FEM Modeling Of Multilayer Canted Cosine Theta (CCT) magnets … · 2019. 9. 17. · 3. 3D periodic CCT model in ANSYS –slicing coordinate z cut (10/13) 4.854 mm 4.5295 mm 0.3245

FEM Modeling Of Multilayer Canted Cosine Theta (CCT) Magnets

with orthotropic material properties

Rafał Ortwein*, Jacek Błocki, Przemysław Wąchal*, Institue of Nuclear Physicsof the Polish Academy of Sciences (IFJ PAN), Cracow, Poland

Glyn Kirby, Jeroen van Nugteren, CERN, Geneva, Switzerland

*Speakers

Contents:1. Motivation for the work2. The MCBXF magnet3. 3D periodic CCT model in ANSYS4. Mesh5. Material properties – homogenization (JB)6. Modeling orthotropic coil properties in ANSYS7. Boundary conditions8. Contacts9. Electromagnetic forces10.2D model – simplified (JB)11.2D model – detailed (PW)12.Comparison of the results13.Conclusions

Speaker – Przemysław Wąchal

2

1. Motivation for the work – MCBXF magnet

Two nested orbit corrector magnets – MCBXF - are needed for the HI-LUMI (HIGH LUMINOSITY) upgrade of the Large Hadron Collider at CERN:- MCBXFB: 2.5 Tm integrated field (peak field 2.5 T)- MCBXFA: 4.5 Tm integrated field

The main design option for the MCBXF is the Cosine-theta and itspursued at CIEMAT (Centre for Energy, Environment and Technology in Spain)

These magnets are low field but present are challenging due to very large torques between the layers – up to 144 000 Nm

Parallel the back-up design is pursued in the Canted Cosine Theta geometry - CERN

𝐵1

𝐵2

𝐵𝑇𝑜𝑡

𝜑𝜑 = 0. . 360

3

2. The MCBXF design (1/3)

Parameter Layer 1 Layer 2 Layer 3 Layer 4Length L [mm] 1400

𝑅𝑧 [𝑚𝑚] 84 93.3 102.6 111.9𝑅𝑤 [𝑚𝑚] 75 84.3 93.6 102.9α𝑖 [

0] -31.793 31.710 -34.170 34.112𝑤𝑖 [𝑚𝑚] 4.854 4.5295n turn [-] 227 237

Cylinder radius [mm] 132.9

4

Courtesy of Glyn Kirby


Layer Torque Mz[Nm]

1 62 0232 77 143 3 -67 2634 -71 799

R

L

𝜏𝑎𝑣 =

𝑀𝑑𝑖𝑓𝑓_𝑚𝑎𝑥

𝑅2𝜋𝑅𝐿

=𝑀𝑑𝑖𝑓𝑓_𝑚𝑎𝑥

2𝜋𝑅2𝐿=

144300

2𝜋 ∙ 0.0933 2∙ 1.4= 1.9 𝑀𝑃𝑎

𝑅 = 0.0933 𝑚

𝐿 = 1.4 𝑚

𝑀𝑑𝑖𝑓𝑓_𝑚𝑎𝑥 = 77.1 + 67.2

= 144.3 𝑘𝑁𝑚

5


Yasuhide Shindo, Interlaminar shear properties of composite insulation systems for fusion magnets at cryogenic temperatures, Cryogenics 50 (2010) 36-42

Interlaminar shear strength of Samples A-C at 4K

𝝉𝒂𝒗_𝑴𝑪𝑩𝑿𝑭 = 𝟏. 𝟗 𝑴𝑷𝒂

This average stress is 20 timesbelow the limit S2-glass+resin+Kapton komposite

𝝉𝒍𝒊𝒎𝒊𝒕 = 𝟒𝟎𝑴𝑷𝒂

6

3. 3D periodic CCT model in ANSYS (1/13)

Initial approach with Boolean operations failed -> ANSYS cannot perform Boolean operations on such complex geometries

[1] Thompson MK. Methods for Generating Rough Surfaces in ANSYS

https://www.ansys.com/-/media/ansys/corporate/resourcelibrary/conference-paper/2006-int-ansys-conf-256.pdf

[2] Alternatives to Boolean Operations https://www.sharcnet.ca/Software/Fluent14/help/ans_mod/Hlp_G_MOD5_4.html

