Electromagnetics and differential forms

Deschamps, G.

doi:10.1109/proc.1981.12048

Cited by 239 publications

(173 citation statements)

References 8 publications

Supporting

Mentioning

170

Contrasting

Unclassified

Order By: Relevance

“…As explained above, X 0 k ( T ) = DP 0 k+1 ( T ), so that the dimension of X 0 k ( T ) is equal to the dimension of the space of multivariate polynomials of degree ≤ k + 1. Thus, the assertion of the theorem for the case l = 0 follows from (15).…”

Section: Theorem 6 For T ⊂ R N the Dimensions Of The Local Ansatz Spmentioning

confidence: 83%

See 1 more Smart Citation

Untitled

1999

Math. Comp.

View full text Add to dashboard Cite

Abstract. The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H(curl;Ω) and H(Div;Ω). Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes.Given a simplicial triangulation of the computational domain Ω, among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for H(Div;Ω) and H(curl;Ω). At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: div, curl and grad can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent.Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar H(Div;Ω)-and H(curl;Ω)-conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

show abstract

Section: Theorem 6 For T ⊂ R N the Dimensions Of The Local Ansatz Spmentioning

confidence: 83%

“…Motivated by the elegant formulation of Maxwell's equations in the calculus of differential forms [2,15], discretizations were sought that fit this point of view. First order schemes, called Whitney forms, were proposed as conforming finite element methods for spaces of differential forms [4,25,30].…”

Section: Introductionmentioning

confidence: 99%

Untitled

1999

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…Following De Rham [6] {see [7] for the physical applications} and Meetz-Engl [8], Deschamps [9] introduced in electromagnetism the notion of exterior differential forms, gene-rating numerous works [10][11][12][13][14][15][16] on this subject. Exterior means that the algebra of differential forms is endowed with the Cartan exterior derivative d assigning to a p-form w (in this Note, differential forms are represented by underlined expressions) a (p + 1)-form dw such that d is linear and…”

Section: Electromagnetic Differential Formsmentioning

confidence: 99%

Spinor and Hertzian Differential Forms in Electromagnetism

Hillion¹

2011

PIER M

View full text Add to dashboard Cite

Abstract-The purpose of this paper is to extend to spinor electromagnetism the differential forms, based on the Cartan exterior derivative and originally developed for tensor fields, in a very compact way. To this end, differential electromagnetic forms are first compared to conventional tensors. Then, using the local isomorphism between the O(3,C) and SL(2,C) groups supplying the well known connection between complex vectors and traceless second rank spinors, they are generalized to spinor electromagnetism and to Proca fields. These differential forms are finally expressed in terms of Hertz potentials.

show abstract

“…And they are both based on the vectorial version of Maxwell's equations. As is well known, differential forms can be used to recast Maxwell's equations in a more succinct fashion, which completely separates metric-free and metric-dependent components [4][5][6][7][8][9]. Discrete exterior calculus (DEC), a discrete version of differential geometry, provides a numerical treatment of differential forms directly [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…It should be noted that the only limitation of our approach is that without Delaunay triangulation, the positivity of Laplacian cannot be guaranteed [13,14], which will lead to instability of time domain analysis. However it is still of interest to investigate if a self-consistent and self-contained discrete electromagnetic theory can be developed on simplicial mesh using DEC, which has been accomplished based on Yee grid [3,4]. In this paper, we show that many electromagnetic theorems still exist in this discrete electromagnetic theory with DEC, such as Huygens' principle, reciprocity theorem, Poynting's theorem and uniqueness theorem.…”

Section: Introductionmentioning

confidence: 99%

Electromagnetic Theory With Discrete Exterior Calculus

Chen¹,

Chew²

2017

PIER

View full text Add to dashboard Cite

Abstract-A self-contained electromagnetic theory is developed on a simplicial lattice. Instead of dealing with vectorial field, discrete exterior calculus (DEC) studies the discrete differential forms of electric and magnetic fields, and circumcenter dual is adopted to achieve diagonal Hodge star operators. In this paper, Gauss' theorem and Stokes' theorem are shown to be satisfied inherently within DEC. Many other electromagnetic theorems, such as Huygens' principle, reciprocity theorem, and Poynting's theorem, can also be derived on this simplicial lattice consistently with an appropriate definition of wedge product between cochains. The preservation of these theorems guarantees that this treatment of Maxwell's equations will not lead to spurious solutions.

show abstract

Electromagnetics and differential forms

Cited by 239 publications

References 8 publications

Untitled

Untitled

Spinor and Hertzian Differential Forms in Electromagnetism

Electromagnetic Theory With Discrete Exterior Calculus

Contact Info

Product

Resources

About