It is pointed out that for a closed cavity should not be solved by the 'vectorial' analog of the classical finite-element method (that is, with node-based finite elements). It is suggested that there are good reasons to use edge-elements instead. Such elements (whose degrees of freedom correspond to the edges of the mesh, not its nodes) not only guarantee the right kind of continuity for fields like h and e, but also prevent the appearance of unwanted, unphysical divergent solutions in the problem of eigenmodes, the so-called spurious modes. >

Solving Maxwell equations in a closed cavity, and the question of 'spurious modes'

The theory and applications of eddy currents induced in conducting materials by time-varying magnetic fields are reviewed. The mathematical methods employed in solving the relevant problems are presented. Both analytical and numerical methods are described. Applications based on effects arising from eddy currents are discussed in detail. These applications are to magnetic levitation, electromagnetic launching, hyperthermia treatment of cancer, and nondestructive testing. >

Eddy currents: theory and applications

The vector parasite problem is studied in the context of finite-element solutions of Maxwell's equations for driven boundary-value problems. An expanded weak form which combines the divergence equation with the conventional weak form of the double-curl equation is introduced. This form is related to penalty methods where the penalty or weighting factor varies with the dielectric constant. The resulting algebraic system is identical to the Galerkin-Helmholtz operator on homogeneous subregions. Normal and tangential boundary conditions arise in terms of the divergence and curl of the field on the boundary which can be reexpressed as equivalent charges and currents. Computational results show the occurrence of two distinct types of parasitic modes in driven problems and their elimination with the formulation presented. Practical observations concerning the conditions which provoke spurious modes in these problems are reported. Spurious solutions arise from improper or unphysical boundary conditions, and the importance of careful specification of boundary-value problems is illustrated. Most conceptual difficulties with boundary conditions per se are removed when hybrid methods are used to couple the interior finite-element solution to the exterior problem. which focuses attention on the physics of the source distribution. >

Elimination of vector parasites in finite element Maxwell solutions

Dispersion relations are derived for conventional finite-element (FE) and finite-difference (FD) approximations for four versions of the Maxwell equations in the plane: the double-curl equation; the vector Helmholtz equation; the penalty equation; and the primitive, coupled Maxwell curl equations. Comparison with their analytic counterparts reveals the presence and origin of vector parasites. For the double-curl and penalty methods, the dispersion relations are double-valued, admitting an extra, spurious dispersion surface of real-valued wavenumbers. As a result, low wavenumbers support well-resolved and poorly resolved vector parasites. The Helmholtz schemes have monotonic, single-valued dispersion relations for divergence-free physical modes. Specification of divergence-free boundary conditions is sufficient to guarantee the absence of parasites. The primitive schemes have single-valued but folded (nonmonotonic) dispersion relations, supporting poorly resolved vector parasites at low wavenumbers. Use of a staggered finite-difference grid eliminates these parasites and results in a dispersion relation identical to that for the Helmholtz scheme. In cases where vector parasites arise, the same essential weakness in the discretized form of either the first or cross-derivative is responsible. >

Origin of vector parasites in numerical Maxwell solutions

The appearance of unphysical "spurious solutions" in addition to the true physical modes of a resonator or waveguide is a problem which pertains to a variety of numerical methods for the electromagnetic field eigenvalue problem e.g. finite-elements, boundary-elements, finite-differences, point-matching and the spectral domain method. The common cause of the problem is shown to be a misconception of the discretization of a parameter dependent indefinite operator. In particular the application of the method of moments (MoM) and equivalent procedures to problems which they are not suited for. The method of least squares with intermediate projection (MLSIP) is described as one reliable approach for discretization of parameter dependent indefinite operators. >

The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems

A novel approach to the solution of three-dimensional micro-wave cavity and waveguide problems by finite elements reduces the number of spurious, nonphysical modes. Solutions are obtained in terms of the field vector H. Three-dimensional vector boundary conditions are implemented in a way that allows arbitrarily-shaped curved boundaries to be modeled. The formulation is based on a subparametric finite element with 27 interpolation nodes.

On the Reduction of the Number of Spurious Modes in the Vectorial Finite-Element Solution of Three-Dimensional Cavities and Waveguides

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On the Reduction of the Number of Spurious Modes in the Vectorial Finite-Element Solution of Three-Dimensional Cavities and Waveguides