Vol. 32
Latest Volume
All Volumes
PIER 176 [2023] PIER 175 [2022] PIER 174 [2022] PIER 173 [2022] PIER 172 [2021] PIER 171 [2021] PIER 170 [2021] PIER 169 [2020] PIER 168 [2020] PIER 167 [2020] PIER 166 [2019] PIER 165 [2019] PIER 164 [2019] PIER 163 [2018] PIER 162 [2018] PIER 161 [2018] PIER 160 [2017] PIER 159 [2017] PIER 158 [2017] PIER 157 [2016] PIER 156 [2016] PIER 155 [2016] PIER 154 [2015] PIER 153 [2015] PIER 152 [2015] PIER 151 [2015] PIER 150 [2015] PIER 149 [2014] PIER 148 [2014] PIER 147 [2014] PIER 146 [2014] PIER 145 [2014] PIER 144 [2014] PIER 143 [2013] PIER 142 [2013] PIER 141 [2013] PIER 140 [2013] PIER 139 [2013] PIER 138 [2013] PIER 137 [2013] PIER 136 [2013] PIER 135 [2013] PIER 134 [2013] PIER 133 [2013] PIER 132 [2012] PIER 131 [2012] PIER 130 [2012] PIER 129 [2012] PIER 128 [2012] PIER 127 [2012] PIER 126 [2012] PIER 125 [2012] PIER 124 [2012] PIER 123 [2012] PIER 122 [2012] PIER 121 [2011] PIER 120 [2011] PIER 119 [2011] PIER 118 [2011] PIER 117 [2011] PIER 116 [2011] PIER 115 [2011] PIER 114 [2011] PIER 113 [2011] PIER 112 [2011] PIER 111 [2011] PIER 110 [2010] PIER 109 [2010] PIER 108 [2010] PIER 107 [2010] PIER 106 [2010] PIER 105 [2010] PIER 104 [2010] PIER 103 [2010] PIER 102 [2010] PIER 101 [2010] PIER 100 [2010] PIER 99 [2009] PIER 98 [2009] PIER 97 [2009] PIER 96 [2009] PIER 95 [2009] PIER 94 [2009] PIER 93 [2009] PIER 92 [2009] PIER 91 [2009] PIER 90 [2009] PIER 89 [2009] PIER 88 [2008] PIER 87 [2008] PIER 86 [2008] PIER 85 [2008] PIER 84 [2008] PIER 83 [2008] PIER 82 [2008] PIER 81 [2008] PIER 80 [2008] PIER 79 [2008] PIER 78 [2008] PIER 77 [2007] PIER 76 [2007] PIER 75 [2007] PIER 74 [2007] PIER 73 [2007] PIER 72 [2007] PIER 71 [2007] PIER 70 [2007] PIER 69 [2007] PIER 68 [2007] PIER 67 [2007] PIER 66 [2006] PIER 65 [2006] PIER 64 [2006] PIER 63 [2006] PIER 62 [2006] PIER 61 [2006] PIER 60 [2006] PIER 59 [2006] PIER 58 [2006] PIER 57 [2006] PIER 56 [2006] PIER 55 [2005] PIER 54 [2005] PIER 53 [2005] PIER 52 [2005] PIER 51 [2005] PIER 50 [2005] PIER 49 [2004] PIER 48 [2004] PIER 47 [2004] PIER 46 [2004] PIER 45 [2004] PIER 44 [2004] PIER 43 [2003] PIER 42 [2003] PIER 41 [2003] PIER 40 [2003] PIER 39 [2003] PIER 38 [2002] PIER 37 [2002] PIER 36 [2002] PIER 35 [2002] PIER 34 [2001] PIER 33 [2001] PIER 32 [2001] PIER 31 [2001] PIER 30 [2001] PIER 29 [2000] PIER 28 [2000] PIER 27 [2000] PIER 26 [2000] PIER 25 [2000] PIER 24 [1999] PIER 23 [1999] PIER 22 [1999] PIER 21 [1999] PIER 20 [1998] PIER 19 [1998] PIER 18 [1998] PIER 17 [1997] PIER 16 [1997] PIER 15 [1997] PIER 14 [1996] PIER 13 [1996] PIER 12 [1996] PIER 11 [1995] PIER 10 [1995] PIER 09 [1994] PIER 08 [1994] PIER 07 [1993] PIER 06 [1992] PIER 05 [1991] PIER 04 [1991] PIER 03 [1990] PIER 02 [1990] PIER 01 [1989]
0000-00-00
Finite Element-Based Algorithms to Make Cuts for Magnetic Scalar Potentials: Topological Constraints and Computational Complexity
By
, Vol. 32, 207-245, 2001
Abstract
This paper outlines a generic algorithm to generate cuts for magnetic scalar potentials in 3-dimensional multiply-connected finite element meshes. The algorithm is based on the algebraic structures of (co)homology theory with differential forms and developed in the context of the finite element method and finite element data structures. The paper also studies the computational complexity of the algorithm and examines how the topology of the region can create an obstruction to finding cuts in O(m20) time and O(m0) storage, where m0 is the number of vertices in the finite element mesh. We argue that in a problem where there is no a priori data about the topology, the algorithm complexity is O(m20) in time and O(m4/30 ) in storage. We indicate how this complexity can be achieved in implementation and optimized in the context of adaptive mesh refinement.
Citation
Paul W. Gross Peter Robert Kotiuga , "Finite Element-Based Algorithms to Make Cuts for Magnetic Scalar Potentials: Topological Constraints and Computational Complexity," , Vol. 32, 207-245, 2001.
doi:10.2528/PIER00080109
http://www.jpier.org/PIER/pier.php?paper=00080109
References

