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

On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries

We present a hierarchical basis preconditioning strategy for the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation considering both simply and multiply connected geometries.To this end, we first consider the direct application of hierarchical basis preconditioners, developed for the Electric Field Integral Equation (EFIE), to the PMCHWT. It is notably found that, whereas for the EFIE a diagonal preconditioner can be used for obtaining the hierarchical basis scaling factors, this strategy is catastrophic in the case of the PMCHWT since it leads to a severly ill-conditioned PMCHWT system in the case of multiply connected geometries. We then proceed to a theoretical analysis of the effect of hierarchical bases on the PMCHWT operator for which we obtain the correct scaling factors and a provably effective preconditioner for both low frequencies and mesh refinements. Numerical results will corroborate the theory and show the effectiveness of our approach

Internal deformation of the subducted Nazca slab inferred from seismic anisotropy

Within oceanic lithosphere a fossilized fabric is often preserved originating from the time of plate formation. Such fabric is thought to form at the mid-ocean ridge when olivine crystals align with the direction of plate spreading1, 2. It is unclear, however, whether this fossil fabric is preserved within slabs during subduction or overprinted by subduction-induced deformation. The alignment of olivine crystals, such as within fossil fabrics, can generate anisotropy that is sensed by passing seismic waves. Seismic anisotropy is therefore a useful tool for investigating the dynamics of subduction zones, but it has so far proved difficult to observe the anisotropic properties of the subducted slab itself. Here we analyse seismic anisotropy in the subducted Nazca slab beneath Peru and find that the fast direction of seismic wave propagation aligns with the contours of the slab. We use numerical modelling to simulate the olivine fabric created at the mid-ocean ridge, but find it is inconsistent with our observations of seismic anisotropy in the subducted Nazca slab. Instead we find that an orientation of the olivine crystal fast axes aligned parallel to the strike of the slab provides the best fit, consistent with along-strike extension induced by flattening of the slab during subduction (A. Kumar et al., manuscript in preparation). We conclude that the fossil fabric has been overprinted during subduction and that the Nazca slab must therefore be sufficiently weak to undergo internal deformation

The Lunar Geophysical Network Mission

The National Academy’s current Planetary Decadal Survey (NRC, 2011) prioritizes a future Lunar Geophysical Network (LGN) mission to gather new information that will permit us to better determine how the overall composition and structure of the Moon inform us about the initial differentiation and subsequent evolution of terrestrial planets

A DC Stable and Large-Time Step Well-Balanced TD-EFIE Based on Quasi-Helmholtz Projectors

The marching-on-in-time (MOT) solution of the time-domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of issues, including the emergence of spurious static currents (dc instability) and ill-conditioning at large-time steps (low frequencies). In this contribution, a space-time Galerkin discretization of the TD-EFIE is proposed, which separates the loop and star components of both the equation and the unknown. Judiciously integrating or differentiating these components with respect to time leads to an equation which is free from dc instability. By choosing the correct temporal basis and testing functions for each of the components, a stable MOT system is obtained. Furthermore, the scaling of these basis and testing functions ensure that the system remains well conditioned for large-time steps. The loop-star decomposition is performed using quasi-Helmholtz projectors to avoid the explicit transformation to the unstable bases of loops and stars (or trees), and to avoid the search for global loops, which is a computationally expensive operation

Hierarchical basis preconditioners and their application to the PMWCHT integral equation

We present an analysis of the application of (general) hierarchical basis preconditioners developed for preconditioning the electric field integral equation (EFIE) to the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation on simply and multiply-connected geometries. First, we discuss the correct choice of the rescaling factors for the solenoidal and non-solenoidal hierarchical bases. Then we consider the harmonic subspace that appears on multiply-connected domains: it turns out that differently from the combined field integral equation (CFIE), the global loops must not be rescaled in frequency. The numerical results confirm the theoretically predicted results

On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries

We present a hierarchical basis preconditioning strategy for the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation considering both simply and multiply connected geometries. To this end, we first consider the direct application of hierarchical basis preconditioners, developed for the Electric Field Integral Equation (EFIE), to the PMCHWT. It is notably found that, whereas for the EFIE a diagonal preconditioner can be used for obtaining the hierarchical basis scaling factors, this strategy is catastrophic in the case of the PMCHWT since it leads to a severely ill-conditioned PMCHWT system in the case of multiply connected geometries. We then proceed to a theoretical analysis of the effect of hierarchical bases on the PMCHWT operator for which we obtain the correct scaling factors and a provably effective preconditioner for both low frequencies and mesh refinements. Numerical results will corroborate the theory and show the effectiveness of our approach