Helicity functionals and metric invariance in three dimensions

Kotiuga, P. Robert

doi:10.1109/20.34293

Cited by 33 publications

(16 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The basic feature underlying this remarkable possibility is the invariance of Maxwell equations under diffeomorphisms of the metric (metric invariance) [1], [2], [3], [4], [5], i.e., the fact that a change on the metric of space can be mimicked by a proper change of the constitutive tensors. In this work, we discuss how this feature of Maxwell equations is obviated using differential forms and the exterior calculus framework [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. We then illustrate how this feature also allows for generic masking of objects (again, under idealized conditions) via appropriate metamaterial coatings.…”

Section: Introductionmentioning

confidence: 99%

Differential form approach to the analysis of electromagnetic cloaking and masking

Teixeira

2007

Micro & Optical Tech Letters

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Differential form approach to the analysis of electromagnetic cloaking and masking

Teixeira

2007

Micro & Optical Tech Letters

View full text Add to dashboard Cite

show abstract

“…Since a p-simplex has p + 1 p − 1-faces, every p-simplex is found in p + 1 of the sets described in (15). This is equivalent to saying that there are p + 1 nonzero entries per column in ∂ T p .…”

Section: Coboundary Data Structuresmentioning

confidence: 99%

“…In this formulation, it is interesting to note that the contribution of the so-called helicity density ω ∧ dω to the finite element "stiffness" matrix is independent of metric and constitutive laws [15]. In particular, the contribution of the helicity on a tetrahedron to the stiffness matrix is…”

Section: Example: the Helicity Functionalmentioning

confidence: 99%

Data Structures for Geometric and Topological Aspects of Finite Element Algorithms

Gross¹,

Kotiuga²

2001

PIER

View full text Add to dashboard Cite

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 Poincaré 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.

show abstract

“…FEM is traditionally based upon seeking solutions by properly weighting the residual of the second-order vector wave equation, with stability and convergence issues being addressed by variational principles. Another route to derive stable FEM discretizations, first suggested by Bossavit and Kotiuga [6]- [8], and increasingly adopted in recent years [9]- [23], is based on a representation of the electromagnetic field in terms of differential forms [24]- [34]. In this geometric discretization approach, Whitney forms (elements) [6]- [9], [14], [35], [36] become interpolants for cochains (discrete differential forms) [35] representing the discretized electromagnetic field.…”

Section: Introductionmentioning

confidence: 99%

Differential Forms, Galerkin Duality, and Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations

Teixeira

2007

IEEE Trans. Antennas Propagat.

View full text Add to dashboard Cite

Abstract-We identify primal and dual formulations in the finite element method (FEM) solution of the vector wave equation using a geometric discretization based on differential forms. These two formulations entail a mathematical duality denoted as Galerkin duality. Galerkin-dual FEM formulations yield identical nonzero (dynamical) eigenvalues (up to machine precision), but have static (zero eigenvalue) solution spaces of different dimensions. Algebraic relationships among the degrees of freedom of primal and dual formulations are explained using a deep-rooted connection between the Hodge-Helmholtz decomposition of differential forms and Descartes-Euler polyhedral formula, and verified numerically. In order to tackle the fullness of dual formulation, algebraic and topological thresholdings are proposed to approximate inverse mass matrices by sparse matrices.Index Terms-Differential forms, finite element methods (FEMs), Maxwell equations, sparse matrices.

show abstract

Helicity functionals and metric invariance in three dimensions

Cited by 33 publications

References 17 publications

Differential form approach to the analysis of electromagnetic cloaking and masking

Differential form approach to the analysis of electromagnetic cloaking and masking

Data Structures for Geometric and Topological Aspects of Finite Element Algorithms

Differential Forms, Galerkin Duality, and Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations

Contact Info

Product

Resources

About