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This article presents a theoretical study of an absolute, ratiometric inductive position sensor (IPS) based on eddy currents. The aim is to describe the working principle of the sensor, having as key components a transmitting coil, the receiving coils, and the conductive target, by introducing area-of-overlap functions. We show that each target–receiver pair needs the adoption of a different reconstruction formula for the identification of the target position, whereas in the literature the usual inverse tangent function is applied for every possible pair. Then, we seek the target–receiver pair that maximizes the amplitude of the induced voltages on the receivers. The results show that to achieve the maximum value of the induced voltages, the best choice is to have a rectangular target and rectangular receivers. To verify the theory, a simulation and optimization method has been applied to the rectangular receiver coils on two rotary IPS realized with printed circuit board (PCB) technology. Measurements performed on the prototypes have shown an increment of the induced voltage of more than 57% with respect to the commonly used sinusoidal receivers. However, a linearity error of 1.5%FS is obtained by using the inverse tangent reconstruction formula. When using the formula provided by the theory, the linearity error becomes 0.6%FS for the nonoptimized prototype and below 0.15%FS for the optimized one. [ABSTRACT FROM AUTHOR]

In this work, the solution to the eddy current problem for slowly moving conductors is investigated with the volume integral formulation of the electric field integral equation (EFIE). On the one hand, the formulation gives the possibility to mesh only the conductors; on the other hand, the inductance matrix is fully populated, thus limiting the size of the problem. Furthermore, the standard solution implies the reassembling of the stiffness matrix for each position assumed by the moving conductors. This results in a limit in the size of the problem to treat and a relevant cost in the computation time. To overcome this limit, in this work, the mutual coupling part of the system with two conductors is computed on the fly, thus giving the possibility to solve problems with a higher number of unknowns. Finally, to speed up the solution of the system, the Gauss–Seidel (GS) iterative techniques and the fast multipole method (FMM) are applied to take into account the mutual effects between the conductors. A comparison of the proposed method with a reference one shows the effectiveness of this technique. [ABSTRACT FROM AUTHOR]

The aim of this article is to describe and model a sawtooth crash in RFX-mod2 and compute the eddy currents induced in the stabilizing shell surrounding the plasma for two alternative gap concepts and the subsequent error field. The sawtooth crash displacement of the plasma column is represented by a vacuum model coupled with a suitable magneto-quasi-static (MQS) volume integral (VI) formulation. [ABSTRACT FROM AUTHOR]

Magic formulas are the geometric identities at the root of modern compatible schemes for polyhedral grids. We present rigorous yet elementary proofs of the magic formulas originating from Stokes theorem. The proofs enlighten new fundamental aspects of the mass matrices produced with the magic formulas. First, the construction of the mass matrices works for an unexpectedly broad type of mesh cells. Second, they show that dual nodes can be arbitrarily positioned thus extending the construction of the dual barycentric grid. [ABSTRACT FROM AUTHOR]

This contribution addresses the exploitation of mirror symmetry (formally, the abelian D1 and D2 dihedral symmetry groups) in various integral formulations for eddy current problems. The novelty of the contribution is in particular how to rigorously treat non-simply-connected conductors when computing the first cohomology group generators on the symmetry cell of the problem only. [ABSTRACT FROM AUTHOR]

In this work, a surface current plasma model is coupled to a volume integral formulation for studying the interaction between tokamak tearing modes (TMs) and the machine metallic structures surrounding the plasma. By evaluating the passive response on a set of pickup probes, TMs’ amplitude is estimated. [ABSTRACT FROM AUTHOR]

This contribution addresses the solution of eddy-current problems by means of a volume integral formulation based on the electric vector potential on a computational domain that exhibits a cyclic symmetry. Even if grids discretizing the domain are typically composed of tetrahedral or hexahedral elements, the proposed approach also works for general polyhedral meshes, such as those ones obtained by subgridding. In this article, an algorithm to compute a set of suitable cohomology generators needed when the conductors are not simply connected is introduced first. Besides being purely combinatorial, with linear-time worst case complexity and suitable with polyhedral meshes, it reuses a code that computes generators for triangular surface meshes, with obvious advantages concerning the implementation effort. Second, the formulation and the algorithm for cohomology computation are tweaked to be able to solve eddy-current problems with cyclic symmetry reserving specific attention to the construction of suitable tree–cotree decomposition for the problem gauging. [ABSTRACT FROM AUTHOR]