- Was not possible to create 3D CCT geometry using ANSYS Design Modeler and ANSYS SpaiceClaim- Decision to go to the ANSYS APDL modeling

7

https://www.ansys.com/-/media/ansys/corporate/resourcelibrary/conference-paper/2006-int-ansys-conf-256.pdf

https://www.sharcnet.ca/Software/Fluent14/help/ans_mod/Hlp_G_MOD5_4.html

Ԧ𝑟𝐴𝑖 𝜃 = Ԧ𝑟𝐶𝑖 𝜃 + Ԧ𝑟𝐶𝐴𝑖 𝜃

Ԧ𝑟𝐶𝑗𝑖 𝜃 = 𝑋𝑟𝑗 Ƹ𝑟 + 𝑋𝑏𝑗 𝑏

Ԧ𝑟𝐶𝑖(𝜃) = 𝑅𝐶𝑖 Ƹ𝑟 + 𝑅𝐶𝑖 cot 𝛼𝑖 sin 𝜃 − 𝜃0𝑖 +𝑤𝑖

2𝜋(𝜃 − 𝜃0𝑖) + 𝑧0𝑖 Ƹ𝑧

i – layer number

[1] Brouwer LN. Canted-Cosine-Theta Superconducting Accelerator Magnets for High Energy Physics and Ion Beam Cancer Therapy. Ph.D. UC Berkeley. 2015.

[2] Brouwer L, Arbelaez D, Caspi S, Felice H, Prestemon S, Rochepault E. Structural Design and Analysis of Canted–Cosine–Theta Dipoles. IEEE T APPL SUPERCON 2014;24(3):1-6

[3] Brouwer L, Caspi S, Prestemon S. Structural Analysis of an 18 T Hybrid Canted–Cosine–Theta Superconducting Dipole. IEEE T APPL SUPERCON 2015;25(3):4000404

[4] Auchmann B, Brouwer L, Caspi S, Gao J, Montenero G, Negrazus M, Rolando G, Sanfilippo S. Electromechanical Design of a 16-T CCT Twin-Aperture Dipole for FCC. IEEE T APPL

SUPERCON 2018;28(3):4000705

3. 3D periodic CCT model in ANSYS – geometry (2/13) 8

Ԧ𝑡(𝜃) =)𝑑 Ԧ𝑟𝐶𝑖(𝜃

𝑑𝜃Ƹ𝑡(𝜃) =

൯Ԧ𝑡(𝜃

൯Ԧ𝑡(𝜃

𝑏 𝜃 = Ƹ𝑡(𝜃) × Ƹ𝑟 𝜃

Ԧ𝑟𝐴𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐴 cos 𝜃 −𝑋𝑏𝐴𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐴 sin 𝜃 +

𝑋𝑏𝐴𝐶 𝜃

𝐷 𝜃cos 𝜃 + 𝑘

𝑅𝐶𝑖 sin 𝜃 − 𝜃0𝑖tg 𝛼𝑖

+𝑤𝑖 𝜃 − 𝜃0𝑖

2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐴𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐵𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐵 cos 𝜃 −𝑋𝑏𝐵𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐵 sin 𝜃 +

𝑋𝑏𝐵𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐵𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐷𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐷 cos 𝜃 −𝑋𝑏𝐷𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐷 sin 𝜃 +

𝑋𝑏𝐷𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐷𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐸𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐸 cos 𝜃 −𝑋𝑏𝐸𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐸 sin 𝜃 +

𝑋𝑏𝐸𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐸𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐹𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐹 cos 𝜃 −𝑋𝑏𝐹𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐹 sin 𝜃 +

𝑋𝑏𝐹𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐹𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐺𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐺 cos 𝜃 −𝑋𝑏𝐺𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐺 sin 𝜃 +

𝑋𝑏𝐺𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐺𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐻𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐻 cos 𝜃 −𝑋𝑏𝐻𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐻 sin 𝜃 +

𝑋𝑏𝐻𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐻𝑅𝐶𝑖𝐷 𝜃

Ԧ𝑟𝐼𝑖 𝜃 = Ƹ𝑖 𝑅𝐶𝑖 + 𝑋𝑟𝐼 cos 𝜃 −𝑋𝑏𝐼𝐶 𝜃

𝐷 𝜃sin 𝜃 + Ƹ𝑗 𝑅𝐶𝑖 + 𝑋𝑟𝐼 sin 𝜃 +

𝑋𝑏𝐼𝐶 𝜃




2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐼𝑅𝐶𝑖𝐷 𝜃

x y z

3. 3D periodic CCT model in ANSYS – geometry (3/13) 9

Single lines only

Radius of the sphere 45 µm

3. 3D periodic CCT model in ANSYS (4/13)

