Edge Finite Element Research Articles

Exact conservation in transient electromagnetics using edge elements

In time domain to solve electromagnetic radiation and scattering problems, several numerical methods are available in literature. Among these numerical methods Finite Difference Time Domain (FDTD) method and Time Domain FEM (TDFEM) method are more popular. In the present work, we have proposed a quantity in continuum, which is conserved under certain boundary conditions. Additionally, we have also proposed a time marching scheme within edge element framework, which satisfy this conservation characteristic in algorithmic sense irrespective of the choice of time step. To facilitate this, we have used some conversion algorithm that transform nodal finite element data to edge finite element data.

A high-performance electromagnetic code to simulate high-temperature superconductors

Superconductivity is a physical phenomenon of some materials that allow them to conduct electrical current without resistance. This property has a wide range of applications. In particular, type-II high-temperature superconductors appear promising for building strong electromagnets carrying large amounts of current in the extreme conditions foreseen for future fusion reactors. However, their fabrication and related experiments are highly expensive and complex. Therefore, there is an increasing need for numerical models to guide the design optimization of superconducting cables and to predict their performance. In this work, we present a new code to simulate high-temperature superconductors (HTS) based on the edge finite element discretization of Maxwell’s equations in the time domain using the widely adopted H-formulation in the superconductor analysis community. This code is integrated in the High-Performance Computing (HPC) Alya suite and obtain an excellent performance up to 1024 processors in MareNostrum 4 supercomputer. This capacity allows to us to solve a wide variate of problems in big domains in relatively reduce amount of computer time, being a promising tool to aboard HTS coils design problems. Furthermore, validations against experimental data are presented and code results for superconducting tapes with different magnetic properties are analyzed.

Meshing strategies for 3d geo-electromagnetic modeling in the presence of metallic infrastructure

In 3D geo-electromagnetic modeling, an adequate discretisation of the modeling domain is crucial to obtain accurate forward responses and reliable inversion results while reducing the computational cost. This paper investigates the mesh design for subsurface models, including steel-cased wells, which is relevant for many exploration settings but still remains a numerically challenging task. Applying a goal-oriented mesh refinement technique and subsequent calculations with the high-order edge finite element method, simulations of 3D controlled-source electromagnetic models in the presence of metallic infrastructure are performed. Two test models are considered, each needing a distinct version of approximation methods to incorporate the conductive steel casings of the included wells. The influence of mesh quality, goal-oriented meshing, and high-order approximations on problem sizes, computational cost, and accuracy of electromagnetic responses is investigated. The main insights of our work are: (a) the applied numerical schemes can mitigate the computational burden of geo-electromagnetic modeling in the presence of steel artifacts; (b) investigating the processes driving the meshing of models with embedded metallic infrastructures can lead to adequate strategies to deal with the inversion of such electromagnetic data sets. Based on the modeling results and analyses conducted, general recommendations for modeling strategies are proposed when performing simulations for challenging steel infrastructure scenarios.

Computations and measurements of the magnetic polarizability tensor characterisation of highly conducting and magnetic objects

PurposeThe ability to characterise highly conducting objects, that may also be highly magnetic, by the complex symmetric rank–2 magnetic polarizability tensor (MPT) is important for metal detection applications including discriminating between threat and non-threat objects in security screening, identifying unexploded anti-personnel landmines and ordnance and identifying metals of high commercial value in scrap sorting. Many everyday non-threat items have both a large electrical conductivity and a magnetic behaviour, which, for sufficiently weak fields and the frequencies of interest, can be modelled by a high relative magnetic permeability. This paper aims to discuss the aforementioned idea.Design/methodology/approachThe numerical simulation of the MPT for everyday non-threat highly conducting magnetic objects over a broad range of frequencies is challenging due to the resulting thin skin depths. The authors address this by employing higher order edge finite element discretisations based on unstructured meshes of tetrahedral elements with the addition of thin layers of prismatic elements. Furthermore, computer aided design (CAD) geometrical models of the non-threat and threat object are often not available and, instead, the authors extract the geometrical features of an object from an imaging procedure.FindingsThe authors obtain accurate numerical MPT characterisations that are in close agreement with experimental measurements for realistic physical objects. The assessment of uncertainty shows the impact of geometrical and material parameter uncertainties on the computational results.Originality/valueThe authors present novel computations and measurements of MPT characterisations of realistic objects made of magnetic materials. A novel assessment of uncertainty in the numerical predictions of MPT characterisations for uncertain geometry and material parameters is included.

