Author: David Max MA DPhil (2016-07-06)
An FEA example: the NHMFL 15T system
Presented below are some results of finite element analysis (FEA) calculations for a superconducting magnet system described in the literature.
The FEA problem was solved using the CalculiX FEA package.
The magnet was built at the National High Magnetic Field Laboratory (NHMFL) in Florida, USA.
Coil dimension data for the magnet are given by Markiewicz et al 1994 [1], who also present detailed results for stress and strain in the coils and external binder.
- Construction of the NHMFL 15T system
- Main results in the Markiewicz et al. paper
- Assumptions for the CalculiX FEA finite element analysis model
- Binder (TEST362) and no-binder (TEST363) model comparison
- R graphics for model results
- Main results of models TEST362, TEST363
- Did the external binder work well?
On devices with wider screens a version of this webpage is available as a pdf document.
CalculiX-FEA input decks are available.
Construction of the NHMFL 15T system
The NHMFL 15 tesla magnet consists of two coaxial solenoids, each of winding length 500 mm.
- The inner section of the magnet is wound from niobium-tin conductor.
- The outer is niobium-titanium.
- The inner section has a (stainless steel) external binder on the outside of the coil to take up some of the electromagnetic stress.
An interesting question is whether the external binder was effective, and that is the subject of the analysis here.
Main results in the Markiewicz et al. paper
Markiewicz et al. state that peak hoop stress in the Nb3Sn wire is around 200 MPa (Figure 2 in the paper, where it is labelled 'tangential stress').
Their analysis found that hoop stress in the external binder is much higher at around 330 MPa. Figure 3 shows maximum hoop strain in the wire was about 3.1 millistrains (3.1e-03). This is around the level at which niobium-tin short-sample wire performance is maximised [2], owing to the relief of the compressive pre-strain that results from the high-temperature reaction process.
The NHMFL design may thus be optimised in this respect. Conditions may have been adjusted by the designers to take maximum advantage of the performance of the expensive niobium-tin wire.
Assumptions for the CalculiX FEA finite element analysis model
For the calculations outlined here, only electromagnetic forces are considered, and thermal contraction effects have been ignored. No information about winding tension is given in the paper so this also must be neglected. For comparison, models were solved using two packages, CalculiX FEA and LISA FEA. Only a brief outline summary of the CalculiX results for the tin section (coils 1a and 1b) will be given here.
Binder (TEST362) and no-binder (TEST363) model comparison
Two models were constructed, one with the reinforcing binder (TEST362), and one without (TEST363), to investigate the effectiveness of fitting the binder.
In these models, the coils, and in TEST362, the binder, were represented as a regular array of eight-node quadrilateral elements (CalculiX type 'CAX8').
A table of Lorentz force per unit volume (axial and radial components) was computed for the elements. The forces on each element were split between nodes by 'mass lumping' [3] and the contributions (from up to four elements) summed for each node. Next the results of the mass lumping process were built into a CalculiX 'input deck' (a .inp file), along with node coordinates, node numbers for each element, materials properties etc. The orthotropic material properties given in the Markiwewicz et al paper can be specified using the CalculiX 'card' *ELASTIC,TYPE=ENGINEERING CONSTANTS.
The work of constructing the CalculiX FEA input deck was automated using Fortran-77 and an R script.
The next stage was to solve the model using the CalculiX ccx command. (A point of interest here is that, behind the scenes, CalculiX converts the eight-node quadrilateral elements into 20-node brick elements with a circumferential thickness of 2 degrees.) After solving the model, the results then require post-processing.
R graphics for model results
The CalculiX graphical post-postprocessor cgx produces nice graphics but these are of fixed dimensions (width and height) and may be too big to place on a webpage. Some post-processing of the CalculiX results was therefore also done in R to produce colour-coded plots using R's grid package [4]. Part of this process involved extracting some of the results tables from the CalculiX .frd and .dat output files.
Main results of models TEST362, TEST363
The results of the main model TEST362, shown in the table below, are in reasonable agreement with those in the paper, though they are not identical.