APDL script -> volumes & mesh

10

z𝐴𝑖 𝜃 = z𝐶𝑢𝑡 , z𝐴𝑖 𝜃 = z𝐶𝑢𝑡 + 𝑤𝑖



2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐴𝑅𝐶𝑖

𝑅𝐶𝑖2 + 𝑅𝐶𝑖 cot 𝛼𝑖 cos 𝜃 − 𝜃0𝑖 +

𝑤𝑖2𝜋

2− z𝐶𝑢𝑡 = 0

(𝐸𝑞. 24)𝑅𝐶𝑖 sin 𝜃 − 𝜃0𝑖

tg 𝛼𝑖+𝑤𝑖 𝜃 − 𝜃0𝑖

2𝜋+ 𝑧0𝑖 −



𝑤𝑖2𝜋

2− z𝐶𝑢𝑡 − 𝑤𝑖 = 0



2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐵𝑅𝐶𝑖


𝑤𝑖2𝜋

2− z𝐶𝑢𝑡 = 0

(𝐸𝑞. 26)𝑅𝐶𝑖 sin 𝜃 − 𝜃0𝑖

tg 𝛼𝑖+𝑤𝑖 𝜃 − 𝜃0𝑖

2𝜋+ 𝑧0𝑖 −

𝑋𝑏𝐵𝑅𝐶𝑖


𝑤𝑖2𝜋

2− z𝐶𝑢𝑡 −𝑤𝑖 = 0

3. 3D periodic CCT model in ANSYS (5/13) 11

Equations coded into ANSYS APDL and solved with bisection method (<40 iteration needed)



2𝜋+ 𝑧0𝑖 −



𝑤𝑖2𝜋

2− z𝐶𝑢𝑡 = 0


- Any number of keypoints (up to 180) can be used to generate the splines- Now the splines are much shorter -> thus more precise and the maximum distance to

the keypoints is 5 µm


𝑧𝐶𝐴𝐷_𝐷 =𝑅𝐶𝑖

tg 𝛼𝑖+𝑤𝑖

4+ 𝑧0𝑖 −

𝑋𝑏𝐷𝑅𝐶𝑖

𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2

𝛼𝑖𝐷 = 𝑎𝑟𝑐𝑡𝑎𝑛1

𝑅𝐶𝑖𝑧𝐶𝐴𝐷_𝐷 −

𝑤𝑖

4− 𝑧0𝑖 +

𝑏𝑅𝐶𝑖

2 𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2

𝛼𝑖𝐷′ = 𝑎𝑟𝑐𝑡𝑎𝑛1

𝑅𝐶𝑖𝑧𝐶𝐴𝐷_𝐷′ −

3𝑤𝑖

4− 𝑧0𝑖 +

𝑏𝑅𝐶𝑖

2 𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2

3. 3D periodic CCT model in ANSYS – angle αi (8/13)

Ԧ𝑟𝐶𝑖(𝜃) = 𝑅𝐶𝑖 Ƹ𝑟 + 𝑅𝐶𝑖 cot 𝛼𝑖 sin 𝜃 − 𝜃0𝑖 +𝑤𝑖

2𝜋(𝜃 − 𝜃0𝑖) + 𝑧0𝑖 Ƹ𝑧

𝑅𝐶𝑖, 𝑋𝑟𝐺 , 𝑋𝑏𝐺, 𝑤𝑖, 𝜃0𝑖, 𝑧0𝑖

14

Layer 4

Point alfa

θ0+Pi/2

D 34.1123241

E 34.1123244

F 34.1123241

G 34.1123244

θ0+3*Pi/

2

D’ 34.1123245

E’ 34.1123242

F’ 34.1123245

G’ 34.1123242

Layer 𝛼𝑖1 -31.79375

2 31.71078

3 -34.17021

4 34.11232

α=31.7 α=31.8

α=31.71 α=31.72

The angle αi needs to be known with 0.001⁰ precision (in the case of the MCBXF geometrical parameters)

3. 3D periodic CCT model in ANSYS – angle αi (9/13) 15

3. 3D periodic CCT model in ANSYS – slicing coordinate zcut (10/13)

