On the Computation of Power in Volume Integral Equation Formulations

We present simple and stable formulas for computing power (including absorbed/radiated, scattered and extinction power) in current-based volume integral equation formulations. The proposed formulas are given in terms of vector-matrix-vector products of quantities found solely in the associated linear system. In addition to their efficiency, the derived expressions can guarantee the positivity of the computed power. We also discuss the application of Poynting's theorem for the case of sources immersed in dissipative materials. The formulas are validated against results obtained both with analytical and numerical methods for scattering and radiation benchmark cases

On the direct evaluation of surface integral equation impedance matrix elements involving point singularities

The direct evaluation method tailored to the 4-D singular integrals over vertex adjacent triangles, arising in the first-kind and second-kind Fredholm surface integral equation formulations, is presented. A combination of singularity cancellation, reordering of the integrations, and one analytical integration results in 3-D integrals of sufficiently smooth functions, allowing a straightforward computation by standard Gaussian rules. Numerical results demonstrate that the uncertainty about the accuracy of the impedance matrix elements associated to the interaction of vertex adjacent triangles is safely removed

Efficient algorithms for computing Sommerfeld integral tails

Sommerfeld-integrals (SIs) are ubiquitous in the analysis of problems involving antennas and scatterers embedded in planar multilayered media. It is well known that the oscillating and slowly decaying nature of their integrands makes the numerical evaluation of the SI real-axis tail segment a very time consuming and computationally expensive task. Therefore, SI tails have to be specially treated. In this paper we compare two recently developed techniques for their efficient numerical evaluation. First, a partition-extrapolation method, in which the integration-then-summation procedure is combined with a new version of the weighted averages (WA) extrapolation technique, is summarized. The previous variants of WA technique are also discussed. Then, a review of double-exponential (DE) quadrature formulas for direct integration of the SI tails is presented. The efficient way of implementing the algorithms, their pros and cons, as well as comparisons of their performance are discussed in detail

Considerations on double exponential-based cubatures for the computation of weakly singular Galerkin inner products

Highly accurate and efficient cubatures based on the double exponential quadrature rules are presented for the computation of weakly singular integrals arising in Galerkin mixed potential integral equation formulations. Due to their unique ability to handle non-smooth kernels, the proposed integration schemes can safely replace (in a "plug-n-play" sense) the traditional Gauss-Legendre rules in the existing singularity cancellation and singularity subtraction methods. Numerical examples using RWG basis functions confirm the excellent performance of the proposed method

On the analytic-numeric treatment of weakly singular integrals on arbitrary polygonal domains

An alternative analytical approach to calculate the weakly singular free-space static potential integral associated to uniform sources is presented. Arbitrary oriented flat polygons are considered as integration domains. The technique stands out by its mathematical simplicity and it is based on a novel integral transformation. The presented formula is equivalent to others existing in literature, being also concise and suitable within a singularity subtraction framework. Generalized Cartesian product rules built on the double exponential formula are utilized to integrate numerically the resulting analytical 2D potential integral. As a consequence, drawbacks associated to endpoint singularities in the derivative of the potential are tempered. Numerical examples within a surface integral equation-Method of Moments framework are finally provided

Fast and accurate computation of hyper-singular integrals in Galerkin surface integral equation formulations via the direct evaluation method

Hypersingular 4-D integrals, arising in the Galerkin discretization of surface integral equation formulations, are computed by means of the direct evaluation method. The proposed scheme extends the basic idea of the singularity cancellation methods, usually employed for the regularization of the singular integral kernel, by utilizing a series of coordinate transformations combined with a reordering of the integrations. The overall algebraic manipulation results in smooth 2-D integrals that can be easily evaluated via standard quadrature rules. Finally, the reduction of the dimensionality of the original integrals together with the smooth behavior of the associated integrands lead up to unmatched accuracy and efficiency