Spectral Stochastic Finite Element Method for Electromagnetic Problems with Random Geometry
DOI:
https://doi.org/10.2478/ecce-2014-0011Keywords:
Eddy currents, Finite element analysis, Stochastic systems, Random variablesAbstract
In electromagnetic problems, the problem geometry may not always be exactly known. One example of such a case is a rotating machine with random-wound windings. While spectral stochastic finite element methods have been used to solve statistical electromagnetic problems such as this, their use has been mainly limited to problems with uncertainties in material parameters only. This paper presents a simple method to solve both static and time-harmonic magnetic field problems with source currents in random positions. By using an indicator function, the geometric uncertainties are effectively reduced to material uncertainties, and the problem can be solved using the established spectral stochastic procedures. The proposed method is used to solve a demonstrative single-conductor problem, and the results are compared to the Monte Carlo method. Based on these simulations, the method appears to yield accurate mean values and variances both for the vector potential and current, converging close to the results obtained by time-consuming Monte Carlo analysis. However, further study may be needed to use the method for more complicated multi-conductor problems and to reduce the sensitivity of the method on the mesh used.References
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J. Lähteenmäki, Design and voltage supply of high-speed induction machines, Acta Polytechnica Scandinavica, Electrical Engineering series no. 108, Espoo 2002.
P.B. Reddy and T.M. Jahns, “Analysis of bundle losses in high speed machines,” 2010 International Power Electronics Conference (IPEC), pp. 2181–2188, 2010. http://dx.doi.org/10.1109/IPEC.2010.5542354
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J. Fang, X. Liu, B. Han and K. Wang, “Design aspects of winding for low-voltage high-speed permanent magnet motor,” IEEE Transactions on Energy Conversion (in review), 2014.
R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements – A Spectral Approach. New York: Springer Verlag, 1991. http://dx.doi.org/10.1007/978-1-4612-3094-6
R. Gaignaire, S. Clenet, B. Sudret, and O. Moreau, “3-D spectral stochastic finite element method in electromagnetism,” IEEE Transactions on Magnetics, vol. 43, no. 4, pp. 1209–1212, 2007. http://dx.doi.org/10.1109/TMAG.2007.892300
K. Beddek, Y. Le Menach, S. Clenet, and O. Moreau, “3-D stochastic spectral finite-element method in static electromagnetism using vector potential formulation,” IEEE Transactions on Magnetics, vol. 47, no. 5, pp. 1250–1253, 2011. http://dx.doi.org/10.1109/TMAG.2010.2076274
R. Gaignaire, S. Clenet, O. Moreau and B. Sudret, “Current calculation in electrokinetics using a spectral stochastic finite element method,” IEEE Transactions on Magnetic, vol. 44, issue 6, pp. 754–757, 2008. http://dx.doi.org/10.1109/TMAG.2008.915801
D.H. Mac, S. Clénet, J.C. Mipo and O. Moreau, “Solution of static field problems with random domains,” IEEE Transactions on Magnetics, vol. 2, no. 8, pp. 3385–3388, 2010.
D.H. Mac, S. Clénet and J.C. Mipo, “Calculation of field distribution in electromagnetic problems with random domains,” IET 8th International Conference on Computation in Electromagnetics (CEM), pp. 1–2, 2011.
B.J. Debusschere, H.N. Najm, P. Pebay, O.M. Knio, R.G. Ghanem and O. Le Maitre, “Numerical challenges in the use of polynomial chaos representations for stochastic processes,” SIAM Journal of Scientific Computing, vol. 26, no. 2, pp. 698–719, 2004. http://dx.doi.org/10.1137/S1064827503427741
A. Arkkio, Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations, Acta Polytechnica Scandinavica, Electrical Engineering series no. 59, Espoo 1987.
M.J. Islam, J. Pippuri, J. Perho and A. Arkkio, “Time-harmonic finite-element analysis of eddy currents in the form-wound stator winding of a cage induction motor,” IET Electric Power Applications, vol. 1, no. 5, pp. 849–846, 2007.
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23.10.2014
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Lehikoinen, A. (2014). Spectral Stochastic Finite Element Method for Electromagnetic Problems with Random Geometry. Electrical, Control and Communication Engineering, 6(1), 5-12. https://doi.org/10.2478/ecce-2014-0011
