the phases of

construction of the geometry and generation of the mesh.

The user will also find in this document useful information concerning the

software: description of the environment, data management, graphic

representation, etc.

* The functionalities of the software related to the following phases - description of

the physical properties, resolution, analysis of the results - are not detailed in this

document.



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1.2. The studied device: a variable reluctance speed sensor

Introduction The device to be analyzed is a speed sensor.

Structure The variable reluctance speed sensor consists of a cogged wheel, a magnet

and a coil connected to a measuring resistance.

Functionality The rotation of the target wheel near the tip of the sensor changes the

magnetic flux, creating an analog voltage signal that can be recovered in

probes.

Typical

applicationsTypical applications are:

ignition system engine speed and position

speed sensing for electronically controlled transmissions

vehicle speed sensing

wheel speed sensing for ABS and traction control systems



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1.3. The device description in FLUX: which strategy?

Problem How to describe the device in FLUX?

Reminder: we only are interested in geometrical construction and generation of the mesh.

Geometric

structureThe device consists of:

one cogged wheel with three teeth

two probes with a magnet and a coil around

PROBE 2

MAGNET 1

COIL 1

COIL 2

WHEEL

MAGNET 2

PROBE 1

Strategy Two strategies of description exist:

one-phase description:

- description of the whole device in only one FLUX project

two-phase description:

- independent description of separated parts of the device in several FLUX

projects

- merging the independent projects into one

The second strategy is selected in this tutorial.

Of course, the geometry can be built in ways other than the presented one. The

sensor geometry is defined in this particular way in order to introduce you to

the most used Preflux features.

Continued on next page



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Process

(general

aspects)

An outline of the general construction process is given in the two following

blocks:

the first process (1) is presented to facilitate comprehension

the second process (2) is the real building process used in this document.

Process (1) An outline of the logical process of the geometry description is given in the

table below.

Phase Description

1 Probe description

2 Cogged wheel description

3 Sensor description

4 Addition of air around the device and closing of the domain

by the technique of the Infinite Box




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Process (2) An outline of the real process of the geometry description, used in this tutorial,

is given in the table below.

1 Probe object description Project: PROBE_3D.FLU

2 Wheel base object description (elementary pattern) Project: WHEEL_BASE_3D.FLU

3 Sensor description Project: SENSOR_3D.FLU

Importation of the wheel base object (elementary pattern) (WHEEL_BASE_3D)

Building of the whole wheel

Importation of a probe object (PROBE_3D)

Rotation of the probe and rotation of the cogged wheel

Importation of a probe object (PROBE_3D)

Addition of an Infinite Box



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FLUX9.20 Part A: General informationGet started with FLUX

2. Get started with FLUX

Introduction This chapter shows how to start working with FLUX and includes a

presentation of the FLUX Supervisor.

It also shows how to start Preflux, the preprocessor for FLUX 2D and FLUX

3D, and includes a brief introduction to Preflux.

More detailed information about Preflux menus and commands is presented

in 3.1.2 About the Preflux window.


Topic See Page

Starting FLUX 11Starting Preflux 15



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Part A: General information FLUX9.20Get started with FLUX



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2.1. Starting FLUX

Introduction FLUX software is managed by a supervisor.

The new Supervisor for FLUX 9 organizes all the modules for both FLUX 2D

and FLUX 3D.


Topic See Page

Start the FLUX Supervisor 12

About the FLUX Supervisor 13



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2.1.1. Start the FLUX Supervisor

Goal Starting FLUX involves opening the FLUX Supervisor.

Action To start FLUX from the Windows taskbar:

1. Point on Start, Programs, Cedrat

(or your FLUX installation

directory)

and click on FLUX 9.2

Result The FLUX Supervisor window opens.



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2.1.2. About the FLUX Supervisor

The FLUX

Supervisor

window

The FLUX Supervisor organizes all the modules for both FLUX 2D and

FLUX 3D.

The FLUX Supervisor window is divided into several areas. These areas are

identified in the following figure and described in the table below.

Programmanager

My programs

ProjectfilesDirectory

manager

Menu bar

Tool bar

Flux3D tab

Flux view(3D only)

Area Function

Program manager to list and launch all the FLUX modules

(Geometry&Physics, Circuit, etc.)

Directory manager to show the computers complete directoryProject files to display all FLUX projects in the selected directory

My programs contains shortcuts to the Dos Shell and the Explorer

Flux view to display a preview of the geometry, if a project is

selected

Some checks

before you

begin

From the FLUX Supervisor you should:

Select the FLUX 3D tab in order to access the specific FLUX 3D programs.

Access your working directory by selecting it in the supervisors directory

manager window. Verify that the title of the Program manager area is the standard version

(Flux3D: Standard). If not, in the menu bar, select Versions and check

Standard.



