An eight-pole superconducting magnet is being developed for soft x-ray magnetic dichroism (XMD) experiments at the Advanced Light Source, Lawrence Berkley National Laboratory (LBNL). Eight conical Nb{sub 3}Sn coils with Holmium poles are arranged in octahedral symmetry to form four dipole pairs that provide magnetic fields of up to 5 T in any direction relative to the incoming x-ray beam. The dimensions of the magnet yoke as well as pole taper, diameter, and length were optimized for maximum peak field in the magnet center using the software package TOSCA. The structural analysis of the magnet is performed using ANSYS with the coil properties derived using a numerical homogenization scheme. It is found that the use of orthotropic material properties for the coil has an important influence in the design of the magnet

Arbelaez, D.

Arenholz, E.

Black, A.

Chen, J.

Prestemon, S.O.

Wang, S.

UNT Digital Library

REPLACE WITH PAPER IDENTIFICATION NUMBER 1Mechanical design and analysis of an eight-polesuperconducting vector magnet for soft x-raymagnetic dichroism measurementsD. Arbelaez, A. Black, S.O. Prestemon, S. Wang, J. Chen, and E. Arenholz,Senior Member, IEEEAbstract—An eight-pole superconducting magnet is being de-veloped for soft x-ray magnetic dichroism (XMD) experimentsat the Advanced Light Source, Lawrence Berkley NationalLaboratory (LBNL). Eight conical Nb 3Sn coils with Holmiumpoles are arranged in octahedral symmetry to form four dipolepairs that provide magnetic fields of up to 5 T in any directionrelative to the incoming x-ray beam. The dimensions of themagnet yoke as well as pole taper, diameter, and length wereoptimized for maximum peak field in the magnet center using thesoftware package TOSCA. The structural analysis of the magnetis performed using ANSYS with the coil properties derived usinga numerical homogenization scheme. It is found that the use oforthotropic material properties for the coil has an importantinfluence in the design of the magnet.Index Terms—soft x-rays, magnetic dichroism, superconduct-ing magnet.I. I NTRODUCTIONI N order to take full advantage of soft x-ray magneticdichroism (XMD) measurements, magnetic fields that canbe oriented in any direction relative to the incoming x-raybeam in an x-ray absorption experiment are required. A resis-tive eight pole magnet that is currently installed at beamline4.0.2 of the Advanced Light Source (ALS), Lawrence BerkleyNational Laboratory (LBNL), has been used to demonstratethat the combination of x-ray magnetic circular and lineardichroism measurements as well as the XMD angular de-pendence provide information not accessible by other means[1]. While this device has been successfully used to studymany materials, its peak magnetic field of 0.9 T is limitingthe systems that can be investigated. In order to align themagnetic moments along the hard magnetization direction inmany oxide hetero- and nanostructures, magnetic fields near5 T are required. An eight-pole superconducting magnet isbeing developed at LBNL and Wang NMR Inc. to overcomethe limitations of the resitive magnet device. The plannedsuperconducting magnet is composed of eight conical Nb3Sncoils with Holmium poles that are arranged in an octahedralpattern to form four dipole pairs. Figure 1 shows a CAD modelof the magnets central set up including the internal supportstructure, the Nb3Sn coils, the poles, and the magnet yokes.The internal monolithic structure contains eight pockets tosupport the Nb3Sn coils. A Holmium pole is inserted into eachManuscript received 3 August 2010. This work was supported by ...D. Arbelaez, A. Black, S. Prestemon and E. Arenholz, are withLawrenceBerkeley National Laboratory, Berkeley, CA, 94720 USA (e-mail: darbe-laez@lbl.gov).S. Wang and J. Chen are with Wang NMR Inc.coil. The magnet yoke consists of eight parts that are attachedto the Ho poles . These pieces are bolted onto the monolithiccentral magnet structure. Figure 2 shows a cross-section ofthemagnet which contains a six-way cross that allows access forthe sample, the x-ray beam, as well as detectors.Fig. 1. CAD model of the magnet’s central structure including the internalmonolithic support piece, the Nb3Sn coils, the Holmium poles, and the yokepieces.Fig. 2. Cross-section of the magnet showing the six-way cross thatallows access to the sample from different directions for magnetic dichroismmeasurements.In this paper the structural analysis of the magnet is dis-cussed. The mechanical analysis of the magnet is performedwith ANSYS with applied magnetic forces that are computedusing TOSCA. The effective material properties for the wirewound coils are computed with a computational homogeniza-tion scheme. In [2], a similar scheme was used for coilsREPLACE WITH PAPER IDENTIFICATION NUMBER 2composed of Rutherford cables. Using the computed proper-ties, analytical and computational