The ROXIE program developed at CERN for the design and optimization of the superconducting LHC magnets has been recently extended in a collaboration with the University of Stuttgart, Germany, with a field computation method based on the coupling between the boundary element (BEM) and the finite element (FEM) technique. This avoids the meshing of the coils and the air regions, and avoids the artificial far field boundary conditions. The method is therefore specially suited for the accurate calculation of fields in the superconducting magnets in which the field is dominated by the coil. We will present the fringe field calculations in both 2d and 3d geometries to evaluate the effect of connections and the cryostat on the field quality and the flux density to which auxiliary bus-bars are exposed

Kurz, S

Russenschuck, Stephan

Siegel, N

CERN Document Server

IEBE TRrWSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 10, NO. 1, MARCH 2000 85 Accurate Calculation of Fringe Fields in the LHC Main Dipoles S. Kurz ITE, University of Stuttgart, Germany S, Russenschuck, N. Siege1 CERN, Geneva, Switzerland Abstmct- The ROXIE program developed at CERN for the design and Optimization of the superconducting LHC magnets has been recently extended in a collaboration with the University of Stiittgart, Germany, with a Add compu- tation method based on the coupling between t h e  boundary element (BEM) and tho flnite element (PEM) technique. This avoids the meshing of the coils and the air regions, and avoids the artlficial far field boundary conditions. The method is therefore specially suited for the accurate calcu- lation of fields in tho superconducting magnets in whlch the field la dominated by the coil. We will present the fringe field calculations in both ad and 3d geometries to evaluate the effect of connections and the cryostat on tho Rald quality nnd the flux density to which auxliiary bus-bars are exposed. I. INTRODUCTION The design and optimization of the LHC magnets is gov- erned by the requirement of an extremely uniform field which is mainly defined by the layout of the superconduct- ing coils. Even very small geometrical effects such as thc keystoning of the cable, the insulation, grading of the cur- rent density in the cable due to  different cable compaction and coil deformations due to  collaring, cool down and elec- tromagnetic forces have to  be considered for the field calcu- lation. For the field optimization of the LHC magnets [I], the usefulness of commercial software has shown to be lim- ited in particular in the three dimensional case, Therefore the ROXIE [Z] program package was dcveloped at CERN for the design and optimization of the LHC superconduct- ing magnets. Furthermore, the application of the BEM- FEM coupling method [3] yiolds the reduced field due to  the magnetization of the iron yoke only. This avoids the meshing of thc coils and the air regions, and avoids the ar- tificial far field boundary conditions. T h e  method is there- fore specially suited for the calculation of fringe fields in the superconducting magnets, both in 2 and 3 dimensions. 11. T H E  BEM-FEM COUPLING METHOD The total magnetic induction in a certain point f i n  the npert2re of the magnet can be decomposed into a contribu- tion Bs due to the superconducting coil and a contribution BR duc to the magnetic yoke. If the field: are extressed in terms of the magnetic vector potential B = curl A ,  then the decomposition into sourcc and reduced contributions gives A= i s  + A R ,  This approach has the following intrinsic advantages: Manuscript received September 28, 1999 The coil field can be t2ken into account in terms of its sourcc vector potentid As ,  which can be obtained easily from thc filamentary currents Is by means of Biot-Savart type integrals without meshing of the coil. 4 The BEM-FEM coupling method allow: a direct compu- tation of the reduced vector potential AR instead of the total vector potential A’. Consequently, numerital errors do not influence the