臺灣博碩士論文加值系統

本文旨在開發與驗證一套利用非結構性網格之平行化靜磁場模擬程式，並與文獻中由電流和磁鐵所產生的磁場為範例進行程式驗證。此外，本文將針對靜磁場方程離散化、靜磁場異質介面條件做詳細推導與討論。在數值方法的部分，我們採用有限體積法離散蒲松方程式，並且在非結構性網格的情況下利用空間泰勒展開處理非正交網格所帶來的影響，並採用最小平方法來估算所有算體中心物理性質的梯度，對於磁化源 (magnetization) 題目則改寫最小平方法使數值解在異質介面更加平滑。另外採用開源軟體GMSH來建造複雜結構的網格，在平行化的部分是利用區域分割方法(domain decomposition)，均勻分配每顆電腦的計算量；在程式方面則採用ultraMPP (ultra-fast Massive Parallel Platform) 平行計算平台，其中解矩陣的部分仰賴PETSc (Portable, Extensible Toolkit for Scientific Computation)，並藉由分散式叢集電腦來進行平行運算。本文將提供五個算例，包括長直導線、帶電流的銅環、螺線管線圈與鐵棒、甜甜圈磁鐵與海爾貝克轉子(Halbach rotor)，這些算例結果則與解析解、開源數值程式openFOAM和商用軟體COMSOL之結果做定量或定性的驗證比較。再來，本文針對最小平方之梯度法在磁鐵邊界上算不準的問題提出原因與討論，並加以改進。接著，本文探討邊界大小、形狀對數直結果的影響。在最後將以長直導線為例子，做平行化效率的測試，包含strong scaling和weak scaling。本文測試的電腦為單節點多處理器(single node multi-processor)，內含40顆真實核心。在最大核心數時40顆時，計算部分的平行效率在strong scaling仍然有60%以上的好成績；而weak scaling有將近65%。

This dissertation aims to develop and validate a parallel magnetostatics code with the cell-centered finite-volume method using an unstructured grid, and to simulate many magnetostatic problems. Furthermore, this dissertation covers detailed derivation and discussion for the discretization of magnetostatic governing equations and boundary conditions. For the numerical methods, the cell-centered finite volume method was employed to discretize all the governing equations. The spatial Taylor expansion was used to deal with the effect of non-orthogonality of the mesh, and the least-square gradient method was applied for calculating the gradient data of all solution variables. Moreover, we modify the least-square gradient method at the interface between different permeabilities when magnetization problem was involved to make the numerical result smoother. An open-source software GMSH was used to build the mesh for problems with complex geometry. ultraMPP (ultra-fast Massive Parallel Platform) was employed for the programming. It was parallelized with domain decomposition method using message passing interface (MPI) that was implemented on a distributed-memory PC cluster. There were five benchmark test cases, which include a long straight wire, a currented copper ring with iron bar, a spiral wire, a donut magnet, and a Halbach rotor. These cases were validated with either exact solutions or results qualitatively from other paper with some numerical software, e.g., OpenFOAM or COMSOL. Next, we discuss about the failure of the traditional least-gradient method on the magnetization problem and how it was resolved in the current study, followed by the discussion of the effect of choice of boundary domain size and shape. Finally, the parallel efficiency of long straight wire was thoroughly investigated, including weak scaling and strong scaling considering a computer with a single-node multi-processor architecture that contains 40 real cores. The results show that the parallel efficiency has at least achieved 60% in strong scaling, while in weak scaling has around 65%.

