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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 introduces a novel method for constructing mass matrices for general tetrahedral elements. By considering the construction of the reluctivity mass matrix as an example, a new derivation of a recipe to geometrically construct a symmetric positive semi-definite and consistent mass matrix is provided. It is shown why such a matrix can be used inside formulations where the mass matrix is right multiplied by the appropriate incidence matrix. Then, a second recipe is introduced to produce a consistent and symmetric positive definite matrix. The proposed method is based on an original algebraic technique that differs fundamentally from what is available in the literature. The proposed recipe opens also the possibility to improve the conditioning of the resulting elementary mass matrices. [ABSTRACT FROM AUTHOR]

This paper first introduces a simplified construction of the constitutive matrices for general polyhedral meshes. These constitutive matrices are geometrically defined by means of simple closed-form expressions involving the geometric elements of the primal and dual meshes. Then, we solve for the first time the Maxwell’s eigenvalue problem on general polyhedral meshes. In particular, we analyze the convergence of the eigenvalues when the mesh is adaptively refined by using the subgridding technique together with the constitutive inconsistency as error indicator. [ABSTRACT FROM AUTHOR]

In this work we propose an adaptive version of the recently introduced Mixed High-Order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism. The latter include, in particular, the numerical modeling of comb-drive and MEMS devices. Mesh adaptation is driven by newly derived, residual-based error estimators. The resulting method has several advantageous features: It supports fairly general meshes, it enables arbitrary approximation orders, and has a moderate computational cost thanks to hybridization and static condensation. The a posteriori-driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions, allowing to fully exploit the high-order of approximation. [ABSTRACT FROM AUTHOR]

In the recent years, reformulating the mathematical description of physical laws in an algebraic form using tools from algebraic topology gained popularity in computational physics. Physical variables are defined as fluxes or circulations on oriented geometric elements of a pair of dual interlocked cell complexes, while physical laws are expressed in a metric-free fashion with incidence matrices. The metric and the material information are encoded in the discrete counterpart of the constitutive laws of materials, also referred to as constitutive or material matrices. The stability and consistency of the method is guaranteed by precise properties (symmetry, positive definiteness, consistency) that material matrices have to fulfill. The main advantage of this approach is that material matrices, even for arbitrary star-shaped polyhedral elements, can be geometrically defined, by simple closed-form expressions, in terms of the geometric elements of the primal and dual grids. That is why this original technique may be considered as a "Discrete Geometric Approach" (DGA) to computational physics. This paper first details the set of vector basis functions associated with the edges and faces of a polyhedral primal grid or of a dual grid. Then, it extends the construction of constitutive matrices for bianisotropic media. [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]

We introduce a novel technique—lean complementarity—that attempts to eliminate any waste of computational resources occurring during the pursuing of complementarity. First, contrarily to the widely used practice of solving the problem two times with a pair of complementary or complementary-dual formulations, lean complementarity requires just one solution with the computationally cheap formulation based on the scalar potential. This result is enabled by a novel and explicit flux equilibration technique that produces tight bounds and is computationally inexpensive, because no system has to be solved. Second, the systems arising during the adaptive mesh refinement procedure are solved inexactly on purpose, by stopping the iterations of the iterative solver when the algebraic error gets negligible with respect to the discretization error. The discretization error is bounded with complementarity, whereas the algebraic error is computed very accurately with a novel and cheap technique. [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]

Taking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system. [ABSTRACT FROM PUBLISHER]

We present a technique to extend the geometric construction of diagonal discrete Hodge operators to arbitrary triangular and tetrahedral boundary conforming Delaunay meshes in the frequent case of piecewise uniform and isotropic material parameters. The technique is based on the novel concept of signed dual complex that originates from a physical argument. In particular, it is shown how the positive definiteness of the mass matrix obtained with the signed dual complex is easily ensured for all boundary conforming Delaunay meshes without requiring—as expected by the common knowledge—that each circumcenter has to lie inside the corresponding element. Eliminating this requirement, whose fulfillment presents otherwise formidable practical difficulties, enables one to easily obtain efficient, consistent, and stable schemes. [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]