dominating contribution As due to the superconducting coil. Since the field in the apcrture is calculated through the integration over all the BEM elements, local field errors in the iron yoke cancel out and the calculated multipole content is sufficiently accurate even for a very sparso mesh. The surrounding air reiion need not be meshed at  all. This simplifies the preprocessing and avoids artifi- cial boundary conditions at some ”far” distance. Since the air region is not meshed, only the iron region mesh- ing is affeckd by geometrical modifications. This strongly supports the feature based, parametric geomctry modelling which is required for mathematical optimization. The elementary model problem for a single aperture model dipole (featuring both Dirichlet and Neumann bounds on the iron yolw) is shown in Fig. 1. When the Fig. 1. Elementary model problem far the numerical ficld calcula, tlon of a superconducting (single aperture) modcl magnet. In the iron domain the total vector potential is displayed. The non- conductive air region n, contains a certain number of conductor sources which do not intersect the iron region f l i t  The finite- element method inslde the magnetic body RI  = RFBM is coupled with the boundary-element mcthod in the domain outside the magnetic material Ra = ~ B E M ,  by means of thc normal deriva- tive of tho vector-potential on the i n t o r h e  rai = rBEMFEM between iron and air. 1051+8223/00$10.00 Q 2000 IEEE 86 BEM-FEM coupling method is applied, only the magnetic sub-domain Q which coincides with the magnetic yoke has to be meshed by finite elements. Iron saturation effects can then be treated within the FEM domain. The non- magnetic sub-domain s1, which represents the surrounding air region and the excitation coiI is treated by the bound- ary element method. Only the common boundary rai needs to  be discretized by boundary elements. If the domain is discretixed into finite elements (Co-continuous, isopara- metric 20-nodod hexahedron elements are used), and the Galerkin method is applied to the weak formulation, then a non linear system of equations is obtained with all nodal values of Anl Ay,, and Qrnl grouped in arrays. As we will see later, Qral i s  the normal derivative of A on rRi (3) which w e  assume as given. The subscripts rai and R, refer to nodes on tho boundary and in the interior of the domain, respectively. The domain and boundary integrals in the weak formulation yield the stiffness matrices [IC] and the boundary matrix [TI. The stiffness matrices depend on the local permeability distribution in tho nonlinear material. All the matrices in (2) are sparse. For the meshing of the boundary FRi with boundary elements Fa,,j, C'-continuous, isoparametric 8-noded quadrilateral boundary elements are used. The discrete analogue of the Fkedholm integral equa- tion can be obtained by successively putting the evaluBtion point r'j at thc location of each nodal point T;. This pro- cedure is cslled poinbwise collocation and yields a linear system of equations, [GI {Oral} + [T-II{A;.,~I = (4) In (4), {As} contains the values of the source vector poten- tial at  the nodal points F3, j = 1,2, I . . . The matrices [GI and [HI are unsymmatric and fully populated. An overall numerical description of the field problem can be obtained by complementing the FEM description (2) by the BEM description (4) resulting in 0 (5) Equation (4) gives exactly the missing relationship between the Dirichlet data {&A,} and the Neumann data {Qrhl} on the boundary ra;. It can be shown [4] that this pra- cedure yields the correct physical interface conditions, the continuity of 6 . B and ii x H across rai. A detailed description of the method and the applicatiori to superconducting magnets can be found in [4], [ 5 ] .  