Tetrahedral Edge Elements Research Articles

Space-Time-Parallel 3-D Finite Element Transformer Model With Adaptive TLM and Parareal Techniques for Electromagnetic Transient Analysis

In this paper, a novel massively space-time parallel finite element (FE) transformer model is proposed for power system electromagnetic transient simulations. The method utilizes the reduced magnetic vector potential (RMVP) formulation to solve the electromagnetic field in the transformer, and the 3-D tetrahedral edge element is applied to discretize the RMVP formulation. The discretized nonlinear system is solved by adaptive transmission line modeling (TLM) and matrix-free conjugate gradient iterator, which allows parallelism at the elemental-level and eliminates the need for Newton-Raphson (NR) iterations over the large-scale global matrix. The paper also integrates the parallel-in-time method with the finite element solver to further parallelize the model at space-time level, and the GPU implementation of the model realized a speedup of over 55 and high accuracy, compared with a commercial FE software.

An edge-based finite element method for 3D marine controlled-source electromagnetic forward modeling with a new type of second-order tetrahedral edge element

This paper presents a second order edge-based finite element method based on a new type of second order edge element for numerical modeling of 3D controlled source electromagnetic data in an anisotropic conductive medium. We adopt the anomalous potential field formulation to avoid the effect of sou rce’s singularity. The computational domain was subdivided into unstructured tetrahedral elements. The sparse linear system was solved using the Induced dimensional reduction (IDR(s)) method with an incomplete LU preconditioner. We have applied the developed algorithm to compute a typical MCSEM response over several 3D models, we compare the numerical results from our method and FEM solution based on conventional first and second order element. The results of comparison shown that our algorithm can significantly reduce the computation cost without loss to much accuracy.

New Type of Second-Order Tetrahedral Edge Elements by Reducing Edge Variables for Quasi-Static Field Analysis

Second-order tetrahedral edge elements afford much better accuracy in magnetic field analysis compared with first-order elements. However, they often entail higher computational cost due to the higher number of unknowns, many of them being redundant. The partial tree-gauging approach eliminates only a portion of the redundant edge variables, often reducing the overall computational cost. In this paper, we propose a new partial tree-gauging approach in second-order tetrahedral edge elements by unifying the two edges on the sides of elements into one, resulting in a new type of second-order tetrahedral edge element. Using the proposed elements in the analysis of two linear quasi-static magnetic field problems reveal the superiority of the edge unification over the conventional partial tree gauging in term of computational load.

Magnetic Field Analysis Using Hybrid Mesh of Linear Hexahedral and Tetrahedral Edge Elements

Linear hexahedral and tetrahedral elements are joined without linking elements of pyramidal shape. Two triangular faces are constrained by a quadrilateral face in irregular meshes with hanging edges. Edge shape functions are not conforming on the interfaces. Test computations showed good accuracies in the results by the method comparing with results using meshes made of linear hexahedral, tetrahedral and quadratic-nodal-linear-edge elements. The hybrid mesh makes mesh generations flexible and overcomes the degradation of results using the linear tetrahedral edge elements with high aspect ratio for conductive regions in eddy current analyses. The quadratic-nodal-linear-edge elements were also studied and remedy the degradation of linear tetrahedral elements.

Thermal analysis of an eddy‐current heated piece by means of the FEM‐DBCI method

PurposeThis paper aims to propose a hybrid method, called finite element method‐Dirichlet boundary condition iteration (FEM‐DBCI), for the computation of time‐harmonic eddy current problems inside a conductor heated by coils in 3D open‐boundary geometry.Design/methodology/approachThe method assumes the electrical field as unknown on a mesh of tetrahedral edge elements. The heating power density inside the conductor is then computed and a steady‐state thermal analysis is performed on the same mesh of nodal tetrahedra to calculate the temperature distribution inside the heated piece, taking radiation and convection into account. A numerical example is also provided.FindingsThe method couples a differential equation for the interior problem in terms of the electric fields with an integral equation for the exterior one. The global algebraic system is efficiently solved in an iterative way.Originality/valueThe paper illustrates the computation of time‐harmonic eddy current problems inside a conductor heated by coils.

Three-dimensional finite element analysis of nonreciprocal phase shifts in magneto-photonic crystal waveguides

This report presents the first three-dimensional characterization of nonreciprocal phase shifts in magneto-photonic crystal (MPC) slab waveguides. We model MPC waveguides using a three-dimensional finite element method with curvilinear tetrahedral edge elements. This study investigates the dependence of nonreciprocal phase shifts on the width and the thickness of the waveguides, and we investigate the dependence of losses on the air hole depth, leading to a guideline for the design of optical isolators. Simulations show that waveguides with reduced width and deep air holes exhibit high nonreciprocal phase shifts and low losses. The study also shows that, compared with two-dimensional calculations, nonreciprocal phase shifts express key similarities, although the frequencies of the guided modes shift.