4.854 mm 4.5295 mm

0.3245 mm difference

0.16 mm

16

𝑧𝐴0𝐼_𝑖 =𝑅𝐶𝑖


4+ 𝑧0𝑖 −

𝑏 ∙ 𝑅𝐶𝑖

2 𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2

𝑧𝐴0𝐼𝐼_𝑖 = −𝑅𝐶𝑖

tg 𝛼𝑖+3𝑤𝑖

4+ 𝑧0𝑖 −


2 𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2

𝑧𝐴𝐼_𝑖(𝑛) = 𝑧𝐴0𝐼_𝑖 + 𝑛 ∙ 𝑤𝑖

𝑧𝐴𝐼𝐼_𝑖(𝑛) = 𝑧𝐴0𝐼𝐼_𝑖 + 𝑛 ∙ 𝑤𝑖

𝑧𝐵𝐼_𝑖 𝑛 =𝑅𝐶𝑖


4+ 𝑧0𝑖 +


2 𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2+ 𝑛 ∙ 𝑤𝑖 = 𝑧𝐵0𝐼_𝑖 + 𝑛 ∙ 𝑤𝑖

𝑧𝐵𝐼𝐼_𝑖 𝑛 = −𝑅𝐶𝑖

tg 𝛼𝑖+3𝑤𝑖

4+ 𝑧0𝑖 +


2 𝑅𝐶𝑖2 +

𝑤𝑖2𝜋

2+ 𝑛 ∙ 𝑤𝑖 = 𝑧𝐵0𝐼𝐼_𝑖 + 𝑛 ∙ 𝑤𝑖

3. 3D periodic CCT model in ANSYS – slicing coordinate zcut (11/13) 17

3. 3D periodic CCT model in ANSYS – slicing coordinate zcut (12/13)

- Thickness of the volume 9.4 µm- Elements are extremely skewed, can cause erroneous results locally

18

140 141140

86858683 83

143 143

75

85

95

105

115

125

135

145

155

n [

-]

zAI_L1

zBI_L1

zAII_L1

zBII_L1

zAI_L2

zBI_L2

zAII_L2

zBII_L2

cut 1-2

middle

151150

151

8586

85 838283

153 154153

75

95

115

135

155

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

n [

-]

z [m]

zAI_L3zBI_L3zAII_L3zBII_L3zAI_L4zBI_L4zAII_L4zBII_L4cut 3-4 v1cut 3-4 v2

3. 3D periodic CCT model in ANSYS – slicing coordinate zcut (13/13) 19

4. Mesh

Transition region

Mapped mesh region

20

5. Material properties - homogenization

Mat. Temp. [K] PropertiesAluminum

6082 T6293, 1.9 E=70 GPa [25], ν=0.33 [26], α293-1.9=13.5e-6 [27]

Fiberglass composite

293 [29]EX=16 GPa, EY=30 GPa, EZ=27 GPa, νxy=0.11, νyz=0.21,νxz=0.42, GXY=5.7 GPa, GYZ=6.8 GPa, GXZ=4.8 GPa

1.9 [29]EX=24 GPa, EY=37 GPa, EZ=35 GPa, νxy=0.11, νyz=0.27,νxz=0.42, GXY=9.7 GPa, GYZ=11.6 GPa, GXZ=8.2 GPa

Thermal [30] αx_293-1.9=20.9e-6, αy_293-1.9=αz_293-1.9=6.8e-6

Mat. Temp. [K] Properties

Strand293 [32] EX=EY=110 GPa, EZ=114 GPa, νxy=0.381, νyz=νxz=0.357,

GXY=41.4 GPa, GYZ=GXZ=41.1 GPaThermal [27] αx_293-1.9=6e-6

Resin CTD-101K

293 [33] E=3.5 GPa, ν=0.38Thermal [30] αx_293-1.9=21e-6

Homogenized coil

293, 1.9EX=EY=15.2 GPa, EZ=64.7 GPa, νxy=0.347,νyz=νxz=0.086, GXY=12.1 GPa, GYZ=GXZ=19.3 GPa