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2.2. Starting Preflux

Introduction Preflux is the preprocessor to describe the geometry, mesh and physical

properties of the studied device.


Topic See Page

Open Preflux 16



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2.2.1. Open Preflux

Goal The preprocessorPreflux will be opened to manage the geometry building of

the device and mesh generation.

Action To open Preflux from the FLUXSupervisor:

1. Click on theFlux3D tab

2. Select the directoryof the project

3. Double-click on

Geometry&Physics

Result The Preflux window for FLUX 3D applications is opened.

There are two menus in the Preflux window: Project and Help*.

Menus bar

Project toolbar

* A new project must be created to see the complete set of Preflux commands.



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FLUX9.20 Part B: Description of the probeGet started with FLUX

PART B: DESCRIPTION OF THE PROBE

Introduction This part B contains the geometry description and mesh generation of the

probe.

Project name The FLUX project is named PROBE_3D.FLU.

Contents This part contains the following topics:

Topic See Page

Geometry description of the probe object 19

Mesh generation of the probe object 85



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Part B: Description of the probe FLUX9.20Geometry description of the probe object



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FLUX9.20 Part B: Description of the probeGeometry description of the probe object

3. Geometry description of the probe object

Introduction This chapter presents the general steps of the geometry construction and the

data required to describe the probe geometry.

The probe object is presented in the figure below.

COIL

MAGNET


Topic See Page

Create a FLUX project for the probe 21

Strategy and tools for geometry description 29

Creation of geometric tools 33

Creation of points and lines of the probe base 53

Building faces and volumes for the probe 63

Check and correct the geometry 79



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3.1. Create a FLUX project for the probe

Introduction Each time that a FLUX program is started, it is possible to open an existing

project or create a new project.


Topic See Page

Create a new project for the probe 22

About the Preflux window 23

About the Help menu / Users guide 24

About the geometry context 26

Name the project 27



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3.1.1. Create a new project for the probe

Goal At the beginning of the geometry description a new project must be created.

Action To create a new project from the

Project menu:

1. Click on New

OR

Project toolbar:

1. Click on the icon

Result FLUX retrieves a great deal of information from the database model in order

to build the proper database of the new project. The new project is

temporarily named ANONYMOUS.

The Preflux project window opens in the Geometry context by default. TheGeometry context icon is depressed, as shown in the following figure.



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3.1.2. About the Preflux window

Preflux window The Preflux project window has the complete set of the tools to build the

geometry of the device, to mesh the computation domain and to visualize the

device during different steps of the construction.

Areas The Preflux project window is divided into three main areas. The different

areas can be resized or hid by using the arrows.

Graphic

sceneData tree

History zone

Area Function

Data tree displays all the problem data in a tree structure that is

expanded using the key

Graphic scene displays the graphic entities

History zone prints Python command instructions

Menus and

toolbarsAll Preflux commands are in the menus. Toolbars include icons that are

shortcuts to the most useful commands.

Menus

Toolbars



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3.1.3. About the Help menu / Users guide

Introduction There are several ways to access the users guide information:

the complete users guide

the on-line help within a dialog

Method 1 To open the complete users guide in the FLUX Supervisor from the

Help menu:

1. Click on

Manual

OR

Help toolbar:


Method 2 To open the complete users guide in Preflux from the Help menu:

1. Click on Contents

Method 3 To open the on-line help about an entity from its dialog box:

1. Click on the button




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Users guide The on-line version of the FLUX users guide is presented in the figure

below. The corresponding sections of the FLUX users guide can be opened

by clicking on the hyperlinks.

More information

about the FLUXSupervisor

Click on FLUX

Index

More information

on Geometry and

mesh

More information

on General tools



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3.1.4. About the geometry context

Presentation There are three contexts in Preflux:

Context FunctionGeometry to build the geometry of the device

Mesh to mesh the computation domain

Physics* to define the materials, sources and to prepare the

regions

* The icon corresponding to the Physics context appears after the definition of the physicalapplication

Tools of the

geometry

context

After having activated the geometry context, toolbars dedicated to the

geometry description appear in the Prefluxwindow.

The different toolbars and their principal roles are briefly described below.

1 2 3 4

6

5

Geometry context toolbars Function

1 to create geometric entities

2 to propagate / extrude points, lines, etc.

3 to build faces, volumes

4 to compute geometric values

5 to check the geometry

6 to display point and line reference numbers



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3.1.5. Name the project

Goal The new project, temporarily named ANONYMOUS, will be renamed and

saved.

Action To rename the project from the

Project menu:

1. Click on Save or

Save as

OR

Project toolbar:


2. Type PROBE_3D

as project name

3. Click on Save

Note:The user can choose another name for the project and change the current project directory

(working directory), displayed in the Save In field at the top.

A periodic data backup is recommended.



