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To describe and extend existing concepts of discrete electromagnetism in a unified formalism; to give examples for the usefulness of the presented ideas for our theoretical work, especially with regard to energy.

After a concise introduction to the mathematical concepts of discrete electromagnetism, we introduce continuous de Rham currents and give their discrete counterpart. We define operators acting upon discrete currents, and apply the theory to electromagnetism.

de Rham current theory yields a mathematical framework for the discussion of discrete electromagnetic problems: The focus is on energy‐balance equations; a discrete Lagrangian can be defined for various modeling problems; the Galerkin approach fits nicely into the proposed formalism; boundary terms in discrete formulations are an implicit feature to the theory.

In this paper, we use the interpolation of discrete fields by Whitney forms on a simplicial cell complex. The resulting discrete formulation is identical to a Galerkin finite‐element method. Other numerical techniques that do not resort to Whitney‐form interpolation can equally be discussed in de Rham‐current terminology.

Rather than a novel numerical technique, the paper presents a unified mathematical framework for the discussion of different practical approaches. We advocate a canonical treatment of energy‐related quantities and of boundary terms in discrete formulations.

The purpose of this paper is to present a geometric approach to the problem of dimensional reduction. To derive (3 + 1) D formulations of 4D field problems in the relativistic theory of electromagnetism, as well as 2D formulations of 3D field problems with continuous symmetries.

The framework of differential‐form calculus on manifolds is used. The formalism can thus be applied in arbitrary dimension, and with Minkowskian or Euclidean metrics alike.

The splitting of operators leads to dimensionally reduced versions of Maxwell's equations and constitutive laws. In the metric‐incompatible case, the decomposition of the Hodge operator yields additional terms that can be treated like a magnetization and polarization of empty space. With this concept, the authors are able to solve Schiff's paradox without use of coordinates.

The present formalism can be used to generate concise formulations of complex field problems. The differential‐form formulation can be readily translated into the language of discrete fields and operators, and is thus amenable to numerical field calculation.

The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.

To introduce a Whitney‐element based coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM); to discuss the algebraic properties of the resulting system and propose solver strategies.

The FEM is interpreted in the framework of the theory of discrete electromagnetism (DEM). The BEM formulation is given in a DEM‐compatible notation. This allows for a physical interpretation of the algebraic properties of the resulting BEM‐FEM system matrix. To these ends we give a concise introduction to the mathematical concepts of DEM.

Although the BEM‐FEM system matrix is not symmetric, its kernel is equivalent to the kernel of its transpose. This surprising finding allows for the use of two solution techniques: regularization or an adapted GMRES solver.

The programming of the proposed techniques is a work in progress. The numerical results to support the presented theory are limited to a small number of test cases.

The paper will help to improve the understanding of the topological and geometrical implications in the algebraic structure of the BEM‐FEM coupling.

Several original concepts are presented: a new interpretation of the FEM boundary term leads to an intuitive understanding of the coupling of BEM and FEM. The adapted GMRES solver allows for an accurate solution of a singular, unsymetric system with a right‐hand side that is not in the image of the matrix. The issue of a grid‐transfer matrix is briefly mentioned.

The purpose of this paper is to establish the mathematical foundations of magnetic measurement methods based on translating-coil and rotating-coil magnetometers for accelerator magnets and solenoids. These field transducers allow a longitudinal scanning of the field distribution, but require a sophisticated post-processing step to extract the coefficients of the Fourier–Bessel series (known as pseudo-multipoles or generalized gradients) as well as a novel design of the rotating coil magnetometers.

Calculating the transversal field harmonics as a function of the longitudinal position in the magnet, or measuring these harmonics with a very short, rotating induction-coil scanner, allows the extraction of the coefficients of a Fourier–Bessel series, which can then be used in the thin lens approximation of the end regions of accelerator magnets.

The extraction of the leading term in the Fourier–Bessel series requires the solution of a differential equation by means of a Fourier transform. This yields a natural way to de-convolute the measured distribution of the multipole content. The author has shown that the measurement technique requires iso-parametric coils to avoid interception of the longitudinal field component. The compensation of the main signal cannot be done with the classical arrangement of search coils at different radii, because no easy scaling law exists. A new design of an iso-perimetric induction coil has been found.

In the literature, it is stated that the pseudo-multipoles can be extracted from field computations or measurements. While this is true for computations, the author shows that the measurement of the field harmonics must be done with iso-parametric coils because otherwise the leading term in the Fourier–Bessel series cannot be extracted.

The author has now established the theory behind a number of field transducers, such as the moving fluxmeter, the rotational coil scanner and the solenoidal field transducer.

This paper brought together the known theory of the orthogonal expansion method with the methods and tools for magnetic field measurements to establish a field description in accelerator magnets.

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.

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.

A precise notion of “relative dual complex”, and certain necessary requirements for discrete Hodge‐operators are found.

The paper includes a set of prerequisites, especially for discrete Hodge‐operators. The prerequisites aid, for example, in verifying new computational methods and algorithms.

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.

The purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.

At the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.

Different field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.

A novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.