致謝 ii
摘要 iii
Abstract v
List of Tables ix
List of Figures x
Nomenclature xiii
Chapter 1. Introduction 1
1.1. Background and Motivation 1
1.2. Literature Survey of Magnetostatics 1
1.3. Software products for Pre-processing, Post-processing and Numerical Calculation 2
1.4. Organization of this Thesis 3
Chapter 2. Theoretical Background 4
2.1. Electric Displacement and Magnetic Field Intensity 4
2.2. Maxwell Equations and Assumption of Magnetostatics 5
2.3. Derivation of Magnetostatics Governing Equations 6
2.3.1. Current Source Governing Equation with Magnetic Vector Potential 6
2.3.2. Magnetic Moment Source Governing Equation with Magnetic Scalar Potential 7
2.4. Interface Boundary Condition 8
2.5. Magnetization Boundary Condition 11
2.6. Outer Boundary Condition 12
Chapter 3. Discretization Using Finite Volume Method 14
3.1. Gradient Calculation 14
3.2. Unstructured Finite Volume Method for Poisson Equation 15
3.3. Outer Boundary Condition Treatment 17
3.4. General Interface Boundary Condition Treatment 17
3.5. Gradient Re-calculation on the Magnet Interface 18
Chapter 4. Results and Discussion 20
4.1. Long Straight Wire 20
4.2. Single Bar with Copper Ring 21
4.3. Spiral Wire 22
4.4. Donut Magnet 23
4.5. Halbach Rotor Magnet 24
4.6. The Detail of Failure of Least-Square Gradient Method on Magnetization Problem 25
4.7. The Boundary Effect 27
4.8. Parallel Efficiency Study 28
4.8.1. Strong Scaling 30
4.8.2. Weak Scaling 30
Chapter 5. Conclusions and Recommendation of Future Work 32
5.1. Conclusions 32
5.2. Recommendation of Future Work 32
References 33
Tables 36
Figures 43

[1] COMSOL, “Static Field Modeling of a Halbach Rotor”, https://www.comsol.com/model/static-field-modeling-of-a-halbach-rotor-14369
[2] Lipnikov, K., Manzini, G., Brezzi, F., and Buffa, A., 2011, “The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes,” Journal of Computational Physics, vol. 230, no. 2, pp. 305–328, https://doi.org/10.1016/j.jcp.2010.09.007
[3] DI Peter Gangl, 2016, "Some Benchmark Problems in Electromagnetics," https://www.numa.uni-linz.ac.at/Teaching/Bachelor/oberndorfer-bakk.pdf
[4] Griffiths, D. J., 2005, "Introduction to electrodynamics," Pearson.
[5] Chubar, O., Benabderrahmane, C., Marcouille, O., Marteau, F., Chavanne, F. J., and Elleaume, P, 2004, "Application of finite volume integral approach to computing of 3D magnetic fields created by distributed iron-dominated electromagnet structures," Proceedings of EPAC 2004 (2004): 1675-1677.
[6] Saravia, M., 2021, "A finite volume formulation for magnetostatics of discontinuous media within a multi-region OpenFOAM framework," Journal of Computational Physics, 433, 110089, https://doi.org/10.1016/j.jcp.2020.110089.
[7] COMSOL, “Permanent Magnet,” https://www.comsol.com/model/permanent-magnet-78
[8] Magdaleno-Adame, S., Olivares-Galvan, J. C., Campero-Littlewood, E., Escarela-Perez, R., and Blanco-Brisset, E., 2010, "Coil systems to generate uniform magnetic field volumes," In Excerpt from the proceedings of the COMSOL conference, Vol. 13, pp. 401-411, COSMOL, Inc, Lindsay Paterson, http://www.comsol.com/paper/coil-systems-to-generate-uniform-magnetic-field-volumes-7772
[9] Huang, Z., 2010, “OpenFOAM Simulation for Electromagnetic Problems,” MS Thesis, in Department of Energy and Environment, Division of Electric Power Engineering (p. 54). Chalmers University of Technology, Chalmers University of Technology.
[10] Alferenok, A., 2013, "Numerical simulation and optimization of the magnet system for the Lorentz Force Velocimetry of low-conducting materials," Doctoral dissertation, TU Ilmenau.
[11] Patidar, B., Saify, M. T., Hussain, M. M., Jha, S. K., and Tiwari, A. P., 2015, "Analytical numerical and experimental validation of coil voltage in induction melting process," International Journal of Electromagnetics, 1(1), 19-31.
[12] Bastos, J. P. A., and Sadowski, N., 2003, "Electromagnetic modeling by finite element methods," CRC press.
[13] Hsieh, M. F., and Hsu, Y. C., 2011, "A generalized magnetic circuit modeling approach for design of surface permanent-magnet machines," IEEE Transactions on Industrial Electronics, 59(2), 779-792. https://doi.org/10.1109/TIE.2011.2161251.