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3.2. Strategy and tools for geometry description of theprobe

Introduction This section shows:

the available tools for geometry building

the analysis carried out for construction of the probe geometry and the

selected strategy


Topic See Page

Available geometric tools and analysis before geometry

description

30

Main phases for geometry description of the probe 32

Reading advice This section presents an outline of the geometry building process of the

probe. Details on the different contents - definition of new concepts,

explanation on the use of different tools, etc.- are given in the following

sections.



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3.2.1. Available geometric tools and analysis before geometrydescription

Available tools The tools available for the geometric construction are: geometric parameters,

coordinate systems and transformations.

Geometric tool Function

geometric parameter to allow the dimensional parameter setting of parts

coordinate system to facilitate the relative positioning of parts

transformation to allow the construction by propagation or extrusion

Device analysis

and choice of

construction

tools

An analysis of the device is necessary to determine the strategy of

construction, and the choice of construction tools.

The analysis of the device and the construction tools chosen within the

framework of this tutorial are summarized in the table below.

The operations

it is planned

to enter the

coordinates of

the points

to create a

PROBE_CS Cartesian

coordinate system

specific to the probe

PROBE_CS

to change

dimensions of

the magnet and

the coil

to create 5 parameters

for setting the magnet

and the coil

dimensions MAG_H

COIL_H

MAG_R

COIL_IR

COIL_OR




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Device analysis and choice of construction tools (continued)

The operations

it is planned

to locate the

probe in the final

project

(anticipation)

to create a MAIN_CSCartesian coordinate

system

(thePROBE_CScoordinate

system will be attached to

this coordinate system)

to create an ANGLEparameter to define the

angular position of the

MAIN_CS coordinatesystem

MAIN_CS

PROBE_CS

ANGLE

to simplify the

geometry

building

to create a ROTX_PROBEtransformation of the

rotation type to build faces

and volumes by extrusion

ROTX_PROBE



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3.2.2. Main phases for geometry description of the probe

Outline An outline of the geometry building process is presented in the table below.

Stage Description

1

Creation of 6

geometric

parameters

Inner radius of the coil: COIL_IR = 2.8 mm

Outer radius of the coil: COIL_OR = 3.5 mm

Height of the coil: COIL_H = 16 mm

Radius of the magnet: MAG_R = 2.5 mm

Height of the magnet: MAG_H = 20 mm

Angle for the probe angular positionin the final device: ANGLE = 0

2

Creation of 2

coordinate

systems

Cartesian coordinate system: MAIN_CS(Global coordinate system for the probe positioningin the final device)

Cartesian coordinate system: PROBE_CS(Local coordinate system for the probe description)

3Creation of points and lines for

the probe base

4Building faces

base

for the probe

5

n

ransformation for the probe:

ROTX_PROBE

Creation of 1

transformatio

Rotation t

generator*)

ROTX_PROBE

6

Building faces and volumes by

extrusion

(and preparation of the mesh

* Explanation concerning this subject is presented in Available meshing tools and analysis

before mesh generation. (Refer to section About meshing tools on Extrusion MeshGenerator)



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3.3. Creation of geometric tools

Introduction The geometry building begins by the creation of geometric tools to build the

probe geometry: geometric parameters and coordinate systems.

The parameters and coordinate systems required to describe the geometry of

the probe are presented in the figure below.

MAG_R

COIL_IR

COIL_OR

ANGLE

MAIN_CS

PROBE_CS

COIL_H

MAG_H


Topic See Page

About creation of an entity 34

About geometric parameters 36

Create the geometric parameters 37

About the Tools menu / toolbar 39

About selection of graphic entities 40

About modification and deletion of an entity 40

About graphic view 45

Change the background color 47

About coordinate systems 48

Create the coordinate systems 50



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3.3.1. About creation of an entity

Definition of

entityAn entity is an object in the database of a FLUX project.

It can be:

a point, a line, a coordinate system, etc. in the Geometry context a mesh point, a mesh line, etc. in the Mesh context

a line region, a volume region, etc. in the Physics context

Creating

processAn outline of the creating process is presented in the table below. The

different steps are detailed in the blocks describing the creation of project

entities.

Step Description

1 Activating the New command

2 Definition of entity attributes

Access the

New

command

The access to the New command can be carried out:

from the Geometry menu bar (1)

using icons from the Geometry toolbar (2)

from the data tree (3)

These three methods to access the New command are presented in the

following figure (with the example of creation of a geometric parameter) and

described in the table below.

1

3

2

Method Description

1 point on the entity-type and click on New

2 click on the corresponding icon

3 double-click on the entity-type or right click and click on New




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Dialog box The interaction with the database is done using dialog boxes. In this box the

user can enter information relating to the data.

Entity-type:

Geometric parameter

Name

Comment

Characteristics

Title bar

On-line help

concerning the entity

The required fields (necessary and sufficient for the definition of the entity) are marked by anasterisk *.



