本研究使用有限單元商用軟體ANSYS，針對螺旋聚頻磁鐵之四磁極陣列模型與雙磁極陣列模型的支撐結構進行優化分析。此聚頻磁鐵由永磁磁鐵所組成，本身所產生之磁力會使整體結構產生相對位移，若變形過大則會改變磁場分佈。本文主要目標以ANSYS分析在磁力作用下整體結構之位移及應力分佈。首先將其不鏽鋼外框進行倒角切削，在兼顧整體結構位移增加量及總重量之目標，進行適當倒角切削，降低總重量達4%且位移增加量只有0.15%。接著使用田口法(Taguchi Methods)優化四磁極陣列模型，藉由改變其支撐結構之鋁塊寬度、鋁塊高度、不鏽鋼寬度、不鏽鋼高度四種變數，設計出相較原始尺寸可降低磁鐵端面尖端位移達59.4%。再者針對雙磁極陣列模型使用反應曲面法(Response Surface Methodology, RSM)進行優化，變數與四磁極陣列模型相同，設計出之模型與原始尺寸相比，磁鐵端面尖端位移降幅可達62.04%。
文中也對聚頻磁鐵模型進行熱-結構模擬，探討由室溫293K降溫至液態氮77K之整體結構溫度分佈與殘留熱應力，並研究更換熱膨脹係數較小之金屬外框材質對熱應力之影響。若將鋁金屬外框材質更換為銅金屬外框，其支撐結構之夾具最大von Mises stress降幅達34%，可避免整體結構在降溫過程因熱應力過大使夾具超過降伏強度進入塑性變形，且對整體結構冷縮位移有正面幫助，整體結構位移值降幅達42%。

The finite element commercial code ANSYS was used to optimize support frames of both the four-array model and two-array model of helical permanent undulator (HPMU). The HPMU is composed of paired permanent magnet blocks and poles and thus generates magnetic force to deform whole structure and magnetic field distribution. This research analyzes the displacement and stress distributions of the supporting frame structure under magnetic force and improve the structure with systematical parameter analysis. First, by chamfering the stainless steel frame, 4% of the total weight is thus achieved, while an increase of 0.15% on maximum displacement is induced. Second, by applying Taguchi method on the four- array model, the maximum displacement of tip of magnet block is reduced by 59.4% as the geometry dimensions of the aluminum block and stainless steel support frame are optimized. Alternatively, the maximum displacement of tip of magnet block of the double-array model is highly reduced to 62.04% of the original design by applying the response surface methodology to optimize the geometry dimensions of the aluminum block and stainless steel support frame.
Considering the operation of HPMU of double-array type at cryogenic temperature, its temperature and stress distributions after cooling down were also computed. It is concluded that the frame material shall be with an effective thermal contraction as close to the magnet material as possible. For example, a reduction of 34% on the maximum von Mises stress and 42% on the maximum displacement can be achieved by replacing the original frame material of aluminum alloy to copper.

1. F. R. Elder, A. M. Gurewitsch, R. V. Langmuir, H. C. Pollock, “Radiation from Electrons in a Synchrotron.” Physical Review, Vol. 71, pp. 829-830, 1947.
2. 國家同步輻射研究中心：http：//www.nsrrc.org.tw/ retrieved on August 20, 2018.
3. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast Imaging of Weakly Absorbing Materials Using Hard X-rays.” Nature, Vol. 373, pp. 595-598, 1995.
4. H. D. Nuhn, S. Anderson, G. Bowden, Y. Ding, G. Gassner, Z. Huang, E. M. Kraft, Y. Levashov, F. Peters, F. E. Reese, J. J. Welch, Z. Wolf, J. Wu, “R&D Towards a Delta-type Undulator for the LCLS,” Proceedings of Free Electron Laser Conference, New York, pp. 348-350, 2013.
5. J. Clarke, Insertion Devices Lecture4 Undulator Magnet Designs, Accelerator Science and Technology Centre, UK, 2007.
6. C. Kitegi, Development of a Cryogenic Permanent Magnet Undulator at the ESRF, Universite Joseph Fourier-Grenoble, Saint-Martin, France, 2008.