111. FIELD QUALITY IN ACCELERATOR MAGNETS The magnetic field errors in the aperture of the magnets are expressed as the coefficients of the Fourier-series ox- pnnsion of' the radial field component a t  D. ,given reference radius (in the 2-dimensional case). In the 3-dimensional case, the transverse field components are given at Q longi- tudinal position LO or integrated over the entire length of the magnet. For a given radial component of the magnetic flux density B, at a reference radius T = ro inside the aperture of a magnet the Fourier-series expansion of the field reads M B T I T O I P )  = C(B,(.ru)sinntp+A,Iro)cosn~), ( 6 )  ~,(m,cp) = B ~ ( T O )  C ( b n ( . r o ) s i n w  + ~ , ( r O ) c o s w ) ,  (7) n=l  E4 n=l  with the main field component B n  ( N  = 1 dipole, N = 2 .quadrupole, etc.). The B, are called the normal and the A ,  the skew components o f  the field given in Tesla, b, the normal relative, and a, the skew relativc field corn- ponents. They are dimensionless and are usually given in units of at a 17 mm reference radius. In prac- tice the D, components arc calculated, in discrete points pk = % - T ,  IC = 0 , 1 , 2 , . . ~ ~ - 1  in the interval [ - x , K )  and a discrete Fourier transform is carried out: (81 (9) 1 2p-1 1 2 p - 1  & ( T O )  M ~ ( w , P ~ ) c o ~ ~ ( P ~ E ,  Bnho) = Br(nh'Pk) s i n w k .  k=O k=O The interpolation-error depends on the number of cvalua- tion points and the amount nf higher order multipolr? crrors in the field. For the calculation of miltipoles up to  the or- der n = 13, 79 evaluation points (P = 40) are sufficient. , IV. RiwuL'i's A .  Field Quality in Collared Coils Space limitations in the existing' LEP tunnel and eco- nomical considerations have dictated a so-called two-in- one magnet design with two scts of coils and beam chan- nels within a common mechanical structure, iron yokc, and cryostat. As a first example of the application of the BEM- FEM method we calculate the magnetic field in a configu- ration as shown in Fig. 2. Using double aperture stainless- steel collars with a relative permeability of fir = 1.0025 creates asymmetries in the magnetic field in the case of warm measurements of the collared coil assembly where only one aperture is powered. The relative field errors at 17 mm reference radius in the aperture are given in Table 1. As w&s mentioned earlier tht? BEM-FEM method does not require the meshing of the coil (which can therefore be modelled with the required accuracy) and does not require a "far field" boundary condition which would influence con- siderably the results of this unbounded field problem. At the position of the support posts of the cold mass n circular opening is made in the cryostat vacuum vessel, closed by a non-magnetic plate. As a first approach, this opening has been modelled as a two-dimensional problem. The fringe field is higher at the opening but stays below 0 . 5 .  IOW3 Tesla, see Fig. 4,  Again, the advantage of the BEM-FEM method lay8 in the fact that the surrounding air regions do not have to be considered and that the dis- connected iron parts can bc meshed independently with the required refinement. The eccentricity of the cryostat can therefore easily be modelled. Tablo 2 gives the nomi- nal, and the additional field errors due to the cryost,at at nominal field of 8.36 T in units of at 17 mm. Fig. 2. Geometric model of one dipole coil power4 for worm mea, surement in the combined collar structure asuming a constant magnetic Rux density in the collnrs. - -. - - .. . -. x i a 4  relative perme&ility of fir = L0025. The figure displays the 4.0 7w 0 0.4 - TABLE I 0.P I . B 1  - ADDITIONAL FIELD ERRORS IN COIL COLLAR ASSEMOLY WITH RIGHT-HAND-SIDE hPERTURE POWERED FOR WARM MEASUREMENTS. (UNITS OF AT 17 MM.) AND KOMINAL VALUES FOR A BARE COIL WITHOUT COLLAR. ha 3.9150 b7 0.7456 b3 -1.6326 b y  -0.0918 9L Nominal Additional bz 0.0000 b~ 0.0000 I b2 -0.0737 bfi 0,0000 bq 0.0000 ba 0,0000 bg -1.0385 bg 0,1224 b4 -0.0018 be 0.0000 a w i o0  160 aw ZY suo am 65 0.4306 bg 0.0081 Awutar ParlUanIdWl Fig. 3. Geometric model of the two-dimensional cross-section of the  intagnet showing iron yokc and cryostat vacuum vessel. a1 0.000 B2 0.G46 a2 0.000 b~ 5.775 a3 0.000 b4 0.217 a4 0.000 b g  -0.914 US 0.000 B. Influence of the Cryostat Eccentricity The cold mass of the two-in-one magnet is placed in the vacuum vessel of the cryostat (made of magnetic steel) with an off-centering of 80 mm in the vertical plane, see Fig. 3. 