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]

The numerical modeling of nanoscale electron devices needs the development of accurate and efficient numerical methods, in particular, for the numerical solution of the Schrödinger problem. If FEMs allow an accurate geometric representation of the device, they lead to a discrete counterpart of Schrödinger problem in terms of a computationally heavy generalized eigenvalue problem. Exploiting the geometric structure behind the Schrödinger problem, we will construct a numerically efficient discrete counterpart of it, yielding to a standard eigenvalue problem. We will also show how the two approaches are only partially akin to each other even when lumping is applied. [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]

This paper presents a novel discrete model for cell membranes and electrodes contact impedances alternative to the widely used finite elements. The finite element approach can be considered as a tool for constructing finite dimensional systems of equations that approximate the specific electroquasistatic biological problem on the discrete level. Although the finite element technique is explained typically in terms of variational or weighted-residual approaches, another, less familiar way is available to reformulate geometrically the same physical problem. This approach, referred to as discrete geometric approach, allows a direct link between geometry and the degrees of freedom describing the specific biological problem. It is straightforward to implement in any finite element open software and it assures a correct modeling of voltages and currents playing a fundamental role in a biological problem. The validation has been performed, as a first step, against analytical solutions; then, we considered impedance measurements regarding erythrocytes in whole blood flowing in microchannels at high shear rates. [ABSTRACT FROM PUBLISHER]

A numerical model of the cathode of a proton-exchange membrane fuel cell (PEMFC) based on the discrete geometric approach is presented. The current density distribution inside a PEMFC cathode is simulated. The discrete formulation of a steady-state conduction problem in nonisotropic media is coupled to nonlinear boundary conditions representing the charge transfer at catalyst layers. The cathode performance is assessed under different load conditions by assuming a constant voltage operation mode. Results are in good agreement with those presented in the literature. [ABSTRACT FROM PUBLISHER]

Abstract: The time independent Schrödinger equation stems from quantum theory axioms as a partial differential equation. This work aims at providing a novel discrete geometric formulation of this equation in terms of integral variables associated with precise geometric elements of a pair of three-dimensional interlocked grids, one of them based on tetrahedra. We will deduce, in a purely geometric way, a computationally efficient discrete counterpart of the time independent Schrödinger equation in terms of a standard symmetric eigenvalue problem. Moreover boundary and interface conditions together with non homogeneity and anisotropy of the media involved are accounted for in a straightforward manner. This approach yields to a sensible computational advantage with respect to the finite element method, where a generalized eigenvalue problem has to be solved instead. Such a modeling tool can be used for analyzing a number of quantum phenomena in modern nano-structured devices, where the accounting of the real 3D geometry is a crucial issue. [ABSTRACT FROM AUTHOR]

Abstract: By exploiting the geometric structure behind Maxwell’s equations, the so called discrete geometric approach allows to translate the physical laws of electromagnetism into discrete relations, involving circulations and fluxes associated with the geometric elements of a pair of interlocked grids: the primal grid and the dual grid. To form a finite dimensional system of equations, discrete counterparts of the constitutive relations must be introduced in addition. They are referred to as constitutive matrices which must comply with precise properties (symmetry, positive definiteness, consistency) in order to guarantee the stability and consistency of the overall finite dimensional system of equations. The aim of this work is to introduce a general and efficient set of vector functions associated with the edges and faces of a polyhedral primal grids or of a dual grid obtained from the barycentric subdivision of the boundary of the primal grid; these vector functions comply with precise specifications which allow to construct stable and consistent discrete constitutive equations for the discrete geometric approach in the framework of an energetic method. [ABSTRACT FROM AUTHOR]

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]