Adaptive edge finite element method and numerical design for metasurface cloak

Metasurfaces are metamaterials with two-dimensional (2D) or planar structures. Compared to traditional metamaterials, metasurfaces have the advantage of being lightweight, easy to control and simple to design. This paper outlines the phase matching principle and constitutive parameter theory of electromagnetic (EM) metasurfaces, as well as the constitutive parameter analysis and model equations for metasurface cloaks. Due to the singularity of the metasurface cloak, an Adaptive Edge Finite Element Method (AEFEM) is developed in this paper. In addition, the convergence theory of the AEFEM for the time-harmonic Maxwell's equations with one sign-changing coefficient is established under the T-coercivity assumption. Numerical simulations of the arc-shaped metasurface cloak have confirmed the accuracy of our model and numerical theory. Furthermore, we have used our numerical method to design a triangular-shaped metasurface cloak with fewer singularities.

Vector Nodal Meshless Method for 3-D Applications

Traditional meshless methods use scalar-based functions and therefore, present difficulties in approximating vector fields. The Vector Nodal Meshless Method (VNMM) constructs its approximations using shape functions based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(curl)</i> spaces and Nédélec’s first type elements polynomial space. In this sense, a set of nodes is distributed in the domain and on its boundary, and each node has an associated unit vector. The VNMM has been applied to solve two-dimensional electromagnetic problems. This paper presents the construction of the VNMM shape functions for use in three-dimensional vectorial problems. The interpolation capacity of shape function and the method convergence rates are shown. Then, the VNMM is applied to an eigenvalue problem and compared to the traditional Edge Finite Element Method (EFEM). The numerical solution for eigenvalues is not corrupted by spurious modes. Finally, a nonlinear magnetostatic problem is solved. A good performance of the method is observed.

Construction of a spanning tree for high‐order edge elements

AbstractThe well‐known tree–cotree gauging method for low‐order edge finite elements is extended to high‐order approximations, within the first family of Nédélec finite element spaces. The key point in this method is the identification of degrees of freedom for edge and nodal finite element spaces, such that the matrix of the gradient operator is the transpose of the all nodes incidence matrix of a directed graph. This is straightforward for low‐order finite elements, and it can be proved that it is still possible in the high‐order case using either moments or weights as degrees of freedom. In the case of weights, a geometrical realization of the graph associated with the gradient operator is very natural. We recall in details the definition of weights and present an algorithm for the construction of a spanning tree of this graph. The starting point of the algorithm is a spanning tree of the graph given by vertices and edges of the mesh (the so‐called global spanning tree, that is the one used in the low‐order case). This global step, interpreted in the high‐order sense, is enriched locally, with a loop over the elements of the mesh, with arcs corresponding to edge, face, and volume degrees of freedom required in the high‐order case.

On the approximation of electromagnetic fields by edge finite elements—Part 4: analysis of the model with one sign-changing coefficient

In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set \(\varOmega \) of \(\mathbb {R}^3\), and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale \((\varvec{H}^{\texttt {s}}(\varOmega ))_{\texttt {s}>0}\). With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived.