Thermal αx_293-1.9=αy_293-1.9=23.5e-6, αz_293-1.9=7.13e-6

Model by Jacek Błocki

21

6. Modeling orthotropic coil properties in ANSYS (1/2)

∆𝜃𝑖_𝑗_𝑚𝑎𝑥= 𝑎𝑟𝑐𝑡𝑔

𝑅𝐶𝑖 + 𝑋𝑟𝑗 sin 𝜃0 +𝑋𝑏𝑗 ∙ 𝑅𝐶𝑖 cot 𝛼𝑖 +

𝑤𝑖2𝜋

𝑅𝐶𝑖2 + 𝑅𝐶𝑖 cot 𝛼𝑖 +

𝑤𝑖2𝜋

2cos 𝜃0

𝑅𝐶𝑖 + 𝑋𝑟𝑗 cos 𝜃0 −𝑋𝑏𝑗 𝑅𝐶𝑖 cot 𝛼𝑖 +

𝑤𝑖2𝜋

𝑅𝐶𝑖2 + 𝑅𝐶𝑖 cot 𝛼𝑖 +

𝑤𝑖2𝜋

2sin 𝜃0

− 𝜃0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-180 -120 -60 0 60 120 180

Angle

dif

fere

nce

Δθj

[⁰]

Angle θ [⁰]

L1_D

L1_E

L1_F

L1_G

L2_D

L2_E

L2_F

L2_G

L3_D

L3_E

L3_F

L3_G

L4_D

L4_E

L4_F

L4_G

22

C

K1 = 𝑥𝐶 Ƹ𝑖 + 𝑦𝐶 + cos(θ) Ƹ𝑗 + 𝑧𝐶 + sin(θ) 𝑘

K2 = 𝑥𝐶 − sin(θ)𝐶 𝜃

𝐷 𝜃Ƹ𝑖 + 𝑦𝐶 + cos(θ)

𝐶 𝜃

𝐷 𝜃Ƹ𝑗 + 𝑧𝐶 −

𝑅𝐶𝐷 𝜃

𝑘

6. Modeling orthotropic coil properties in ANSYS (2/2) 23

7. Boundary conditions (1/2)

Ux=0

Uz=0

Uy=0

Uz=0Uy=0

Uz=0

Ux=0

Uz=0

Uz=0

24

Ux_left=Ux_rightUy_left=Uy_right

7. Boundary conditions (2/2) 25

Periodic boundary conditions remove large out of plane deformations that would be present without them

8. Contacts

The most important setting for the contacts to work: Initialpenetration/gap -> Exclude

26

Contacts between layers are necessary – as crating coincident geometry seems very difficult

9. Electromagnetic forces

Layer Torque Mz[Nm]

1 62 0232 77 143 3 -67 2634 -71 799

27

Courtesy of Jeroen van Nugteren

The overall sum of forces in x, y and z directions is close to zero

10. Simplified 2D model in the xy plane (1/2)

Homogenized material for the coil (NbTi strands + cured resin)

Aluminum

S2-glass fiber composite

• Rotational symmetry of order 60 (360 deg/6 deg = 60)

• PLANE183 elements with plane stress option

• Element systems of the interlayer insulation were rotated to the global cylindrical system

• First and fast estimation of the magnet behavior

• Assumption of regularstructure

• In the same way ortotropy in coils material was included

• The model is placed in the center of the magnet (z = 0 plane)

z = 0 plane

10. Simplified 2D model in the xy plane (2/2)

• The electromagnetic forces were applied in the center nodes of the homogenized coils

• Values of EM forces have been calculated based on the nodal force densities of the full 3D EM model

The considered length in z-direction:

𝐿2𝐷 =1

214 ∙ 𝑤1 + 15 ∙ 𝑤3

where w1 and w3 are the layer pitches

• Respective sums of EM forces in the X and Y directions for each 6 deg piece in the azimuthal direction

• Three load steps: Cooling down from 293 to 1.9K Cooling down + EM forces EM forces only at 293K

=6deg

Ux=0

Uy=0

Uy=0

11. Detailed 2D model in the xy plane (1/7)

• A slice was cut from the 3D CAD model

• Model was prepared and cleaned in SpaceClaim and developed in Ansys APDL

• Geometry was imported to ANSYS APDL, where only the key points from the z=0 plane were kept

• The interlayer insulation was added

CCT magnet slice

z=0 plane

• The areas of the coils, formers and interlayerinsulation were recreated


φ1

φ3

φ2

• EM forces in x and y-direction were calculatedfrom the full 3D model

• Angular selections betweencoil turns were made

• Summation of the force in the range z=-7∙w1..7∙w1 and for the layer 3-4 for z=-7.5∙w3..7.5∙w3

