A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

In this article, to solve the frequency-domain airborne electromagnetic (EM) modeling problem, we express the electric and magnetic field by the vector magnetic potential and the scalar electric potential and derive the governing equation using the Galerkin weighted residual method under the Coulomb gauge. To avoid the strong singularity of the solution near the transmitting source, we directly solve the relatively slow-changing secondary potential. By introducing the spectral element method (SEM) based on unstructured tetrahedral grids, in which the EM field in each element is characterized by high-order Proriol–Koornwinder–Dubiner (PKD) orthogonal polynomials, we can obtain stable and accurate numerical solutions. By establishing the mapping relationship between the physical domain, the right-angled tetrahedral reference domain, and the orthogonal hexahedral domain, we can easily accomplish the element matrix analysis and calculation. The numerical examples show that, when the SEM is used to simulate the airborne EM response, a high accuracy can be obtained even if a very coarse grid is used. Furthermore, since the tetrahedral grids can easily model the complex boundaries, the SEM based on unstructured grids has significant advantages for simulating complex underground structures.

3D minimum-structure inversion of controlled-source EM data using unstructured grids

Abstract. Rapid changes in the ionospheric current system give rise to induction currents in the conducting ground that can significantly contribute to magnetic and especially electric fields at the Earth's surface. Previous studies have concentrated on the surface fields, as they are important in, for example, interpreting magnetometer measurements or in the studies of the Earth's conductivity structure. In this paper we investigate the effects of induction fields at the ionospheric altitudes for several realistic ionospheric current models (Westward Travelling Surge, Ω-band, Giant Pulsation). Our main conclusions are: 1) The secondary electric field caused by the Earth's induction is relatively small at the ionospheric altitude, at most 0.4 mV/m or a few percent of the total electric field; 2) The primary induced field due to ionospheric self-induction is locally important, ~ a few mV/m, in some "hot spots", where the ionospheric conductivity is high and the total electric field is low. However, our approximate calculation only gives an upper estimate for the primary induced electric field; 3) The secondary magnetic field caused by the Earth's induction may significantly affect the magnetic measurements of low orbiting satellites. The secondary contribution from the Earth's currents is largest in the vertical component of the magnetic field, where it may be around 50% of the field caused by ionospheric currents. Keywords. Geomagnetism and paleomagnetism (geomagnetic induction) – Ionosphere (electric fields and currents)

A generalised three-dimensional solver based on unstructured grid methodology has been developed for electrohydrodynamic flows. The Navier-Stokes equations with electrical field source term are solved in conjunction with the coupled electrical potential and ion charge density equations in a finite volume framework. An iterative boundary condition for updating the value of charge density on the discharge electrode is used while solving the electrical field equations. The solver is validated with available literature and the good agreement is found.

A new agglomeration algorithm to generate coarse grids for multigrid methods on unstructured and hybrid grids is proposed in this paper. The algorithm, which we call a global coarsening algorithm, is based on the edge coloring of the grids so that it can agglomerate any type of grids. The agglomeration procedure is automatic and generates a globally optimum shape in contrast to the conventional greedy-type frontal algorithm. The method is applied to compute the 2D/3D Euler and 2D Navier-Stokes equations on triangular, tetrahedral and hybrid meshes, respectively. The present agglomeration multigrid method shows better convergence than the existing method owing to the better quality of the coarse grids.

On overset interpolation strategies and conservation on unstructured grids in OpenFOAM

