Analytical computation of impedance integrals with power-law Green's functions

Abstract

-In this contribution, a method is presented for reducing the number of subsequent integrations that occur in impedance integrals with Green's functions of the form R ν , with R the distance between source and observation point. The method allows the number of integrations to be reduced to 1 in the two dimensional case and 2 in the three dimensional case, irrespective of the number of subsequent integrations that were originally present. These last integrations can be done analytically using well-known results if ν ∈ Z, resulting in a computation that is free of numerical integrations. The dynamic Green's function can be treated in a semianalytical way, by expanding it into a Taylor series in the wavenumber. The method can be applied if both the basis and test functions are polynomial functions with polygonal support and if certain non-parallelity conditions are satisfied.

On the analytical treatment of singular impedance integrals in electromagnetics

“…The fact that this reduction is possible is because the dimension reduction method presented in [14] leaves power-law Green's functions invariant. This allows the summation operator S to be applied to a function that has a similar structure as the original Green's function, which enables the analytical evaluation of the sum in this case.…”

“…In [14], the impedance integral between a constant test function on a line support Γ t = {t 1 , t 2 } and a constant basis function on a triangular support Γ b = {b 1 , b 2 , b 3 } was considered:…”