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3.3.2. About geometric parameters

Principle of use Geometric parameters are entities that can be used for the geometry building

of the device, i.e. for the definition of points, coordinate systems, geometric

transformations, infinite box dimensions and other geometric entities.

Defining parameters simplifies the construction of the geometry and enables

modifications to be made more easily later. Many changes can be made by

modifying only the definition of the parameters instead of modifying all the

individual points, lines or nodes that might be built using the parameters.

Parameters also can modify the scale of the geometry through their

relationship with coordinate systems.

Definition of

parametersThe geometric parameters are defined by the name and the algebraic

expressions.The algebraic expressions may contain:

constants

arithmetic operators (+, -, *, /, **)

arithmetic functions allowed in FORTRAN (SQRT, LOG, SIN, etc.)

other parameters

combinations of any of these

Parameters and

measurement

units

Please note that parameters are independent of any unit of measurement. In

other words, the numerical value entered for a parameter is not changed when

the unit of measurement is changed. Any measurement unit associated with a

parameter derives from the coordinate system in which the parameter is used.

For example, a parameter's value may be 10 in a coordinate system with

millimeters as units. This parameter's value is still 10 whether the coordinate

system's units are changed to inches or meters or kilometers or any other

available unit. Thus, when you use parameters, you can also modify the scale

of a geometric feature without reentering each point or item.



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3.3.3. Create the geometric parameters

Goal Six parameters, required to describe the geometry of the probe, are presented

in the figure below.

MAG_H

COIL_H

MAG_R

COIL_IR

COIL_OR

ANGLEMAGNET base

COIL base

Data The table below contains the values of the geometric parameters.

Geometric parameters

Name Comment Expression

COIL_IR Inner radius of the coil 2.8

COIL_OR Outer radius of the coil 3.5

COIL_H Height of the coil 16

ANGLE Angle of the probe position 0

MAG_R Radius of the magnet 2.5

MAG_H Height of the magnet 20




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Action To create the geometric parameters from the

Data tree:

1. Double-click

on Geometric parameter

OR

Geometry toolbar:


2. Type COIL_IRas name

3. Type Inner radius of the coil as

comment4. Type 2.8 as algebraic expression for

the parameter

5. Click on OK

6. Repeat steps 2 to 5 in the new dialog,entering data for the remaining entities.

(see the table on the previous page)

7. Click on Cancel to quit the sequence

Result The geometric parameters are listed in the data tree:

Notice too, that as you move your cursor over the parameter names, the comments are

displayed to help you to identify the parameters.



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3.3.4. About the Tools menu / toolbar

Undo command There is a FLUX command to undo operations. The user can use this

command if an error was made.

There are two possibilities described in the table below.

Method Function

1 to undo the previous operation to undo the last action

2 to undo several operations to undo all actions up to the indicated

action

Method 1 To undo the previous operation from the Tools toolbar:


Method 2 To undo several operations from the

Tools menu:

1. Click on Undo

OR

Tools toolbar:


2. Click on the last operation to undo



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3.3.5. About selection of graphic entities

Overview of

selection modesSelection of entities can be done with the following selection modes:

graphic selection (with the mouse)

- in the data tree for all entities- in the graphic scene for graphic entities

identifier selection (by name / by number)

advanced selection (by criterion / by choice)

Graphic

selection

process

An outline of the selection process for graphic entities is presented in the

table below.

Step Description

1 Activating of the selection filter2 Selection of the entity in the graphic scene

Selection filter A selection filter makes possible to identify the selectable entity-type.

For the graphic entities, the selection filter can be activated by the

commands from the Selection menu or from the Selection toolbar.

Selection menu/

toolbarThe choices in the Selection menu or in the Selection toolbar relate to the

graphic entities; they are presented in the figure and described in the table

below.

No

selection

Free

selection

Select

points / lines / faces / volumes

Selectface regions / volume regions

Choice Description

No selection nothing selectable

Free selection

all is selectable

The first entity, selected by the user, determines the

entity-type selectable

Select points the points are selectable




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Step 1:

activating of

the selection

filter

The activating of the selection filter can be carried out:

from the Select menu (1)

using icons from the Select toolbar (2)

These two methods to activate the selection filter are presented in the

following figure and described in the table below.

1

2

Step 2:

selection in the

graphic scene

Click on the specific graphic entity to select the entity in the graphic scene.

The selected entity is highlighted.



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3.3.6. About modification and deletion of an entity

Modification /

deletion processAn outline of the modification / deletion process is presented in the table

below.

Step Description

1 Activating the command (Edit, Edit array, Delete, Force delete)

and selection of entities

2 Modification of the entity characteristics /

Validation of the entity deletion

Access the

commandsFor the commands Edit / Edit array / Delete / Force delete, which require

data selection, the access to the command, can be carried out:

from the menu

- activation of the command and then selection via a selection box (1)

from the data tree:

- activation of the command and then selection via a selection box (2)

- direct selection of an entity and then activation of the command(2)

from the graphic scene (only for graphic entities)

These methods to access the command are presented in the following figure

(with the example of editing the ANGLE geometric parameter) and described

in the table below.