1. Adkins, W. A. and S. H. Weintraub, Algebra: An Approach via Module Theory, 307-327, Springer-Verlag, New York, 1992.

2. Armstrong, M., Basic Topology, Springer-Verlag, New York, 1983.
doi:10.1007/978-1-4757-1793-8

3. Balabanian, N. and T. A. Bickart, Electrical Network Theory, 80, John Wiley and Sons, New York, 1969.

4. Bamberg, P. and S. Sternberg, A Course in Mathematics for Students of Physics: 2, Ch. 12, Cambridge U. Press, NY, 1990.

5. Bossavit, A., A. Vourdas, and K. J. Binns, "Magnetostatics with scalar potentials in multiply connected regions," IEE Proc. A, Vol. 136, 260-261, 1989.

6. Bott, R. and L. W. Tu, Differential Forms in Algebraic Topology, 40-42, 51, 234, 258, 240, Springer-Verlag, New York, 1982.
doi:10.1007/978-1-4757-3951-0

7. Brown, M. L., "Scalar potentials in multiply connected regions," Int. J. Numer. Meth. Eng., Vol. 20, 665-680, 1984.
doi:10.1002/nme.1620200406

8. Cohen, H., A Course in Computational Algebraic Number Theory, Springer-Verlag, New York, 1993.
doi:10.1007/978-3-662-02945-9

9. Coleman, T. F., A. Edenbrandt, and J. R. Gilbert, "Predicting fill for sparse orthogonal factorization," Journal of the Association for Computing Machinery, Vol. 33, 517-532, 1986.
doi:10.1145/5925.5932

10. Croom, F. H., Basic Concepts of Algebraic Topology, Chaps. 2, 7.3, 4.5, Springer-Verlag, New York, 1978.
doi:10.1007/978-1-4684-9475-4

11. Deschamps, G. A., "Electromagnetics and differential forms," IEEE Proc., Vol. 69, 676-696, 1981.
doi:10.1109/PROC.1981.12048

12. Greenberg, M. J. and J. R. Harper, Algebraic Topology, 235, 63–66 Benjamin/Cummings, Reading, MA, 1981.

13. Gross, P. W., "The commutator subgroup of the first homotopy group and cuts for scalar potentials in multiply connected regions,", Master’s thesis, Dept. of Biomed. Eng., Boston U., September 1993.

14. Gross, P. W. and P. R. Kotiuga, "Data structures for geometric and topological aspects of finite element algorithms,", this volume.

15. Gross, P. W. and P. R. Kotiuga, "A challenge for magnetic scalar potential formulations of 3-d eddy current problems: Multiply connected cuts in multiply connected regions which necessarily leave the cut complement multiply connected," Electric and Magnetic Fields: From Numerical Models to Industrial Applications, A. Nicolet and R. Belmans (eds.), Plenum, 1–20, New York, 1995. Proc. of the Second Int. Workshop on Electric and Magnetic Fields.