A Parallel Eigensolver for Photonic Crystals Discretized by Edge Finite Elements

A Parallel Eigensolver for Photonic Crystals Discretized by Edge Finite Elements

Numerical simulation methods for motion-induced eddy current testing signals based on Ar formulation and edge finite elements

In this paper, numerical methods for simulation of the motion-induced eddy current testing (MIECT) signals were developed based on the reduced magnetic vector potential formulation (Ar method) and the edge finite element. According to the principle of the MIECT problem, numerical methods with reference frame referring to the transducer or the inspection object were developed respectively to solve the eddy current and pickup magnetic field signals. In addition, a quasi-static numerical scheme was also adopted to simulate the MIECT problem for inspecting crack along the moving direction. Based on the proposed numerical schemes and the conventional Ar code, numerical programs of the 3 methods were developed and validated for simulating MIECT signals. Through comparing the numerical results and efficiencies of the three simulation schemes each other, the validity of the proposed methods and the corresponding numerical codes were theoretically verified at first. Then, a MIECT experimental system and a prototype probe were developed and the validity of the proposed numerical methods were further proved experimentally. It is found that the moving probe-fixed object method is the most efficient scheme for MIECT signal simulation among the three proposed numerical methods. The numerical methods and codes developed in this paper provide numerical tools for design and optimization of the MIECT probes and testing system used in inspection of rails or wheels system etc, as well as the quantitative defect evaluation based on the MIECT signals through model based inverse analyses.

Numerical solution of anH(curl)-elliptic hemivariational inequality

AbstractThis paper is concerned with the analysis and numerical solution of an $\boldsymbol {H}({\textbf {curl}})$-elliptic hemivariational inequality (HVI). One source of the HVI is through a temporal semidiscretization of a related hyperbolic Maxwell equation problem. An equivalent minimization principle is introduced, and the solution existence and uniqueness of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI are proved. Numerical analysis of the HVI is provided with a general Galerkin approximation, including a Céa’s inequality for convergence and error estimation. When the linear edge finite element method is employed, an optimal-order error estimate is derived under a suitable solution regularity assumption. A fully discrete scheme based on the backward Euler difference in time and a mixed finite element method in space is also analyzed, and stability estimates are derived for first-order terms of the fully discrete solution. Numerical results are reported on linear edge finite element solutions of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI for numerical evidence of the theoretically predicted convergence order.

On the Tree Gauge in Magnetostatics

We recall the classical tree-cotree technique in magnetostatics. (1) We extend it in the frame of high-order finite elements in general domains. (2) We focus on its connection with the question of the invertibility of the final algebraic system arising from a high-order edge finite element discretization of the magnetostatic problem formulated in terms of the magnetic vector potential. With the same purpose of invertibility, we analyse another classically used condition, the Coulomb gauge. (3) We conclude by underlying that the two gauges can be naturally considered in a high order framework without any restriction on the topology of the domain.

3D magnetotelluric modeling using high-order tetrahedral Nédélec elements on massively parallel computing platforms

We present a routine for 3D magnetotelluric (MT) modeling based upon high-order edge finite element method (HEFEM), tailored and unstructured tetrahedral meshes, and high-performance computing (HPC). This implementation extends the PETGEM modeller capabilities, initially developed for active-source electromagnetic methods in frequency-domain. We assess the accuracy, robustness, and performance of the code using a set of reference models developed by the MT community in well-known reported workshops. The scale and geological properties of these 3D MT setups are challenging, making them ideal for addressing a rigorous validation. Our numerical assessment proves that this new algorithm can produce the expected solutions for arbitrarily 3D MT models. Also, our extensive experimental results reveal four main insights: (1) high-order discretizations in conjunction with tailored meshes can offer excellent accuracy; (2) a rigorous mesh design based on the skin-depth principle can be beneficial for the solution of the 3D MT problem in terms of numerical accuracy and run-time; (3) high-order polynomial basis functions achieve better speed-up and parallel efficiency ratios than low-order polynomial basis functions on cutting-edge HPC platforms; (4) a triple helix approach based on HEFEM, tailored meshes, and HPC can be extremely competitive for the solution of realistic and complex 3D MT models and geophysical electromagnetics in general.