1

2

Selectionvia

a selection box

2

Selectionvia

a selection box

Method Description

1 point on the entity-type and click on the command

select entities via a Selection box

2 right click on the entity-type and click on the command

select entities via a Selection box

2 double-click on the entity

or right click on the entity and click on the command

3 right click on the graphic entity* and click on the command

* The corresponding selection filter must be first activated.




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Edition mode To check the data, the user needs to edit (and modify if necessary) the entities

created.

There are two modes of edition:

the edition in a dialog box is used to check and to modify the

characteristics ofone entity

Entity-type

Name

Comment

Type (1)

Characteristics

Entity

Type (2)

On-line help

concerning the entity

the edition in a data array is used to check and to modify the

characteristics of a group of entities

Entity-type

Name

Comment

Type (1)

Characteristics

Entities:[CORE], [MAIN]

Type (2)

Structure(Database)

Informationrelating to the

group of entities

Informationrelating to theentity[CORE]

Information

relating to theentity[MAIN]




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Deletion mode The user sometimes needs to delete entities. He can easily delete an entity if it

is an independent entity. However, very often, the entity is connected to other

entities and the deletion of the entity can cause the deletion of all the

connected entities.

There are thus two modes of deletion:

the simple deletion:

is carried out on independent entities (non connected with other entities)

the in force deletion :

is carried out on any entity.

These two modes are described in the table below:

Mode Destroyable entity What is destroyed

simple independent selected entity

in force any selected entity + entities connected to it



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3.3.7. About graphic view

Introduction When referring to the graphic representation of a device, we are interested in:

the different entities and their appearance: points and their visibility, lines

and their color, faces, surface elements, etc. the type of displayed view: side view, top view, bottom view, global view,

etc. and its position and dimensions in the graphic display zone.

How to modify

a viewThere are three methods to modify the view in the graphic scene. The settings

can be made:

from the View menu (1)

using icons from the View toolbar (2)

using the mouse (3)

1

3

2

Using the View

menu / iconsPreflux offers modes to modify the view using commands from the View

menu oricons from the View toolbar. They are described in the table below.

Command Icon Mode Mode activation

Viewdirection

To rotate, translateand resize the view

click on the command / iconand fill out the dialog box

Zoom in - To enlarge the view click on the command

Zoom out - To reduce the view click on the command

Zoom all To set total view click on the command / icon

Zoom

region

To enlarge a part of

view

click on the command / icon

and select the rectangular zone

to enlarge using the mouse




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Using the

mousePreflux offers modes to modify the view using the mouse, described in the

table below. User can determine the active mode by the different cursors.

Mode Mode activation Cursor

2D planar rotation around

the center of the view

mouse is far from the center of the

view, click on the graphic scenewith the leftbutton of the mouse

and move the mouse, keeping the

left buttonpressed

3D rotation around the

center of the object

mouse is close to the center of the

object, click on the view with the

left button of the mouse and move

the mouse, keeping the left button

pressed

3D rotation around the

point defined by mouse

cursor

while holding the Shift key, click

on the view with left button of the

mouse and move the mouse,

keeping the left buttonpressed

Displacement (to

translate the view)

click on the view with the right

button of the mouse and drag the

view to the new location, keeping

the rightbuttonpressed

Dimension (to resize the

view)

click on the graphic scene with the

leftbutton of the mouse and resize

the view with the scrollingwheel

of your mouse

Predefined

viewsIt is possible to choose one view from predefined views available in FLUX.

The different commands to set predefined views and their corresponding

icons are presented in the table below.

View command Icon Description

Standard view n1 First Preflux 3D predefined view (default one)

Standard view n2 Second Preflux 3D predefined view

Opposite view View perspective from the opposite direction

View direction View defined by the user

Four views mode Four different views graphic display

X plane view View of the device so that the X axis is

perpendicular to the graphic scene

Y plane view View of the device so that the Y axis is


Z plane view View of the device so that the Z axis is


Backgroundcolor

It is possible to swap the background color from black to white and vise versaby using the Reverse video command.



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3.3.8. Change the background color

Goal To better visualize the geometry, the background color will be changed.

Action To change the background color from the View menu:

1. Click on

Reverse video



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3.3.9. About coordinate systems

Introduction All geometric features are defined within a specific coordinate system.

Defining our own coordinate systems enables us to describe and modify the

geometry much more easily.

Types of coord.

systemsThe different types of coordinate systems for 3D domain and associated

coordinates are presented below.