models for the deformationand stress state in the coil are presented. The effect of theorthotropic properties of the coil on the contact state betwe nthe coil and the support structure and poles is also discussed.II. EFFECTIVE COIL PROPERTIESIn this section, a computational homogenization schemethat is used to determine the effective properties of the “coilmaterial” is presented. Figure 3 shows an example of acalculation over the periodic unit cell that is used to determinethe effective properties. The unit cell includes the strand(inner circle), the strand insulation (outer ring surroundingthe strand), and the interstitial epoxy. The “coil material” sFig. 3. Example of a unit cell calculation for the coil material. The unit cellincludes the strand (inner circle), the strand insulation (uter ring), and theinterstitial epoxy. The colors indicate different strain states.created by superimposing unit cells in an infinite space, whichis expected to be a good approximation for a coil with alarge number of windings. For this model, periodic boundaryconditions are appropriate. The periodic boundary conditionis described byu+ − u− = E(X+ − X−), (1)whereu+ andu− represent the displacements of two pointsdirectly opposite to one another on the periodic surfaces,X+andX− are their corresponding positions, andE is the averagestrain imposed over the unit cell. In order to compute the effc-tive properties, six different appropriate loading conditions canbe applied to determine the effective elasticity tensor. Writingthe stress strain linear elastic relation in engineering notation:σxσyσzτxyτyzτzx=C11 C12 . . . C16C12 C22 C26C13 C23 C36C14 C24 C46C15 C25 C56C16 C26 . . . C66εxεyεzγxyγyzγzx(2)This equations shows that if only one non-zero value of thestrain components is chosen, then the corresponding columnof the elasticity matrix can be determined by computingthe stress tensor for that applied strain state. To extract theeffective properties of the unit cell, six different average strainstates (imposed byE in equation 1) over the defined domainare applied, and the corresponding average stress states arecalculated. For example, to compute the first column of theaverage elasticity matrix, a value of〈εx〉Ω is imposed withthe other strain components set to zero and the average stress〈σ〉Ω is computed. The operator〈·〉Ω denotes the average overthe domainΩ which corresponds to the unit cell.Table I shows the material properties that are used tocalculate the effective coil properties, where the abbreviationsStr., Ep., and Ins. stand for strand, epoxy, and insulationrespectively. For the insulation, the subscripts for the Young’smoduli,E, the Poisson’s ratios,ν, the shear moduli,G, and thetotal thermal strain from 295 K to 4.2 K,εth, correspond tothe three principal directions. For the epoxy and the strand,the materials are assumed to be isotropic; therefore, onlytwo elastic constants are needed. The Poisson’s ratios,νijare defined as the negative of the transverse strain in thej-direction over the strain in thei-direction, when stress isapplied only in thei-direction. The remaining Poisson’s ratioscan be found using the relationνji = νijEjEi. (3)The strand properties are determined using a rule of mix-tures approach with the individual material properties forCuand Nb3Sn obtained from the work by Mitchell [3]. The epoxyproperties are obtained from the CTD-101 data sheet [4] wherethe shear modulus is given at 77 K and 295 K. Assuming aPoisson’s ratio of 0.3, the Young’s modulus is determined.Note that the 77 K shear modulus is used at 4.2 K sinceno measured value is given for this temperature. The valuefor the thermal expansion of epoxy was obtained from theprogram CRYOCOMP. For the insulation layer, properties forglass fiber and epoxy composites, which are obtained fromtwo different sources, are used. For the shear moduli and theYoung’s modulus in the3 direction, data was taken from theCTD-101 data sheet for a combination of 50% and 60% s-glassreinforced epoxies. Unfortunately, data is not provided for theYoung’s moduli in the fiber directions. Therefore, for the1and 2 directions, data for G10 is used. The Young’s moduliare taken to be equal to the average of the moduli in the filland warp directions, where this data was obtained from theUSPAS 2001 notes on superconducting accelerator magnets byGoodzeit [5]. The data for thermal expansion of the insulationlayer is taken from CRYOCOMP using G10 as the material.Clearly the accuracy of the individual material propertieswillhave a large impact on the calculated homogenized properties.Using the properties in Table I, the effective propertiesof the coil can be determined. Since the insulation layer isanisotropic, the principal directions have to be specified withinan element. In this case the3 direction corresponds to theradial direction from the center of the strand on which