Different finite element formulations of 3D magnetostatic fields

Preis, Kurt; Bardi, Istvan; Bíró, Oszkár; Magele, Christian; Vrisk, G.; Richter, K.R.

doi:10.1109/20.123863

Cited by 78 publications

(25 citation statements)

References 7 publications

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“…The transformation of the terms (3b)-(3c) of A − V formulation into term (12b) makes the system not compatible [6]. Then A − V formulation (12)-(4) needs a gauge condition, defined here by a tree of edges [7]. According to Table I, computation times of source A j are similar on the reduced domain Ω s and on the complete domain Ω.…”

Section: Magnetodynamic a − V Formulationmentioning

confidence: 86%

Computation of Source for Non-Meshed Coils in a Reduced Domain With <inline-formula> <tex-math notation="LaTeX">${A}$ </tex-math> </inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math> </inline-formula> Formulation

et al. 2016

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The discretized source magnetic vector potential A j , interpolated from source field Hs with H(curl, Ω) semi-norm, is studied for non-meshed coils with magnetic vector potential A and electric scalar potential V formulation. As a novelty, source potential A j is computed in a reduced domain Ω red instead of the complete domain Ω. For domains with fixed and moving parts, potential A j can be computed on each part, for each of the related current sources, with no need to ensure its continuity between these parts.

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Section: Magnetodynamic a − V Formulationmentioning

confidence: 86%

Computation of Source for Non-Meshed Coils in a Reduced Domain With <inline-formula> <tex-math notation="LaTeX">${A}$ </tex-math> </inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math> </inline-formula> Formulation

et al. 2016

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“…Finite element discretizations of these formulations are based on nodal shape functions. An alternative to this approach consists in representing the field generated by the current sources in terms of edge element shape functions, reducing cancellation and allowing the use of reduced scalar potential everywhere [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 the case when the algorithm returns INFINITE LOOP, since it is known that the topology of the domain is trivial, one can theoretically use the strategy explained in [41] and [42] to solve the problem. The raw idea is as follows:…”

Section: ) Is a Given Electric Current 2-cocycle The Values Of I S Amentioning

confidence: 99%

Critical Analysis of the Spanning Tree Techniques

Dłotko¹,

Specogna²

2010

SIAM J. Numer. Anal.

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Abstract.Two algorithms based upon a tree-cotree decomposition, called in this paper spanning tree technique (STT) and generalized spanning tree technique (GSTT), have been shown to be useful in computational electromagnetics. The aim of this paper is to give a rigorous description of the GSTT in terms of homology and cohomology theories, together with an analysis of its termination. In particular, the authors aim to show, by concrete counterexamples, that various problems related with both STT and GSTT algorithms exist. The counterexamples clearly demonstrate that the failure of STT and GSTT is not an exceptional event, but something that routinely occurs in practical applications.Key words. algebraic topology, scalar potential in multiply connected regions, tree-cotree decomposition, belted tree, computational topology, homology theory, cohomology theory, homology and cohomology generators, homology-cohomology duality AMS subject classifications. 65N30, 78M10, 78M25, 55N99, 55M05, 55N33 DOI. 10.1137/0907663341. Introduction. The tree-cotree decomposition arises from graph theory and consists in partitioning the edges of a graph into a spanning tree and its complement, referred to as the cotree. The idea of taking advantage of the tree-cotree decomposition is at the root of electric network theory [1], [2]. It has been used, for example, to generate a maximal set of independent Kirchhoff's equations for the network analysis (see, for example, [3] [10]. The tree-cotree decomposition became popular in computational electromagnetics after [11]. It had been widely used as a gauging technique to set well-posed magnetostatic and magneto-quasi-static boundary value problems (BVP). Nowadays, such gauging techniques have lost their importance, since the ungauged formulations were shown to be more effective, improving the condition number of the matrix for the linear system of equations; see, for example, [12].More recently, two algorithmic techniques based upon the tree-cotree decomposition have been shown to be quite useful in computational electromagnetics. The first one, introduced in [13] (see also [14], [15], [16] and referred to as spanning tree technique (STT) in this paper, is commonly employed in order to compute the socalled generalized source magnetic fields, needed to enforce the source currents when solving magnetostatic and magneto-quasi-static BVP formulated by using a magnetic scalar potential. In this application, the STT is used to compute a 1-cochain when its coboundary 2-cochain is given as input.

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Different finite element formulations of 3D magnetostatic fields

Cited by 78 publications

References 7 publications

Computation of Source for Non-Meshed Coils in a Reduced Domain With <inline-formula> <tex-math notation="LaTeX">${A}$ </tex-math> </inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math> </inline-formula> Formulation

Computation of Source for Non-Meshed Coils in a Reduced Domain With <inline-formula> <tex-math notation="LaTeX">${A}$ </tex-math> </inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math> </inline-formula> Formulation

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

Critical Analysis of the Spanning Tree Techniques

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