A semi-analytical method to evaluate the effect of coil de-formations on the field-shape imperfections of the LHC dipole is presented. The deformation induced by the collaring procedure and by the thermal stresses is evaluated numerically with a finite element code. The vector field of mechanical displacements is approximated with truncated Taylor and Fourier series. The fitting function agrees with the numerical data to within less that 10 mm. The decom-position in modes of the truncated series permits identification of displacements which are dangerous for the multi-polar content and how they could be cured. An application to compare two designs of the LHC dipole is given

Ferracin, P

Scandale, Walter

Todesco, Ezio

Tropea, P

CERN Document Server

A METHOD TO EVALUATE THE FIELD-SHAPE MULTIPOLESINDUCED BY COIL DEFORMATIONSP. Ferracin, W. Scandale∗, E. Todesco, P. Tropea, CERN - LHC Division, Geneva, SwitzerlandAbstractA semi-analytical method to evaluate the effect of coil de-formations on the field-shape imperfections of the LHCdipole is presented. The deformation induced by the col-laring procedure and by the thermal stresses is evaluatednumerically with a finite element code. The vector field ofmechanical displacements is approximated with truncatedTaylor and Fourier series. The fitting function agrees withthe numerical data to within less that 10 µm. The decom-position in modes of the truncated series permits identifi-cation of displacements which are dangerous for the multi-polar content and how they could be cured. An applicationto compare two designs of the LHC dipole is given.1 THE LHC DIPOLE MODELIn this paper we deal with the field quality evaluation inthe LHC main superconducting dipoles [1]. The magnetcross section should optimize field quality to obtain a suffi-ciently constant magnetic field to provide a wide dynamicaperture for the beam [2]. Indeed, the nominal design maybe difficult to reach under realistic conditions, and coils canbe deformed by either manufacturing forces or tolerancesin the mechanical parts (see for instance [3]). In this paperwe analyse the effect of deformations due to manufacturingand we develop a formalism to work out the dependence ofthe multipolar coefficients of the field expansion on the de-formations.Mechanical models have been already used to evaluatethe displacement of the LHC dipole coils induced by manu-facturing and thermal effects. The present model considersone quarter of the two-in-one dipole, i.e. half aperture, withan up-down symmetry that allows only normal multipoles,and a left-right symmetry that imposes the same multipoleson both apertures. Computations are carried out at 1.9 K ,and at the injection energy where the problem of the fieldquality is more relevant. The coil prestress in the midplane,at 1.9 K is fixed at 40 MPa.The finite element code ANSYS [4] provides the field ofdeformations induced in the superconducting coil. A For-tran code (ANSIA [5]) produces displaced coordinates foreach conductor in a format compatible with ROXIE [6].The latter evaluates the magnetic field as a multipolar ex-pansion. The model used to carry out the magnetic com-putations only considers the effect of deformations on thenominal geometry; the contribution due to iron deforma-tions can be assumed to be negligible.The mechanical model includes the nonlinear elasticityof the coil [7, 8] as obtained from experimental data [9].∗Email: walter.scandale@cern.chParticular care was taken in preparing the representationof the elastic modulus of the copper wedges and insula-tion, that are considered a unique material in the model.Our computations show that the multipoles exhibit a strongdependence on mechanical deformations. Experiments toverify our results are in preparation.2 TOOLS FOR THE DEPENDENCE OFMULTIPOLES ON DEFORMATIONSThe deformation is a field of 2D vectors that is evaluatedthrough the finite element code ANSYS on a finite set ofnodes in a 2D space (the magnet cross section). Since thisvector field is rather smooth, we propose to parametrize thedeformations by interpolating them with low-order Taylorand Fourier series. In this way all the information relativeto the deformations can be reduced to a few coefficients,thus simplifying the analysis of the relation between de-formations and multipoles. We numerically estimate theinfluence of single deformation modes on the multipoles,thus identifying the most dangerous.Since deformations are implemented in ROXIE as shiftsin the position of the inner part and of the outer part ofthe conductor, we consider a field of deformation definedover four radii (r1, r2, r3, r4)=(28, 43.4, 43.9, 59.3) mmthat surround the inner and the outer layers of the coils; thefield then depends on the discrete variable ri and on thecontinuous angular variable φ. Deformations are then de-composed into a radial dr(ri, φ) and a tangential dp(ri, φ)component.Radial deformations are well interpolated by the sum ofthe first orders of an even Fourier series in the azimuthalvariable φdr(ri, φ) =K∑k=0ak,icos(kφ) i = 1, 4.Here the index i runs over the four concentric circles thatsurround inner and outer coils, and k runs over the modes.The mode zero a0,i gives rise to a contraction or to a dilata-tion of the coil; the mode one a1,i produces an horizontalshift in the coil, whilst higer order modes correspond toradial waves of deformations with decreasing period. Theup-down symmetry is responsible of the absence of sineterms in the Fourier series.Tangential deformations are well interpolated by low or-der odd Taylor series in each half of the aperturedφ(ri, φ) =J∑j=0bj,iφ2j+1 i = 1, 4 φ 2 [−pi/2, pi/2]0-7803-5573-3/99/$10.00@1999 IEEE. 3206Proceedings of the 1999 Particle Accelerator Conference, New York, 1999dφ(ri, φ) =J∑j=0cj,iφ2j+1 i = 1, 4 