• PLANE183 elements with plane stress option

-7∙w1

-7.5∙(w3)7∙w1

7.5∙(w3)

Ux=0

Uy=0Uy=0

• Three load steps: Cooling down from 293 to 1.9K Cooling down + EM forces EM forces at 293K only

• The electromagnetic forces were applied in the center nodes of the homogenized coils


Homogenized material for the coil (NbTi strands + cured resin)

Aluminum

S2-glass fiber composite

• The basis for determining anisotropic material properties for each cross section of coil turn are the homogenized properties of NbTistrands with cured resin

Mat. Temp. [K] Properties

Aluminum 293, 1.9 E=70 GPa [25], ν=0.33 [26], α293-1.9=13.5e-6 [27]

Fiberglass composite

293 EX=16 GPa, EY=30 GPa, EZ=27 GPa, νxy=0.11, νyz=0.21, νxz=0.42,GXY=5.7 GPa, GYZ=6.8 GPa, GXZ=4.8 GPa

1.9 EX=24 GPa, EY=37 GPa, EZ=35 GPa, νxy=0.11, νyz=0.27, νxz=0.42,GXY=9.7 GPa, GYZ=11.6 GPa, GXZ=8.2 GPa

Thermal αx_293-1.9=20.9e-6, αy_293-1.9=αz_293-1.9=6.8e-6

Homogenized coil

293, 1.9 EX=EY=15.2 GPa, EZ=64.7 GPa, νxy=0.347, νyz=νxz=0.086,GXY=12.1 GPa, GYZ=GXZ=19.3 GPa

Thermalαx_293-1.9=αy_293-1.9=23.5e-6,αz_293-1.9=7.13e-6

Material properties used in the analysis

• Anisotropy of coils mateial shouldbe considered

• Element systems of the interlayer insulation were rotated to the global cylindrical system

• Each turn of coils intersects with cutting plane under different angle


• Steps in determining anisotropic properties for each area of coils

1. Obtain θ angle for each coil area

θ

• Angles are taken from ANSYS

Ortogonal stiffness matrix

Slawinski, M. A., 2010, Waves and Rays in Elastic Continua: 2nd Ed., World Scientific.

Parameter Layer 1 Layer 2 Layer 3 Layer 4

1 𝒃 [𝒎𝒎], 𝒉 [𝒎𝒎], Magnet’s length L [mm] 2, 5, 1400

2 𝑹𝒛 [𝒎𝒎] 84 93.3 102.6 111.9

3 t [mm] (interlayer composite) 0.3 0.3 0.3 0.3

4 𝑹𝒛𝒓 = 𝑹𝒛 + 𝒕 [𝒎𝒎] 84.3 93.6 102.9 112.2

5 𝑹𝒘 [𝒎𝒎] 75 84.3 93.6 102.9

6 𝑹𝑪𝒊 = 𝑹𝒛 − 𝒉/𝟐 𝒎𝒎 81.5 90.8 100.1 109.4

7 𝒘𝒊 [𝒎𝒎] 4.854 4.5295

8 𝜶𝒊 [𝟎] -31.79375 31.71078 -34.17021 34.11232

9 𝜽𝟎𝒊 [𝟎] 180 270

10 𝒛𝟎𝒊 [𝒎𝒎] -550.93014 -536.746055

11 n_turn [-] 227 227 237 237

12 Outer cylinder radius [mm] Rw=112.2, Rz=132.9

Paramters of the 4-layer MCBXF orbit corrector

Homogenized properties of NbTi strands with cured resin---> basis for determining anisotropic material properties

[C]=



1. Obtain θ angle for each coil area2. Calculation of directional cosines aij

From xyz to rbt

r b t

x cos 𝜃 − sin 𝜃𝐶 𝜃

𝐷 𝜃

−sin 𝜃 𝑅𝐶𝑖𝐷 𝜃

y sin 𝜃 cos 𝜃𝐶 𝜃

𝐷 𝜃

𝑅𝐶𝑖 cos 𝜃

𝐷 𝜃

z 0 −𝑅𝐶𝑖𝐷 𝜃

𝐶(𝜃)

