Example#7

Example # 7 : ESRF Quadrupole in 2D and 3D using Automatic Triangulation

Version : 0.6 of 13th April 2008 by J. Chavanne, O. Chubar, P. Elleaume
This notebook example requires Mathematica 6.0 and Radia 4.25 or higher to execute. It makes use of the automatic mesh generation of extruded Triangle borrowed from Jonathan Richard Shewchuk (http://www.cs.cmu.edu/~quake/triangle.html).
It computes in 2D and 3D the field from the storage ring quadrupole of the ESRF. The simulation of any other quadrupole can be derived from this notebook by changing the routines / parameters in the "Iron, Coil, Yoke and Build Functions" section. Dipole and sextupole could also be bilt in a similar fashion.
To run, one must first execute all cells of the next section (Iron, Coil, Yoke and Build Functions) which execute rapidly. Graphics production and field solving as well as post-processing takes more time and are triggered by the execution of the cells in the other sections ("2 - Dimensional Simulation" and "3 - Dimensional Simulation"). Some of the cells may take a lot of time to execute. Take your time to explore this notebook. Start with 2D computation and then move to 3D when you fill at ease. Read all comments written in each cell. This example is longer and quite more involved than any one of the examples 1 to 6.

Iron, Coil, Yoke and Build Functions

Introduction

The quadrupole is defined through the:
- "B vs H" ("M vs H") dependence of the iron type material;
- Coil geometrical parameters;
- Yoke geometrical parameters.
The figure below defines a number of variables used in the following functions to built the quadrupole.

Load and Initialize Radia

Iron Functions

Defining non-linear "M vs H" dependence for iron-type material

Coil Functions

Auxiliary function to be used for the definition of coils

Yoke Functions

Auxiliary functions defining the yoke boundary

Draw the contor of the lamination

Build 2D Functions

Defining the "Build" function for 2D Geometry

Defining the general "Solve" function and auxiliary functions for 2D case

Build 3D Functions

Defining the "Build" function for 3D Geometry

Auxiliary functions for 3D case

2 - Dimensional Simulation

Introduction

To simulate the qaudrupole in 2D, the yoke is created with a single layer of very large thickness.

Show in 2D

Instantiating the Quadrupole and compute the memory requirement

Show the Geometry

Solve in 2D

Solving

The following computation assumes a linear magnetic material with relative permeability mu. This is usefull to compare with the prediction of the Ampere Theorem. Note that with infinite permeability, the only difference with the Ampere Theorem comes from the multipole content and imperfect segmentation.

Post-Processing in 2D

Harmonic Analysis

The Harmonic Analysis to compute the multipole content around an axis requires the calculation of the field at equidistant points over a circle of radius R. The precision depends on the value of the radius R in relation to rint (the inscribed radius of the quadrupole). The optimal precision is reached if one uses R value not in the immediate vicinity of 0 or rint. A value between 1/4 rint and 3/4 rint is safe. One may want to check that the harmonic content weakly depends on R

Field Plot

Graphics:Mid-Plane Vert. Field [T] vs Horiz. Pos.[mm] Computed Directly (line) Reconstructed from multipoles (dots)

Gradient Homogeneity

Graphics:Mid-Plane Rel. Gradient Variation vs Horizontal Position [mm] Computed Directly (line) Reconstructed from multipoles (dots)

Field and Magnetization Line Plots

Graphics:None

Energy and Inductance in 2D

Auxiliary Functions

Energy and Self-Inductance

Gradient vs Current in 2D

Graphics: 2D On-Axis Gradient [T/m] vs Current [A]

3 - Dimensional Simulation

Introduction

This part concerns the 3D simulation of the same quadrupole. The yoke is created with its actual length. The extremities of the pole are machined to generate the chamfer. The model is created with a number of layers along the beam axis (longitudinal segmentation). At the end, the multipole expansion is computed as well as the magnetic length of the quadrupole. The accurate computation requires a large longitudinal segmentation and the memory requirement is significant. For this type of computation, the time longer than 10 minutes, as well as memory well above 200 MB are required.

Show in 3D

Instantiating the Quadrupole and compute the memory requirement

Show the Geometry

Solve in 3D

Solving

The solving time and memory requirement largely depends on nz (the longitudinal segmentation).To debug and accelerate the computation use e.g. nz = {2, 1}, the price is a poor absolute precision. For lmag = 400, some stability of the gradient versus horizontal position as well as magnetic length is obtained with nz = {32, 1}, which need quite some cpu and memory. In practice one must increase nz[[1]] to check the stability of the result.

Post-Processing in 3D

Harmonic Analysis

The Harmonic Analysis to compute the multipole content around an axis requires the computation of the field on equidistant points over a circle of radius R. The precision depends on the value of the radius R in relation to rint (the inscribed radius of the quadrupole). The optimum precision is reached if one uses a R value not in the immediate vicinity of 0 or rint. A value between 1/4 rint and 3/4 rint is safe. One may want to check that the harmonic content weakly depends on R.