CSM: ideal solenoid magnet - Analytical expression

[Trophime]

Description of the example

In this example, we consider a solenoid conductor with finite thickness and infinite length.
This allow us to ignore the z components in our equations.
We admit that there is only a radial expansion.

Geometry

The conductor $\Omega$ consists in a rectangular cross section torus.
The geometry also contains an external domain which is an approximation of $\mathbf{R}^3/\Omega$.

Name	Description	Value	Unit
$r_1$	internal radius	$1.10^{-3}$	m
$r_2$	external radius	$2.10^{-3}$	m
dz~	height	$2.10^{-1}$	m

Input parameters

current density: $\textbf{j}$ in $A/m^2$.
magnetic field: $\textbf{b}$ in $T$.

Model & Toolbox

• From the momentum conservation equation, we have:
  [
  div\sigma+\textbf{j}\times\textbf{b}=0
  ]
  With the hypothesys specified in introduction, this equation may be rewritten :
  [
  -\sigma_{\theta}+\frac{\partial}{\partial r}(r\sigma_{r})=-rj_{\theta}b_{z}
  ]
  For a solenoid conductor with finite thickness and infinite length, we have for a constant current density
  $j_{\theta}$:
  [

• b_{z} = ...
  ]
  With these hypothesys, we can show that (see [REF002]):
  [

• u_{r} = ...
  ]

• toolbox: elasticity

Materials

|Name |Description | Value | Unit |
|$E$ |Young modulus||$128.10^{9}$|$Pa=kg.m^{-1} .s^{-2}$|
|$\nu$|Poisson's ratio|0.33|- |

Boundary conditions

• entry/exit: $\textbf{u} \dot \texbf{n} = 0$, displacements only in the perpendical plane,
• top:bottom: clamped

Outputs

The output is describe the output set of the example

Fields

add scalar vectorial and matricial fields to be visualized

Measures

add measures, scalar quantities, mean values, performance metrics

Benchmark

Describe Benchmark type:
[X] Verification
[] Validation
[] Performance

The computed values of the displacements on r-axis in the solenoidal mid-plane are compared
with the analytical expression given by Montgomery.

References (articles, papers, reports...)

• https://github.com/feelpp/hifimagnet/blob/develop/MagnetTools/docs/Stress/stress.tex
• [REF001] Montgomery, Solenoid Magnet Design, Wiley-Interscience, New-York. 1969.
• [REF002] M N. Wilson, Superconducting Magnets, Oxford University Press, London, 1987.

[Trophime]

This example may also be used as a validation for the complete model of a sodenoidal magnet in hifimagnet coupling data from thermoelectric and magnetostatic anaytical benchmarks.

[Trophime]

This example seems to be already existing in CSM benchmarks.
We should only make it look more like the others