Abstract. The geoelectric field drives geomagnetically induced currents (GICs) in technological conductor networks, which can affect the performance of critical ground infrastructure such as electric power transmission grids. The three-dimensional (3-D) electric field at the Earth's surface consists of an external divergence-free (DF) part due to temporally and spatially varying ionospheric and magnetospheric currents, an internal DF part due to temporally and spatially varying telluric currents, and a curl-free (CF) part due to charge accumulation at ground conductivity gradients. We have developed a new method for estimating these contributions. The external and internal parts of the DF electric field are calculated from the time derivative of the external and internal parts of the observed ground magnetic field, respectively, using DF two-dimensional (2-D) spherical elementary current systems (SECSs). The horizontal surface CF electric field is calculated from the known surface DF electric field using coefficients that linearly relate the DF electric field to the CF electric field. The coefficients were obtained from the 3-D induction model PGIEM2G (Polynomial Galerkin Integral Equation Modelling in ElectroMagnetic Geophysics). The calculations are carried out in the time domain, and only two consecutive time steps of the observed magnetic field are needed to compute the surface electric field. The external part of the DF electric field is valid at and below the ionosphere, the internal part at and above the Earth's surface, and the CF part at the Earth's surface. A dense magnetometer network is a requirement for reliable results. The external and internal parts of the DF electric field are generally oppositely directed and have comparable amplitudes, both on the ground and in the ionosphere, indicating that both contributions are significant for the total DF electric field. The largest peaks of the total DF electric field tend to occur when either the external or internal contribution is temporarily suppressed. At a given location, a DF electric field with a given amplitude can result in a total surface electric field amplitude with a difference of orders of magnitude depending on the direction of the DF electric field with respect to the locally dominant conductivity gradient structure. The electric field calculation is computationally light, facilitating operational implementation of a near-real-time 3-D surface electric field monitoring and derivation of long electric field time series.

Numerical simulation of the CERN LINAC4 H(-) source 2 MHz RF system has been performed taking into account a realistic geometry from 3D Computer Aided Design model using commercial FEM high frequency simulation code. The effect of the plasma has been added to the model by the approximation of a homogenous electrically conducting medium. Electric and magnetic fields, RF power losses, and impedance of the circuit have been calculated for different values of the plasma conductivity. Three different regimes have been found depending on the plasma conductivity: (1) Zero or low plasma conductivity results in RF electric field induced by the RF antenna being mainly capacitive and has axial direction; (2) Intermediate conductivity results in the expulsion of capacitive electric field from plasma and the RF power coupling, which is increasing linearly with the plasma conductivity, is mainly dominated by the inductive azimuthal electric field; (3) High conductivity results in the shielding of both the electric and magnetic fields from plasma due to the skin effect, which reduces RF power coupling to plasma. From these simulations and measurements of the RF power coupling on the CERN source, a value of the plasma conductivity has been derived. It agrees well with an analytical estimate calculated from the measured plasma parameters. In addition, the simulated and measured impedances with and without plasma show very good agreement as well demonstrating validity of the plasma model used in the RF simulations.

Steady, laminar three-dimensional (3D) magnetohydrodynamic (MHD) flow of an electrically conducting fluid in a circular pipe under the both magnetic and electrical field are studied. External magnetic and electrical fields are applied perpendicular to the flow direction and each other while the fluid motion is subjected to constant pressure gradient along axial-direction in the present paper. Fluent 14.0, the finite element software based on the finite volume approach was used to calculate the 3D fluid dynamics and electromagnetic field partial differential equations iteratively. The originality of this work is that, in addition to magnetic field; the effect of electrical field on MHD flows is being examined with help of user defined function (UDF) code. The magnetic field leads to decrease in the velocity of flow, whereas the electrical field applied with magnetic field acted to increase and decrease the velocity of flow depending on the direction of applied external electrical field. The obtained results were depicted graphically and discussed.

A method and results of numerical modeling of radiative heating of the back surface of the MSRO (Mars Sample Return Orbiter) space vehicle of the European Space Agency are presented. To determine radiation heat fluxes, the method of discrete ordinates on unstructured tetrahedral grids is used. The radiative model is based on the radiation-transfer equation in a multigroup approximation. Numerical calculation has been performed for the most thermally stressed point of the assumed trajectory of the entry of an MSRO-type space vehicle into the Mars atmosphere. A comparison with the discrete ordinates method on structured grids is made. Good agreement between the results of calculations on structured and unstructured grids is demonstrated. The level of radiation heat fluxes to the back surface of the MSRO space vehicle is predicted.

