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The purpose of this paper is to review the mutual coupling of electromagnetic fields in the magnetic vector potential formulation with electric circuits in terms of (modified) nodal and loop analyses. It aims for an unified and generic notation.

The coupled formulation is derived rigorously using the concept of winding functions. Strong and weak coupling approaches are proposed and examples are given. Discretization methods of the partial differential equations and in particular the winding functions are discussed. Reasons for instabilities in the numerical time domain simulation of the coupled formulation are presented using results from differential-algebraic-index analysis.

This paper establishes a unified notation for different conductor models, e.g. solid, stranded and foil conductors and shows their structural equivalence. The structural information explains numerical instabilities in the case of current excitation.

The presentation of winding functions allows to generically describe the coupling, embed the circuit equations into the de Rham complex and visualize them by Tonti diagrams. This is of value for scientists interested in differential geometry and engineers that work in the field of numerical simulation of field-circuit coupled problems.

The simulation of magnetic fields with geometric discretization schemes using magnetic vector potentials involves the solution of very large discrete consistently singular curl‐curl systems of equations. Geometric and algebraic multigrid schemes for their solution require intergrid transfer operators of restriction and prolongation that achieve the discrete conservation of integral quantities serving as state‐variables of geometric discretization methods. For non‐conservative restriction operations, a consistency error correction operator related to an algebraic filtering is proposed. Numerical results show the effects of the consistency correction for a non‐nested geometric multigrid method.

Transient calculation of currents in brain tissue induced during a transcranial magnetic stimulation treatment.

Because of the short pulses used in this technique a time‐harmonic approximation is no longer valid, and transient effects have to be considered. We have performed a Fourier analysis of the induced currents calculated in a high‐resolution model of the brain using the extended scalar potential finite differences (Ex‐SPFD) approach.

The peak induced currents in the transient development of the pulse are higher by a factor of approximately seven than the time harmonic solutions at the fundamental frequency. Furthermore, an analysis of the impact of the conductivity dispersion revealed an increase in the peak induced currents by 17.3 percent for white matter and by 20.8 percent for gray matter.

Using the numerically efficient Ex‐SPFD approach, along with a high performance cluster, the current densities inside the brain can be calculated incorporating more details than previous calculations of this type.

Inductive charging systems for electrically powered cars produce a magneto-quasistatic field and organism in the vicinity might be exposed to that field. Magneto-quasistatic fields induce electric fields in the human body that should not exceed limits given by the International Commission of Non-Ionizing Radiation protection (ICNIRP) to ensure that no harm is done to the human body. As these electric fields cannot be measured directly, they need to be derived from the measured magnetic flux densities. To get an almost real-time estimation of the harmfulness of the magnetic flux density to the human body, the electric field needs to be calculated within a minimal computing time. The purpose of this study is to identify fast linear equations solver for the discrete Poisson system of the Co-Simulation Scalar Potential Finite Difference scheme on different graphics processing unit systems.

The determination of the exposure requires a fast linear equations solver for the discrete Poisson system of the Co-Simulation Scalar Potential Finite Difference (Co-Sim. SPFD) scheme. Here, the use of the AmgX library on NVIDIA GPUs is presented for this task.

Using the AmgX library enables solving the equation system resulting from an ICNIRP recommended human voxel model resolution of 2 mm in less than 0.5 s on a single NVIDIA Tesla V100 GPU.

This work is one essential advancement to determine the exposure of humans from wireless charging system in near real-time from in situ magnetic flux density measurements.

Transient eddy current formulations based on the Finite Integration Technique (FIT) for the magneto‐quasistatic regime are extended to include motional induction effects of moving conductors with simple geometries by different approaches. A new regularization of the formulation using discrete grad‐div augmentation of the curlcurl formulation is presented and tested. To improve the implicit time integration process, several schemes for an error controlled variable time step selection are presented and for the repetitive solution of the arising large sparse systems of equations a sparse direct solver is compared to iterative methods such as a preconditioned conjugate gradient method and a new algebraic multigrid solver, which is aware of the curlcurl nullspace.

To propose an air‐gap element for electrical machine simulation which accounts for static and dynamic rotor eccentricity.