The cell method (CM) or discrete geometric approach (DGA) in the time domain, already introduced by Codecasa et al. in 2008 for the coupled Ampere–Maxwell and Faraday equations, is here recast as a Galerkin Method similar to the finite-element method (FEM). In particular, it is shown to be a mixed method comprising an explicit scheme and two discontinuous Galerkin (DG) FEM spaces formulated on dual meshes, in which each of the two function spaces provides a continuous numerical flux choice for its dual mesh counterpart. The implemented version is shown to compare favorably in terms of accuracy and efficiency with respect to the classic conforming FEM scheme using Whitney elements. When tested on the same tetrahedral mesh, the Courant–Friedrichs–Lewy (CFL) condition for the proposed approach is a factor of 2 less restrictive on the time step with respect to the curl-conforming FEM scheme. [ABSTRACT FROM AUTHOR]

This contribution introduces a novel method for constructing mass matrices for general tetrahedral elements. By considering the construction of the reluctivity mass matrix as an example, a new derivation of a recipe to geometrically construct a symmetric positive semi-definite and consistent mass matrix is provided. It is shown why such a matrix can be used inside formulations where the mass matrix is right multiplied by the appropriate incidence matrix. Then, a second recipe is introduced to produce a consistent and symmetric positive definite matrix. The proposed method is based on an original algebraic technique that differs fundamentally from what is available in the literature. The proposed recipe opens also the possibility to improve the conditioning of the resulting elementary mass matrices. [ABSTRACT FROM AUTHOR]

There are two ways of solving eddy current problems with an edge element $h$ -conforming formulation in the case of non-simply-connected conductors: first employs a “fake” artificial conductivity in part of the insulator. Second uses the cohomology theory to define a scalar magnetic potential in the whole insulator. This contribution provides, for the first time, a thorough comparison between these methods. We concentrate on the accuracy and computational effort required to solve a problem using both of the two methods. [ABSTRACT FROM AUTHOR]

This contribution addresses the exploitation of cyclic symmetry in boundary integral formulations for eddy current problems formulated with a magnetic scalar potential. The novelty of the contribution is how to rigorously treat non-simply connected conductors. In particular, a general algorithm to compute a set of suitable cohomology generators is introduced. The algorithm is purely combinatorial and exhibits a linear-time worst-case complexity. [ABSTRACT FROM AUTHOR]

A fundamental extension for a recently introduced numerical scheme for the time-domain solution of Maxwell’s equations on tetrahedral meshes is introduced: the algorithm is here shown to be able to handle materials with finite electric resistivity, without losing any of its amenable properties. A theoretic stability analysis, valid for the extended algorithm, is presented for the first time, and a Courant–Friedrich–Lewy sufficient condition on the maximum time step allowed for the scheme to be stable is derived analytically. This result is completely novel for finite-difference time-domain (FDTD)-like approaches on tetrahedral grids. Finally, the accuracy of the extended algorithm is tested with respect to the well-known implicit finite-element method scheme on tetrahedral grids, to the canonical FDTD scheme on Cartesian orthogonal grids, and to a commercial code implementing FDTD with staircasing mitigation techniques. [ABSTRACT FROM AUTHOR]

A recently introduced time-domain method for the numerical solution of Maxwell’s equations on unstructured grids is reformulated in a novel way, with the aim of implementation on graphical processing units (GPUs). Numerical tests show that the GPU implementation of the resulting scheme yields correct results, while also offering an order of magnitude in speedup and still preserving all of the main properties of the original finite-difference time-domain algorithm. [ABSTRACT FROM AUTHOR]

