PIER | |

Progress In Electromagnetics Research | ISSN: 1070-4698, E-ISSN: 1559-8985 |

Home > Vol. 32 > pp. 151-169
## DATA STRUCTURES FOR GEOMETRIC AND TOPOLOGICAL ASPECTS OF FINITE ELEMENT ALGORITHMSBy P. W. Gross and P. R. Kotiuga
Abstract:
This paper uses simplicial complexes and simplicial (co)homology theory to expose a foundation for data structures for tetrahedral finite element meshes. Identifying tetrahedral meshes with simplicial complexes leads, by means of Whitney forms, to the connection between simplicial cochains and fields in the region modeled by the mesh. Furthermore, lumped field parameters are tied to matrices associated with simplicial (co)homology groups. The data structures described here are sparse, and the computational complexity of constructing them is O(n) where n is the number of vertices in the finite element mesh. Non-tetrahedral meshes can be handled by an equivalent theory. These considerations lead to a discrete form of Poincar´e duality which is a powerful tool for developing algorithms for topological computations on finite element meshes. This duality emerges naturally in the data structures. We indicate some practical applications of both data structures and underlying theory.
2. Bamberg, P. and S. Sternberg, 3. Bossavit, A., 4. Brisson, E., "Representing geometric structures in d dimensions: Topology and order," 5. Brown, M. L., "Scalar potentials in multiply connected regions," 6. Chammas, P. and P. R. Kotiuga, "Sparsity vis a vis Lanczos methods for discrete helicity functionals," 7. Croom, F. H., 8. Dodziuk, J., "Combinatorial and continuous hodge theories," 9. Dodziuk, J., "Finite-difference approach to the hodge theory of harmonic forms," 10. Eilenberg, S. and N. Steenrod, 11. Gross, P. W. and P. R. Kotiuga, "Finite element-based algorithms to make cuts for magnetic scalar potentials: Topological constraints and computational complexity,", this volume.
12. 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," 13. Kotiuga, P. R., "Hodge decompositions and computational electromagnetics,", Ph.D. thesis, 123, McGill University, Montreal, 1984.
14. Kotiuga, P. R., "An algorithm to make cuts for scalar potentials in tetrahedral meshes based on the finite element method," 15. Kotiuga, P. R., "Helicity functionals and metric invariance in three dimensions," 16. Kotiuga, P. R., "Analysis of finite-element matrices arising from discretizations of helicity functionals," 17. Kotiuga, P. R., "Topological duality in three-dimensional eddycurrent problems and its role in computer-aided problem formulation," 18. Kotiuga, P. R., "Essential arithmetic for evaluating three dimensional vector finite element interpolation schemes," 19. Kron, G., "Basic concepts of multidimensional space filters," 20. Maxwell, J. C., 21. Muller, W., "Analytic torsion and r-torsion of riemannian manifolds," 22. Munkres, J. R., 23. Rotman, J. J., 24. Silvester, P. and R. Ferrari, 25. Whitney, H., "On matrices of integers and combinatorial topology," 26. Whitney, H., "r-Dimensional integration in n-space," 27. Whitney, H., |

© Copyright 2014 EMW Publishing. All Rights Reserved