Evaluation of 4-D Reaction Integrals in the Method of Moments: Coplanar Element Case

Recently, the benefits of simultaneously treating source and testing integrals in the numerical evaluation of 4-D reaction integrals have been reported. The reported schemes usually first transform the reaction integral to parametric coordinates, and some combination of radial, angular, and/or line segment integrals is then used to treat coincident, edge-adjacent, or vertex-adjacent triangular source and test element pairs. However, advantages of the reported approaches are tempered by their lack of generality and severely degraded performance on poorly shaped elements, the latter caused primarily by the parametric transformations' severe distortion of the kernel's circularly concentric level contours. Here, for coplanar element pairs and kernels with 1/R singularities, we apply the surface divergence theorem twice to obtain a novel formula for 4-D reaction integrals, generalizing earlier schemes while retaining their benefits and without distorting the original configuration space. Numerical results illustrate the method's efficiency, which is improved by employing appropriate transformations to further smooth the resulting integrands and hence accelerate their convergence. The reaction integral formula can be extended to noncoplanar elements.