As a first step, we will investigate the extent of the consistency error for a hexahedral primal grid due to the use mass matrices of finite elements within the discrete geometric approach (DGA) as discrete counterparts of the constitutive relations. Such matrices are constructed by resorting to mixed elements edge and face vector functions. Then, we will propose a novel set of geometric vector basis functions for hexahedra that guarantee consistency and stability of the constitutive matrices for the DGA constructed according to the so-called energetic approach. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a Discrete Geometric Approach to solving 2-D non-linear magnetostatic problems. In particular, an efficient algorithm will be presented to solve magnetostatics in isotropic non-linear media by means of a Newton-Raphson scheme, in which the Jacobian is calculated analytically. Results on a reference configuration (TEAM Workshop Problem 25) are reported and discussed. [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]

The aim of the paper is to highlight some of the innovative methodologies, techniques and systems for nondestructive electromagnetic testing, which have been developed in the framework of the applications of methods of a diagnostics electromagnetic project partially funded by the Italian Ministry of University and Research. In particular, we will present the feasibility design of a suitable exciting-receiving coils configuration able to detect long defects by means of eddy-currents. To solve the forward eddy-currents problem, advanced analysis tools have been developed and validated. [ABSTRACT FROM AUTHOR]

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]

We propose a geometric reinterpretation of the Nodal Force Method in the framework of a pair of discrete formulations for magneto- statics on complementary meshes. [ABSTRACT FROM AUTHOR]

We propose a discrete geometric formulation based on a magnetic scalar potential and on the circulation of an electric vector potential to solve eddy-current problems for nondestructive evaluation of steel bars with longitudinal defects. [ABSTRACT FROM AUTHOR]

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 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 focuses on the characterization of radio propagation, and data communication in a marine environment. More specifically, we consider signal propagation when three different sub-gigahertz industrial, scientific, and medical (ISM) bands, i.e., 169 MHz, 434 MHz, and 868 MHz, are used. The main focus of the paper is to evaluate the path loss (PL), i.e., the power loss that a propagation radio wave would experience when communication occurs between a sail boat and a buoy. We describe the measurement results obtained performing three different radio power measurement campaigns, at the three different aforementioned ISM sub-gigahertz bands. We also want to correlate the radio propagation quality with the weather conditions present in the measurement areas. The obtained results show that higher distances are achieved by transmitting at lower frequencies, i.e., 169 MHz, and, on average, the propagation is directly dependent from the dew point index. [ABSTRACT FROM AUTHOR]

Magic formulas are the geometric identities at the root of modern compatible schemes for polyhedral grids. We present rigorous yet elementary proofs of the magic formulas originating from Stokes theorem. The proofs enlighten new fundamental aspects of the mass matrices produced with the magic formulas. First, the construction of the mass matrices works for an unexpectedly broad type of mesh cells. Second, they show that dual nodes can be arbitrarily positioned thus extending the construction of the dual barycentric grid. [ABSTRACT FROM AUTHOR]

We propose a novel technique to estimate the total volume of unknown insulating inclusions in an electrically conducting body from voltage measurements. Unlike conventional Electrical Impedance Tomography (EIT) systems that usually exhibit low spatial resolution and accuracy, the proposed device is composed of a pair of driving electrodes which, supplied with a known sinusoidal voltage, create a current density field inside a region of interest. The electrodes are designed to generate a current density field in the region of interest that is uniform, to a good approximation, when the inclusions are not present. A set of electrodes with a polygonal geometry is used for four-wires resistance measurements. The proposed technique has been tested designing a low cost prototype, where all electrodes are on the bottom of the conducting body, showing good performances. Such a device may be used to monitor the volume of biological cells inside cell culture dishes or the volume of blood clots in micro-channels in lab-on-a-chip biosensors. [ABSTRACT FROM AUTHOR]

The cell method (CM) or discrete geometric approach (DGA) in the time domain, already introduced by Codecasa et al. in 2008 for the coupled Ampere–Maxwell and Faraday equations, is here recast as a Galerkin Method similar to the finite-element method (FEM). In particular, it is shown to be a mixed method comprising an explicit scheme and two discontinuous Galerkin (DG) FEM spaces formulated on dual meshes, in which each of the two function spaces provides a continuous numerical flux choice for its dual mesh counterpart. The implemented version is shown to compare favorably in terms of accuracy and efficiency with respect to the classic conforming FEM scheme using Whitney elements. When tested on the same tetrahedral mesh, the Courant–Friedrichs–Lewy (CFL) condition for the proposed approach is a factor of 2 less restrictive on the time step with respect to the curl-conforming FEM scheme. [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]

