Tree-Cotree Decompositions for First-Order Complete Tangential Vector Finite Elements

Manges, John; Cendes, Z.J.

doi:10.1002/(sici)1097-0207(19970515)40:9<1667::aid-nme133>3.0.co;2-9

Cited by 17 publications

(19 citation statements)

References 16 publications

(13 reference statements)

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“…These spurious solutions have to be suppressed by the explicit introduction of the divergence-free condition over the solution domain of the discretized problem. Thus the modified problem now has the form in in on PEC on PMC on (5) Following the analysis of Manges and Cendes [18] and Venkatarayalu and Lee [1], the divergence condition of the electric field can be imposed by exploiting the graph characteristics of the FEM mesh. Dividing appropriately the mesh in tree and cotree edges, the finite-element base is decomposed into its solenoidal and irrotational part.…”

Section: )mentioning

confidence: 99%

“…First-order ABCs, (17) and second-order ABCs, (18) where the subscript denotes the normal, while the subscript denotes the tangential components, respectively.…”

Section: Energy Leakage-radiation Lossesmentioning

confidence: 99%

“…The secondorder ABCs equation (18) consists of three terms, the first which is identical to the term of first-order ABCs and two additional terms. For the first term, the constraint enforcement has been shown in Section IV-A.1.…”

Section: ) Constrained Second-order Abcs Equationmentioning

confidence: 99%

See 2 more Smart Citations

DC and Imaginary Spurious Modes Suppression for Both Unbounded and Lossy Structures

Zekios

Allilomes

Kyriacou

2015

IEEE Trans. Microwave Theory Techn.

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A generalization of the tree-cotree technique for the removal of imaginary and dc spurious modes in finite-element-based eigenanalysis of 3-D lossy unbounded structures is introduced. Five frequently encountered types of polynomial eigenvalue problems are tackled including: 1) closed structures with finite metals conductivity losses; 2) closed structures with material losses due to migrating charge carriers; 3) open-radiating structures using the absorbing boundary conditions of both first-and second-order; 4) open-radiating structures with finite conductivity metallic objects; and 5) any combination of the aforementioned cases. The resulting polynomial eigenvalue problems are linearized utilizing both the companion and the symmetric approaches. The different linearization techniques are being compared for their efficiency and robustness.

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Section: )mentioning

confidence: 99%

“…First-order ABCs, (17) and second-order ABCs, (18) where the subscript denotes the normal, while the subscript denotes the tangential components, respectively.…”

Section: Energy Leakage-radiation Lossesmentioning

confidence: 99%

See 1 more Smart Citation

DC and Imaginary Spurious Modes Suppression for Both Unbounded and Lossy Structures

Zekios

Allilomes

Kyriacou

2015

IEEE Trans. Microwave Theory Techn.

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“…Discretizations involving edge shape functions have been introduced in this context, since only the continuity of the tangential component of the vector potential is required [7]. In this case the gauge condition is more conveniently imposed by removing the line integrals along the edges of a tree of the graph spanned by the mesh [1] (see also [2,20,22,27]). Numerical experiments with ungauged vector potential have been also carried out [21,27].…”

Section: Introductionmentioning

confidence: 99%

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Perugia

1999

Numerische Mathematik

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A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates.

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“…In other words, for a given physical domain X and any sufficiently smooth function : X → R, ∇ turns out to satisfy (2) for = 0 without satisfying the divergence terms of (1) (as ∇ ∧∇ = 0). An immediate observation at this point is that such solutions can be identified as ker(∇∧, V), or the kernel of the curl operator in the space of admissible functions V, defined in (3).…”

mentioning

confidence: 99%

Dual‐grid‐based tree/cotree decomposition of higher‐order interpolatory H(∇∧, Ω) basis

Beig

Leong

2010

Numerical Meth Engineering

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This work extends the zeroth-order tree/cotree (TC) decomposition method into higher order (HO) interpolatory elements and develops the constraints operator required for the elimination of spurious solutions for general HO spectral basis. Earlier methods explicitly enforce the divergence condition that requires a mixed finite element (FE) formulation with both H 1 and H (∇∧) expansions and involves repeated solutions of the Poisson equation. A recent approach, which avoids the mixed formulation and the Poisson problem, uses TC decomposition of edge DoF over the primal graph and construction of integration and gradient matrices. The approach is easily applied to HO hierarchical elements but becomes quite complex for HO spectral elements. In the presence of internal DoF, it is difficult to utilize the primal graph for an explicit decomposition of the spectral DoF. In contrast, this work utilizes the dual grid, resulting in an explicit decomposition of DoF and construction of constraint equations from a fixed element matrix. Thus, mixed formulation and the Poisson problems are avoided while eliminating the need for evaluation of integration and gradient matrices. The proposed constraints matrix is element-geometry independent and possesses an explicit sparsity formulation reducing the need for dynamic memory allocation. Numerical examples are included for verification. 16953. The transformations are developed for a different purpose, i.e. explicit imposition of divergence conditions. 4. The subsequent fast implementations are based on the nice properties of first-order polynomial basis, i.e. for the first-order polynomial basis the resulting G and Q are made of simple {0, ±1} entries only but this property does not generalize to HO cases. ON A SINGLE ELEMENTThis section covers the following matters:

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Tree-Cotree Decompositions for First-Order Complete Tangential Vector Finite Elements

Cited by 17 publications

References 16 publications

DC and Imaginary Spurious Modes Suppression for Both Unbounded and Lossy Structures

DC and Imaginary Spurious Modes Suppression for Both Unbounded and Lossy Structures

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Dual‐grid‐based tree/cotree decomposition of higher‐order interpolatory H(∇∧, Ω) basis

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