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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]

This paper introduces a novel methodology to optimize the design of a ratiometric rotary inductive position sensor (IPS) fabricated in printed circuit board (PCB) technology. The optimization aims at reducing the linearity error of the sensor and amplitude mismatch between the voltages on the two receiving (RX) coils. Distinct from other optimization techniques proposed in the literature, the sensor footprint and the target geometry are considered as a non-modifiable input. This is motivated by the fact that, for sensor replacement purposes, the target has to fit a predefined space. For this reason, the original optimization technique proposed in this paper modifies the shape of the RX coils to reproduce theoretical coil voltages as much as possible. The optimized RX shape was obtained by means of a non-linear least-square solver, whereas the electromagnetic simulation of the sensor is performed with an original surface integral method, which are orders of magnitude faster than commercial software based on finite elements. Comparisons between simulations and measurements performed on different prototypes of an absolute rotary sensor show the effectiveness of the optimization tool. The optimized sensors exhibit a linearity error below 0.1% of the full scale (FS) without any signal calibration or post-processing manipulation. [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]

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]

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]

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]

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]

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 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]

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]

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]

We present a technique to efficiently solve 3D eddy current problems in fusion devices, whose structures exhibit a geometrical symmetry. The proposed approach is based on the exploitation of symmetry via harmonic analysis and it is suitable to treat also problems where the field sources (plasma or external coils) do not share the domain symmetry. A simple test case is presented to describe the methodology. Then, a more complicated geometry is considered, which represents a realistic vacuum vessel of an ITER like fusion device (double layer structure with several portholes). The results are discussed for both axisymmetric and non-axisymmetric excitations in frequency domain. [ABSTRACT FROM AUTHOR]

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]

The development of methods for the active control of magnetohydrodynamic (MHD) instabilities and the correction of error fields is mandatory in view of robust magnetic confinement in future fusion reactors. Two numerical codes (CAFE, CARIDDI) have been used to simulate the dynamic response of MHD control systems in large fusion devices. Their main pros and cons are discussed. As reference test case, the frequency domain characterization of a key system for MHD control in the large JT-60SA fusion device, namely the resistive wall mode active coil set, is considered. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to present an automatic and efficient algorithm to find cohomology generators suitable for 2-D eddy-current problems formulated by means of complementary formulations. The algorithm is general, straightforward to implement, exhibits a linear worst-case computational complexity and produces optimal representatives of generators. By optimal we mean the representatives that minimize in practical cases the fill-in of the system of equations matrix and guarantee that the current flowing in each conductor is in one-to-one correspondence with a generator. As a numerical example, the complementary formulations are used to compute the frequency-dependent per-unit-length impedance in integrated circuits. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a 3-D time-domain eddy-current A formulation based on the discrete geometric approach (DGA) over unstructured and nonorthogonal hexahedral dual grids. The resulting differential algebraic system of equation, solved by means of a singly-diagonally implicit Runge-Kutta (SDIRK) variable step-size solver, leads to very accurate results at reduced computational costs, as shown by numerical analysis. [ABSTRACT FROM AUTHOR]

We examine the construction of a symmetric positive definite conductance matrix for eddy-current problems, using a discrete approach. We construct a new set of piecewise uniform basis vector functions on both the primal and the dual complex. We define these vector functions for both tetrahedra and prisms. [ABSTRACT FROM AUTHOR]

In magnetic confinement fusion devices, close-fitting passive conducting structures surrounding the plasma are an efficient way to slow down the growth rate of magnetohydrodynamic (MHD) instabilities, enabling the possibility of an active stabilization by means of external actuators (coils). Nevertheless, a fully quantitative investigation, under realistic 3D boundary conditions, is still a very hard challenge. Moreover, the inclusion of fine geometrical details of actual passive structures might easily lead to impractical memory and computational time requirements if the problem is not carefully addressed. In this work we present an effective technique to solve eddy current problems in thin conductors of arbitrary topology by a boundary element method (BEM) based on a stream function. The proposed technique is applied to study the dynamic response of the stabilizing shell of RFX-mod, both to axisymmetric and non-axisymmetric external fields. [ABSTRACT FROM AUTHOR]

