482
Views
10
CrossRef citations to date
0
Altmetric
Articles

Beyond DIV, CURL and GRAD: modelling electromagnetic problems using algebraic topology

Pages 121-149 | Received 26 Sep 2016, Accepted 26 Oct 2016, Published online: 22 Nov 2016

References

  • Deschamps GA. Electromagnetics and differential forms. Proc IEEE. 1981;69:676–696.
  • Baldomir D. Differential forms and electromagnetism in 3-dimensional Euclidean space ℝ3. IEE Proc. 1986;133:139–143.
  • Burke WL. Div, grad, curl are dead, version 2.0, October 1995. Accessed online August 2016. Available from: https://people.ucsc.edu/ rmont/papers/Burke_DivGradCurl.pdf.
  • Burke WL. Applied differential geometry. Cambridge University Press; 1985.
  • Roth JP. An application of algebraic topology to numerical analysis: on the existence of a solution to the network problem. Proc Nat Acad Sci PNAS. 1955;41:518–521.
  • Langefors B. Algebraic topology and networks. Technical Report TN 43. Svenska Aeroplan Aktiebolaget; 1959.
  • Branin FH Jr. The algebraic-topological basis for network analogies and the vector calculus. In: Symposium on Generalized Networks; April; Polytechnic Institute of Brooklyn; 1966.
  • Tonti E. Finite formulation of the electromagnetic field. Progress in electromagnetics research. 2001;32:1–44.
  • Tarhasaari T, Kettunen L. Topological approach to computational electromagnetism. Progr Electromagn Res. 2001;32:189–206.
  • Gross PW, Kotiuga PR. Electromagnetic theory and computation: a topological approach. Cambridge University Publications; 2004.
  • Miyazaki Y. Direct and intuitive analysis of electric networks. In: Kondo K, editor. RAAG memoirs: unifying study of the basic problems in engineering sciences by means of geometry. Vol. 1. Gakujutsu Bunken Fukyu-Kai; 1955. p. 113–171.
  • Kondo K, Iri M. On the theory of trees, cotrees, multi-trees and multi-cotrees. In: Kondo K, editor. RAAG memoirs: unifying study of the basic problems in engineering sciences by means of geometry. Vol. 2. Gakujutsu Bunken Fukyu-Kai; 1958. p. 220–261.
  • Mizoo Y, Iri M, Kondo K. On the torsion and linkage characteristics and the duality of electric, magnetic and dielectric networks. In: Kondo K, editor. RAAG memoirs: unifying study of the basic problems in engineering sciences by means of geometry. Vol. 2. Gakujutsu Bunken Fukyu-Kai; 1958. p. 262–295.
  • Seshu S, Balabanian N. Linear network analysis. Wiley; 1959.
  • Koenig HE, Blackwell WA. Linear graph theory – a fundamental engineering discipline. I R E Trans Edu. 1960;E-3:42–49.
  • Seshu S, Reed MB. Linear graphs and electrical networks. Addison-Wesley; 1961.
  • Karunakaran T. Algebraic structure of network topology. Proc Indian Nat Sci Acad. 1975;41:213–215.
  • Greenspan D. Discrete models. Addison-Wesley Company; 1973.
  • Sommerfeld A. Electrodynamics – lectures on theoretical physics. Vol. 3. Academic Press; 1952.
  • Vagner ID, Lembrikov BI, Wyder P. Electrodynamics of magnetoactive media. Springer-Verlag; 2004.
  • Stratton JA. Electromagnetic theory. McGraw-Hill Book Company; 1941.
  • Smythe WR. Static and dynamic electricity. 2nd ed. McGraw-Hill Book Company; 1950.
  • Pohl RW. Electricity and magnetism. Blackie & Son Limited; 1930.
  • Patterson EM. Topology. Oliver and Boyd Ltd.; 1959.
  • Fomenko A. Visual geometry and topology. Springer-Verlag; 1994.
  • Branin FH Jr. The network concept as a unifying principle in engineering and the physical sciences. In: Husseyin K, Branin FH Jr, editors. Problem analysis in science and engineering. Academic Press; 1977. p. 41–111.
  • Bamberg P, Sternberg S. A course in mathematics for students of physics. Vol. 1 & 2. Cambridge University Press; 1991.
  • Tonti E. The mathematical structure of classical and relativistic physics. Springer; 2013.
  • Grady LJ, Polimeni JR. Discrete calculus – applied analysis on graphs for computational science. Springer-Verlag; 2010.
  • Schouten JA. Tensor analysis for physicists. Clarendon Press; 1954.
  • Mattiussi C. An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J Comput Phys. 1997;133:289–309.
  • Gerritsma M, Hiemstra R, Kreeft J, et al. Lattice electromagnetic theory from a topological viewpoint. In: Lecture notes in computational science and engineering, spectral and high order methods for partial differential equations – ICOSAHOM 2012. Vol. 95. Springer International Publishing; 2014. p. 17–35.
  • Teixeira FL, Chew WC. Lattice electromagnetic theory from a topological viewpoint. J Math Phys. 1999;40:169–185.
  • Sankaran K. Accurate domain truncation techniques for time-domain conformal methods [doctoral dissertation]. Swiss Federal Institute of Technology ETH Zurich; 2007.
  • Madsen NK. Divergence preserving discrete surface integral methods for Maxwell’s curl equations using nonorthogonal unstructured grids. J Comput Phys. 1995;119:34–45.
  • Tonti E. Why starting with differential equations for computational physics. J Comput Phys. 2014;257:1260–1290.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.