7. J. C. Huang, H. Kitamura, C.Y. Kuo, C.K. Yang, C. H. Chang, Y. T. Yu, Y. Y. Lin, C. S. Hwang, “Design of a Magnetic Circuit Cryogenic Undulator in Taiwan Photon Source,” AIP Conference Proceedings, Vol. 1741, New York, pp. 1-4, 2016.
8. T. Hara, T. Bizen, T. Seike, R. Tsuru, X. Marechal,H. Hirano, M. Morita, H. Teshima, S. Nariki, N. Sakai, I. Hirabayashi, M. Murakami, H. Kitamura, “Development of Cryogenic Permanent Undulators Operating Around Liquid Nitrogen Temperature,” New Journal of Physics, Vol. 8, pp. 1-16, 2006.
9. 王昭晴，4.5-T 低溫永磁移頻磁鐵研究，碩士論文，國立清華大學先進光源科技學位學程，新竹，台灣，2016。
10. K. J. Kim, “Characteristics of Synchrotron Radiation,” AIP Conference, Berkeley, Vol.184, California, pp. 565-631, 1989.
11. T. Shiozawa, Classical Relativistic Electrodynamics, 1st ed, Springer-Verlag, Berlin Heidelberg, Germany, 2004.
12. H. Wiedemann, Particle Accelerator Physics, 3rd ed, Springer-Verlag, Berlin Heidelberg, Germany, 2007.
13. 陳建德，劃時代的科學研究利器-超高亮度的台灣光子源，國家同步輻射研究中心，物理雙月刊，2011。
14. S. Sasaki, “Analyses for a Planar Variably-polarizing Undulator,” Nuclear Instruments and Methods in Physics Research Section A, Vol. 347, pp. 83–86, 1994.
15. 黃清鄉、張正祥，橢圓偏聚頻磁鐵研發概述，同步輻射研究中心籌建處，科學發展月刊，2000。
16. S. H. Kim, C. L. Doose, “Development of a MODEL Superconducting Helical Undulator for the ILC Positron Source,” Proceeding of Particle Acclerator Conference, Albuquerque, USA, pp. 1136–1138, 2007.
17. A. B. Temnykh, “Delta Undulator for Cornell Energy Recovery Linac,” Physical Review Special Topic, Vol. 11, pp. 324-326, 2008.
18. L. N. P. Vilela, R. Basílio, J. F. Citadini, J. R. Furaer, F. Rodrigues, “Advances in the Sirius Delta-type Undulator Project,” International Particle Accelerator Conference , Canada, pp. 1491-1493, 2018.
19. C. Y. Kuo, C. H. Chang, T. Y. Chung, J. C. Jan, C. S. Hwang, C. H. Chang, “Design of a Short Period Helical Permanent Magnet Undulator,” IEEE Transactions on Applied Superconductivity, Vol. 28, pp. 1-4, 2018.
20. P. Sharma, A. Verma, R. K. Sidhu, O.P. Pandey, “Effect of Processing Parameters on the Magnetic Properties of Strontium Ferrite Sintered Magnets Using Taguchi Orthogonal Array Design ,” Journal of Magnetism and Magnetic Materials, Vol. 307, pp. 157-164, 2006.
21. Y. K. Kim, J. P. Hong, G. H. Lee, and Y. S. Jo, “Application of Response Surface Methodology to Robust Design for Racetrack Type High Temperature Superconducting Magnet,” IEEE Transactions on Applied Superconductivity, Vol. 12, pp. 1434-1437, 2002.
22. ANSYS : http://www.ansys.com/
23. J. Goupy, Methods for Experimental Design, 1st ed, Elsevier Science, Vol. 12, France, 1993.
24. R. L. Mason, R. F. Gunst, J. J. Hess, Statistical Design and Analysis of Experiments with Applications to Engineering and Science, 2nd ed, Wiley, New York, 2003.
25. 黎中正譯(1993)，穩健設計之品質工程，原作者： M. S. Phadke(1989)，Quality Engineering Using Robust Design，台北圖書有限公司，台北。
26. C. R. Rao, “Factorial Experiments Derivable from Combinatorial Arrangements of Arrays,” Supplement to the Journal of the Royal Statistical Society, Vol. 9, pp.128-139, 1947.
27. G. E. P. Box, K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Wiley for the Royal Statistical Society, New Jersey, 1951.
