Multiscale Finite Element Modeling of Nonlinear Magnetoquasistatic Problems Using Magnetic Induction Conforming Formulations

In this paper we develop magnetic induction conforming multiscale formulations for magnetoquasistatic problems involving periodic materials. The formulations are derived using the periodic homogenization theory and applied within a heterogeneous multiscale approach. Therefore the fine-scale problem is replaced by a macroscale problem defined on a coarse mesh that covers the entire domain and many mesoscale problems defined on finely-meshed small areas around some points of interest of the macroscale mesh (e.g. numerical quadrature points). The exchange of information between these macro and meso problems is thoroughly explained in this paper. For the sake of validation, we consider a two-dimensional geometry of an idealized periodic soft magnetic composite.Comment: Paper accepted for publication in the SIAM MMS journa

A Structural Analysis of Field/Circuit Coupled Problems Based on a Generalised Circuit Element

In some applications there arises the need of a spatially distributed description of a physical quantity inside a device coupled to a circuit. Then, the in-space discretised system of partial differential equations is coupled to the system of equations describing the circuit (Modified Nodal Analysis) which yields a system of Differential Algebraic Equations (DAEs). This paper deals with the differential index analysis of such coupled systems. For that, a new generalised inductance-like element is defined. The index of the DAEs obtained from a circuit containing such an element is then related to the topological characteristics of the circuit's underlying graph. Field/circuit coupling is performed when circuits are simulated containing elements described by Maxwell's equations. The index of such systems with two different types of magnetoquasistatic formulations (A* and T-$\Omega$) is then deduced by showing that the spatial discretisations in both cases lead to an inductance-like element

Variational Inequalities in Critical-State Problems

Similar evolutionary variational inequalities appear as convenient formulations for continuous quasistationary models for sandpile growth, formation of a network of lakes and rivers, magnetization of type-II superconductors, and elastoplastic deformations. We outline the main steps of such models derivation and try to clarify the origin of this similarity. New dual variational formulations, analogous to mixed variational inequalities in plasticity, are derived for sandpiles and superconductors.Comment: Submitted for publicatio

Cohomology in electromagnetic modeling

Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for potential design. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problems which involve conductive regions with holes, \textit{cuts} are needed to make the boundary value problem well defined. While an intimate connection with homology theory has been quickly recognized, heuristic definitions of cuts are surprisingly still dominant in the literature. The aim of this paper is first to survey several definitions of cuts together with their shortcomings. Then, cuts are defined as generators of the first cohomology group over integers of a finite CW-complex. This provably general definition has also the virtue of providing an automatic, general and efficient algorithm for the computation of cuts. Some counter-examples show that heuristic definitions of cuts should be abandoned. The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem

Lorenz gauged vector potential formulations for the time-harmonic eddy-current problem with L∞-regularity of material properties

In this paper we consider some Lorenz gauged vector potential formulations of the eddy-current problem for the time-harmonic Maxwell equations with material properties having only L1-regularity. We prove that there exists a unique solution of these problems, and we show the convergence of a suitable finite element approximation scheme. Moreover, we show that some previously proposed Lorenz gauged formulations are indeed formulations in terms of the modified magnetic vector potential, for which the electric scalar potential is vanishing

Efficient solution of 3D electromagnetic eddy-current problems within the finite volume framework of OpenFOAM

Eddy-current problems occur in a wide range of industrial and metallurgical applications where conducting material is processed inductively. Motivated by realising coupled multi-physics simulations, we present a new method for the solution of such problems in the finite volume framework of foam-extend, an extended version of the very popular OpenFOAM software. The numerical procedure involves a semi-coupled multi-mesh approach to solve Maxwell's equations for non-magnetic materials by means of the Coulomb gauged magnetic vector potential and the electric scalar potential. The concept is further extended on the basis of the impressed and reduced magnetic vector potential and its usage in accordance with Biot-Savart's law to achieve a very efficient overall modelling even for complex three-dimensional geometries. Moreover, we present a special discretisation scheme to account for possible discontinuities in the electrical conductivity. To complement our numerical method, an extensive validation is completing the paper, which provides insight into the behaviour and the potential of our approach.Comment: 47 pages, improved figures, updated references, fixed typos, reverse phase shift, consistent use of inner produc

Analysis of electromagnetic interference from power system processing and transmission components for Space Station Freedom

The goal is to analyze the potential effects of electromagnetic interference (EMI) originating from power system processing and transmission components for Space Station Freedom.The approach consists of four steps: (1) develop analytical tools (models and computer programs); (2) conduct parameterization studies; (3) predict the global space station EMI environment; and (4) provide a basis for modification of EMI standards