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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]

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]

Abstract: An electromagnetic problem can be discretized on a pair of interlocked primal–dual grids according to discrete geometric approaches like the Finite Integration Technique (FIT) or the Cell Method (CM). The critical aspect is however the construction of the discrete counterparts of the constitutive relations assuring stability and consistency of the overall discrete system of algebraic equations. Initially only orthogonal Cartesian grids where considered; more recently primal grids of tetrahedra and oblique prisms with triangular base can be handled. With this paper a novel set of edge and face vector functions for general polyhedral primal grids is presented, complying with precise specifications which allow to construct stable and consistent discrete constitutive equations in the framework of an energetic approach. [Copyright &y& Elsevier]

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]

The geometric reinterpretation of the Finite Element Method (FEM) shows that Raviart–Thomas and Nédélec mass matrices map from degrees of freedoms (DoFs) attached to geometric elements of a tetrahedral grid to DoFs attached to the barycentric dual grid. The algebraic inverses of the mass matrices map DoFs attached to the barycentric dual grid back to DoFs attached to the corresponding primal tetrahedral grid, but they are of limited practical use since they are dense. In this paper we present a new geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids and possibly inhomogeneous and anisotropic materials, debunking the conventional wisdom that the barycentric dual grid prohibits a sparse representation for inverse mass matrices. In particular, we provide a unified framework for the construction of both edge and face mass matrices and their sparse inverses. Such a unifying principle relies on novel geometric reconstruction formulas, from which, according to a well-established design strategy, local mass matrices are constructed as the sum of a consistent and a stabilization part. A major difference with the approaches proposed so far is that the consistent part is defined geometrically and explicitly, that is, without the necessity of computing the inverses of local matrices. This provides a sensible speedup and an easier implementation. We use these new sparse inverse mass matrices to discretize a three-dimensional Poisson problem, providing the comparison between the results obtained by various formulations on a benchmark problem with analytical solution. • First construction of sparse inverses of edge and face Finite Elements mass matrices for arbitrary tetrahedral grids. • We provide a unified framework for the design of both edges and face mass matrices and their sparse inverses. • Our construction handles inhomogeneous and anisotropic materials. • The entries of the sparse inverse mass matrices are geometrically defined with closed form expressions. [ABSTRACT FROM AUTHOR]