A fundamental extension for a recently introduced numerical scheme for the time-domain solution of Maxwell’s equations on tetrahedral meshes is introduced: the algorithm is here shown to be able to handle materials with finite electric resistivity, without losing any of its amenable properties. A theoretic stability analysis, valid for the extended algorithm, is presented for the first time, and a Courant–Friedrich–Lewy sufficient condition on the maximum time step allowed for the scheme to be stable is derived analytically. This result is completely novel for finite-difference time-domain (FDTD)-like approaches on tetrahedral grids. Finally, the accuracy of the extended algorithm is tested with respect to the well-known implicit finite-element method scheme on tetrahedral grids, to the canonical FDTD scheme on Cartesian orthogonal grids, and to a commercial code implementing FDTD with staircasing mitigation techniques. [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]

The frequency-domain wave propagation problem is notoriously difficult to solve through iterative methods because it leads to a symmetric but indefinite linear system. For this reason, direct methods are usually employed at the expense of great memory usage. Convergence of iterative methods, however, could be obtained by regularizing the wave equation. We introduce such regularization in discrete geometric approach framework on polyhedral grids. Moreover, we extend the regularization to the impedance boundary condition. [ABSTRACT FROM AUTHOR]

We present a numerical method named mixed high order (MHO) to obtain high order of convergence for electrostatic problems solved on general polyhedral meshes. The method, based on high-order local reconstructions of differential operators from face and cell degrees of freedom, exhibits a moderate computational cost thanks to hybridization and static condensation that eliminate cell unknowns. After surveying the method, we assess its effectiveness for 3-D problems by comparing, for the first time in literature, its performances with classical conforming finite elements. Moreover, we emphasize the algebraic equivalence of MHO in the lowest order with the analog formulation obtained with the discrete geometric approach or the finite-integration technique. [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]

We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse theory. We introduce the new concept of complete acyclic matchings and we show that they give the same end result of Gaussian elimination. Basically, instead of doing costly row and column operations on a sparse matrix, we compute equivalent cheap combinatorial operations that preserve the underlying sparsity structure. Currently, the most efficient algorithms proposed in literature to find discrete vector potentials make use of tree-cotree techniques. We show that they compute a special type of complete acyclic matchings. Moreover, we show that the problem of computing them is equivalent to the problem of deciding whether a given mesh has a topological property called collapsibility. This fact gives a topological characterization of well-known termination problems of tree-cotree techniques. We propose a new recursive algorithm to compute discrete vector potentials. It works directly on basis elements of 1- and 2-chains by performing elementary Gaussian operations on them associated with acyclic matchings. However, the main novelty is that it can be applied recursively. Indeed, the recursion process allows us to sidetrack termination problems of the standard tree-cotree techniques. We tested the algorithm on pathological triangulations with known topological obstructions. In all tested problems we observe linear computational complexity as a function of mesh size. Moreover, the algorithm is purely graph-based so it is straightforward to implement and does not require specialized external procedures. We believe that our framework could offer new perspectives to sparse matrix computations. • Novel graph-theoretic algorithm to compute discrete potentials. • Characterization of termination problems of tree-cotree techniques. • Discrete Morse theory framework applicable to general sparse matrix computations. • Linear computational complexity for meshes having known topological obstructions. [ABSTRACT FROM AUTHOR]

This paper presents a general technique to apply excitations in the framework of discrete geometric numerical methods using dual grids, such as discrete geometric approach and finite integration technique. The technique overcomes some limitations of the impedance boundary condition we proposed in a previous work, especially when dealing with waveguides, where specific excitation modes must be applied. The proposed approach is based on a scattering/total field decomposition, which, if needed, allows to study scatterings due to objects. [ABSTRACT FROM AUTHOR]