φ 2 [pi/2, 3pi/2]The mode zero b0,i and c0,i corresponds to tangential com-pression or dilatation, uniform along the azimuth. Higherorder modes correspond to tangential waves of compres-sions and dilatations. Also in this case, the up-down sym-metry implies the presence of odd terms only.Figure 1: Plot of radial deformations for each of the fourcoil contours, stainless steel collars. Dots: ANSYS values.Lines: interpolation.In Figs. 1 and 2 we show the radial and tangential defor-mations evaluated through ANSYS (dots) and the interpo-lation (solid line) with seven radial modes and two tangen-tial modes (K=6 and J=1). If the displacements evaluatedthrough ANSYS are replaced by the interpolating field, theresulting multipoles agree with the ANSYS results (see Ta-ble 1, last two rows). This means that low order Taylor andFourier series can take into account the part of the defor-mations that are relevant to evaluate the multipoles.The contribution of each mode to the multipoles is givenin Table 1. For symmetry reasons, even radial modes onlycontribute to odd multipoles, and odd radial modes to evenmultipoles. One can distinguish two main effects. Radial deformation of order 2 (i.e., circles are de-Figure 2: Plot of tangent deformations for each of the fourcoil contours, stainless steel collars. Dots: ANSYS values.Lines: interpolation.Figure 3: Plot of radial deformations, mode 2, in carte-sian coordinates. Undeformed circle (solid line) versus de-formed circle (dotted line).3207Proceedings of the 1999 Particle Accelerator Conference, New York, 1999Table 1: Contribution of radial and tangential modes to themultipoles shift induced by deformation, total effect of 7radial and 2 tangential modes, and comparison with AN-SYS results. Multipole shifts are given in units of 10−4 atRref =17 mm.b2 b3 b5 b7radialk=0 0.00 0.11 -0.06 0.01k=1 0.05 0.00 0.00 0.00k=2 0.00 -2.98 0.18 -0.04k=3 0.08 0.00 0.00 0.00k=4 0.00 0.53 0.69 0.01k=5 0.00 0.00 0.00 0.00k=6 0.00 0.02 -0.05 -0.07tang.j=0 0.04 0.17 -0.05 0.02j=1 -0.01 0.03 -0.01 0.00total 0.16 -2.12 0.70 -0.07ANSYS 0.16 -2.19 0.59 -0.08formed into ellipses, see fig. 3) have a strong effecton the sextupole ( −3 units) and on the decapole( 0.2 units). Radial deformation of order 4 gives rise to 0.5 units ofb3 and 0.7 units of b5.Simulations have also shown that the radial deformationsare rather well decoupled with the tangential one. In Ta-ble 1 the effect of the tangential modes seems to be verylimited (only some contribution to b3): in our computationswe fixed the coil length to an optimal value that allows torecover the nominal length under deformations, at 1.9 K .Indeed, a non optimized coil length produces tangential de-formations that strongly affects the field multipoles. Tangent deformations due to a coil length differentfrom the nominal one have a large influence on b3, b5and b7. Simulations show that a 0.1 mm longer coilgives rise to around −3 units of sextupole, and 0.5 ofdecapole.Therefore, the determination of the exact coil length un-der manufacturing forces and at cold is crucial to obtaina correct estimate of the absolute value of the multipoles.Other contributions of the deformations are not very rele-vant, giving rise to multipole shifts of less than 0.1 units.In particular, the following effects are negligible: The effect of the first order (see Table 1, mode j=1)tangential mode (tangetial waves fo compression anddilatation) is negligible. The radial mode of order zero (see Table 1, modek=0), i.e. the radial shrinkage due to low temperature,is negligible. Odd radial modes (see Table 1, mode k=1,3,5), i.e.modes related to left-right asymmetries, are very low.Therefore, deformations have negligible effect oneven multipoles.3 CONCLUSIONSWe have analysed the variation of the field quality in thesuperconducting LHC dipoles due to deformations of thecoils with respect to the nominal design.We propose a method, based on interpolation, to reducethe deformation field to a few coefficients of radial and tan-gential modes. This leads to a reduction in the complex-ity of the problem and to relate modes to multipoles. Weare able to identify which modes are more dangerous andwhich are negligible. We showed that the effect of defor-mation on multipoles is relevant, and that the main contri-butions come from radial deformations of order two andfour (see for instance fig. 3). Moreover, a systematic differ-ence in the coil length between the nominal and the real one(i.e., at 1.8 K and under deformations) leads to additionalmultipole shifts. These effects are uncoupled.We wish to acknowledge A. Desirelli, P. Fessia, D.Leroy, M. Modena, D. Perini, Prof. Rebora, G. Spigo andR. Vuillermet for their help in the definition and in the un-derstanding of the mechanical model. We also thank S.Russenschuck for providing the ROXIE code and the filesof the magnetic model.4 REFERENCES[1] The LHC study group, CERN 95-05 (1995).[2] C. Wyss et al., CERN LHC-MMS 98-198 IT 2325 (1998).[3] R. Bartolini, P. Ferracin, P. Fessia, W. Scandale, E. Tode-sco, ’, in Sixth European Accelerator Conference, editedby C. Petit-Jean Genaz (IoP, New York, 1998) pp. in press.[4] ANSYS 5.4, licensed by and trademark of Swanson Anal-ysis Inc., Houston, PA, USA.[5] E. Todesco, private communication.[6] S. Russenschuck, LHC Project Report 159 (1997).[7] W. Scandale, E. Todesco, P. Tropea, in preparation.[8] P. Tropea, A. Rebora, E. Todesco, CERN LHC-MMS In-ternal Note 99-04 (1999).[9] D. Tommasini, S. Russenschuck, L. Bottura, in: Proceed-ings of the first international conference of Roxie users,CERN Yellow Report, in press.3208Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

A Method to Evaluate the Field-Shape Multipoles Induced by Coil Deformations

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