The air‐gap element technique is extended to account for a non‐centered rotor. The consistency, stability and convergence of the discretisation error are studied. A specialized efficient solution technique combining the conjugate gradient algorithm with fast Fourier transforms is developed.

The eccentric air‐gap technique offers better discretisation properties than the classical techniques based on remeshing. Thanks to the specialized solver, the computation times remain comparable.

The introduction of eccentricity in the air‐gap element used for finite element electrical machine simulation is a new development.

To propose trigonometric interpolation in combination with the sliding‐surface technique for modeling rotation in electrical machine models discretised by the finite integration technique (FIT).

Locked‐step, linear and trigonometric interpolation techniques are developed for coupling the stator and rotor model parts of an electrical machine model.

Linear and trigonometric interpolation should be preferred over the locked‐step approach. Three‐machine models with sliding‐surface coupling discretised by the FIT result in efficient and reliable models.

The introduction of sliding‐surface techniques in the FIT, the trigonometric interpolation used in combination, the application of the FIT for simulating electrical machines.

To provide a numerical technique for the quick and simple determination of the switching time instants for field‐circuit coupled problems with switching elements.

3D magnetic vector potential formulation coupled to an electrical circuit with switching elements, for example, diodes, is presented. The change of the state of the switching elements is implemented as a modification of the model topology.

Since every step of the singly diagonally implicit Runge‐Kutta methods delivers not only the solution of this time step but also its stage derivatives, they can be efficiently employed to construct a dense‐output‐based interpolation polynomial, with their roots approximating the switching time instants.

This paper presents a computationally cheap interpolation approach for quick and accurate determination of switching time instances for field‐circuit coupled problems with switching elements. The proposed technique can be successfully incorporated into software packages designed to model coupled problems of different nature, where sudden changes of quality may take place.

The space discretization of eddy‐current problems in the magnetic vector potential formulation leads to a system of differential‐algebraic equations. They are typically time discretized by an implicit method. This requires the solution of large linear systems in the Newton iterations. The authors seek to speed up this procedure. In most relevant applications, several materials are non‐conducting and behave linearly, e.g. air and insulation materials. The corresponding matrix system parts remain constant but are repeatedly solved during Newton iterations and time‐stepping routines. The paper aims to exploit invariant matrix parts to accelerate the system solution.

Following the principle “reduce, reuse, recycle”, the paper proposes a Schur complement method to precompute a factorization of the linear parts. In 3D models this decomposition requires a regularization in non‐conductive regions. Therefore, the grad‐div regularization is revisited and tailored such that it takes anisotropies into account.

The reduced problem exhibits a decreased effective condition number. Thus, fewer preconditioned conjugate gradient iterations are necessary. Numerical examples show a decrease of the overall simulation time, if the step size is small enough. 3D simulations with large time step sizes might not benefit from this approach, because the better condition does not compensate for the computational costs of the direct solvers used for the Schur complement. The combination of the Schur approach with other more sophisticated preconditioners or multigrid solvers is subject to current research.

The Schur complement method is adapted for the eddy‐current problem. Therefore, a new partitioning approach into linear/non‐linear and static/dynamic domains is proposed. Furthermore, a new variant of the grad‐div gauging is introduced that allows for anisotropies and enables the Schur complement method in 3D.

To propose trigonometric interpolation in combination with both sliding‐surface and moving‐band techniques for modelling rotation in finite‐element electrical machine models. To show that trigonometric interpolation is at least as accurate and efficient as standard stator‐rotor coupling schemes.

Trigonometric interpolation is explained concisely and put in a historical perspective. Characteristic drawbacks of trigonometric interpolation are alleviated one by one. A comparison with the more common locked‐step linear‐interpolation and mortar‐element approaches is carried out.

Trigonometric interpolation offers a higher accuracy and therefore can outperform standard stator‐rotor coupling techniques when equipped with an appropriate iterative solver incorporating Fast Fourier Transforms to reduce the higher computational cost.

The synthetic interpretation of trigonometric interpolation as a spectral‐element approach in the machine's air gap, the efficient iterative solver combining conjugate gradients with Fast Fourier Transforms. The unified application to both sliding‐surface and moving‐band techniques.