Cartesian coordinate

system

Coordinates (x, y, z)

Cylindrical coordinate

system

Coordinates (r, , z)Spherical coordinate

system

Coordinates (r, , )

y

zp

x

zp

r

p

r

Reference

coordinate

systems

It is possible to distinguish the following coordinate systems:

The global coordinate system is the coordinate system where all

computations are performed. It is inaccessible to the user. The global

coordinate system is a universal Cartesian coordinate system using meters

as the length unit and degrees as the angle unit. The working coordinate systems are coordinate systems created by the

user to cover the study needs.

The working coordinate systems are defined:

- with respect to the Global coordinate system, when they refer to the

global coordinate system

- with respect to a Local coordinate system, when they refer to other

coordinate systems.

All entities are defined in the working coordinate systems (users coordinate

systems) and are evaluated in the global coordinate system for calculations.

Coordinate

system unitsThe user can define the length and angle units for a coordinate system defined

with respect to the global coordinate system (millimeter and degree by

default).

A coordinate system defined with respect to the local coordinate system

inherits the units of the reference coordinate system (parent coordinate

system).




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Predefined

coordinate

systems

To assist the user, FLUX provides some default coordinate systems. They are

created for every new project. It is possible to rename them, to modify them

or to delete them.

The predefined working coordinate systems provided to the user are:

ofCartesian type

defined with respect to the global coordinate system

The coordinate systems are distinguished one from each other by their

positioning as presented in the table below.

Coordinate system Characteristics

XYZ1Z

Xy

z

x

Origin of coordinate system:

first component: 0

second component: 0

third component: 0

Rotation angle: about X axis: 0

about Y axis: 0

about Z axis: 0

Z_ON_OX

Y

X

Z

z

y

x


first component: 0

second component: 0

third component: 0

Rotation angle:

about X axis: 90 about Y axis: 90

about Z axis: 0

Z_ON_OY

Z

X


first component: 0

second component: 0

third component: 0

Rotation angle:

about X axis: 90

about Y axis: 0

about Z axis: 0x

y

z



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3.3.10. Create the coordinate systems

Goal Two coordinate systems, required to describe the geometry of the probe, are

presented in the figure below.

MAIN_CS

PROBE_CS

32 mm

Data The tables below describe the coordinate systems.

Cartesian coordinate system typedefined with respect to the Global system

Origin coord. Rotation angleName Comment Units

X Y Z About X About Y About Z

MAIN_CSMaincoordinatesystem

millimeter/degree

0 0 0 0 0 ANGLE

Cartesian coordinate systemtypedefined with respect to the Local system

Origin coord. Rotation angleName Comment

Parent

coord.

systemX Y Z About X About Y About Z

PROBE_CSProbecoordinatesystem

MAIN_CS 32 0 0 0 0 0




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Action To create the coordinate systems from the

Data tree:

1. Double-click

on Coordinate systemOR

Geometry toolbar:


2. Type MAIN_CS as name of

coordinate system3. Type Maincoordinate system

as associated comment4. Select Cartesian as type of

coordinate system5. Select Global as definition of

coordinate system

6. Select MILLIMETERas lengthunit

7. Select DEGREE as angle unit

8. Type 0 as first coordinate

9. Type 0 as second coordinate10. Type 0 as third coordinate

11. Type 0 as rotation angle about X

axis12. Type 0 as rotation angle about Y

axis13. Type ANGLE as rotation angle

about Z axis

14. Click on OK




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15. Type PROBE_CS as name of

coordinate system16. Type Probecoordinate system

as comment17. Select Cartesian as type

18. Select Local as definition ofcoordinate system

19. Select MAIN_CS as parentcoordinate system

20. Type 32 as first coordinate21. Type 0 as second coordinate22. Type 0 as third coordinate

23. Type 0 as rotation angle about Xaxis

24. Type 0 as rotation angle about Yaxis

25. Type 0 as rotation angle about Zaxis

26. Click on OK


Result The two new coordinate systems are

listed in the data tree: displayed in the graphic scene*:

PROBE_CS

MAIN_CS

* use the Zoom all command or (see About graphic view).



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3.4. Creation of points and lines of the probe base

Introduction The next step of the geometry description is the creation of points and lines to

build the probe base.

The next figure describes the planar geometry of the probe base.

MAG_H

COIL_H

MAG_R

COIL_IR

COIL_OR


Topic See Page

Change to the plane view 54

About points 55

Create points for the probe base 56

About display of entities in the graphic scene 58

Display point numbers 59

About lines 60

Create lines for the probe base 61



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3.4.1. Change to the plane view

About view See 3.3.7 About graphic view.

Goal To better visualize the planar geometry of the probe base in the XOY plane,

the view will be changed.

Action To change the view from the

View menu:

1. Click on Z plane view

OR

View toolbar:




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3.4.2. About points

Points A point can be created

as a set of coordinates in a specified coordinate system

as an image of an existing point through a geometric transformation within the propagation or extrusion from other entities

Point

coordinatesA point could be defined by its coordinates in a coordinate system (see 3.3.9

About coordinate systems).

