Edge Finite Element Research Articles

A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements

An advanced A-V method employing edge-based finite elements for the vector potential A and nodal shape functions for the scalar potential V is proposed. Both gauged and ungauged formulations are considered. The novel scheme is particularly well suited for efficient iterative solvers such as the preconditioned conjugate gradient method, since it leads to significantly faster numerical convergence rates than pure edge element schemes. In contrast to nodal finite element implementations, spurious solutions do not occur and the inherent singularities of the electromagnetic fields in the vicinity of perfectly conducting edges and corners are handled correctly. Several numerical examples are presented to verify the suggested approach.

Multigrid computation of vector potentials

In this paper we analyze the use of edge finite elements and the multigrid method to approximate the problem of computing a static magnetic field in a cavity. We show how the mixed finite element discrete problem may be reformulated as a symmetric positive-definite system. This involves the proof of a discrete analogue of the second kind Friedrichs' inequality. Then we show how the multigrid method may be applied to compute the magnetic vector potential. The difficulty here is that the bilinear forms on different meshes are not related in the standard way. Two different intergrid transfer operators are defined in the paper for the multigrid method. The convergence theory and the numerical tests include both algorithms.

A mixed solving procedure for ungauged 3D edge finite element analysis

A new mixed procedure for solving the system of algebraic equations generated by the edge finite element method (EFEM) for ungauged vector potential formulation is proposed. The singularity of the matrix of the system obtained by ungauged vector formulation results in unstable and nonuniform convergence rates of the Preconditioned Conjugate Gradient (PCG) iteration procedure. The proposed method uses the excellent initial convergence rate of the PCG procedure, and enables switching from the PCG procedure to the Gauss-Seidel procedure when numerical instability of the former method occurs, continuing with a slower but more stable convergence rate. The proposed mixed PCG-Gauss-Seidel procedure requires monitoring of the PCG iteration process, however, it does not require generation of tree graphs, rearranging of unknowns inside the matrix of the system nor any additional memory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Shape optimization of H-plane waveguide tee junction using edge finite element method

The shape of conducting post in H-plane waveguide tee-junction is optimized to maximize the transmitting power. The method employs an edge finite element technique with magnetic fields as state variables. Sensitivity analysis and steepest descent algorithm are used for the optimization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Superconvergence of finite element approximations to Maxwell's equations

AbstractWe study superconvergence of edge finite element approximations to the magnetostatic problem and to the time‐dependent Maxwell system. We show that in special discrete norms there is an increase of one power in the order of convergence of the finite element method compared to error estimates in standard Sobolev norms. Our results are restricted to an orthogonal grid in R3, but the grid may be nonuniform. © 1994 John Wiley &amp; Sons, Inc.

Modelling, analysis and validation of electromagnetic-mechanical coupling on clean test pieces

A 3D numerical modelling, analysis and validation of the coupling effects between eddy currents and conductor motion, during transient electromagnetic phenomena is presented. The method involves a finite edge element computation from the electromagnetic point of view, coupled with a mechanical modal analysis. The approach is first implemented into clean benchmark problems concerning cantilever plates in flexion and torsion. The numerical predictions are in agreement with experiments. An application of the method is presented after, using a box-type component to simulate an in-vessel segment of a fusion reactor. Finally, the ELBA testing device, installed at the TESLA Laboratory and the first experimental validation studies are described, in order to confirm the applied numerical techniques. >

Analysis of induction skull melting furnace by edge finite element method excited from voltage source

To optimize the production of high-efficiency induction skull melting furnace, we analyzed magnetic flux density, eddy current and electromagnetic force distributions using the 3D edge-based finite element method excited from a voltage source. Changing the number of copper rods and, therefore, the distance between them, we analyzed both the intensity and direction of the electromagnetic forces and the amount of power consumed by the molten alloy and the rods. In this paper, we present our analysis method, the obtained results and conclusions. >

On the p- and hp-extension of Nédélec's curl-conforming elements

We establish interpolation error estimates for the hp-extension of Nédélec's curl- and divergence-conforming three-dimensional finite elements. Using these interpolation estimates, we investigate the use of higher-order edge finite elements for the linear three-dimensional magnetostatic problem. Our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p.

An improved method for magnetic flux density visualization using three-dimensional edge finite element method

Visualization of a magnetic flux density distribution in three-dimensional (3-D) finite element analysis (FEA) is very important in order to grasp the real behavior of the magnetic field, especially in the design process. In this paper, we present an improved method for visualizing magnetic flux density distributions calculated by 3-D first-order edge FEA. Compared with traditional methods, this method provides more accurate, smoother magnetic flux density distribution inside the analyzed region and satisfies the proper boundary conditions across intermaterial boundaries. The usefulness of our algorithm is demonstrated with several examples.

Efficient mode analysis with edge elements and 3-D adaptive refinement

Efficient mode analysis of 3-D inhomogeneously loaded cavity resonators of arbitrary shape by the finite edge element method is presented in this paper. Two weak formulations involving field vectors E and H are derived from a Galerkin weighted scheme resulting in a sparse symmetric generalized eigenvalue problem, the solution of which is obtained by a sparse eigenvalue technique. Edge elements with divergence free shape functions guarantee the continuity of the tangential components of the field variables E or H, but not of their normal components, across element interfaces in contrast with node based elements that impose full continuity. The discontinuity of the normal component of D or B, present in the numerical model, is proposed as an error estimator suitable for adaptive mesh refinement of 3-D tetrahedral meshes with edge elements. Application to a dielectrically loaded cavity is given with full documentation and by way of illustration.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Coupling between edge finite elements, nodal finite elements and boundary elements for the calculation of 3-D eddy currents

Coupling between edge finite elements, nodal finite elements, and boundary elements for the calculation of 3-D eddy currents is presented. The objective is to allow an adequate treatment of different formulations using scalar and vector variables. Three kinds of first order edge elements are studied: tetrahedron, hexahedron, and prism. The hexahedron is the most interesting element: its shape functions are bilinear for each edge. The prism is interesting too: its shape functions are bilinear for its horizontal edges whereas they are linear for its vertical edges. The tetrahedron is the least interesting element as far as accuracy is concerned: its shape functions are all linear.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nodal and edge element analysis of inhomogeneously loaded waveguides

Comparisons of nodal and edge finite element analyses of inhomogeneously loaded waveguides are made. For nodal elements, the vector potential together with a scalar potential description is applied. For edge elements, the field description is used. Both descriptions are free of spurious solutions. The case of the treatment of the propagation constant as an eigenvalue is also discussed. A modified version of the bisection method is used when the matrices of the eigenvalue problem are not positive definite. Examples including waveguides with sharp metal edges are presented to compare the different methods. For problems with sharp edges, the edge element approximation is favorable. >

Integral formulation for 3D eddy-current computation using edge elements

An integral formulation for eddy-current problems in nomagnetic structures is presented. The solenoidality of the current density is assured by introducing a current vector potential T. This potential possesses only two scalar components, as the gauge chosen to ensure its uniqueness is T. u = 0, where u is a prescribed vector field. The discrete analogue of this gauge and the boundary conditions are directly imposed by the shape functions, exploiting the use of edge finite elements and the methods of network theory. In the frame of the integral methods, this approach seems the most adequate to analyse the eddy currents induced in both massive conductors and thin shells. In massive structures, the two degrees of freedom are to be compared to four of the usual integral methods which exploit the presence of a scalar potential to ensure solenoidality. On the other hand, the procedure naturally reduces to the stream function approach when applied to thin shells. Finally, an integration procedure which guarantees symmetry and positive-definiteness of the inductance matrix is proposed.