The differences may reflect differences in model assumptions. For example, the binder is assumed here to be fixed in the axial and radial directions relative to the outer surface of the underlying niobium-tin coil. A better model might be to allow the two surfaces to slide relative to each other, but this would entail introducing contact into the model and would be a significant complication.
| field | units | TEST362 | TEST363 | ||||
|---|---|---|---|---|---|---|---|
| in binder | in coils | in coils | |||||
| min. | max. | min. | max. | min. | max. | ||
| sxx (radial stress) | MPa | -10.0 | 2.9 | -14.8 | 4.8 | 0.1 | 11.8 |
| syy (axial stress) | MPa | -200.1 | 7.2 | -37.2 | 2.7 | -68.2 | -0.2 |
| szz (hoop stress) | MPa | 141.0 | 287.9 | 47.1 | 215.8 | 86.5 | 254.9 |
| sxy (shear stress) | MPa | -2.5 | 8.5 | -1.6 | 9.0 | -2.3 | 0.3 |
| svm (von Mises stress) | MPa | 138.5 | 426.1 | 51.6 | 218.3 | 86.7 | 274.1 |
| EXX (radial strain) | (none) | -0.0005 | -0.0001 | -0.0018 | -0.0005 | -0.0019 | -0.0007 |
| EYY (axial strain) | (none) | -0.0015 | -0.0002 | -0.0017 | -0.0003 | -0.0026 | -0.0006 |
| EZZ (hoop strain) | (none) | 0.0007 | 0.0018 | 0.0008 | 0.0033 | 0.0013 | 0.0041 |
Hoop and axial stress and strain
.png)
For peak hoop stress and hoop strain, finite element analysis model TEST362 and the Markiewicz et al paper are in fair agreement. (Hoop stress: model TEST362: 215.8 MPa; Markiewicz et al 1994: ~205 MPa. Hoop strain: model TEST362: 0.0033; paper: ~0.0031). The overall pattern of radial stress and strain on the magnet centreplane is also similar.
Agreement is not so good for computed axial stress. Model TEST362 shows a peak of -200 MPa in the binder, whereas the paper gives a peak of -150 MPa. The high computed axial stress is a significant contributor to the high level of von Mises stress in the binder (426 MPa). (See the adjacent plot, where svm is an abbreviation for von Mises stress; the (z, r) scale is in metres.) Peak von Mises stress in the adjacent outer layers of the coil is much lower at around 140 MPa.
Shear stress at coil-binder interface
.png)
In one region near the end of the coil, shear stress sxy near the coil-binder interface is quite high (9.0 MPa). Just possibly, this may be high enough to cause local slippage between the coil and external binder, if the two were not bonded together in any way.
If slippage did occur, local frictional heating might be sufficient to initiate quenches in this region, although this is a low-field, and thus relatively stable, part of the coil. Shear stress near either side of the centreplane is negligible, even though there is a very sharp discontinuity in axial stress between coil and binder in this region.
For superconducting magnet designs, peak magnet von Mises stress may be a good indicator of overall design viability.
Did the external binder work well?
Comparing models TEST362 and TEST363, several points indicate that the external binder was very successful and served its purpose well.
- The binder has reduced peak hoop strain in the coil from 4.1 millistrains to 3.3 millistrains. Thus the tin section operates under approximately 'optimal strain' conditions [2].
- Peak von Mises stress in the coil falls from 274 MPa (without binder) to 218 MPa (with binder). Even the resulting level with the binder would be quite high for, say, an MRI magnet.
- Hoop stress falls from 254.9 MPa (without binder) to 215.8 MPa (with binder).
- Peak axial stress in the coil has almost halved (without binder: -68.2 MPa; with binder: -37.2 MPa).
CalculiX-FEA input decks
You can run this problem if you have CalculiX FEA installed.
Archive files below contain CalculiX FEA input decks (.INP files) for the two cases (TEST362, TEST363).
The files can be downloaded as either tar or zip archives, as below:
- FEA-NHFML.tar (for Unix, Linux etc; tar is also available for Windows)
- FEA-NHFML.zip (for Windows)
References
[1] Markiewicz, W.D., Vaghar, M.R., Dixon, I.R. & Garmestani, H. 1994. IEEE Transactions on Magnetics, 30(4), 2233-2236.
[2] Cheggour, N. & Hampshire, D.P. 1999. IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY 9(2), 2517-2520.
[3] Zienkiewicz, O.C., Taylor, R.L. & Zhu, J.Z. 2013. The Finite Element Method: Its Basis and Fundamentals. Elsevier, Amsterdam.
[4] Murrell, P. 2011. R Graphics Second Edition. CRC Press, Boca Raton.