Electromagnetic eigen analysis: Performance comparison of four node and four edge quadrilateral elements with the effect of distortion

In edge finite elements, electric fields are obtained across edges of each element instead of nodal output as obtained in nodal finite element. In the nodal framework, the potential formulation provides required tangential continuity and normal discontinuity for electromagnetic analysis whereas this continuity requirement is directly attained in the edge framework. In this work, we have used the conversion technique to transfrom four node quadrilateral element to four edge quadrilateral element. Then, we have compared the performance of these elements by performing the eigenvalue analysis for two standard examples. Finally, we also analyzed the effect of distortion in those elements.

Two-level iterative algorithms for two-dimensional H(curl)-elliptic problem with a symmetry interior penalty discontinuous Galerkin method

In this paper, we first present a symmetry interior penalty discontinuous Galerkin (SIPDG) discretization for two-dimensional H(curl)-elliptic problem, then develop the corresponding two-level iterative algorithms. The main idea of the construction of algorithms is to decompose the DG finite element spaces into the high frequency component and edge finite element spaces. We also present some numerical experiments to show the efficiency of our algorithms.

High order edge-based finite elements for 3D magnetotelluric modeling with unstructured meshes

We present a 3D magnetotelluric (MT) forward modeling program which computes the distribution of the electric field on an unstructured mesh. We employ high order edge elements and implement the Edge Finite Element method to accurately simulate magnetotelluric data in the form of impedance tensor. The program is open-source and optimized for both high-performance computing (HPC) architectures and non-HPC architectures. The paper details the theory, method and code work-flow. We evaluate the accuracy of our algorithm with various synthetic test models and compare our results with other Finite Element (FE) and Finite Difference (FD) algorithms. The experiments show that the implemented high order elements are effective in providing highly accurate solutions for multiple frequencies with few elements independently of the model’s geometry. Hence, leading to reduced computing times.

An adaptive edge finite element DtN method for Maxwell’s equations in biperiodic structures

Abstract We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell’s equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method.

A review on methods to simulate three dimensional rotating electrical machine in magnetic vector potential formulation using edge finite element method under sliding surface principle

AbstractIn recent years edge finite element method has become quite popular in modeling the three‐dimensional rotating electrical machine due to its ability to represent the physical behavior of field along interface correctly. However, modeling becomes complex when significant mesh distortion occurs along a sliding surface due to movement. A remeshing step is needed, which increases computational cost. In order to avoid remeshing, efficient numerical methods have already been developed to model movement in the three‐dimensional rotating electrical machine under edge finite element method. Nevertheless, very little research has been carried out on comparing these methods. The main objective of this paper is to compare the edge‐based interpolation and mortar edge element method. To improve the accuracy of edge‐based interpolation for tetrahedral mesh, we propose an improved strategy. Numerical results illustrate the accuracy, robustness and performance of the methods. A detailed comparison is presented based on the criteria mentioned above.

The effect of quadrature rules on finite element solutions of Maxwell variational problems

We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete a priori error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditions with respect to mesh refinement and precision for the quadrature rules so as to guarantee convergence rates following that of exact numerical integration. On curved domains, we isolate the error contribution of numerical quadrature rules.

Edge finite element method for periodic structures using random meshes

ABSTRACT This paper introduces a new method to impose the periodic boundary conditions on the arbitrary mesh when we use the edge finite element method. By using the Floquet theorem, we reduce the analysis of an infinite bi-periodic structure to an elementary cell with pseudo-periodic conditions on the lateral sides. In this method, we establish a relationship between the electric field circulation along an edge on a face and the circulation of the associated edges on the opposite face. After that, we introduce periodic relations in the finite element system. To show the possibilities of the proposed method, we present two numerical studies and the obtained numerical results are compared to those produced by the existing methods.