Edge Element Basis Research Articles

Loops and Meshes Formulations for 3-D Eddy-Current Computation in Topologically Non-Trivial Domains With Volume Integral Equations

The volume integral equation method based on the current vector potential approximated by means edge element basis function is a well-established approach for 3-D eddy currents computation. The application of the method is straightforward when simply connected geometries and no connection with external circuits are involved. In this case, in fact, the solving system is easily obtained based on tree-cotree decomposition of the primal graph. However, when multiply connected geometries or external generators are considered, “additional” degrees of freedom, not related to interior cotree edges of the primal graph, need to be identified and involved to assure the consistency of the numerical solution. In this article, the link between the volume integral equation method and the circuit is investigated in detail, and the circuit view is used as a guide for systematically finding the additional degrees of freedom arising in case of multiply connected geometries and/or external circuits. In particular, the dual graph is introduced as the support of the circuit and it is shown that this is the natural frame for taking the topology into account. For multiply connected geometries, the additional degrees of freedom are related to loop currents crossing one only time (or an odd number of times) any cutting surface making the domain simply connected, and are found by applying a minimum path algorithm on the dual graph forming the circuit. In case of conducting domain connected to an external generator, an extended dual graph is introduced for finding the further additional degrees of freedom. This article also researches into the possibility to replace the usual current density of the elements, obtained via facet-element shape functions and exactly matching the current of the faces, with a uniform current density obtained by means of a minimum error procedure and approximately matching the current of the faces. The use of this uniform current density, besides improving the calculation time and the accuracy of the coupling coefficients, also allows the extension of the volume integral equation method to discretizations of the problem domain made of arbitrarily shaped polyhedral elements.

Robust full‐wave Maxwell solver in time‐domain using magnetic vector potential with edge elements

Magnetic vector potential (MVP) formulations are widely used in low-frequency eddy-current computation. For full-wave Maxwell problems with coupled resistive, inductive and capacitive effects, existing MVP formulations are either ungauged or with too complicated stabilisation procedure or with non-symmetric system matrix. Since sophisticated preconditioners are necessary to ensure good convergence rates of iterative solvers when used to solve the resultant linear system, it is also important to investigate symmetric gauged formulations which can be solved by more robust state-of-the-art sparse direct solvers. Traditional Coulomb gauged formulation using penalty technique is not feasible for edge elements since the divergence of the edge element basis function is zero within each element. In this study, a novel MVP formulation with Coulomb Gauge for full-wave Maxwell problems using edge elements is proposed. The proposed formulation is symmetric and stable for all frequencies, easy to implement and uniquely solvable. Numerical results obtained using the proposed MVP formulation are demonstrated to showcase its stability and accuracy. The method is very promising in further engineering applications.

A New Stable Full-Wave Maxwell Solver for All Frequencies

Magnetic vector potential (MVP) formulations are widely used in the electromagnetic (EM) field computation. For problems with both inductive and capacitive effects, existing full-wave MVP formulations are mostly nonuniquely solvable, and sophisticated preconditioners are indispensable when solving these resultant linear matrix equations. To find a stable formulation with a unique solution, it is important to incorporate the necessary physics equations and the gauge condition properly. Traditional Coulomb-gauged formulation using the penalty technique does not work for edge elements, since the divergence of the edge element basis function is zero within each element. In this paper, a novel MVP formulation with Coulomb gauge for full-wave Maxwell equations using edge elements is proposed in the frequency domain. Extensive numerical examples are computed to show that the proposed formulation is stable even for tiny frequencies, which is very useful in full-wave engineering EM field computation.

Algebraically constrained extended edge element method (eXFEM-AC) for resolution of multi-material cells

Surface effects are critical to the predictive simulation of electromagnetics as current tends to concentrate near material interfaces. There are two principal difficulties in the accurate representation of these effects in discrete models. First, many applications of interest operate at large deformations, where body-fitted meshes are impractical. Second, physics-compatible discretizations of the governing equations require curl-conforming edge elements, for which no practical alternatives to body-fitted meshes exist. The main purpose of this paper is to develop such an alternative that avoids remeshing the problem.Our approach uses the existing edge element basis to dynamically construct an interface-conforming basis. We show that in the case of triangular grids in two dimensions, our approach generates a basis that spans the same space as edge elements on an interface-fitted mesh. We also demonstrate the efficacy of the approach computationally.

MIXED FINITE ELEMENT METHOD FOR 2D VECTOR MAXWELL'S EIGENVALUE PROBLEM IN ANISOTROPIC MEDIA

It is well known that the conventional edge element method in solving vector Maxwell's eigenvalue problem will lead to the presence of nonphysical zero eigenvalues. This paper uses the mixed flnite element method to suppress the presence of these nonphysical zero eigenvalues for 2D vector Maxwell's eigenvalue problem in anisotropic media. We introduce a Lagrangian multiplier to deal with the constraint of divergence-free condition. Our method is based on employing the flrst-order edge element basis functions to expand the electric fleld and linear nodal element basis functions to expand the Lagrangian multiplier. Our numerical experiments show that this method can successfully remove all nonphysical zero and nonzero eigenvalues. We verify that when the cavity has a connected perfect electric boundary, then there is no physical zero eigenvalue. Otherwise, the number of physical zero eigenvalues is one less than the number of disconnected perfect electric boundaries.

Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators

When using edge element basis functions for the solution of eigenmodes of the vector wave equation, "dc spurious modes" are introduced. The eigenvalues of these modes are zero and their corresponding eigenvectors are in the null space of the curl operator. These modes arise due to the irrotational vector space spanned by the edge element basis functions and lead to nonzero divergence of the electric flux. We introduce a novel method to eliminate the occurrence of such solutions using "divergence-free" constraint equations. The constraint equations are imposed efficiently by tree-cotree partitioning of the finite-element mesh and does not require any basis functions other than the edge elements. The constraint equations can be directly incorporated into any Krylov-subspace-based eigenvalue solver, such as the Lanczos/Arnoldi algorithm used widely for the solution of generalized sparse eigenvalue problems.

Shell elements for the computation of magnetic forces

The local application of the principle of virtual work for the computation of magnetic forces requires a layer of elements around the object. It fails to work when the object is touching other objects or when the boundary element method is applied on its boundary. To overcome this difficulty, a shell element is introduced on the boundary of the object. Derivation of magnetic co-energy (or energy) is performed on the shell element with the help of edge element basis and the local Jacobian derivative technique. The force expression derived from this method shows a close similarity to the Maxwell stress tensor. Efficiency of the method is shown through the computation of magnetic force on the moving core of an electromagnet when it touches the fixed yoke.

Solving 3D eddy current problems using second order nodal and edge elements

Several second order nodal and edge elements have been applied in a potential formulation to solve 3D eddy current problems. The asymmetry of the facet related functions in the edge element basis is discussed. A new basis is proposed. Application of a gauge condition for the uniqueness of vector potential is cumbersome in the case of high order elements. This work shows that the system converges without explicit gauge condition when using the bi-conjugate gradient method. The performance of different elements is compared through an example.

Hierarchic "Edge" elements for high-frequency problems

The edge element has become an indispensable tool in the h-version FEM for high-frequency electromagnetic problems. To investigate a p-method based on this technology, hierarchic higher-order edge elements are desirable. We establish such families of 3D elements by mixing the edge element basis and the hierarchic p-shape functions. These elements are then employed in modal analysis of electromagnetic cavities.