[14] Прачуковска, А. П., Новицки, М. С., Коробийчук, И. В., Шевчик, Р. Ю., and Салах, Я. Л., 2015, "Modeling and validation of magnetic field distribution of permanent magnets," Eastern-European Journal of Enterprise Technologies, 6(5), 4–11, https://doi.org/10.15587/1729-4061.2015.55323
[15] Miltat, J. E., and Donahue, M. J., 2007, "Numerical micromagnetics: Finite difference methods," Handbook of magnetism and advanced magnetic materials, 2, 742-764. https://doi.org/10.1002/9780470022184.hmm202.
[16] Syrakos, A., Varchanis, S., Dimakopoulos, Y., Goulas, A., and Tsamopoulos, J., 2017 "A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods," Physics of Fluids 29(12), 127103. https://doi.org/10.1063/1.4997682.
[17] Akishin, P. G., and Sapozhnikov, A. A., 2019, "The volume integral equation method in magnetostatic problem," Discrete and Continuous Models and Applied Computational Science, 27(1), 60-69, https://doi.org/10.22363/2658-4670-2019-27-1-60-69.
[18] Bermúdez, A., Rodríguez, R., and Salgado, P., 2008, "A finite element method for the magnetostatic problem in terms of scalar potentials," SIAM Journal on Numerical Analysis, 46(3), 1338-1363, https://doi.org/10.1137/06067568X.
[19] Sarma, M., 1976, "Magnetostatic field computation by finite element formulation," IEEE Transactions on Magnetics, 12(6), 1050-1052, https://doi.org/10.1109/TMAG.1976.1059162.
[20] Silva, E. J., and Mesquita, R. C., 1997, "Three dimensional magnetostatics using the magnetic vector potential with nodal and edge finite elements," Applied Computational Electromagnetics Society Journal, 12, 153-156. https://doi.org/10.1016/B978-0-08-037191-7.50024-9
[21] Biro, O., Preis, K., and Richter, K. R., 1996, "On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems," IEEE Transactions on magnetics, 32(3), 651-654, https://doi.org/10.1109/20.497322.
[22] Saravia, M., 2021, "A finite volume formulation for magnetostatics of discontinuous media within a multi-region OpenFOAM framework," Journal of Computational Physics, 433, 110089, https://doi.org/10.1016/j.jcp.2020.110089.
[23] Zou, J., Yuan, J. S., Ma, X. S., Cui, X., Chen, S. M., and He, J. L., 2004, "Magnetic field analysis of iron-core reactor coils by the finite-volume method," IEEE transactions on magnetics, 40(2), 814-817, https://doi.org/10.1109/TMAG.2004.824773.
[24] Nikitenko, G., Konoplev, E., Salpagarov, V., Konoplev, P., and Bobryshev, A., 2020, " Method of calculating magnetic system using finite difference method," Engineering for Rural Development, (19), 1373-1380, https://doi.org/10.22616/erdev.2020.19.tf339.
[25] Nagel, J. R., 2014, "Numerical solutions to Poisson equations using the finite-difference method [education column]," IEEE Antennas and Propagation Magazine 56(4), 209-224, https://doi.org/10.1109/MAP.2014.6931698.
[26] Chung, T. S., and Zou, J., 2001, "A finite volume method for Maxwell's equations with discontinuous physical coefficients," International Journal of Applied Mathematics, 7(2), 201-224.
[27] Weiss, J., Garg, V., Shah, M., and Sternheim, E., 1984, "Finite element analysis of magnetic fields with permanent magnet," IEEE Transactions on Magnetics 20(5), 1933-1935,
https://doi.org/10.1109/TMAG.1984.1063462.
[28] Vanoost, D., De Gersem, H., Peuteman, J., Gielen, G., and Pissoort, D., 2013, "2D magnetostatic finite element simulation for devices with a radial symmetry," IEEE Trans. Magn.,
https://doi.org/10.1109/TMAG.2013.2292672
[29] Strømmen, E. N., 2014, “Structural Dynamics,” springer, pp. 161–204, https://doi.org/10.1007/978-3-319-01802-7_4
[30] Chávez-González, A. F., Aguila-Muñoz, J., Perez-Benitez, J. A., and Espina-Hernandez, J. H., 2013, March, "Finite differences software for the numeric analysis of a non-destructive electromagnetic testing system," CONIELECOMP 2013, 23rd International Conference on Electronics, Communications and Computing (pp. 82-86), IEEE, https://doi.org/10.1109/CONIELECOMP.2013.6525764.