Integral methods for solving eddy current problems use Biot–Savart law to produce non-local constitutive relations that lead to fully populated generalized mass matrices, better known as inductance matrices. These formulations are appealing because—unlike standard Finite Element solutions—they avoid the generation of a mesh in the insulating regions. The aim of this paper is to alleviate the three main problems of volume integral methods. First, the computation of the inductance matrix elements is slow and also delicate because of the singularity in the integral equation. This paper introduces novel face basis functions that allow a much faster inductance matrix construction with respect to the standard one based on the Rao–Wilton–Glisson (RWG) or Raviart–Thomas (RT) basis functions. Second, our basis functions work for polyhedral elements formed by any number of faces (including prisms, hexahedra and pyramids), while producing the same results as RWG and RT basis functions for tetrahedral meshes. Third, the new basis functions allow to factorize the inductance matrix and to introduce a novel family of groundbreaking low-rank inductance matrix compression techniques that show several orders of magnitude improvement in memory occupation and computational effort than state-of-the-art alternatives, allowing to solve problems that otherwise cannot be faced. • Self-contained refoundation of compatible integral methods for eddy currents. • It is based on volume uniform basis functions that work for polyhedral elements. • Solves the singularity and speeds up the computation of the double integrals. • A factorization of the inductance matrix provides a groundbreaking speedup and memory reduction. • It enables the use of fast techniques to speedup the matrix–vector product like FMM. [ABSTRACT FROM AUTHOR]

A major refurbishment of the toroidal complex of the RFX-mod device is in progress and will include the removal of the Inconel vacuum vessel and a modification of the stainless steel supporting structure to be made vacuum tight. The plasma facing graphite tiles will be mounted onto the inner surface of the copper shell so as to increase the plasma proximity factor. New operation regimes are expected to provide a significant reduction of the amplitude of RFP tearing modes with magnetic chaos mitigation and confinement improvement. On the other hand, due to the shorter distance from the passive structures, the plasma is expected to be even more sensitive to magnetic field errors at the plasma boundary, produced by the induced currents near the cuts of the same structures. In preparation to the new RFX-mod2 experiments, a thorough revision of the conditions triggering the error fields due to the eddy currents was undertaken. An analysis of the static and dynamic equilibrium magnetic field has been carried out in RFX-mod and RFX-mod2 to estimate the magnetic field driving the shell eddy currents. A lower equilibrium magnetic field should be required in RFX-mod2. Analyses of different configurations of the poloidal gap were also carried out by a specialized computational tool. A solution with a spaced poloidal gap and a more extended overlapping was envisaged capable of maintaining the same error fields at the closer plasma boundary with the same forcing field and meeting the more stringent insulation and assembly requirements of RFX-mod2. [ABSTRACT FROM AUTHOR]

After several years of successful RFX-mod operations, both in RFP and Tokamak configurations, an upgrade of the machine has been conceived. In particular, the vacuum vessel might be removed with the aim of improving passive MHD control and enabling plasma rotation at higher currents. In this paper, a detailed analysis of the new magnetic front-end is presented, with particular emphasis on the computation of the magnetic field errors generated at the poloidal gaps during transient phases of the discharge. At this purpose, a non-linear MHD equilibrium code, MAXFEA, has been used to provide the input data for the 3D electromagnetic analyses in the time domain carried out with the CAFE code. [ABSTRACT FROM AUTHOR]

RFX-mod is a device equipped with a state-of-the-art system for active control of MHD instabilities, which consists of 192 saddle coils and 192 radial field sensors. In order to further extend its operational space an upgrade of its magnetic front-end is now being studied. In this paper we focus on the implementation of fast and efficient algorithms for the 3D electromagnetic analysis of the effect of the designed magnetic front-end on the MHD control system in RFX-mod2, in the presence of complex conducting structures surrounding the plasma. [ABSTRACT FROM AUTHOR]