Purpose: – 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. Design/methodology/approach: – 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. Findings: – 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. Originality/value: – 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.

Aiming to better understand and model the different loss mechanisms occurring within electric machines during operation, we present a method to consider the degradation of steel during manufacturing due to punching in finite-element modeling of electric machines. The approach introduces a certain number of boundary layers along the punched edges, with the relative permeability decreasing toward the edge. The method, its implementation, and two example applications are presented and supported with experimental results.

The application of high-temperature superconductors to accelerator magnets for future particle colliders is under study. Numerical methods are crucial for an accurate evaluation of the complex dynamical behavior of the magnets, especially concerning the magnetic field quality and thermal behavior. We present a coupled A-H field formulation for the analysis of magneto-thermal transients in accelerator magnets. The magnetic field strength H accounts for the eddy current problem in the source regions containing the superconducting domains, while the magnetic vector potential A represents the magnetoquasistatic problem in the normal and non-conducting domains. Furthermore, we include a thin shell approximation for the source regions, making the formulation suitable for large scale models composed of thousands of tapes. In this work, the relevant equations are derived and discussed, with emphasis on the coupling conditions. The weak formulation is derived, and numerical results are provided in order to both, verify the formulation and scale it to the size of an accelerator magnet.

The integration of machine learning (Keplerian paradigm) and more general artificial intelligence technologies with physical modeling based on first principles (Newtonian paradigm) will impact scientific computing in engineering in fundamental ways. Such hybrid models combine first principle-based models with data-based models into a joint architecture. This paper will give some background, explain trends and showcase recent achievements from an applied mathematics and industrial perspective. Examples include characterization of superconducting accelerator magnets by blending data with physics, data-driven magnetostatic field simulation without an explicit model of the constitutive law, and Bayesian free-shape optimization of a trace pair with bend on a printed circuit board.

This paper proposes a framework of waveform relaxation methods to simulate electromagnetic fields coupled to electric networks. Within this framework, a guarantee for convergence and stability of Gauß-Seidel-type methods is found by partial differential algebraic equation (PDAE) analysis. It is shown that different time step sizes in different parts of the model can be automatically chosen according to the problem's dynamics. A finite-element model of a transformer coupled to a circuit illustrates the efficiency of multirate methods.