We present an effective technique to solve eddy current problems on the conductors of arbitrary topology by an iterative geometric formulation suitable for an arbitrary polyhedral mesh. The formulation is based on the reformulation of the volume integral formulation in a form suitable to be solved by fixed point iterations. Given that nowadays, the computation of Biot–Savart fields can be performed efficiently on graphics processing units, the right-hand side of the systems is constructed at each fixed point iteration without computing the inductance matrix, which is fully populated. The proposed technique is useful in eddy current problems at industrial frequency where meshing the complement of conductors is hard, thus representing a sound alternative to integral formulations. [ABSTRACT FROM AUTHOR]

An innovative approach has been recently proposed for the voltage breakdown prediction in high-voltage systems, insulated by large vacuum gaps. This approach is based on complementary geometric formulations for electrostatics coupled to the analytical solution of the equations of motion for charged particles. In this paper, a goal-oriented local mesh refinement technique is introduced, which allows to increase the rate of convergence of the solution, enabling an effective voltage breakdown prediction also in large-scale systems with complex geometries. [ABSTRACT FROM AUTHOR]

Solving eddy current problems formulated by using a magnetic scalar potential in the insulator requires a topological pre-processing to find the so-called first cohomology basis of the insulating region, which may be very time-consuming for challenging industrially driven problems. The physics-inspired Dłotko–Specogna (DS) algorithm was shown to be superior to alternatives in performing such a topological pre-processing. Yet, the DS algorithm is particularly fast when it produces as output not a regular cohomology basis but a so-called lazy one, which contains the regular one but it keeps also some additional redundant elements. Having a regular basis may be advantageous over the lazy basis if a technique to produce it would take about the same time as the computation of a lazy basis. In the literature, such a technique is missing. This paper covers this gap by introducing modifications to the DS algorithm to compute a regular basis of the first cohomology group in practically the same time as the generation of a lazy cohomology basis. The speedup of this modified DS algorithm with respect to the best alternative reaches more than two orders of magnitudes on challenging benchmark problems. This demonstrates the potential impact of the proposed contribution in the low-frequency computational electromagnetics community and beyond. [ABSTRACT FROM AUTHOR]

Integral formulations are attractive for solving eddy-current problems in complex electromagnetic systems, since they do not require the discretization of the complement of the conducting structures. This paper addresses the coupling of volume-integral and surface-integral formulations for eddy-current problems on general star-shaped polyhedral and polygonal meshes. [ABSTRACT FROM AUTHOR]

This paper first introduces a simplified construction of the constitutive matrices for general polyhedral meshes. These constitutive matrices are geometrically defined by means of simple closed-form expressions involving the geometric elements of the primal and dual meshes. Then, we solve for the first time the Maxwell’s eigenvalue problem on general polyhedral meshes. In particular, we analyze the convergence of the eigenvalues when the mesh is adaptively refined by using the subgridding technique together with the constitutive inconsistency as error indicator. [ABSTRACT FROM AUTHOR]

The frequency-domain wave propagation problem is notoriously difficult to solve through iterative methods because it leads to a symmetric but indefinite linear system. For this reason, direct methods are usually employed at the expense of great memory usage. Convergence of iterative methods, however, could be obtained by regularizing the wave equation. We introduce such regularization in discrete geometric approach framework on polyhedral grids. Moreover, we extend the regularization to the impedance boundary condition. [ABSTRACT FROM AUTHOR]

We present a numerical method named mixed high order (MHO) to obtain high order of convergence for electrostatic problems solved on general polyhedral meshes. The method, based on high-order local reconstructions of differential operators from face and cell degrees of freedom, exhibits a moderate computational cost thanks to hybridization and static condensation that eliminate cell unknowns. After surveying the method, we assess its effectiveness for 3-D problems by comparing, for the first time in literature, its performances with classical conforming finite elements. Moreover, we emphasize the algebraic equivalence of MHO in the lowest order with the analog formulation obtained with the discrete geometric approach or the finite-integration technique. [ABSTRACT FROM AUTHOR]

This paper presents a volume integral formulation to solve eddy current problems on general polyhedral meshes. The required topological preprocessing is efficiently addressed in linear worst case complexity by using the idea of lazy cohomology generators. [ABSTRACT FROM AUTHOR]