Fast calculation of the sensitivity matrix in magnetic induction tomography by tetrahedral edge finite elements and the reciprocity theorem

Magnetic induction tomography of biological tissue is used to reconstruct the changes in the complex conductivity distribution by measuring the perturbation of an alternating primary magnetic field. To facilitate the sensitivity analysis and the solution of the inverse problem a fast calculation of the sensitivity matrix, i.e. the Jacobian matrix, which maps the changes of the conductivity distribution onto the changes of the voltage induced in a receiver coil, is needed. The use of finite differences to determine the entries of the sensitivity matrix does not represent a feasible solution because of the high computational costs of the basic eddy current problem. Therefore, the reciprocity theorem was exploited. The basic eddy current problem was simulated by the finite element method using symmetric tetrahedral edge elements of second order. To test the method various simulations were carried out and discussed.

Symbolic algebra as a tool for understanding edge elements

The paper describes a methodology for systematic comparison of tetrahedral edge element spaces using symbolic algebra. Various known edge element families, with their respective spaces and irrotational subspaces, are studied. In addition, symbolic algebra facilitates analysis of some novel hexahedral elements and of a new "prolongation of local gradients" condition for spectral convergence (the absence of "spurious modes").

Characterization of spurious‐free finite element methods in electromagnetics

From a theoretical point of view the question of spurious modes has been regarded as a closed problem. However, in this paper we show that even a precise definition of spurious‐free approximation was lacking. Hence, a sound definition of spurious‐free finite element method is given and a set of necessary and sufficient conditions ensuring that a finite element method is spurious‐free in the defined sense is stated. A critical comparison between the proposed theory and the currently accepted one is then carried out and existing counterexamples to the latter are pointed out. Comparison with an older theory leads to another set of necessary and sufficient conditions providing a better grasp of the key feature a finite element space must have to rule out spurious modes. The impact of the proposed theory is stressed, showing that Nedelec's tetrahedral edge elements of all orders provide spurious‐free approximations in all conditions of practical interest. Finally, it is shown, for the first time to the best of authors’ knowledge, that also many high‐order edge elements, recently proposed in the engineering literature for the analysis of electromagnetic problems, provide the same kind of reliable approximation.

Computation results of the TEAM workshop Problem 7 by finite element methods using tetrahedral and hexahedral elements

In this paper, two types of finite element codes are applied to Problem 7 of TEAM workshop benchmark problems and the computation results are compared with the experimental results which were presented at the workshop. The computation model consists of an asymmetrical conductor with hole and an exciting coil. The model is solved as a three-dimensional problem by finite element methods using edge elements. A tetrahedral edge element and a hexahedral edge element are introduced to the two types of codes. The characteristics of the two codes are discussed by using the computation results of the benchmark problem.

Fortin operator and discrete compactness for edge elements

The basic properties of the edge elements are proven in the original papers by Nedelec [22,23] In the two-dimensional case the edge elements are isomorphic to the face elements (the well-known Raviart–Thomas elements [24]), so that all known results concerning face elements can be easily formulated for edge elements. In three-dimensional domains this is not the case. The aim of the present paper is to show how to construct a Fortin operator which converges uniformly to the identity in the spirit of [5,4]. The construction is given for any order tetrahedral edge elements in general geometries. We relate this result to the well-known commuting diagram property and apply it to improve the error estimate for a mixed problem which involves edge elements. Finally we show that this result can be applied to the analysis of the approximation of the time-harmonic Maxwell's system.

Design sensitivities for scattering-matrix calculation with tetrahedral edge elements

The rates of change of the scattering matrix of a microwave device with respect to geometric or material parameters are of interest to designers, and useful in automatic optimization. They are most efficiently calculated by the adjoint variable method, but this usually entails the additional cost of funding the adjoint solutions. In the finite-element formulation presented here, no new solutions are needed beyond those required to calculate the complete scattering matrix. Explicit expressions are given for the calculation of the required matrix for a tetrahedral edge element. Results are presented for three rectangular waveguide problems: a uniform length terminated in a short-circuit; an H-plane miter bend; and a two-step waveguide impedance transformer.

Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements

New vector finite elements are proposed for electromagnetics. The new elements are triangular or tetrahedral edge elements (tangential vector elements) of arbitrary polynomial order. They are hierarchal, so that different orders can be used together in the same mesh and p-adaption is possible. They provide separate representation of the gradient and rotational parts of the vector field. Explicit formulas are presented for generating the basis functions to arbitrary order. The basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning. Matrix assembly for the frequency-domain curl-curl equation is conveniently carried out by means of universal matrices. Application of the new elements to the solution of a parallel-plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient, for tetrahedral elements to order 4. In particular, the full-order elements have only the same asymptotic convergence rate as elements with a reduced gradient space (such as the Whitney element). However, further tests reveal that the optimum balance of the gradient and rotational components is problem-dependent.

Three-dimensional finite element analysis of nondestructive testing of metallic tubular product

This paper presents the analysis of the remote field eddy current phenomenon for the nondestructive testing of metallic tubular products. A finite element code using linear tetrahedral edge elements is used for the analysis. The calculated results for different frequencies of exciting coil are evaluated. In the model without a hole in the pipe, the absolute value of the phase of the remote field depends on the frequency of the exciting coil. The phase varies around the hole and the phase difference compared with the case without hole also depends on the frequency. The calculated results and the developed code can be used for the design of the remote field nondestructive testing probe.

Eddy current analysis for the pipe welding

The pipe welder is one of the applications of induction heating. In this paper, the eddy current distribution and the Joule loss on the pipe surface were calculated by a finite element method. A tetrahedral linear edge element and an impedance boundary condition were incorporated in the finite element method for the three-dimensional eddy current analysis. The impedance boundary condition was applied to the pipe surface because the penetration depth was very small. The computation results of eddy current and Joule loss distributions of a pipe welder model are shown.

Comparison of accuracy criteria for approximation of conservative fields on tetrahedra

A new maximum eigenvalue criterion for a scaled stiffness matrix and a related singular value criterion for the element 'edge shape matrix', as well as conventional conditions of approximation accuracy, are applied to tetrahedral node and edge elements and compared. The singular value criterion is the most precise and least restrictive a priori condition and is recommended for engineering practice.

3-D FEM/BEM-hybrid approach based on a general formulation of Huygens' principle for planar layered media

Huygens' principle is used to exclude inhomogeneous regions and ideal conducting regions from planar layered structures. The fields in the inhomogeneous regions are modeled by the finite-element method (FEM) with tetrahedral edge elements in terms of the electric-field strength E/spl I.oarr/. The fields in the layered structure are described by an integral representation of the electric-field strength E/spl I.oarr/ in terms of equivalent electric and magnetic Huygens' surface current densities for the inhomogeneous regions, and in terms of electric Huygens' surface current densities for ideal conducting regions. It is formulated with the help of electromagnetic (EM) potentials resulting in low-order singular integral kernels to facilitate the numerical handling of the integral representation. A general formulation of the integral representation is given for observation points lying in the Huygens' surface. As compared to the homogeneous-space case, additional terms in the integral representation have to be considered if parts of the Huygens' surface lie in an interface of layers with different material properties. An integral equation is formulated and discretized by a Galerkin testing procedure (boundary-element method), together with the finite-element (FE) system resulting in an unequivocal discretized description of the entire field problem. The method is validated with the help of a canonical test problem. Further numerical results are presented for dielectric resonators coupled to microstrip circuits.

A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements

Starting from a nodal pyramidal finite element, the paper presents first and second order edge pyramidal elements. These new elements can be very useful to link meshes of different type, that is to say hexahedral, tetrahedral and prismatic edge elements used in magnetic field computation.

Development and implementation of second and third order vector finite elements in various 3-D electromagnetic field problems

A generalized theory of higher order vector finite elements is presented. The methodology, that is introduced, is used for the construction of new second and third order generalizations of the concept of tetrahedral edge elements. Basic theoretical issues about their construction, such as the proper choice and placement of degrees of freedom and the correct modeling of irrotational fields are discussed. The presented theory and methodology are also extended to the hexahedral element case. Some practical problems and difficulties related to the numerical implementation of higher order vector finite elements are also dealt with. Finally, both second and third order elements are successfully implemented in several electromagnetic field or potential formulations.

The advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite-element method

In this paper, we examine the numerical dispersion properties of the tetrahedral edge element in comparison to other edge and nodal elements. For all nodal elements as well as hexahedral edge elements, the phase error is always either positive or negative for waves propagating at any incidence angle, which means that the error accumulates as the wave propagates from element to element. This effect can produce large errors for electrically large geometries. On the other hand, the tetrahedral edge elements can produce either a negative or positive phase error, depending on the direction of propagation through the element. For an unstructured mesh, phase cancellation occurs since the orientation of the tetrahedral elements are arbitrary. Because of the cancellation, the numerical dispersion error is very low for meshes composed of well-shaped tetrahedral edge elements. Numerical results are presented to demonstrate this point.