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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 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 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]

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]

Solving lossy multiconductor transmission line (MTL) equations is of fundamental importance for the design and signal integrity verification of interconnections in VLSI systems. It is well established that the critical issue is the efficient and accurate electrical characterization of the MTLs through the determination of their per-unit-length parameters. In this respect, the so-called complementarity has the potential to become a fast and accurate method for the extraction of these parameters. Besides the value of the parameters, in fact, complementarity provides rigorous error bounds for them. Despite this important feature, commercial software do not use complementarity yet, due to the fact that there are unsolved theoretical issues related to the nonstandard formulation based on the electric vector potential. Some attempts to fill this gap have been already reported. The aim of this paper is to fill this gap by introducing a general formulation based on the electric vector potential highlighting the advantages of complementarity with respect to the standard first- and second-order finite element formulations. [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]