When solving eddy-current problems containing topologically non-trivial conductors with formulations using the magnetic scalar potential in the insulators, cohomology generators are necessary to obtain a well-defined problem. The Dłotko–Specogna (DS) algorithm is a simple and efficient tool to compute the lazy generators of the first cohomology group of the insulator that can be used in such potential design. This paper introduces an upgrade in the DS algorithm that speeds up the execution for very complicated geometries. Moreover, this paper provides, for the first time, a detailed comparison of computational resources needed for the topological pre-processing by our toolbox and by the tool to compute a standard cohomology basis available in the mesh generator GMSH. In addition, we make our implementation of DS algorithm available for download on request for non-profit use as a Topoprocessor package. [ABSTRACT FROM AUTHOR]

This paper presents a novel technique to perform the topological pre-processing needed in formulations based on an electric vector potential and a magnetic scalar potential when the eddy-current problem being solved is subject to periodic boundary conditions. In this case, in fact, all the techniques to produce the generators for the first cohomology group introduced in the literature cannot be used directly on the input mesh. The novel technique is very fast and it is used to model cables for power delivery. [ABSTRACT FROM AUTHOR]

Efficient boundary integral formulations based on stream functions for solving eddy current problems in thin conductors, which are modeled by the orientable combinatorial two-manifold with boundary, need generators of the first relative cohomology group to make the problem well defined. The state-of-the-art technique is to compute directly the relative cohomology generators with a combinatorial algorithm having linear worst-case complexity. In this paper, we propose to compute the relative cohomology generators from the homology generators, introducing a novel and general algorithm whose running time is again linear in the worst case. The advantage is that one may use an off-the-shelf software to compute the homology generators and implement only a simple and cheap procedure to obtain the required relative cohomology generators. Although the presented applications relate to ac power systems, the proposed technique is of general interest, and may be used for other applications in computational science and engineering. [ABSTRACT FROM PUBLISHER]

Numerical simulation of anechoic chambers is an important topic nowadays, since it can provide useful data about the performance of an electromagnetic compatibility site. However, the mathematical nature of the problem, the physical dimensions of the simulated sites, and the frequency ranges pose nontrivial challenges to the numerical simulation. Computational requirements, in particular, will quickly become unmanageable if adequate techniques are not employed. In this paper, we describe a novel approach, based on equivalent elements, that enables the simulation of large chambers with modest computational resources. The method is then validated against 558 real measurement results. [ABSTRACT FROM AUTHOR]

We introduce a novel technique that speeds up the computation of a frequency sweep or some parametric change of material properties—assumed uniform over the entire domain—around a nominal value in electroquasi-static problem or magnetoquasi-static problem. In place of using the usual practice of solving the complex systems arising at each frequency and at each material parameter value independently, our technique requires only one factorization of a real, symmetric, and positive definite matrix. The solution at each frequency and each value of material parameter is, then, found with a few back-substitutions only. The obtained speedup is sensible and the implementation is straightforward, showing the usefulness of the proposed technique in practical applications. [ABSTRACT FROM AUTHOR]

We introduce a novel technique—lean complementarity—that attempts to eliminate any waste of computational resources occurring during the pursuing of complementarity. First, contrarily to the widely used practice of solving the problem two times with a pair of complementary or complementary-dual formulations, lean complementarity requires just one solution with the computationally cheap formulation based on the scalar potential. This result is enabled by a novel and explicit flux equilibration technique that produces tight bounds and is computationally inexpensive, because no system has to be solved. Second, the systems arising during the adaptive mesh refinement procedure are solved inexactly on purpose, by stopping the iterations of the iterative solver when the algebraic error gets negligible with respect to the discretization error. The discretization error is bounded with complementarity, whereas the algebraic error is computed very accurately with a novel and cheap technique. [ABSTRACT FROM AUTHOR]

This paper presents a general technique to apply excitations in the framework of discrete geometric numerical methods using dual grids, such as discrete geometric approach and finite integration technique. The technique overcomes some limitations of the impedance boundary condition we proposed in a previous work, especially when dealing with waveguides, where specific excitation modes must be applied. The proposed approach is based on a scattering/total field decomposition, which, if needed, allows to study scatterings due to objects. [ABSTRACT FROM AUTHOR]