0.422 ba 0.099 a2 -0.093 b3 0.015 a3 0.016 ba 0.005 (t4 -0.002 b g  0.000 US 0.000 TABLE I1 ADDITIONAL FIELD EnIIORS DUE TO ASYMMETRY OF THE VACUUM VESSEL AT NOhIINAL FIELD I N  UNITS OP AT 17 M M  AND NOMINAL ERRORS FOR A COLD-MASS WITHOUT VACUUM VESSEL. Nominal Additional C. Fringe Field in the GoiGend Region In order to reduce the peak field in the coil-end and thus increase the quench margin in the region with a weaker mechanical structure, tho magnetic iron yoke ends approx- imately SO mm from the onset of the coil-end at thc non- connection side and about 300 mm from the onset of thc coil-end at the connection end. This asymmetry results from the aim of a further reduction of the field in the ramp and splice region. The BEM-FEM coupling method is used for thc calculation of the end-fields. In particular in 3D, the importance for not having to mesh the coil becomcs obvious from Fig. 5. Tho ends can be modelled with the required 88 accuracy as the source field can be calculated directly by means of Biot-Savart’s law. The two-hone magnet end configuration with the common yoke used for the computa, Lion is shown in Fig. 6. Fig. 7 shows the field components Y l P  f Fig. 5 .  Geometric madcl of the dipole coil-end with local coordinate system. Y Pig. 6. Full 3D modal of coils and two-hone iron yoke. The location of the buebar is at x = 285 mm and y = -67 mm. along a line in the end-region of the twin-aperture dipale prototype magnet (MBPP) from z = - 500 mm (inside the magnet yoke) to 2 = 400 mm outside the yoke. The onset of the coil-ends are at z = 0. On the non-connection side the iron yoke ends at z = -80 mm, on the connection side the iron yoke ends at z = -300 mm. This calculation is im- portant far the evaluation of the farces acting on the 600 A and 6 kA auxiliary bus bars. V. CONCLUSION The BEM-FEM method is specially well suited for the calculation of fringe fields in superconducting magnets, as the coils and the air ‘regions do not have to be represented in the finite-element mesh, discretization errors only influence the calculation of the yoke magnetization and there is not an artificial ‘Lfar field” boundary condition. The method lias been illustrated by three examples, the results of which Ofl6 0 4.w 4.1 FIg. 7. Fringe field outside the dipole cold m w  near the coil-end region, Top: connection end, Bottom: return end. can be summarized aa follows: 1) For warm measurements, an additional field error due to the stainless-steel collar of about -1.6 units in the relative b3 component has to  be considered. 2) The fringe field outside the cryostat btays below 5 I IOp4 T and 3) the  field seen hy the 600 A and 6 kA auxiliary bus bars stays below 0.21 T in the coil-end region. REFERENCES [I] The LHC study group. LBC, The Large Hadron Collider, Con- ceptuhl Design. CEnN/AC/96-05 (LHC), 19915. [Z] Editor: Stephan Russenachuck. RUX1B: Routine for the Opti- mization of magnet X-sections, tnverae field calculation and coil End deaign. Yellow Report, CERN s9-ol(tSBN=92-9o83-140-~), 1999. (3) J. Fetzer, S. Rbele, and G. Lehner. Die Kopplung der Rand- slementmethode und der Methoda der finiten Elemente zur Losung dreidimenaionalsr elektromagnetischer Feldprobleme auf unendlichem Grundgebiet. A r c h  Jur Elektrotechndk, 76(5):361- 368, 1993. 141 S. KurZ;, J. Fetzer, and W. Rucker. Coupled REM-FEM methods for 3D field calculations with iron saturation. Proceedings of the First International ROXlE users meeting and workshop, 1999. 151 S. Kucz and S. Ruesenschudt. The application of the BEM-PEM coupling method far the accurate calculation of fiolds in super- conducting magnets. Accepted for publication in Electrical Engi- neering. 

Accurate Calculation of Fringe Fields in the LHC Main Dipoles

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