Basis Functions With Divergence Constraints for the Finite Element Method

Pinciuc, C. M.; Konrad, A.; Lavers, J.D.

doi:10.1109/tmag.2013.2288608

Cited by 3 publications

(2 citation statements)

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“…40 Within the HFEM framework for 3D calculations, we now have the luxury of explicitly imposing a zero-divergence condition at each node while using Hermite interpolation polynomials since we have derivative degrees of freedom there. [41][42][43] While this does not ensure the complete removal of the divergence in the interior of the finite element through interpolation, it reduces it substantially, especially as the size of the element is reduced. In this article, we use a constant penalty factor, and impose zero-divergence at all nodes to identify the spurious solutions for elimination.…”

Section: The Proposed Hermite Finite Element Methodsmentioning

confidence: 99%

“…We resolve this issue by explicitly imposing the divergencefree condition at each node, using the derivative degrees of freedom. 41,42,49 At the matrix level, one of the terms in the zero-divergence condition, ∇ • ε r E = 0, is eliminated in favor of the other two. The procedure is demonstrated in Fig.…”

Section: B the Penalty Methods And The Zero-divergence Constraintmentioning

confidence: 99%

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Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Pham,

Bharadwaj,

Ram-Mohan

2019

Preprint

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Present day electromagnetic field calculations have basic limitations since they employ techniques based on edge-based discretization methods. While these vector finite element methods (VFEM) solve the issues of tangential continuity of fields and the removal of spurious solutions, the resulting fields do not have a unique directionality at nodes in the discretization mesh. This review presents three calculations of electromagnetic fields: (i) waveguides, (ii) cavity fields, and (iii) photonic crystals. We have developed Hermite interpolation polynomials and node-based finite element methods in the framework of variational principles. We show that the Hermite-finite element method (HFEM) better accuracy with lower computational cost, and provides field directional continuity across the discretized space. It also permits multiscale calculations with mixed physics. For example, the nanoscale modeling of quantum well semiconductor laser structures has to be done together with cavity electrodynamics at the micron scale in vertical-cavity surface emitting lasers and other optoelectronic systems. These can be treated with high accuracy with the new HFEM, which is also applicable to quantum mechanical simulations in meso-and nano-scale systems. We use group representation theory to derive the HFEM polynomial basis set in two dimensions. In three dimensions we derive these polynomials using usual methods. We show that degeneracies in the frequency spectrum in a cubic cavity can be denumerably large even though the symmetry of the cube, O h , supports only singlets, doublets, or triplets. The additional operators available for the problem explains the origin of this "accidental degeneracy." We discuss this remarkable degeneracy and its reduction in detail. We consider photonic crystals corresponding to a 2D checkerboard superlattice structure, and the Escher drawing of "The Horsemen" which satisfies the nonsymmorphic group pg. We show that HFEM is able to deliver high accuracy in such spatially complex examples with far less computational effort than Fourier expansion methods. Finite element analysis employs geometric discretization and hence transcends geometrical limitations. Techniques explained here can be immediately extended to realistic and geometrically complex structures. The new algorithms developed here hold the promise of successful modeling of multi-physics systems. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.

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Section: The Proposed Hermite Finite Element Methodsmentioning

confidence: 99%

Section: B the Penalty Methods And The Zero-divergence Constraintmentioning

confidence: 99%

Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Pham,

Bharadwaj,

Ram-Mohan

2019

Preprint

View full text Add to dashboard Cite

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Cavity electrodynamics with Hermite interpolation: Role of symmetry and degeneracies

Pandey

Bharadwaj

Santia

et al. 2018

Journal of Applied Physics

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We show that the present approaches for the solution of Maxwell’s equations in complex geometries have limitations that can be overcome through the use of C(1)-continuous Hermite interpolation polynomials. Our approach of calculating fields using the Hermite finite element method yields better accuracy by several orders of magnitude than comparable applications of the edge-based vector finite element method. We note that the vector finite element that is widely used yields pixelated solutions and ill-defined vector solutions at nodes. Our solutions have a smooth representation within and across the elements and well defined directions for the fields at the nodes. We reexamine the issue of removing spurious zero-frequency solutions. We investigate fields in an empty cubic metallic cavity and explain the level degeneracy that is larger than what is to be expected from the geometrical Oh symmetry of the cube. This behavior is identified as an example of “accidental degeneracy” and is explained in detail. We show that the inclusion of a smaller dielectric cube of relative permittivity ϵ2 within the cubic cavity leads to the removal of this accidental degeneracy so that the eigenfields have the symmetry Oh. A further reduction of symmetry is obtained by allowing the dielectric function in the enclosed cube to have a linear dependence on z. The proposed method should be effective in obtaining results for scalar-vector coupled field problems such as in modeling quantum well cavity lasers and in plasmonics modeling, while allowing multi-scale physical calculations.

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Physical DC Modes in the Microwave Resonator With Complex Geometric Topology

Jiang

2019

IEEE Trans. Magn.

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Basis Functions With Divergence Constraints for the Finite Element Method

Abstract: Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.iii DedicationTo Professor Konrad and Professor Lavers, and to Michael.iv

Cited by 3 publications

References 49 publications

Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Cavity electrodynamics with Hermite interpolation: Role of symmetry and degeneracies

Physical DC Modes in the Microwave Resonator With Complex Geometric Topology

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