Complementary geometric formulations for electrostatics

Specogna, Ruben

doi:10.1002/nme.3089

Cited by 23 publications

(14 citation statements)

References 57 publications

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“…4 One may also use weighted least squares, for example by assigning bigger weights to closer nodes, but the reconstruction is not improved sensibly. Thus, V n = v(x n , y n , z n ) = w(1) = a is found by solving the fundamental equations [12] …”

Section: B Vertex Reconstruction Techniquesmentioning

confidence: 99%

“…Typically, an irrotational electric field and a solenoidal current density are found by solving a pair of boundary value problems (BVPs) [1]- [4]. The irrotational electric field is easily and efficiently obtained by the standard nodal finiteelement (FE) formulation-referred to as V in this paperbased on the electric scalar potential V sampled on the nodes of the cell complex K represented by the usual G, C, and D incidence matrices [5].…”

Section: Introductionmentioning

confidence: 99%

“…Section IV shows the numerical results. The conclusions are drawn in Section V. potential T [3], [4]. We note that the association of physical variables and role of physical laws are interchanged with respect to the V formulation.…”

Section: Introductionmentioning

confidence: 99%

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One Stroke Complementarity for Poisson-Like Problems

Specogna

2015

IEEE Trans. Magn.

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Taking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system.

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Section: B Vertex Reconstruction Techniquesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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One Stroke Complementarity for Poisson-Like Problems

Specogna

2015

IEEE Trans. Magn.

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“…Finite integration technique (FIT) [1], cell method [2], generalized finite differences [3], discrete geometric approach [4], or discrete geometric methods [5] are instances of this philosophy. Global electromagnetic variables are associated to oriented geometric elements of a pair of dual grids in such a way that electromagnetic laws are naturally casted as exact topological equations.…”

Section: Introductionmentioning

confidence: 99%

Diagonal Discrete Hodge Operators for Simplicial Meshes Using the Signed Dual Complex

Specogna

2015

IEEE Trans. Magn.

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We present a technique to extend the geometric construction of diagonal discrete Hodge operators to arbitrary triangular and tetrahedral boundary conforming Delaunay meshes in the frequent case of piecewise uniform and isotropic material parameters. The technique is based on the novel concept of signed dual complex that originates from a physical argument. In particular, it is shown how the positive definiteness of the mass matrix obtained with the signed dual complex is easily ensured for all boundary conforming Delaunay meshes without requiring-as expected by the common knowledge-that each circumcenter has to lie inside the corresponding element. Eliminating this requirement, whose fulfillment presents otherwise formidable practical difficulties, enables one to easily obtain efficient, consistent, and stable schemes.Index Terms-Diagonal discrete Hodge operator, finite integration technique (FIT), M-matrices, signed dual complex.

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“…As other benefits, it helps to couple the problem with circuit models, provides efficient and accurate post-processing possibilities, and in general gives a structural insight into such problems. Methods that exploit homology and cohomology in electromagnetics are also presented in [11,10,32]. As heat conduction is in many respects analogous to electrostatics, the presented homology and cohomology solver can be exploited in heat conduction boundary value problems as well.…”

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confidence: 99%

Homology and Cohomology Computation in Finite Element Modeling

Pellikka¹,

Suuriniemi²,

Kettunen³

et al. 2013

SIAM J. Sci. Comput.

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Abstract. A homology and cohomology solver for finite element meshes is represented. It is an integrated part of the finite element mesh generator Gmsh. We demonstrate the exploitation of the cohomology computation results in a finite element solver, and use an induction heating problem as a working example. The homology and cohomology solver makes the use of a vectorscalar potential formulation straightforward. This gives better overall performance than a vector potential formulation. Cohomology computation also clarifies the lumped parameter coupling of the problem and enables the user to obtain useful post-processing data as a part of the finite element solution.Key words. homology computation, cohomology computation, finite element method, lumped parameter coupling, electromagnetics AMS subject classifications. 57R19, 58Z05, 65M60, 78M101. Introduction. We present a tool for the homology and cohomology computation of domains tessellated with finite element meshes. The tool is an integrated part of the finite element mesh generator Gmsh [17]. Homology and cohomology computation can be exploited to exhaustively fix the so-called cohomology class of the solution of a boundary value problem that is solved with the finite element method. As a concrete application, we demonstrate how such computations greatly benefit the modeling of an induction heating machine.In boundary value problems that involve the Hodge-Laplace operator, one often needs to choose the cohomology class of the solution. Such problems are usual in electromagnetics, which is why our working example is chosen from that field. The cohomology classes of the problem are generated by the choice of the boundary conditions and by the homology of the problem domain. Informally, homology is about the quantity and the quality of holes in an object, whether it has voids or tunnels or both. Relative homology captures whether the object has holes when one "disregards" a part of the model. In the finite element method, the disregarded part is a subdomain where the solution is fixed by a boundary condition. Cohomology can be characterized by saying that it assigns quantities to these holes. In boundary value problems such assignments fix the cohomology class of the solution. For the technical definitions of homology and cohomology spaces, see appendix A.In the finite element method, typically only a bounded portion of the device and the surrounding space is modeled. The modeling domain may contain holes, and boundary conditions are assigned to confine the fields and couple them with external phenomena outside the domain. Further, the domain can be split into many coupled regions where different approximations and potential formulations are being employed. These modeling aspects give rise to homology and cohomology and their relative forms in numerical models, since one is required to assign source quantities to entities that are absent from the model.For electrical engineers, an evident manifestation of homology and cohomology are Maxwell's equations in their ...

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Complementary geometric formulations for electrostatics

Cited by 23 publications

References 57 publications

One Stroke Complementarity for Poisson-Like Problems

One Stroke Complementarity for Poisson-Like Problems

Diagonal Discrete Hodge Operators for Simplicial Meshes Using the Signed Dual Complex

Homology and Cohomology Computation in Finite Element Modeling

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