Point defined

by propagationA point could be defined by propagation from another point using a

transformation.

translation

origin point

created point

Point number The number to identify the point is automatically allocated by FLUX during

the point creation.



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3.4.3. Create points for the probe base

Goal Ten points are required to build the probe base, as presented in the figure

below.

Point 1MAG_H

COIL_H

Point 2 Point 5

Point 6

PROBE_CS

Point 7

Point 8 Point 9Point 10

MAG_R

COIL_IR

COIL_ORPoint 3 Point 4

Data The table below describes the points for the probe base.

Points defined by its parametric coordinates

CoordinatesNo Coordinate system

X Y Z

1 -MAG_H/2 0 0

2 -MAG_H/2 MAG_R 0

3 -COIL_H/2 MAG_R 0

4 COIL_H/2 MAG_R 0

5 MAG_H/2 MAG_R 0

6 MAG_H/2 0 07 -COIL_H/2 COIL_IR 0

8 -COIL_H/2 COIL_OR 0

9 COIL_H/2 COIL_OR 0

10

PROBE_CS

COIL_H/2 COIL_IR 0




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Action To create the points from the

Data tree:

1. Double-click on Point

OR

Geometry toolbar:


2. In the Geometric Definition tab

select Point defined by itsparametric coordinates as type

of point3. Select PROBE_CS as

coordinate system

4. Type -MAG_H/2 as firstcoordinate

5. Type 0 as second coordinate6. Type 0 as third coordinate

7. Click on OK

8. Repeat steps 4 to 7 in the new dialog,

entering data for the remainingentities



Result The points are

listed in the data tree: displayed in the graphic scene:



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3.4.4. About display of entities in the graphic scene

Introduction The graphic representation of objects is not the same during the different

steps of building the device model.

From one step to the next, we are interested in:

representation of points and lines during geometry building

representation of nodes and surface elements during mesh generation

Possibilities to

modify the

visualization

To control the graphic representation, FLUX provides default settings, but the

user can also modify this representation.

To do so, the following commands are available:

the Display commands, which manages the list of entities to display,

the Edit command, which allows the modification of the entity appearance

(characteristics of visibility and color)

How to display

entitiesThere are two methods to display entities in the graphic scene. The settings

can be made:

from the Display menu (1)

using icons from the Display toolbar (2)

1

2



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3.4.5. Display point numbers

Goal The display of point numbers will be activated to see the reference point

numbers allocated by FLUX.

Action To display the point (reference) numbers from the

Display menu:

1. Click on Display

point numbers

OR

Display toolbar:


Result The points, labeled with reference numbers, are displayed in the graphic

scene.



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3.4.6. About lines

Lines Lines can be created:

manually (choice of line type segment orarc - and entering extremity

points)by propagation from existing lines using a transformation

by extrusion from existing points using a transformation

within the propagation or extrusion from other entities

Segments Segments are defined by starting and ending points. It does not matter if you

swap the starting and ending points.

Circle arcs Circle arcs can be defined in different ways:

eitherin a coordinate system:

The arc is included in a plane parallel to the XOY plane. It is counter-

clockwise oriented around an axis parallel to the OZ axis.

starting point

ending point

center

radius

angle

orby three points:

The arc is included in a plane defined by three points. It is oriented in the

direction imposed by three points.

ending point

starting point

middle point

There are some predefined coordinate systems (XYZ1, Z_ON_OX, Z_ON_OY) to simplifythe circle arcs description in the different main planes.

Number The number to identify the line is automatically allocated by FLUX during

the line creation.



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3.4.7. Create lines for the probe base

Goal Twelve straight segments are required to connect each point and create closed

outlines of the magnet, coil and air-gap bases.

The order to create the lines is presented in the figure below.

Line 1Line 2

Line 5

Line 6

Line 7

Line 9Line 10

Line 12

MAGNET base

COIL base

Line 3 Line 4

Line 8Line 11

AIR-GAP base

Data The table below describes the lines for the probe base.

Segment defined by starting and ending points

No Starting point Ending point

1 1 2

2 2 3

3 3 4

4 4 5

5 5 6

6 6 1

7 3 7

8 7 8

9 8 9

10 9 10

11 10 4

12 10 7




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Action To create the lines from the

Data tree:

1. Double-click on Line

OR

Geometry toolbar:


2. In the Geometric Definition tab

select Segment defined by starting

and ending points as type of the line

3. Click on Point1 in the graphic scene

=> its reference number enters asstarting point

4. Click on Point2 in the graphic scene

=> its reference number enters as

ending point

5. Repeat steps 3 to 4 in the new reduceddialog, entering data for the remainingentities



Result The lines are




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3.5. Building faces and volumes for the probe

Introduction The next step of the geometry description is building faces and volumes for

the probe.

