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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]

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]

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 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]

This paper addresses the calculation of the resistive distribution of halo currents in three-dimensional structures of large magnetic confinement fusion machines. A Neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error. An irrotational electric field is obtained by a geometric formulation based on the electric scalar potential, whereas three geometric formulations are compared to obtain a solenoidal current density: a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed-hybrid Finite Elements. The electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre-processing it requires. To solve this challenging problem, we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state-of-the-art solutions proposed in literature. Believing that our results are useful in other fields of scientific computing, the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented. [ABSTRACT FROM AUTHOR]

This paper presents a methodology to efficiently calculate the resistive distribution of halo currents in 3-D conductive structures that surround the plasma in magnetic confinement fusion devices. The domain of the problem is so complicated that two complementary formulations are used to monitor the discretization error. It turns out that thousands of cohomology generators are needed by the electric vector potential electrokinetic formulation, which would require an enormous amount of memory and computing power to retrieve them even using state-of-the-art algorithms. To solve this challenging problem, we present a novel algorithm to generate the absolute second cohomology group generators exploiting the idea of lazy cohomology generators stored as sparse vectors. The proposed algorithm allows a saving of between four and five orders of magnitude computational time. [ABSTRACT FROM AUTHOR]

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]