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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]

This paper presents in detail the extension of the T-Ω formulation for eddy currents based on higher-order hierarchical basis functions so that it can automatically deal with conductors of arbitrary topology. To this aim, we supplement the classical hierarchical basis functions with nonlocal basis functions spanning the first de Rham cohomology group of the insulating region. Such nonlocal basis functions may be efficiently and automatically found in negligible time with the recently introduced Dłotko–Specogna (DS) algorithm. The approach presented in this paper merges techniques together which to some extent already existed in literature but they were never grouped together and tested as a single unit. [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]

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]

Computing the first cohomology group generators received great attention in computational electromagnetics as a theoretically sound and safe method to produce cuts required when eddy-current problems are solved with the magnetic scalar potential formulations. This paper exploits the novel concept of lazy cohomology generators and a fast and general algorithm to compute them. This graph-theoretic algorithm is much faster than all competing ones being the typical computational time in the order of seconds even with meshes formed by millions of elements. Moreover, this paper introduces the use of minimal boundary generators to ease human-based basis selection and to obtain representatives of generators with compact support. We are persuaded that this is the definitive solution to this long-standing problem. [ABSTRACT FROM AUTHOR]

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]

Abstract: The problem of computing cohomology generators of a cell complex is gaining more and more interest in various branches of science ranging from computational physics to biology. Focusing on engineering applications, cohomology generators are currently used in computer aided design (CAD) and in potential definition for computational electromagnetics and fluid dynamics. The aim of this paper is to introduce a novel technique to effectively compute cohomology generators focusing on the application involving the potential definition for h-oriented eddy-current formulations. This technique, which has been called Thinned Current Technique (TCT), is completely automatic, computationally efficient and general. The TCT runs in most cases in linear time and exhibits a speed up of orders of magnitude with respect to the best alternative documented implementation. [Copyright &y& Elsevier]

The aim of the paper is to present an automatic and general technique, suitable with the A-X geometric eddy-current formulation, to impose sources over massive conductors of any shape. For this purpose, the localized source approach is used, which does not require the solutions of steady-state conduction problems in the preprocessing stage. Nevertheless, this approach needs a thick cut in each active conductor, which is usually found "by hand." In this paper, an automatic and general algorithm to compute such thick cuts is introduced. Some benchmark problems are presented to demonstrate the generality and the robustness of the algorithm. [ABSTRACT FROM AUTHOR]

A perturbation method for the T - Ω geometric formulation to solve eddy-current problems is introduced. The proposed formulation is applied to the feasibility design of a nondestructive evaluation device suitable to detect "long" longitudinal flaws in hot steel bars. [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]

Abstract: In this paper a Geometric T–Ω formulation to solve eddy-current problems on a tetrahedral mesh is presented. When non-simply-connected conducting regions are considered, the formulation requires the so-called thick cuts, while, in the literature, more attention is usually given to the so-called thin cuts. While the automatic construction of thin cuts has been theoretically solved many years ago, no implementation of an algorithm to compute the thick cuts which can be used in practice exists so far. In this paper, we propose how to fill this gap by introducing an algorithm to automatically compute the thick cuts on real-sized meshes, based on a belted tree and a tree–cotree decomposition. The belted tree is constructed by means of a homology computation by exploiting efficient reduction methods. A number of benchmarks are presented to demonstrate the generality and the robustness of the algorithm. A rigorous definition of thick cuts, which necessarily has to rely on cohomology, is presented in addition. [Copyright &y& Elsevier]

We examine the construction of reluctivity and Ohm's constitutive matrices for discrete geometric approaches using nonregular hexa-hedra as primal volumes. The effect of element deformation on the representation of uniform fields have been investigated for static fields. Then convergence and accuracy of the proposed matrices has been carried out using an eddy-current problem as working example. [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]

Using a geometric formulation for eddy currents, we present a geometric approach to constructing approximations of the discrete magnetic and Ohm's constitutive matrices. In the case of Ohm's matrix, we also show how to make it symmetric. We compared the impact on the solution of the proposed Ohm's matrices, and an iterative technique to obtain a consistent right-hand-side term in the final system is described. [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]

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]