Vol. 148
Latest Volume
All Volumes
PIER 179 [2024] PIER 178 [2023] PIER 177 [2023] 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]
2014-07-21
Lattice Maxwell's Equations (Invited Paper)
By
Progress In Electromagnetics Research, Vol. 148, 113-128, 2014
Abstract
We discuss the ab initio rendering of four-dimensional (4-d) spacetime of Maxwell's equations on random (irregular) lattices. This rendering is based on casting Maxwell's equations in the framework of the exterior calculus of differential forms, and a translation thereof to a simplicial complex whereby fields and causative sources are represented as differential p-forms and paired with the oriented p-dimensional geometrical objects that comprise the set of spacetime lattice cells (simplices). We pay particular attention to the case of simplicial spacetime lattices because these can serve as building blocks of lattices made of more generic cells (polygons). The generalized Stokes' theorem is used to construct discrete calculus operations on the lattice based upon combinatorial relations depending solely on the connectivity and relative orientation among simplices. This rendering provides a natural factorization of (lattice) 4-d spacetime Maxwell's equations into a metric-free part and a metric-dependent part. The latter is encoded by discrete Hodge star operators which are built using Whitney forms, i.e., canonical interpolants for discrete differential forms. The derivation of Whitney forms is illustrated here using a geometrical construction based on the concept of barycentric coordinates to represent a point on a simplex, and the generalization thereof to represent higher-dimensional objects (lines, areas, volumes, and hypervolumes) in 4-d. We stress the role of the primal lattice, the barycentric dual lattice, and the barycentric decomposition lattice in achieving a complete description of the lattice theory. Lattice Maxwell's equations based on the exterior calculus of differential forms and on the use of Whitney forms as field interpolants inherits the symplectic structure and discrete analogues of conservation laws present in the continuum theory, such as energy and charge conservation. This framework also provides precise localization rules for the degrees of freedom associated with the different fields and sources on the lattice, and design principles for constructing consistent numerical solution methods that are free from spurious modes, spectral pollution, and (unconditional) numerical instabilities. We also brie y consider the relationship between lattice 4-d Maxwell's equations and some incarnations of discretization schemes for Maxwell's equations in (3+1)-d, such as finite-differences and finite-elements.
Citation
Fernando Lisboa Teixeira, "Lattice Maxwell's Equations (Invited Paper)," Progress In Electromagnetics Research, Vol. 148, 113-128, 2014.
doi:10.2528/PIER14062904
References

1. Bott, R., "On some recent interactions between mathematics and physics," Canad. Math. Bull., Vol. 28, No. 2, 129-164, 1985.
doi:10.4153/CMB-1985-016-3

2. Gockeler, M. and T. Schuker, Differential Geometry, Gauge Theories, and Gravity, Cambridge University Press, 1987.
doi:10.1017/CBO9780511628818

3. Burgess, M., Classical Covariant Fields, Cambridge University Press, 2002.
doi:10.1017/CBO9780511535055

4. Zee, A., Quantum Field Theory in a Nutshell, Princeton University Press, Princeton, NJ, 2003.

5. Teixeira, F. L. and W. C. Chew, "Lattice electromagnetic theory from a topological viewpoint," J. Math. Phys., Vol. 40, No. 1, 169-187, 1999.
doi:10.1063/1.532767

6. Teixeira, F. L., "Differential forms in lattice field theories: An overview," ISRN Math. Phys., Vol. 2013, 487270, 2013.

7. Tarhasaari, T., L. Kettunen, and A. Bossavit, "Some realizations of a discrete Hodge operator: A reinterpretation of finite element techniques," IEEE Trans. Magn., Vol. 35, No. 3, 1494-1497, 1999.
doi:10.1109/20.767250

8. Misner, C. W., K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman and Co., New York, 1973.

9. Deschamps, G. A., "Electromagnetics and differential forms," Proc. IEEE, Vol. 69, 676-696, 1982.

10. Schenberg, M., "Electromagnetism and gravitation," Braz. J. Phys., Vol. 1, 91-122, 1971.

11. Warnick, K. F. and P. Russer, "Two, three, and four-dimensional electromagnetics using differential forms," Turk. J. Elec. Engin., Vol. 14, No. 1, 153-172, 2006.

12. Gross, P. W. and P. R. Kotiuga, "Data structures for geometric and topological aspects of finite element algorithms," Progress In Electromagnetics Research, Vol. 32, 151-169, 2001.
doi:10.2528/PIER00080106

13. Teixeira, F. L., "Geometrical aspects of the simplicial discretization of Maxwell’s equations," Progress In Electromagnetics Research, Vol. 32, 171-188, 2001.
doi:10.2528/PIER00080107

14. Tonti, E., "Finite formulation of the electromagnetic field," Progress In Electromagnetics Research, Vol. 32, 1-44, 2001.
doi:10.2528/PIER00080101

15. Gross, P. W. and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, Cambridge University Press, 2004.
doi:10.1017/CBO9780511756337.002

16. Adams, D. H., "R-torsion and linking numbers from simplicial Abelian gauge theories," High Energy Physics — Theory, 9612009, 1996.

