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This work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.

The proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.

The approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L₂ norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.

The work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.

The Reduced Basis Method (RBM) generates low-order models of parametrized PDEs to allow for efficient evaluation of parametrized models in many-query and real-time contexts. The purpose of this paper is to investigate the performance of the RBM in microwave semiconductor devices, governed by Maxwell's equations.

The paper shows the theoretical framework in which the RBM is applied to Maxwell's equations and present numerical results for model reduction under geometry variation.

The RBM reduces model order by a factor of $1,000 and more with guaranteed error bounds.

Exponential convergence speed can be observed by numerical experiments, which makes the RBM a suitable method for parametric model reduction (PMOR).

The purpose of this paper is to present the advantageous applicability of the bicomplex analysis in the context of the Finite Element Method (FEM). This method can be applied for wave propagation problems in various environments.

In this paper, the bicomplex number system is introduced and accordingly the differential equation for time-harmonic Maxwell’s equations in homogeneous media is derived in detail. Besides that, numerical simulations of wave propagation are performed and compared to the traditional approach based on classical FEM related to the Helmholtz equation. The appropriate error norm is investigated for different discretizations.

The results show that the use of bicomplex analysis in FEM leads to the higher accuracy of the electromagnetic field determination compared to the traditional Helmholtz approach. By using the bicomplex-valued formulation, the complex-valued electric and magnetic fields can be found directly and no additional FEM calculations are necessary to get the whole field.

The direct bicomplex formulation overcomes the use of the second order derivatives, which leads to the higher accuracy. In general, accurate calculations of the wave propagation in FEM is still an open problem and the approach described in this paper is a contribution to this class of problems.

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.

Quality of the weld joint produced by high-frequency induction (HFI) welding of steel tubes is attributed to a number of process parameters. There are several important process parameters such as the speed of the welding line, the angle of the approaching strip edges, the physical configuration of the induction coil, impeder, formed steel strip and weld rolls with respect to each other, the pressure of the weld rolls and frequency of the high-frequency current in the induction coil. The purpose of this paper is to develop a 3D model of tube welding process that incorporates realistic material properties and movement of the strip.

3D numerical simulation by the finite element method (FEM) can be used to understand the influence of these process parameters. In this study, the authors have developed a quasi-steady model along with the coupling of electromagnetic and thermal model and incorporation of non-linear electromagnetic and thermal material properties.

In this study, 3D FEM model has been established which gives results in accordance with previously published work on induction tube welding. The effect of the Vee-angle and frequency on the temperature profile created in the strip edge during the electromagnetic heating is studied.

The authors are now able to simulate the induction tube welding process at a more reasonable computational cost enabling an analysis of the process.

A 3D model has been developed for induction tube welding. A non-linearly coupled system of Maxwell’s electromagnetic equation and the heat equation is implemented using the fixed point iteration method. The model also takes into account non-linear magnetic and thermal material properties. Adaptive remeshing is implemented to optimise mesh size for the electrical skin depth of induced current in the strip. The model also accounts for the high welding-line speeds which influence the mode of heat transfer in the strip.

This paper investigates new technological devices to be utilised in future optical communications, by means of variational method (FEM) and multipole scattering approach (Rayleigh method). This last one provides interesting asymptotic results in the long‐wavelength limit. The so‐called photonic crystal fibres (PCF) possess radically different guiding properties due to photonic band gap guidance: removing a hole within a macro‐cell leads to a defect state within the gap. In the case of multi‐core PCF, the localised modes start talking to each other which possibly leads to a new generation of multiplexer/demultiplexers.

We present in this paper an efficient finite element analysis for different types of waveguides . The computations are performed with “edge elements” . These finite elements avoid all the “spurious modes”, the non‐physical numerical fields obtained from the solution of eigenvalue problems with classical finite elements formulations.Both dielectric‐loaded waveguides and ridged waveguides (waveguides with sharp edges) are analyzed. Comparisons of our results with previously published ones show theaccuracy of the numerical technique.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

The divergence equations in Maxwell’s theory are Gauss law and the statement that magnetic monopoles do not exist; from this point of view these two equations are fundamental. However, from a practical point of view, it is generally believed that Maxwell’s divergence equations are redundant and may be ignored provided that they are satisfied at some time t‐0. It has been proved recently that this idea is not correct for boundary‐initial value problems. To make mathematical arguments more accessible, we analyse here the role of the divergence equations in the framework of 2D‐electromagnetism.

If a scattering problem is solved by finite element, finite volume or finite difference methods, it is necessary to predict the far field pattern (or radar cross‐section) by using a near field to far field transformation. Usually this is done using the Stratton‐Chu integral relations, which give the far field pattern in terms of a near field surface integral. When volume‐based methods are used this is unnatural, and it may be necessary to employ interpolation procedures to provide the necessary surface data. In this paper an alternative method based on volume integrals is proposed. The main advantage of the new procedure is that it allows the use of discrete quantities that are naturally available from the numerical scheme. However, it is now necessary to perform volume integrals. The error in the new procedure is examined, and a simple numerical example provided.