The 2-dimensional program
Poisson/Superfish
is used to simulate the Hall D superconducting solenoid. The geometry of the coils and iron
is an interpretation of the post-installation measurements from 2013 and 2014,
outlined in the
Technical Report on the solenoid. The model
included certain simplifications, in order to adapt it to
the 2-D POISSON program and to finite mesh size:
Thin layers (shims) were attached to one of the adjacent iron pieces
covering the full cross section (in fact the shims are plates of a
smaller surface).
The structure of the conductor was ignored. The subcoils' cross sections
were simulated by rectangular areas with uniform current densities.
Some filler plates between the yoke rings are not uniform in φ.
These plated are presented as azimuthally-symmetric rings of the same
volume as the filler plates and of the same thickness, however
of a different radial dimensions, depending on the actual volume.
Components
18 subcoils (combined into 4 coils). The current through each subcoil is
defined by the full number of turns in the subcoil.
4 barrel iron rings, 2 iron endcaps and several filler plates.
2 cladding rings around the yoke. The gap in the cladding at the
bottom of the magnet was ignored.
The geometry is shown in the following pictures:
The full area used for the Poisson simulation. The mesh is of a variable size.
A pdf picture (large!) is available.
The central area used for the Poisson simulation. The mesh triangles are not drawn.
The picture shows the iron, the conductor and the field/potential lines calculated for 1300A.
The vacuum boxes drawn around the 4 coils are added just for viewing and play no role
in the magnetic calculations.
A pdf picture is available.
The full area selection
The Dirichlet conditions were used on the boundaries (no field perpendicular
to the boundary). The field drops to about 0.1 mT at a ≈10 m
distance:
The field contour plots, calculated for 1350A.
A pdf picture is available.
A large full area requires more nodes on the mesh and longer calculations.
A smaller area is less accurate far outside of the magnet but
may be more accurate inside since a finer mesh can be used.
Z -900cm:1300cm; R 0:1380cm (large area)
Z -500cm: 800cm; R 0:400cm (small area)
The mesh
Poisson allows several areas in Z (and, independently, in R)
with different steps. However, in each projection there could
be not more than 8 areas. It is recommended that the steps in the
adjacent areas differ by a factor not greater than 2-2.5.
The results may depend strongly both on the number of nodes and
on the configurations of the mesh.
The following cases have been observed:
Good result: the calculations converge and the solution is reasonably
smooth.
Fair result: the calculations converge, the full energy estimate
is correct, but there are large point-to-point
fluctuations of the field in some areas.
Poor result: the calculations failed to converge reaching the used limit
for the number of iterations (200000), the full energy estimate
is incorrect and there are very large point-to-point
fluctuations of the field.
No result: POISSON.EXE crashes with a kind of a segmentation fault error.
The Poisson program prints warnings if the conversion speed is low,
suggesting to modify the "relaxation parameter" rhogam in a range
of 0.0005 - 0.08. I found that using a smaller parameter 0.08->0.0005
could improve the convergence and the number of iterations needed,
but would not improve the point-to-point fluctuations
(converting a case #3 to a case #2).
In order to simulate the effects of a small conductor motion I tried to
use as fine mesh as possible, at least in the areas around the conductor.
The full area used was -500cm:800cm in Z and 0:400cm in R. The smallest
steps producing a smooth solution were of 2.5mm in R and 5.0mm in Z,
in the area around the conductor.
The field "fluctuations" were checked in the areas where they were most
noticeable and where the field dependence on the radius
could be approximated with a 2-nd order polynomial function.
The deviations from the fitted functions was calculated along the lines:
line #1 Z1,R1->Z2,R2 (in cm): 30,165->30,180 in iron
line #2 Z1,R1->Z2,R2 (in cm): 30,115->30,130 in air outside of the coils
.am file Min steps: 2.5mm in R and 2.5mm in Z (small area)
.am file Min steps: 2.5mm in R and 2.5mm in Z (larger area)
.am file Min steps: 2.5mm in R and 1.25mm in Z (small area)
The mesh structure is given in the following table
(the units are cm):
The results are shown in the next table. The calculated RMS of BZ
are shown in the last two columns. Although in all 4 cases the calculations converged
after about 20000 iterations, only the first case produced a relatively small
fluctuations along the line #2 - about 20 Gauss. Decreasing the step size
in Z from 0.5cm to 0.25cm, even in a small area around Coil#2, increases the
field RMS to about 200 Gauss. A 0.125cm step in Z leads to strong fluctuations
of about 2000 Gauss.
The field plots for the cases #1-2 and #4 are shown on the following plots:
Case #1: The fields BZ, BR (Gauss) interpolated with a 1mm step
along a Z=30cm line are shown. No significant field fluctuations are seen.
A pdf picture is available.