17. Sen, S., S. Sen, J. C. Sexton, and D. H. Adams, "Geometric discretization scheme applied to the Abelian Chern-Simons theory," Phys. Rev. E, Vol. 61, No. 3, 3174-3185, 2000.
doi:10.1103/PhysRevE.61.3174

18. Clemens, M. and T. Weiland, "Discrete electromagnetism with the finite integration technique," Progress In Electromagnetics Research, Vol. 32, 65-87, 2001.
doi:10.2528/PIER00080103

19. Schuhmann, R. and T. Weiland, "Conservation of discrete energy and related laws in the finite integration technique," Progress In Electromagnetics Research, Vol. 32, 301-316, 2001.
doi:10.2528/PIER00080112

20. He, B. and F. L. Teixeira, "On the degrees of freedom of lattice electrodynamics," Phys. Lett. A, Vol. 336, No. 1, 1-7, 2005.
doi:10.1016/j.physleta.2005.01.001

21. Kheyfets, A. and W. A. Miller, "The boundary of a boundary in field theories and the issue of austerity of the laws of physics," J. Math. Phys., Vol. 32, No. 11, 3168-3175, 1991.
doi:10.1063/1.529519

22. Guth, A. H., "Existence proof of a nonconfining phase in four-dimensional U(1) lattice field theory," Physical Review D, Vol. 21, No. 8, 2291-2307, 1980.
doi:10.1103/PhysRevD.21.2291

23. Whitney, H., Geometric Integration Theory, Princeton University Press, Princeton, NJ , 1957.

24. Bossavit, A., "Generalized finite differences’ in computational electromagnetics," Progress In Electromagnetics Research, Vol. 32, 45-64, 2001.
doi:10.2528/PIER00080102

25. He, B. and F. L. Teixeira, "Geometric finite element discretization of Maxwell equations in primal and dual spaces," Phys. Lett. A, Vol. 349, No. 1–4, 1-14, 2006.
doi:10.1016/j.physleta.2005.09.002

26. Schwarz, A. S., Topology for Physicists, Springer-Verlag, New York, 1994.
doi:10.1007/978-3-662-02998-5_1

27. Bossavit, A., "Whitney forms: A new class of finite elements for three-dimensional computations in electromagnetics," IEE Proc. A, Vol. 135, 493-500, 1988.

28. Salamon , J., J. Moody, and M. Leok, "Geometric representations of Whitney forms and their generalization to Minkowski spacetime," Numerical Analysis, 1402.7109, 2014.

29. Buffa, A. and S. Christiansen, "A dual finite element complex on the barycentric refinement," Math. Comput., Vol. 76, 1743-1769, 2007.
doi:10.1090/S0025-5718-07-01965-5

30. Osterwalder, K. and R. Schrader, "Axioms for Euclidean Green’s functions," Comm. Math. Phys., Vol. 31, No. 2, 83-112, 1973.
doi:10.1007/BF01645738

31. Montvay, I. and G. Munster, Quantum Fields on a Lattice, Cambridge University Press, 1994.
doi:10.1017/CBO9780511470783

32. Ambjorn, J., J. Jurkiewicks, and R. Loll, "Emergence of a 4D world from causal quantum gravity," Phys. Rev. Lett., Vol. 93, 131301, 2004.
doi:10.1103/PhysRevLett.93.131301

33. Ambjorn, J., A. Gorlich, J. Jurkiewicks, and R. Loll, "Nonperturbative quantum gravity," Phys. Rep., Vol. 519, 127, 2012.
doi:10.1016/j.physrep.2012.03.007

34. Jordan, S. and R. Loll, "Causal dynamical triangulations without preferred foliation," High Energy Physics — Theory, 1305.4582, 2013.

35. Erickon, J., D. Guoy, J. M. Sullivan, and A. Ungor, "Buliding space-time meshes over arbitrary spatial domains," Engg. Computers, Vol. 290, 342-353, 2005.
doi:10.1007/s00366-005-0303-0

36. Thite, S., "Adaptive spacetime meshing fod discontinuous Galerkin methods," Comp. Geom., Vol. 42, No. 1, 20-44, 2009.
doi:10.1016/j.comgeo.2008.07.003

37. Stern, A., Y. Tong, M. Desbrun, and J. E. Mardsen, "Variational integrators for mMxwell’s equations with sources," PIERS Online, Vol. 4, No. 7, 711-715, 2008.
doi:10.2529/PIERS071019000855

38. Kim, J. and F. L. Teixeira, "Parallel and explicit finite-element time-domain method for Maxwell’s equations," IEEE Trans. Antennas Propagat., Vol. 59, No. 6, 2350-2356, 2011.
doi:10.1109/TAP.2011.2143682

39. Tarhasaari, T., L. Kettunen, and A. Bossavit, "Some realizations of the discrete Hodge operator: A reinterpretation of finite element techniques," IEEE Trans. Magn., Vol. 35, No. 3, 1494-1497, 1999.
doi:10.1109/20.767250

40. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell’s equation is isotropic media," IEEE Trans. Antennas Propagat., Vol. 14, No. 3, 302-307, 1969.

41. Taflove, A., Computational Electrodynamics: The Finite-difference Time-domain Method, Artech House, Norwood, MA, 1995.

42. Mattiussi, C., "The geometry of time-stepping," Progress In Electromagnetics Research, Vol. 32, 123-149, 2001.
doi:10.2528/PIER00080105