A Fast Quasi-Conformal Mapping Preconditioner for Electromagnetic Integral Equations

Boundary Element Methods (BEMs) are efficient strategies to numerically solve electromagnetic radiation and scattering problems. Unfortunately, however, classical BEM formulations suffer from ill-conditioning when the frequency is low, or the discretization density is high. In the past, several remedies have been presented for these ill-conditioning problems including preconditioners based on Calderón identities, hierarchical bases, and current decompositions. While effective, these strategies however require ad-hoc procedures including mesh-refinements, new basis function definitions, and adapted fast methods that, if not implemented properly, can become computationally cumbersome

On a Frequency-Stabilized Single Current Inverse Source Formulation

Several strategies are available for solving the inverse source problem in electromagnetics. Among them, many have been focusing in retrieving Love currents by solving, after regularization, for Love’s electric and magnetic currents. In this work we present a dual-element discretization, analysis, and stabilization of an inverse source formulation providing Love data by solving for only one current. This results in substantial savings and allows for an effective quasi-Helmholtz projector stabilization of the resulting operator. Theoretical considerations are complemented by numerical tests showing effectiveness and efficiency of the newly proposed method

On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

In this article, we present a new regularized electric flux volume integral equation (D-VIE) for modeling high-contrast conductive dielectric objects in a broad frequency range. This new formulation is particularly suitable for modeling biological tissues at low frequencies, as it is required by brain epileptogenic area imaging, but also at higher ones, as it is required by several applications, including, but not limited to, deep brain stimulation (DBS). When modeling inhomogeneous objects with high complex permittivities at low frequencies, the traditional D-VIE is ill-conditioned and suffers from numerical instabilities that result in slower convergence and less accurate solutions. In this work, we address these shortcomings by leveraging a new set of volume quasi-Helmholtz projectors. Their scaling by the material permittivity matrix allows for the rebalancing of the equation when applied to inhomogeneous scatterers and, thereby, makes the proposed method accurate and stable even for high complex permittivity objects until arbitrarily low frequencies. Numerical results, canonical and realistic, corroborate the theory and confirm the stability and the accuracy of this new method both in the quasi-static regime and at higher frequencies

On preconditioning electromagnetic integral equations in the high frequency regime via helmholtz operators and quasi-helmholtz projectors

Fast and accurate resolution of electromagnetic problems via the boundary element method (BEM) is oftentimes challenged by conditioning issues occurring in three distinct regimes: (i) when the frequency decreases and the discretization density remains constant, (ii) when the frequency is kept constant while the discretization is refined and (iii) when the frequency increases along with the discretization density. While satisfactory remedies to the problems arising in regimes (i) and (ii), respectively based on Helmholtz decompositions and Calderon-like techniques have been presented, the last regime is still challenging. In fact, this last regime is plagued by both spurious resonances and ill-conditioning, the former can be tackled via combined field strategies and is not the topic of this work. In this contribution new symmetric scalar and vectorial electric type formulations that remain well-conditioned in all of the aforementioned regimes and that do not require barycentric discretization of the dense electromagnetic potential operators are presented along with a spherical harmonics analysis illustrating their key properties

Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following shortcomings: 1) ill-conditioning when the frequency is low; 2) ill-conditioning when the discretization density is high; 3) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); 4) incorrect solution at low frequencies due to a loss of significant digits; and 5) the presence of spurious resonances. In this article, quasi-Helmholtz projectors are leveraged to obtain magnetic field integral equation (MFIE) that is immune to drawbacks 1)-4). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector-combined field integral equation (CFIE) is obtained that also is immune to 5). The numerical results corroborate the theory and show the practical impact of the newly proposed formulations

Single-crystal diffraction at the high-pressure Indo-Italian beamline Xpress at Elettra, Trieste

In this study the first in situ high-pressure single-crystal X-ray diffraction experiments at Xpress, the Indo-Italian beamline of the Elettra synchrotron, Trieste (Italy), are reported. A description of the beamline experimental setup and of the procedures for single-crystal centring, data collection and processing, using diamond anvil cells, are provided. High-pressure experiments on a synthetic crystal of clinoenstatite (MgSiO3), CaCO3 polymorphs and a natural sample of leucophoenicite [Mn7Si3O12(OH)2] validated the suitability of the beamline experimental setup to: (i) locate and characterize pressure-induced phase transitions; (ii) solve ab initio the crystal structure of high-pressure polymorphs; (iii) perform fine structural analyses at the atomic scale as a function of pressure; (iv) disclose complex symmetry and structural features undetected using conventional X-ray sources

Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory

We derive an operator identity which relates tight-binding Hamiltonians with arbitrary hopping on the Bethe lattice to the Hamiltonian with nearest-neighbor hopping. This provides an exact expression for the density of states (DOS) of a non-interacting quantum-mechanical particle for any hopping. We present analytic results for the DOS corresponding to hopping between nearest and next-nearest neighbors, and also for exponentially decreasing hopping amplitudes. Conversely it is possible to construct a hopping Hamiltonian on the Bethe lattice for any given DOS. These methods are based only on the so-called distance regularity of the infinite Bethe lattice, and not on the absence of loops. Results are also obtained for the triangular Husimi cactus, a recursive lattice with loops. Furthermore we derive the exact self-consistency equations arising in the context of dynamical mean-field theory, which serve as a starting point for studies of Hubbard-type models with frustration.Comment: 14 pages, 9 figures; introduction expanded, references added; published versio