Geometric finite element discretization of Maxwell equations in primal and dual spaces

He, Bo; Teixeira, F.L.

doi:10.1016/j.physleta.2005.09.002

Cited by 65 publications

(68 citation statements)

References 14 publications

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“…The indices in (53) and (54) run over the edges and faces of the (spatial-slice) primal lattice, respectively. Discretization of (46) and (47) follows the same basic tenets discussed before in the 4-d case and leads to [5,6,25] …”

Section: (3 + 1)-d Theory With Degeometrized Timementioning

confidence: 92%

“…to the time axis. In the Riemannian case and using the parlance of finite elements, the vector representations of Whitney 1-forms and 2-forms recover the so-called edge and face vector basis functions respectively, see e.g., [25,27]. On the other hand, in the Lorentzian case, the vector representation of a Whitney 1-form is exemplified for the (1 + 1)-d Minkowski spacetime in [28].…”

Section: Interpolants For Discrete Differential Formsmentioning

confidence: 99%

“…as a discrete realization of the Hodge star operator [6,25] on a 4-d simplicial lattice with Minkowski spacetime background. Note that, from (32) and (35), one can equivalently write…”

Section: Discrete Action and Discrete Hodge Representationsmentioning

confidence: 99%

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LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Teixeira¹

2014

PIER

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Abstract-We discuss the ab initio rendering 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 pdimensional 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. It 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 briefly 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.

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Section: (3 + 1)-d Theory With Degeometrized Timementioning

confidence: 92%

Section: Interpolants For Discrete Differential Formsmentioning

confidence: 99%

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LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Teixeira¹

2014

PIER

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“…In differential forms [22][23][24][25][26][27][28][29][30], Maxwell's equations (source-free case) are written as…”

Section: Differential Formsmentioning

confidence: 99%

Differential Forms Inspired Discretization for Finite Element Analysis of Inhomogeneous Waveguides

Dai¹,

Chew²,

Jiang³

2013

PIER

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Abstract-We present a differential forms inspired discretization for variational finite element analysis of inhomogeneous waveguides. The variational expression of the governing equation involves transverse fields only. The conventional discretization with edge elements yields an unsolvable generalized eigenvalue problem since one of the sparse matrix is singular. Inspired by the differential forms where the Hodge operator transforms 1-forms to 2-forms, we propose to discretize the electric and magnetic field with curl-conforming basis functions on the primal and dual grid, and discretize the magnetic flux density and electric displacement field with the divergence-conforming basis functions on the primal and dual grid, respectively. The resultant eigenvalue problem is well-conditioned and easy to solve. The proposed scheme is validated by several numerical examples.

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“…However, the computation of interpolation coefficients of global variables E and B has to solve two matrix equations at each time step in this approach, which makes the calculation extremely expensive in a long period time simulation. Although reference [6] offered an improved scheme that only one matrix equation is required to be solved, this defect still limits its development and application.…”

Section: Introductionmentioning

confidence: 99%

A Fast Explicit Fetd Method Based on Compressed Sensing

Qi¹,

Chen²,

Huang³

et al. 2017

PIER M

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Abstract-Linear equations must be solved at each time step as the explicit finite element time-domain (FETD) method is used to solve time dependent Maxwell curl equations, which leads to a huge amount of computational cost in a long period time simulation. A new scheme to accelerate the iteration solution for matrix equation is proposed based on compressed sensing (CS), in which a low rank measurement matrix is established by randomly extracting rows from mass matrix. Meanwhile, to reduce the number of measurements required, a sparse transform is constructed with the help of prior knowledge offered by the solution results of previous time steps. Numerical results of homogeneous cavity and inhomogeneous cavity are discussed to validate the effectiveness and accuracy of the proposed approach.

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Geometric finite element discretization of Maxwell equations in primal and dual spaces

Cited by 65 publications

References 14 publications

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Differential Forms Inspired Discretization for Finite Element Analysis of Inhomogeneous Waveguides

A Fast Explicit Fetd Method Based on Compressed Sensing

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