volume | PIER Journals

Abstract

Software systems designed to solve Maxwell's equations need abstractions that accurately explain what different kinds of electromagnetic problems really do have in common. Computational electromagnetics calls for higher level abstractions than what is typically needed in ordinary engineering problems. In this paper Maxwell's equations are described by exploiting basic concepts of set theory. Although our approach unavoidably increases the level of abstraction,it also simplifies the overall view making it easier to recognize a topological problem behind all boundary value problems modeling the electromagnetic phenomena. This enables us also to construct an algorithm which tackles the topological problem with basic tools of linear algebra.

1. Yosida, K., Functional Analysis, Springer-Verlag,Berlin, 1980.

2. Pierce, J. R., Symbols, Signals and Noise, Harper Torchbooks, New York, 1965.

3. Bossavit, A., "On the geometry of electromagnetism. (1): Euclidean space," J. Japan Soc. Appl. Electromagn. & Mech., Vol. 6, No. 1, 17-28, 1998. Google Scholar

4. Bossavit, A., "On the geometry of electromagnetism. (2): Geometrical objects," J. Japan Soc. Appl. Electromagn. & Mech., Vol. 6, 114-123, 1998. Google Scholar

5. Bossavit, A., "On the geometry of electromagnetism. (3): Integration,Stok es’,F araday’s laws," J. Japan Soc. Appl. Electromagn. & Mech., Vol. 6, 233-240, 1998. Google Scholar

6. Bossavit, A., "On the geometry of electromagnetism. (4): Maxwell’s house," J. Japan Soc. Appl. Electromagn. & Mech., Vol. 6, 318-326, 1998. Google Scholar

7. Meyer, B., Object-Oriented Software Construction, Prentice Hall, Hertfordshire, UK, 1988.

8. Bossavit, A., Computational Electromagnetism, Variational Formulations, Edge Elements, Complementarity, Academic Press, Boston, 1998.

9. de Rham, G., Differentiable Manifolds, Springer,1984. (Translated from the French).

10. Flanders, H., Differential Forms with Applications to the Physical Sciences, Dover Publications, 1989.

11. Bossavit, A., "On non-linear magnetostatics: dualcomplementary models and ‘mixed’ numerical methods," Proc. of European Conf. on Math. in Ind., Glasgow, August 1988, J. Manby (ed.), 3–16, Teubner, Stuttgart, 1990. Google Scholar

12. Bossavit, A., "Differen tial forms and the computation of fields and forces in electromagnetism," European J. Mech., B/Fluids, Vol. 10, No. 5, 474-488, 1991. Google Scholar

13. Lefschetz, S., Algebraic Topology, No. 27, A.M.S. Colloquim Publication, New York, 1942.

14. Cartan, H. and S. Eilenberg, Homological Algebra, Princeton Univ. Press, 1952.

15. Hilton, P. J. and S. Wylie, "Homology Theory, An Introduction to Algebraic Topology," Cambridge University Press, Cambridge, 1965. Google Scholar

16. Hocking, J. G. and G. S. Young, "Topology," Addison-Wesley Pub. Co., Reading, Mass., 1961. Google Scholar

17. Halmos, P. R., "Naive Set Theory," Springer-Verlag, 1974. Google Scholar

18. Kotiuga, P. R., "Hodge decompositions and computational electromagnetics,", Ph.D. thesis, McGill University, Montreal, Canada, 1984. Google Scholar

19. Bossavit, A., "Most general ‘non-local’ boundary conditions for the Maxwell equations in a bounded region," Proceedings of ISEF’99 -9th Int. Symp. Elec. Magn. Fields. in Elec. Eng., Pavia, Italy, 1999. Google Scholar

20. Connor, P. E., "The Neumann’s problem for differential forms on Riemannian manifolds," Memoirs of the AMS, No. 20, 1956. Google Scholar

21. Vick, J. W., Homology Theory, Academic Press, New York, 1973.

22. Stillwell, J., Classical Topology and Combinatorial Group Theory, Springer-Verlag, New York, 1980.
doi:10.1007/978-1-4684-0110-3

23. Markus, H. and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn & Bacon, 1964.

24. Kondo, K. and M. Iri, "On the theory of trees, cotrees, multitrees, and multicotrees," RAAG Memoirs, II, No. Article A-VII, 220–261, 1959. Google Scholar

Dr. Timo Tarhasaari

Dr. Lauri Kettunen