theinsulation lies. Since the properties are taken as transverselyisotropic, defining only one direction is sufficient. Table IIshows the calculated effective properties for the coil. TheseREPLACE WITH PAPER IDENTIFICATION NUMBER 3TABLE IASSUMED PROPERTIES FOR THE CONSTITUENT MATERIALS IN THE COIL295 K 4.2 KStr. Ep. Ins. Str. Ep. Ins.E1 (GPa) 130 3.6 25 130 6.2 32E2 (GPa) - - 25 - - 32E3 (GPa) - - 13 - - 16ν12 0.34 0.30 0.30 0.34 0.30 0.30ν23 - - 0.30 - - 0.30ν31 - - 0.16 - - 0.16G12 (GPa) - - 23 - - 29G23 (GPa) - - 5.2 - - 6.6G31 (GPa) - - 5.2 - - 6.6εth1(%) 0.25 1.4 0.27εth2(%) - - 0.27εth3(%) - - 0.76TABLE IICALCULATED PROPERTIES FOR THE COIL MATERIAL295 K 4.2 KEL (GPa) 88 90ET (GPa) 31 40ν12 0.28 0.29ν31 0.33 0.33G31 (GPa) 13 17εthL(%) 0.26εthT(%) 0.41properties are transversely isotropic; therefore, only longitudi-nal (L or 3) and transverse (T or 1-2 plane) components arenecessary to fully describe the material behavior.III. F REE THERMAL CONTRACTION OF THE ORTHOTROPICCOILThe Nb3Sn coil has orthotropic properties in a cylindricalcoordinate system defined by the three directionsr, θ, andz. After cool down, due to the different thermal expansioncoefficients in ther and θ directions, the coil is no longerin a stress free state even if there are no loads applied on theboundaries. This is due to the geometry of the coil which onlyallows radial displacements due to its axisymmetric nature.The solution for the displacement and stress in an orthotropiccoil with negligible axial stress can be determined analyticallyby using the strain displacement relations, the stress strainrelation, and enforcing that the stress tensor is divergence free.The radial displacement in the orthotropic coil, as a functionof radial position, is determined to beu(r) = D1rk−D2r−k+(1 − νθr)εthr + (νθr − k2)εthθ1 − k2r, (4)whereD1 andD2 are constants determined from the boundaryconditions andk2 = Eθ/Er. For the coil geometry, theproperties in theθ andz directions in equation 4 correspond tothe properties in the3 and1 directions in table II respectively.The stress state in the coil is given byσr(r) = Eθ[D1k−νθrrk−1 + D2k+νθrr−k−1 +εthr−εthθ1−k2]σθ(r) = kEθ[D1k−νθrrk−1 − D2k+νθrr−k−1 +εthr−εthθk(1−k2)],(5)For a free thermal contraction, the constantsD1 and D2 arefound by enforcing that the radial stress vanish at the innerand outer dimensions of the coil. For a detailed analysis oforthotropic coils in solenoids, the reader is referred to workby Markiewicz [6] and Gray [7].Figure 4 shows the radial displacement of the coil base dueto cooling from 295 K to 4.2 K with zero applied externalloads. For this example the dimensions of the plane stress coilare chosen to be those at the top of the vector magnet coilwhere the inner radius is 22 mm and the outer radius is 71.5mm. For an isotropic material, if the radial displacement curveis extrapolated to zero radial distance, the radial displacementshould vanish. For the orthotropic material considered here,this is not the case. The dashed line in figure 4 representsthe effective slope of the radial displacement curve for theinner surface of the coil, which is considerably smaller thanthe slope of the displacement curve inside of the coil. In theinner surface the thermal contraction corresponds to that of nisotropic material with a thermal contraction of0.15% between295 K and 4.2 K. On the outer surface the thermal contractioncorresponds to that of an isotropic material with a thermalcontraction of0.31% between 295 K and 4.2 K. This resultis important for the design of magnet since it is desirableto have the coil remain in contact with the surroundingstructure after the magnet is cold. The thermal deformationof the orthotropic coil is clearly different that that of a coilwith assumed isotropic properties (where either the radialorazimuthal thermal strain is assumed). Figure 5 shows the stressin the radial and azimuthal directions due to the cool down ofthe coil under the plane stress assumption. The radial stress istensile throughout the coil with a peak value of 19 MPa whilethe azimuthal stress has a maximum tensile value (95 MPa) atthe inner radius and a maximum compressive value (-46 MPa)at the outer radius.0 20 40 60 80−250−200−150−100−500Radial distance (mm)Radial displacement (µm)Fig. 4. Radial displacement of the coil base due to for cool down with zeroapplied external loads. The dashed line represents the effective slope of theradial displacement curve for the inner surface of the coil.For comparison, figure 6 shows the FEM solution for theradial and azimuthal stress in the coil where the plane stressand cylindrical shape assumptions are removed. The FEM andanalytical plane stress calculations are seen to match closelynear the top of the coil where the inner and outer radius matchREPLACE WITH PAPER IDENTIFICATION NUMBER 420 30 40 50 60 70 80−50050100Radial distance (mm)Stress (MPa)  rθFig. 