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.
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
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