We study the evolution of drops of a very viscous and conducting fluid under the influence of an external electric field. The drops may be neutral or may be charged with some amount of electric charge. If both the external electric field and total drop charge are sufficiently small, then prolate spherical shapes develop according to Taylor’s observations. For sufficiently large charge and/or external field a self-similar conelike singularity develops in a mechanism different from Taylor’s prediction. The opening semiangle of the cones both for uncharged and charged drops in a constant electric field is typically around 30° with a very slight dependence on the viscosity ratio and independence from both total charge and external field. We also discuss the structure of electric and velocity fields near the tip.

Metal/insulator interfaces are essential structures in most of electronic and optical devices, and thus the stability of such interfaces is important for the reliability of devices. It has long been known that the metal atoms in a gate electrode often penetrate into insulators by thermal diffusion during the annealing processes in device fabrication. In addition, metal atoms are ionized and such diffusion is remarkably enhanced under electric fields during device operation. Diffusing metal atoms often produce defect electronic states, promote leakage currents, and cause serious damage in electronic devices, particularly in recent nanoscale devices such as metal nanodot memories. On the other hand, such diffusion behavior is positively utilized to produce metal nanowires in insulators for resistive switching memory applications. However, the microscopic understanding of ionization and diffusion of metal atoms has not been obtained yet. In this work, we employ the first-principles calculations and clarify how metal atoms are ionized and diffuse under electric fields around metal/insulator interfaces. We consider metal/insulator interfaces made of a variety of metals (Al, Ag, Au, Pt, Ta, W, etc) and crystalline/amorphous SiO2. Electronic structures, ionic charges, and adiabatic diffusion potentials of metal atoms are evaluated using the first-principles calculations in the density functional theory. Electric field is applied perpendicular to the interface by supplying positive/negative electric charges to the interface-slab systems.With respect to the ionization, we found that the ionic charge of metal atom in SiO2 changes depending on the distance from the interface [1]. In a 0-6Å region near the interface, the diffusing atom is sharply ionized with leaving the interface. This occurs owing to the breakdown of orbital hybridization between electronic states of metal atom and metal induced gap states (MIGS) of metal electrode [2,3]. When the substrate is changed to semiconducting Si and Sn, this region is extended to 10-15Å. Outside this MIGS-penetration region, on the other hand, the ionic charge of metal atom still varies. This variation occurs due to the electron transfers from diffusing metal atom to not only surrounding SiO2 but also metal electrode. In this way, we can conclude that the ionization of the metal atom first begins quantum mechanically in the MIGS-penetration region and then its ionization charge changes electromagnetically outside the MIGS-penetration region.With respect to the diffusion under electric field, we found that the diffusion barrier linearly decreases with increasing the electric field. This result indicates that the electric field remarkably enhances the metal-atom penetration probability into SiO2, like an exponential function of electric field. Such enhancement works effectively for metal atoms with small electronegativity and mainly in the case of positive voltage application, which is in good agreement with observations. This is because most metal atoms prefer to ionized positively in insulators and small-negativity atoms have large ionic charges. These results are discussed in details, together with reviewing recent experiments, comparing to the cases of metal/semiconductor interfaces, and focusing on the fundamental physics of ionization and diffusion. <references> [1] Y. Asayama et al., Materials Science in Semiconductor Processing, DOI: 10.1016/j.mssp.2016.09.010 (2016). [2] T. Hiramatsu et al., Jpn. J. Appl. Phys. 53, 058006 (2014).[3] S. Sasaki and T. Nakayama, Jpn. J. Appl. Phys. 55, 111302 (2016). Figure 1

Dynamic behaviors and self-cleaning property of droplet on superhydrophobic coating in uniform DC electric field

When studying electromagnetic field problems like lightning phenomena, the efficiency of the adopted model relies on the accurate and rapid calculation of Sommerfeld type Integrals (SI). In the present work we propose an efficient method for calculating them, based on their numerical evaluation along a deformed path of integration. The method is combined with an interpolation technique in order to reduce the number of frequencies required in the Fourier synthesis of the time domain electric field. The result is a fast and straightforward tool for the calculation of aboveground lightning field, without the use of specially developed numerical algorithms or analytical approximations. The proposed approach is also validated through comparison with results obtained with a standard computational methodology.