5. Radial and azimuthal stress due to the cooling of the orthotropic coilunder the plane stress assumption.those used in the analytical result. As the coil becomes thinnertowards the bottom the effect of the residual stress due to theorthotropic nature of the coil diminishes. With respect to thedesign and structural analysis of the magnet, this effect willlead to a shape change between the coil and the pocket in themonolithic structure during cool down.Fig. 6. Radial (left) and azimuthal (right) components of stre s in the coildue to cool down from 295 K to 4.2 K with zero tractions on all boundaries.IV. STRUCTURAL MAGNET MODELFor the structural analysis of the vector magnet the fol-lowing components are considered: the coil, the pole, theyoke, as well as the internal monolithic structure. In thefinite element model, all material interfaces are meshed withcontact elements. The contact pairs that are modeled are: coil-support structure, coil-pole, coil-yoke, pole-yoke, pole-supportstructure, and yoke-support structure. The behavior of theelements can be specified as either a stick-slip interface, anon-separating interface (i.e. only motion in the tangential directionis allowed), or a fully glued interface. For the case where themagnetic field is oriented in one of the principal directions, aone-eighth model can be used due to symmetry considerations.A zero normal displacement boundary condition on the threeflat lateral surfaces (x, y, and z planes) of the one-eighth modelis used. The magnetic forces are calculated using TOSCAand applied to the structural ANSYS model on the pole andcoil elements. A homogeneous temperature load is applied onevery element in the model to simulate the cool down fromroom temperature to 4.2K. Figure 7 shows the results obtainedfor the contact pressure on the coil due to both the cool downand energization of the magnet. For this case, the entire base ofthe coil is in compression from the differential thermal strainbetween the Aluminum support structure and the coil. Onlycertain regions of the conical face of the coil remain in contactwith the support structure after cool down and energization.The region of highest pressure on the coil is normal to thedirection in which the field is oriented. This occurs since thcoils are pulled towards each other in this direction.Fig. 7. Finite element solution for the contact pressure betwe n the coiland the monolithic support structure after cool down and energization of themagnet. The units for pressure in the figure are in MPa.V. CONCLUSIONIn this paper, the structural analysis for an eight polesuperconducting magnet was presented. Calculations wereperformed to determine the orthotropic material propertiesof the coil. Through the plane stress analysis it was shownthat the thermal deformation of the inner and outer radiiof the coil does not correspond to that seen in an isotropicmaterial where the thermal strain state is equivalent in bothlocations. Therefore, the inclusion of the orthotropic behavioris a critical component in order to determine the contactbehavior of the coil-structure and coil-pole interfaces. TheANSYS model of the magnet allows for different contactscenarios at these interfaces. The optimal choice for havingc rtain interfaces glued or allowed to slip is currently beinginvestigated. The accuracy of the model is clearly dependenton the material properties of all of the components of themagnet. This includes the constituent materials in the coil.Since there is a fair amount of uncertainty in these properties,accurate measurements over the temperature range of interestare necessary.REFERENCES[1] E. Arenholz and S.O. Prestemon, “Design and performanceof an eight-pole resistive magnet for soft x-ray magnetic dichroism measurements”,Review of Scientific Instruments, vol. 76, 083908, 2005.REPLACE WITH PAPER IDENTIFICATION NUMBER 5[2] D. Arbelaez et al., “Cable deformation simulation and a hierarchicalframework for Nb3Sn rutherford cables”, J. Phys.: Conf. Ser., vol. 234,022002, 2010.[3] N. Mitchell, “Finite element simulations of elasto-plastic processes inNb3Sn strands”, Cryogenics, vol. 45, no. 7, pp. 501-515, 2005.[4] CTD-101 epoxy resin system data sheet, Composite Technology Devel-opment Inc., Lafayette, CO, 2002.[5] C.L. Goodzeit, Superconducting accelerator magnets USPAS lecturenotes, USPAS January 2001.[6] w.D. Markiewicz et al., “Generalized plane strain analysis of supercon-ducting solenoids”, Journal of Applied Physics, vol. 86, no. 12, pp. 7039- 7051, 1999.[7] W.H. Gray and J.K. Ballou, “Electromechanical stress analysis of trans-versely isotropic solenoids”, Journal of Applied Physics,vol. 48, no. 7,pp. 3100 - 3109, 1977.

Mechanical design and analysis of an eight-pole superconducting vector magnet for soft x-ray magnetic dichroism measurements

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