𝐷 𝜃

Ƹ𝑟 𝜃 = Ƹ𝑖 cos 𝜃 + Ƹ𝑗 sin 𝜃

𝑏 𝜃 = Ƹ𝑖 − sin 𝜃𝐶 𝜃

𝐷 𝜃+ Ƹ𝑗 cos 𝜃

𝐶 𝜃

𝐷 𝜃+ 𝑘 −

𝑅𝐶𝑖𝐷 𝜃

Ƹ𝑡 𝜃 = Ƹ𝑖− sin 𝜃 𝑅𝐶𝑖

𝐷 𝜃+ Ƹ𝑗

𝑅𝐶𝑖 cos 𝜃

𝐷 𝜃+ 𝑘

𝐶(𝜃)

𝐷 𝜃

𝐶(𝜃) = 𝑅𝐶𝑖 cot 𝛼𝑖 cos 𝜃 − 𝜃0𝑖 +𝑤𝑖

2𝜋

𝐷 𝜃 = 𝑅𝐶𝑖2 + 𝑅𝐶𝑖 cot 𝛼𝑖 cos 𝜃 − 𝜃0𝑖 +

𝑤𝑖

2𝜋

2

= 𝑅𝐶𝑖2 + 𝐶(𝜃)2

Local and global coordinate systems with CCT axis

From xyz to

rbt r b t

x𝐴11 𝐴12 𝐴13

y𝐴21 𝐴22 𝐴23

z𝐴31 𝐴32 𝐴33




3. Calculation of matrix elements A𝜎 and A𝜖


The transformation matrix of a stress tensor can be expressed by:

The transformation matrix of a strain tensor can be expressed by:

[A𝜀]𝑇=[A𝜎]−1

[𝐶] =[A𝜀]𝑇[𝐶][A𝜎]−1

[A𝜀]=

[A𝜎]=


• Steps in determining anisotropic properties for eacharea of coils


3. Create matrix of [C], A𝜎, A𝜀 and calculate matrix elements A𝜎−1 and Aε performing matrix multiplication

4. Resulting stiffness matrix should be symmetric


The transformation matrix of a stress tensor can be expressed by:

The transformation matrix of a strain tensor can be expressed by:

ANSYS Mechanical APDL Theory Reference, ANSYS, Inc. Release 2019 R1, Southpointe January 2019, 2600 ANSYS DriveCanonsburg, PA 15317

[𝐶]=

Implementation in ANSYS

[A𝜀]𝑇=[A𝜎]−1 [𝐶] =[A𝜀]𝑇[𝐶][A𝜎]−1

[A𝜎]=

[A𝜀]=

12. Comparison between the models (1/4)

3D periodic 2D simplified (preliminary) 2D detailed (preliminary)

37

Total deformation (USUM) after cool-down to 1.9 K

Total deformation (USUM) after powering at 1.9 K (both dipoles at maximum current)



38

Peak 486 MPa not shown Von Mises stress after cool-down to 1.9 K

Total deformation (USUM) only due to powering (no thermal effects)



39

Peak 733 MPa not shown


Von Mises stress after cool-down to 1.9 K

Von Mises stress after powering at 1.9 K (both dipoles at maximum current)

12. Comparison between the models (4/4)3D periodic 2D simplified (preliminary) 2D detailed (preliminary)

40



Shear stress after cool-down to 1.9 K

Shear stress after powering at 1.9 K

Shear stress due to EM forces only (no thermal effects)

13. Summary and conclusions

1. 3 FEM models of the nested orbit corrector MCBXF have been made2. The results indicate that the stresses in the coils and the formers are well below

yield limits3. The shear stresses in the interlayer composite show peak values over 40 MPa

(the strength of this material), indicating that the design with castellations might be the only solution – final check with the full 3D model

4. The boundary conditions in the 3D periodic model are not considered fully valid, only after creating the full 3D model and comparing the solutions in the straight section further progress can be made.

5. In the next step the full 3D models with and without castellations will be developed.

6. The 2D detailed model will be developed via parametric creation of the geometry and crating a separate 2D electromagnetic model which will solve the problems with the Lorentz forces now imported from the 3D model.

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Documents

FEM Modeling Of Multilayer Canted Cosine Theta (CCT) magnets … · 2019. 9. 17. · 3. 3D periodic CCT model in ANSYS –slicing coordinate z cut (10/13) 4.854 mm 4.5295 mm 0.3245