16. Guillemin, V. and A. Pollack, Differential Topology, 21, Prentice- Hall, Englewood Cliffs, New Jersey, 1974.

17. Gunning, R. C. and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965.

18. Harrold, C. S. and J. Simkin, "Cutting multiply connected domains," IEEE Trans. Magn., Vol. 21, 2495-2498, 1985.
doi:10.1109/TMAG.1985.1064142

19. Kotiuga, P. R., "Hodge decompositions and computational electromagnetics,", Ph.D. thesis, McGill University, Montreal, 1984.

20. Kotiuga, P. R., "On making cuts for magnetic scalar potentials in multiply connected regions," J. Appl. Phys., Vol. 61, 3916-3918, 1987.
doi:10.1063/1.338583

21. Kotiuga, P. R., "An algorithm to make cuts for scalar potentials in tetrahedral meshes based on the finite element method," IEEE Trans. Magn., Vol. 25, 4129-4131, 1989.
doi:10.1109/20.42544

22. Kotiuga, P. R., "Topological considerations in coupling scalar potentials to stream functions describing surface currents," IEEE Trans. Magn., Vol. 25, 2925-2927, 1989.
doi:10.1109/20.34326

23. Kotiuga, P. R., "Analysis of finite-element matrices arising from discretizations of helicity functionals," J. Appl. Phys., Vol. 67, 5815-5817, 1990.
doi:10.1063/1.345973

24. Kotiuga, P. R., "Topological duality in three-dimensional eddycurrent problems and its role in computer-aided problem formulation," J. Appl. Phys., Vol. 67, 4717-4719, 1990.
doi:10.1063/1.344812

25. Kotiuga, P. R., "Essential arithmetic for evaluating three dimensional vector finite element interpolation schemes," IEEE Trans. Magn., Vol. 27, 5208-5210, 1991.
doi:10.1109/20.278789

26. Kotiuga, P. R., A. Vourdas, and K. J. Binns, "Magnetostatics with scalar potentials in multiply connected regions," IEE Proc. A, Vol. 137, 231-232, 1990.

27. Maxwell, J. C., A Treatise on Electricity and Magnetism (1891), Chap. 1, Art. 18–22, Dover, New York, 1954.

28. Munkres, J. R., Elements of Algebraic Topology, 377-380, Addison-Wesley, Reading, MA, 1984.

29. Murphy, A., "Implementation of a finite element based algorithm to make cuts for magnetic scalar potentials,", Master’s thesis, Dept. of ECS Eng., Boston U., 1991.

30. Pothen, A. and C.-J. Fan, "Computing the block triangular form of a sparse matrix," ACM Transactions on Mathematical Software, Vol. 16, 303-324, 1990.
doi:10.1145/98267.98287

31. Ren, Z. and A. Razek, "Boundary edge elements and spanning tree technique in three-dimensional electromagnetic field computation," Int. J. Num. Meth. Eng., Vol. 36, 2877-2893, 1993.
doi:10.1002/nme.1620361703

32. Rotman, J. J., An Introduction to Algebraic Topology, Springer- Verlag, NY, 1988.
doi:10.1007/978-1-4612-4576-6

33. Saitoh, I., "Perturbed H-method without the Lagrange multiplier for three dimensional nonlinear magnetostatic problems," IEEE Trans. Magn., Vol. 30, 4302-4304, 1994.
doi:10.1109/20.334068

34. Silvester, P. and R. Ferrari, Finite Elements for Electrical Engineers, 2nd Edition, Cambridge U. Press, NY, 1990.

35. Stillwell, J., Classical Topology and Combinatorial Group Theory, Second Edition, Ch. 3,4, Springer-Verlag, NY, 1993.
doi:10.1007/978-1-4612-4372-4

36. Thurston, W. P., Three-Dimensional Geometry and Topology, Princeton University Press, Princeton, New Jersey, 1997.
doi:10.1515/9781400865321

37. Vourdas, A., K. J. Binns, and , "Magnetostatics with scalar potentials in multiply connected regions," IEE Proc. A, Vol. 136, 49-54, 1989.