Integral formulations can be convenient for computing eddy currents in complicated electromagnetic systems. However, large-scale problems may quickly exceed the memory capacity even of very large machines, since the matrices are fully populated. We aim at illustrating how \mathcal H -matrices with adaptive cross approximation can provide an effective method to increase the size of the largest solvable problems by means of boundary element methods based on stream functions with modest implementation effort. The method is first validated on a benchmark problem for which an analytical solution is available and then applied to a complex problem of engineering interest related to the computation of the currents induced in the stabilizing shell of a nuclear fusion device. [ABSTRACT FROM AUTHOR]

We introduce a novel technique to solve eddy current problems in non-manifold thin conductors by a boundary integral method based on a stream function. The idea is to perform a surgery on the non-manifold surface representing the thin conductor to reduce it into a minimal number of manifold parts. Then, manifold parts are stitched together by adding constraints, which ensure that the discrete current continuity law holds. [ABSTRACT FROM AUTHOR]

An electromagnetic wave propagation problem can be formulated according to a pair of complementary formulations, called the e -formulation and the h -formulation. The two formulations are linked to each other by Maxwell’s curl equations, and, in the continuous setting, they are perfectly equivalent in describing the wave propagation phenomenon. However, this is not true in the discrete setting, where the two formulations, in general, give different solutions. In the past decades, complementary formulations were widely studied for static problems and eddy-current problems, where they were exploited as error estimators for adaptive refinement schemes. Moreover, the so-called bilateral energy bounds arise for some problems whether theoretically or at least numerically. However, to the best of our knowledge, little attention has been given to complementarity in the wave propagation problems. In this paper, we propose an adaptive refinement scheme using the constitutive error as an estimator, and then, we investigate the behavior in terms of bilateral energy bounds. [ABSTRACT FROM AUTHOR]

Taking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system. [ABSTRACT FROM PUBLISHER]

We present a technique to extend the geometric construction of diagonal discrete Hodge operators to arbitrary triangular and tetrahedral boundary conforming Delaunay meshes in the frequent case of piecewise uniform and isotropic material parameters. The technique is based on the novel concept of signed dual complex that originates from a physical argument. In particular, it is shown how the positive definiteness of the mass matrix obtained with the signed dual complex is easily ensured for all boundary conforming Delaunay meshes without requiring—as expected by the common knowledge—that each circumcenter has to lie inside the corresponding element. Eliminating this requirement, whose fulfillment presents otherwise formidable practical difficulties, enables one to easily obtain efficient, consistent, and stable schemes. [ABSTRACT FROM PUBLISHER]

We present an effective technique to solve eddy current problems in thin conductors of arbitrary topology by a boundary element method based on a stream function. By considering a mesh of the thin conductor, which we assume to be a surface (i.e., an orientable combinatorial two-manifold embedded in \mathbb R^3 ), the aim of this paper is to introduce a novel technique to render the stream function single valued when the thin conductor is not topologically trivial. In particular, a novel combinatorial algorithm to compute the appropriate cohomology generators in linear time worst case complexity is introduced, providing an effective and rigorous solution for the required topological preprocessing. [ABSTRACT FROM PUBLISHER]

We present a technique to efficiently compute optimal cuts required to solve 3-D eddy current problems by magnetic scalar potential formulations. By optimal cuts, we mean the representatives of (co)homology generators with minimum support among the ones with a prescribed boundary. In this paper, we obtain them by starting from the minimal (co)homology generators of the combinatorial two-manifold representing the interface between conducting and insulating regions. Optimal generators are useful because they reduce the fill-in of the sparse matrix and ease human-guided basis selection. In addition, provided that the mesh is refined enough to allow it, they are not self-intersecting. The proposed technique is based on a novel graph-theoretic algorithm to solve a maximum circulation network flow problem in unweighted graphs that typically runs in linear time. [ABSTRACT FROM PUBLISHER]