Topological Approach to Computational Electromagnetism

Tarhasaari, T.; Kettunen, Lauri

doi:10.2528/pier00080108

Cited by 21 publications

(21 citation statements)

References 10 publications

(10 reference statements)

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“…The chain space C p ðK; EÞ can be decomposed similarly where › is the key operator (Tarhasaari and Kettunen, 2001) which related cuts on a dual complex to finding appropriate generators for primal cohomology groups (Kettunen et al, 1998). Let us next check whether there exists isomorphisms between subspaces of primal cochains and dual chains.…”

Section: A Decompositions On (Co)chain Spacesmentioning

confidence: 99%

Algebraic structures underneath geometric approaches

Kangas

Suuriniemi

Kettunen

2011

COMPEL

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Purpose -The purpose of this paper is to study algebraic structures that underlie the geometric approaches. The structures and their properties are analyzed to address how to systematically pose a class of boundary value problems in a pair of interlocked complexes. Design/methodology/approach -The work utilizes concepts of algebraic topology to have a solid framework for the analysis. The algebraic structures constitute a set of requirements and guidelines that are adhered to in the analysis.Findings -A precise notion of "relative dual complex", and certain necessary requirements for discrete Hodge-operators are found. Practical implications -The paper includes a set of prerequisites, especially for discrete Hodge-operators. The prerequisites aid, for example, in verifying new computational methods and algorithms. Originality/value -The paper gives an overall view of the algebraic structures and their role in the geometric approaches. The paper establishes a set of prerequisites that are inherent in the geometric approaches.

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Section: A Decompositions On (Co)chain Spacesmentioning

confidence: 99%

Algebraic structures underneath geometric approaches

Kangas

Suuriniemi

Kettunen

2011

COMPEL

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“…1). They are related to the homology group (10) This is a conceptually tricky, but practically straightforward, notion. The elements of a homology group are equivalence classes of chains, but in practical computations we just take one chain-a representative-from such a class and operate to it.…”

Section: Classification Of Chainsmentioning

confidence: 99%

“…Assume we have the basis matrices and , according to Problem 3. We decompose basis as follows: This algorithm [10] cumulates a basis of to matrix . The vectors of are linearly independent of the vectors in 3 The null space of matrix A 2 from the QR-decomposition of A costs 2m (n 0 m=3) flops with Householder QR or 3m (n 0 m=3) flops with Givens QR [9].…”

Section: B Chain By Chain Computationmentioning

confidence: 99%

Generalization of the spanning-tree technique

2002

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“…where a p and b p+1 are called cochains [25,28]. Cochains therefore constitute the discrete representation of the electromagnetic fields.…”

Section: The General Finite Formulationmentioning

confidence: 99%

A New Element-Oriented Model for Computational Electromagnetics

Ziyyat¹

2012

PIER B

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Abstract-In this paper, we present a new model using a Fourdimensional (4D) Element-Oriented physical concepts based on a topological approach in electromagnetism.Its general finite formulation on dual staggered grids reveals a flexible Finite-Difference Time-Domain (FDTD) method with reasonable local approximating functions. This flexible FDTD method is developed without recourse to the traditional Taylor based forms of the individual differential operators. This new formulation generalizes both the standard FDTD (S-FDTD) and the nonstandard FDTD (NS-FDTD) methods. Moreover, it can be used to generate new numerical methods. As proof, we deduce a new nonstandard scheme more accurate than the S-FDTD and the known nonstandard NS-FDTD methods. Through some numerical examples, we validate this proposal, and we show the power and the advantage of this Element-Oriented Model.

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Topological Approach to Computational Electromagnetism

Cited by 21 publications

References 10 publications

Algebraic structures underneath geometric approaches

Algebraic structures underneath geometric approaches

Generalization of the spanning-tree technique

A New Element-Oriented Model for Computational Electromagnetics

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