Analytic preconditioners for decoupled potential integral equations for wideband analysis of scattering from PEC objects

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the behavior of CFIE at high frequencies. This paper addresses this deficiency. We do so by deriving new Calder\'{o}n-type identities through the Vector Potential Integral Equation (VPIE) and Scalar Potential Integral Equation (SPIE), and constructing novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets

Survey of nuclear pasta in the intermediate-density regime: Shapes and energies

Background: Nuclear pasta, emerging due to the competition between the long-range Coulomb force and the short-range strong force, is believed to be present in astrophysical scenarios, such as neutron stars and core-collapse supernovae. Its structure can have a high impact e.g. on neutrino transport or the tidal deformability of neutron stars. Purpose: We study several possible pasta configurations, all of them minimal surface configurations, which are expected to appear in the mid-density regime of nuclear pasta, i.e. around 40% of the nuclear saturation density. In particular we are interested in the energy spectrum for different pasta configurations considered. Method: Employing the density functional theory (DFT) approach, we calculate the binding energy of the different configurations for three values of the proton content XP = 1/10, 1/3 and 1/2, by optimizing their periodic length. We study finite temperature effects and the impact of electron screening. Results: Nuclear pasta lowers the energy significantly compared to uniform matter, especially for $X_P \geq 1/3$. However, the different configurations have very similar binding energies. For large proton content, $X_P \geq 1/3$, the pasta configurations are very stable, for lower proton content temperatures of a few MeV are enough for the transition to uniform matter. Electron screening has a small influence on the binding energy of nuclear pasta, but increases its periodic length. Conclusion: Nuclear pasta in the mid-density regime lowers the energy of the matter for all proton fractions under study. It can survive even large temperatures of several MeV. Since various configurations have very similar energy, it is to expected that many configurations can coexist simultaneously already at small temperatures.Comment: 10 pages, 11 figure