The use of parametric elements in the moment method solution of static and dynamic volume integral equations

The use of two- and three-dimensional parametric curved subdomains (elements) in the numerical solution of electromagnetic problems formulated in terms of volume integro-differential equations is considered, and a technique to evaluate the static and dynamic moment integrals on these domains is presented. In the source region, the evaluation of the moment integrals is performed by separating the integrands into a singular and a regular part. The regular integrals are numerically evaluated in a parameter space where the integration domain assumes a very simple form. The singular integrals are shown to be always reducible to nonsingular boundary integrals by use of a nonlinear transformation for the integration variables and an analytic integration on one variable. The reduced singular integrals are then evaluated numerically in a second parameter space, where the boundary of the original curved element is expressed in a simple form. Two simple two-dimensional scattering problems in the frequency domains are considered to compare the results obtained using quadratic isoparametric curved elements against the results given by linear expansion functions on triangular elements. The use of curved parametric elements permit a better geometrical description of the problem, a reduced number of unknowns, shorter computation times, and more accurate results with respect to the usually employed linear elements