The probe geometry is presented in the figure below.


Topic See Page

About automatic construction 64

Build faces of the probe base 65

Change to the standard view 66About transformations 67

Create the geometric transformation 69

About propagation and extrusion 71

About selection by criterion 73

Extrude faces 74

Complete the construction by automatic building 77



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3.5.1. About automatic construction

Introduction The faces and volumes are automatically created and identified using the

algorithms of automatic construction.

Principle:

overviewThe principle of automatic face construction:

First, FLUX computes all the existing surfaces and determines which

surfaces the points and the lines belong to. (A surface contains faces but it

is not limited. A surface is defined by three points linked by two lines.)

Next, the automatic face construction is carried out by a method of

identification of closed contours.

About faces The faces created by FLUX using the automatic construction algorithms are

faces contained by planar, cylindrical or conical surfaces. These faces arenamed automatic faces.



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3.5.2. Build faces of the probe base

Goal The faces will be automatically built by Preflux.

Action To build faces from the

Geometry menu:

1. Point on Build and click on Build faces

OR

Geometry toolbar:


Result The faces are




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3.5.3. Change to the standard view

About view See 3.3.7 About graphic view.

Goal To better visualize the future 3D geometry of the probe, the view will be

changed.

Action To change the view from the

View menu:

Click on Standard view

OR

View toolbar:

Click on the icon

Result The next view is displayed in the graphic scene.



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3.5.4. About transformations

Principle of use Transformations are geometric functions that create new objects from existing

objects.

Various

functionsThe various available functions are:

translation

rotation

affinity

helix

composed

Note: Only the transformation functions used in this tutorial are described here. Refer to theUsers guide for more information about transformations.

Translation A translation is defined by a direction and a distance.

The figures below describe the creation of a new face using two different

translation transformations:

Translation vector

DZ

DX

createdrectangle

basicrectangle

vector

DY

direction and distance are

defined by:

- a working coordinate system

- a vector (the DX, DY and DZ

components define thedirection and the amplitude of

the vector)

Translation defined by 2 points and a ratio

vectorhead

vectortail

basic

rectangle

createdrectangle

direction is defined by twopoints (a vector head point and a

vector tail point)

distance is proportional to the

distance between the two points

calculated with the scaling factor

(ratio)




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Rotation A rotation is defined by a rotation axis and a rotation angle.

The figures below describe the creation of a new point using two different

rotation transformations:

Rotation defined by angles and a pivot point (its coordinates or reference

number)

rotation axis is defined by:

- a working coordinate system

- and a pivot point

rotationangle is defined about X, Y or Z

axis

createdpoint

originalpoint

pivotpoint

rotationangle

y

z

x

Rotation defined by 3 points and 1 angle

headpoint

tailpoint

z

y

rotation axis is defined by direction and

position:

- a head point and a tail point give the

direction

- a pivot point defines the position.

rotation angle is defined in the plane

perpendicular to the axisoriginalpoint

pivotpoint

createdpoint

rotationaxis

x

Note: The positive value of an angle corresponds to a counter-clockwise rotation



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3.5.5. Create the geometric transformation

Goal A transformation ofrotation type is required to build the probe geometry.

The pivot point and the angle, needed to create the transformation, are shown

in the following figure:

Point 1

90

Data The characteristics of the transformation are shown in the following table:

Rotation defined by angles and existing pivot point

Rotation aboutName Comment

Coord.

system

Pivot

point X axis Y axis Z axis

ROTX_PROBERotation

transformationfor the probe

PROBE_CS 1 90 0 0




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Action To create the transformation from the

Data tree:

1. Double-click

on Transformation

OR

Geometry toolbar:


2. Type ROTX_PROBE as name

3. Type Rotation transformationfor the probe as comment

4. Select Rotation defined by

angles and existing pivot pointas type

5. Select PROBE_CS ascoordinate system

6. Select point 1 in the list or inthe graphic scene as pivot point

7. Type 90 as rotation angle about

X axis8. Type 0 as rotation angle about

Y axis9. Type 0 as rotation angle about

Z axis

10. Click on OK

11. Click on Cancel to quit the

sequence

Result The transformation is listed in the data tree:



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3.5.6. About propagation and extrusion

Definition The constructionby propagation / extrusion is a buildingmethod that

constructs s new geometric entities, based on existing entities, by using a

geometric transformation like translation, rotation, etc.

We deal with:

propagation, when the image object, generated by transformation, is not

connected by lines to the source object

extrusion, when the image object, generated by transformation, is

connected by lines to the source object

Examples In the figures below, the parallelepiped is built by propagation / by extrusion

of the existing parallelogram (source) using a translation vector.

Construction by propagation:

translation

source face

image face

Constructio

f3d 920 generic tutorial

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