J.P. Webb

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.

Edge elements and what they can do for you

The various types of edge elements and their useful properties are reviewed. Continuity, singularities, and separation are discussed, with particular reference to tetrahedral elements. The use of triangular and quadrilateral edge elements for 2-D problems is briefly considered. >

Hierarchal scalar and vector tetrahedra

A novel set of scalar and vector tetrahedral finite elements are presented. The elements are hierarchical, allowing mixing of polynomial orders. Scalar orders up to three and vector orders up to two are defined. The vector elements impose tangential continuity on the field but not normal continuity, making them suitable for representing the vector electric or magnetic field. The scalar and vector elements can easily be used in the same mesh, a requirement of many quasi-static formulations. Results are presented for two 50-Hz problems: the Bath cube and the TEAM Workshop problem 7. >

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.

A single scalar potential method for 3D magnetostatics using edge elements

It is demonstrated that it is possible to use a single, continuous, scalar potential to solve magnetostatic problems in three dimensions without the loss of accuracy associated with the reduced potential. This method makes use of tetrahedral edge elements and does not require the initial calculation of the field of the currents in the absence of magnetic materials. The method requires no numerical integration, has no restrictions on the current flow or the iron topology, and needs the solution of only one matrix problem. Results obtained by the method for translational and axisymmetric problems agree well with those computed by two-dimensional analysis with a vector potential. The method can be used to obtain accurate flux densities in both air and iron regions of three-dimensional problems. >

Optimization of microwave devices using 3-D finite elements and the design sensitivity of the frequency response

Direct optimization of microwave devices is more efficient if sensitivities of the cost function are available. An earlier method for finding sensitivities of the scattering matrix efficiently over a range of frequencies, from a finite-element analysis, is applied here to the optimization problem. The method is tested on rectangular waveguide components with 1-4 design parameters: a right-angle miter bend, an E-plane U-bend, and a waveguide impedance transformer.

Finite element analysis of dispersion in waveguides with sharp metal edges

The dispersion characteristics of arbitrarily shaped waveguides with sharp metal edges are found by a finite-element method in which the usual polynomials are supplemented by singular trial functions. As in recent approaches, the method solves for the three components of the magnetic field and can thereby avoid spurious modes. Results for a rectangular waveguide with two double ridges and for shielded microstrip on isotropic and anisotropic substrates are presented. >

Covariant-projection quadrilateral elements for the analysis of waveguides with sharp edges

Covariant-projection elements are shown to be a good way of finding the dispersion characteristics of arbitrarily shaped waveguides. They have been demonstrated to produce no spurious modes, and because only tangential continuity is imposed between elements, either the electric field or the magnetic field may be calculated in the presence of dielectric and magnetic materials. Waveguides with sharp metal edges may be analyzed more efficiently than with other methods. Results are presented for a rectangular waveguide half loaded with dielectric, a double-ridged waveguide, a shielded microstrip line, and coupled microstrip lines on a cylindrical substrate. The matrices generated are sparse. and the number of zero eigenvalues produced is predictable. It therefore seems likely that the algebraic problem can be solved by sparse techniques, which would make the method applicable to even more complicated geometries at a modest computational cost. >

A tunable volume integration formulation for force calculation in finite-element based computational magnetostatics

A generalized formulation for net magnetostatic loading force calculation is derived from the Maxwell stress expression. The formulation yields a combined surface and volume integration method based on the magnetic flux density and an arbitrary scalar function g. The most interesting feature of the technique is its flexibility. For one choice of g, the method reduces to a distributed Maxwell stress scheme; for another, it yields a generalized version of the Coulomb virtual work implementation. With the introduction of an intelligent g-function based on local field-error, the new formulation yields a fully automatic method suitable for extracting accurate and consistent forces from imperfect numerical solutions. It is implemented for two-dimensional first-order finite elements, and two illustrative test problems are analyzed. The performance of the scheme is compared to the Maxwell stress and Coulomb approaches. >

A comparison of formulations for the vector finite element analysis of waveguides

The principal formulations that have been proposed for finding the modes of waveguides by the finite element method are reviewed and compared. In each case, it is shown how Maxwells equations may be reduced to matrix form using the method of weighted residuals. The formulations are compared from several points of view: their ability to handle spurious modes, lossy materials, and reentrant corners; the number of field components; and the properties of the matrices. Three benchmark problems are described and used to compare the formulations: a rectangular waveguide partially loaded with lossless dielectric; an air-filled, double-ridged waveguide; and a shielded image guide with either lossless or lossy dielectric. >