Case #2: The fields BZ, BR (Gauss) interpolated with a 1mm step
along a Z=30cm line are shown. Some field fluctuations are seen.
A pdf picture is available.
Case #4: The fields BZ, BR (Gauss) interpolated with a 1mm step
along a Z=30cm line are shown. Strong field fluctuations are seen.
A pdf picture is available.
I checked if a finer mesh could be used for a simplified conductor geometry. The motivation
was to simulate the effect of a small motion of a a part of the coil, for example a pancake.
Assuming no saturation effects in steel one could simulate just the field generated by one
pancake at different locations, without the other conductor.
Simulation of the field produced by one pancake, at 3 different locations.
This plot presents the results for the central pancake at a 1500A current
(the pancakes at the sides are at zero current).
A pdf picture is available.
With a 0.5mm step in Z the field curve is smooth. However, if a 0.25cm step is used
in a range of 36<Z<68cm the fluctuations emerge again:
One pancake simulation at 750A: The fields BZ, BR (Gauss) interpolated with a 1mm step
along a Z=48.5cm line are shown. Field fluctuations are seen.
A pdf picture is available.
In summary, one can use a 0.25cm mesh step in R in the area of the conductor,
but only a 0.5cm step in Z.
The yoke is built of the AISI 1006 steel. The magnetic properties of the yoke steel have been measured at SLAC (see
a copy
of relevant pages from the SLAC documentation). They pointed out that the measured permeability was somewhat different
(lower) than the data they obtained from other sources. Floyd Martin at JLab made an ANSYS model of the solenoid. He extracted
the SLAC results for the B-H dependence in a relatively narrow range of H, and extrapolated them to lower and stronger fields
using some other data. The Poisson program can either use a user-provided B-H table, or a default table characterizing
the AISI 1010 steel. We have considered the following B-H tables (in CGS units):
The narrow range SLAC data extracted by Floyd Martin: file
The SLAC data extrapolated to a wide range by Floyd Martin: file
Data for the AISI 1006 steel (source unknown): file
Data for the AISI 1018 steel (source unknown): file
Poisson material type=0 - data for the steel AISI 1010 : file
These data are displayed in the following plot:
Comparison of the B-H data in SI units. The vertical axis present the magnetization B-μoH.
A pdf picture is available.
The field maps of the solenoid were calculated:
Poisson default (type=0, for steel 1010 data) for everything.
The extended SLAC B-H for the old yoke, Poisson default for new pieces.
The aisi 1006 B-H for the old yoke, Poisson default for new pieces.
Poisson default for the old yoke, steel 1018 for new pieces.
For the new steel pieces the Poisson default material was used. For the old yoke pieces
calculations have been done for the B-H types 1-3.
The fields along
a line R=(Z-60)·0.1481 are compared:
Comparison of the calculated fields along a line.
A pdf picture is available.
Also, the fields along the lines R=0, R=30cm, and R=60cm are compared:
Comparison of the calculated fields along a line.
A pdf picture is available.
The largest deviation from the map 1 (Poisson) occurs at Z≈60cm, of about 0.05% for
the map 2 (SLAC) and 0.1% for map 3 (1006 AISI). For the map 4 (fillers of steel 1018) the largest
deviation of 0.1% is observed at Z≈300cm. The field integrals along the same line Z,R (60,0) - (350,80)
Bdl=∫[L×B]φdL are are:
(Bdl)2 / (Bdl)1-1 ≈ 0.004%,
(Bdl)3 / (Bdl)1-1 ≈ 0.07%,
(Bdl)4 / (Bdl)1-1 ≈ -0.06%.
We use the case 3 (Poisson default for AISI 1010) for the final calculations of the magnetic field.
Using the case 1 (SLAC measurements) would increase the field by <0.05%, which is an acceptable
uncertainty.
A crystal insert into the Forward Calorimeter (FCAL) will require a
re-evaluation of the magnetic shielding of the PMTs in that area.
TOSCA(Opera) is used for these 3-d calculations. TOSCA can apply
only a uniform external field, while in the area of interest
(Z=655-670cm, R=6-60cm: B≈5.8mT) there is a gradient of
dB/dZ≈-0.05mT/cm. At R=60cm the transverse field component is
1.7mT (30% of the longitudinal component).
One can hardly apply the full solenoid geometry for the TOSCA calculations
with the PMT shields (the size of the allowed TOSCA model is limited).
It might be easier to use a much simplified model - an ideal solenoid
without an iron yoke providing a right field map in the area of the
PMTs. The suggested model:
Geometry symmetric with respect to Z=0
Coil Z:0-300cm; R:102-110cm; full current = 28300A
The area to be used is at Z=300cm. It has the field
and the gradient close to those of the real solenoid.
See the maps for the regular field (Z: 655-670cm) and the model field in
this directory.