28. 呂政冀，應用部分因子實驗設計進行LED磊晶之MOCVD製程最佳參數之研究，碩士論文，國立成功大學工業與資訊管理學系，台南，台灣，2009。
29. M. Muthukumar, D. Sargunamani, N. Selvakumar, J.V. Rao, “Optimisation of Ozone Treatment for Colour and COD Removal of Acid Dye Effluent Using Central Composite Design Experiment,” Dyes Pigments, Vol. 63, pp. 127-134, 2004.
30. G. E. P Box and D. W. Behnken, “Some New Three Level Designs for the Study of Quantitative Variables,” Technometrics, Vol. 2, pp. 455-476, 1960.
31. D. C. Montgomery and G. C. Runger, Applied Statistics and Probability for Engineers Reference, 3rd ed, Wiely, New York, 2002.
32. R. G. Lomax and D. L. Hahs-Vaughn, Statistical Concepts: A Second Course, 4rd ed, Lawrence Erlbaum Associates, New York, 2007.
33. R. H. Byrd, J. C. Gilbert, J. Nocedal, “A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming,” Mathematical Programming, Vol. 89, pp. 149-185, 2000.
34. 洪東銘，內嵌穿晶片導線三微晶片模組之熱應力分析及最佳化，碩士論文，國立清華大學，新竹，台灣，2008。
35. J. H. Lienhard Ⅳ and J. H. Lienhard Ⅴ, A Heat Transfer Textbook, 3rd ed., Phlogiston Press, Cambridge, 2003.
36. Matweb: http://www.matweb.com/ retrieved on March 18, 2018.
37. P. Vedrine, P. Tixador, Y. Brunet, J. C. Bobo, A. Fevrier, A. Leriche, “Mechanical Characteristics of NdFeB Magnets at Low Temperature,” Cryogenics, Vol. 31, pp. 51-53, 1991.
38. J. Kinast, E. Hilpert, N. Lange, A. Gebhardt, R. R. Rohloff, S. Risese, R. Eberehardt, A. Tünnermann, “Minimizing the Bimetallic Bending for Cryogenic Metal Optics Based on Electroless Nickel,” Proceedings of SPIE, Vol. 9151, Canada, 2014.
39. P. Duthil, “Material Properties at Low Temperature,” Proceedings of CAS-CREN Accelerator , pp. 77-95, 2015.
40. NIST: https://www.nist.gov/ retrieved on December 05, 2018.
41. N. J. Simon, E. S. Drexler and R. P. Reed, Properties of Copper and Copper Alloys at Cryogenic Temperatures, United States Government Publishing Office, 1992.
42. R. D. Cook, D. S. Malkus, M. E. Plesha, R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed, Wiley, Danvers, New York, 2002.
43. ANSYS User’s Manual, ANSYS, Inc.
44. 康淵，陳信吉，ANSYS入門，全華科技圖書，新北市，台灣，2005。
45. K. L. Johnson, Contact Mechanics, 2nd ed, Cambridge University Press, UK , 1985.
46. P. Elleaume, “A Flexible Planar/Helical Undulator Design for Synchrotron Sources,” Physica Scripta, Vol. 31, pp. 67-71, 1990.
47. P. Elleaume, Insertion Devices, European Synchrotron Radiation Facility, France, pp. 83-119, 2003.
48. 胡耀倫，質子交換膜燃料電池變形對其效率之影響，碩士論文，國立中山大學機械與機電工程學系，高雄，台灣，2015。
49. Y. T. Yu, C. C. Wang, H. H. Chen, J. C. Huang, C. S. Hwang, C. H. Chang , “Design of a 4.7-T Wavelength Shifter With Cryogenic Permanent Magnets at NSRRC,” IEEE Transactions on Applied Superconductivity, Vol. 26, 2016.
50. Matlab, Version 2016a, The Language of Technical Computing, Available from MathWorks Inc : http://www.mathworks.com/ retrieved on August 22, 2018.
51. Meet Minitab, Version 15, Minitab Inc. Pennsylvania, 2007.
52. X. Tang, V. Prakash, J. Lewandowski , “Dynamic Tensile Deformation of Aluminum 6061-T6 and 6061-OA,” Proceedings of SEM Annual Conference and Exposition on Experimental and Applied Mechanics , Saint Louis, 2006.
53. Americanelements: https://www.americanelements.com/ retrieved on December 22, 2018.
54. Total Material: https://www.totalmateria.com/ retrieved on December 08, 2018.