Electric Field Formulation Research Articles

Controlled-source electromagnetic modeling using a versatile secondary electric-field formulation and efficient multigrid-based preconditioner

Frequency-domain controlled-source electromagnetic (CSEM) surveying is a widely used geophysical electromagnetic exploration method. To avoid the singularity near the transmitting sources and improve the accuracy of three-dimensional (3D) CSEM forward modeling, we adopt the vector finite element method and secondary electric field formulation combined with hexahedral grids to solve the frequency-domain Maxwell equations. Primary field computation applies the fast Hankel transform using digital linear filters to quickly calculate the electromagnetic field generated by different electromagnetic dipoles or loops in the layered medium. Therefore, our forward modeling algorithm is highly versatile and compatible with various electromagnetic sources and scenarios of CSEM application (e.g., airborne, offshore, or marine). To improve the convergence rate, we use an algebraic multigrid solver based on the established Hiptmair–Xu decomposition. Using direct solvers requires a vast amount of memory, while conventional iterative solvers usually fail or converge only very slowly. Our new iterative scheme does not require the divergence corrections commonly applied in finite difference or finite volume electromagnetic modeling to accelerate the convergence. We implement complete parallelism in our finite element programming to efficiently solve the CSEM forward problems on massively parallel clusters. We use different numerical methods to obtain high-precision comparison results for one-dimensional and typical 3D models to demonstrate our algorithm's effectiveness and correctness. Finally, a public model of 3 million cells is used to simulate the responses of marine CSEM surveying. The proposed new solver shows robust scalability when using nearly 1000 cores on the cluster, demonstrating that our algorithm is particularly suitable for large-scale 3D forward modeling.

Effect of Displacement Current on the Finite-Difference Modeling of Natural Source Electromagnetic Diffusion

For electromagnetic (EM) modeling based on the electric-field formulation at low frequencies, the quasi-static approximation (i.e., only the conduction current is considered and the displacement current is ignored) is commonly applied, and a small conductivity value for the air layer is chosen subjectively. Actually, in the air layer, due to the use of the small conductivity value, the quasi-static approximation is ubiquitously violated. However, the effect of the violation of the quasi-static approximation in the air on EM modeling is not well examined in the literature. In this paper, we investigate this issue by comparing the finite-difference modeling results from the calculation with the quasi-static approximation and those considering both the conduction and displacement currents. For the quasi-static approximation, the conductivity in the air is set to be different small values, and zero air conductivity is used for the modeling with both the conduction and displacement currents considered. Several simple models are designed to verify the numerical solution and study how the assigned conductivity for the air affects the modeling accuracy. One flat model and two models with topography are designed to examine the effect of the quasi-static approximation on the EM modeling results. For frequencies used in typical geophysical applications of EM diffusion, using the quasi-static approximation is able to produce accurate modeling results for models with typical earth conductivity. However, if the rough surface topography is considered, the use of the quasi-static approximation can reduce the EM modeling accuracy substantially at much lower frequencies (as low as several hundred Hz), which is probably due to the inaccurate description of EM waves in the air, and poses problems for applications based on direct EM field interpretation.

GEMM3D: An Edge Finite Element program for 3D modeling of electromagnetic fields and sensitivities for geophysical applications

This paper presents the 3D modeling program GEMM3D (Geophysical ElectroMagnetic Modeling 3D), a Fortran program for modeling frequency domain electromagnetic responses of geophysical sources in three-dimensional models. The program is an implementation of the Edge Finite Element method, in an electric field formulation, to simulate any electromagnetic source of interest, at all frequencies that are used in the geophysical methods in the quasi-static regime. Its current version includes routines for plane waves and simulation of magnetotelluric data; electrical and magnetic dipoles; circular loops. Other sources and configurations may easily be included with the addition of routines for simulating primary (layered earth) fields. The program includes routines for calculating sensitivities using the Adjoint State method, which makes efficient use of direct solvers for factoring once and solving for multiple systems with the same coefficient matrix. The paper details the methods, the implementation, and the usage of the program. Validation tests are presented for different sources. The program is freely available for use in research and teaching.

Nonlinear impedance boundary condition for 2D BEM

PurposeThe purpose of this paper is the derivation and efficient implementation of surface impedance boundary conditions (SIBCs) for nonlinear magnetic conductors.Design/methodology/approachAn approach based on perturbation theory is proposed, which expands to nonlinear problems the methods already developed by the authors for linear problems. Differently from the linear case, for which the analytical solution of the diffusion equation in the semi-infinite space for the magnetic field is available, in the nonlinear case the corresponding nonlinear diffusion equation must be solved numerically. To this aim, a suitable smooth map is defined to reduce the semi-infinite computational domain to a finite one; then the diffusion equation is solved by a Galerkin method relying on basis functions constructed via the push-forward of a Lagrangian polynomial basis whose degrees of freedom are collocated at Gauss–Lobatto nodes. The use of such basis in connection with a suitable under-integration naturally leads to mass-lumping without impacting the order of the method. The solution of the diffusion equation is coupled with a boundary element method formulation for the case of parallel magnetic conductors in terms of E and B fields.FindingsThe results are validated by comparison with full nonlinear finite element method simulations showing very good accordance at a much lower computational cost.Research limitations/implicationsLimitations of the method are those arising from perturbation theory: the introduced small parameter must be much less than one. This implies that the penetration depth of the magnetic field into the magnetic and conductive media must be much smaller than the characteristic size of the conductor.Originality/valueThe efficient implementation of a nonlinear SIBC based on a perturbation approach is proposed for an electric and magnetic field formulation of the two-dimensional problem of current driven parallel solid conductors.

Microwave Nondestructive Imaging of Buried Objects Using Improved Scattered‐Field Calibration Technique

AbstractA novel microwave subsurface imaging scheme based on an improved electric field calibration technique is proposed in this paper. The proposed imaging scheme requires RF measurement system comprising of a uniform linear transmitter‐receiver antenna array wherein the region under examination is illuminated sequentially to measure the scattering parameters of the test. An appropriate imaging frequency is selected so as to minimize the errors introduced due to coupling among the coresident antennas. The electric field required for the imaging algorithm is obtained from the measured scattering parameters using the proposed calibration methodology in two subsequent steps. The first step of calibration removes the systematic errors inherent in the measured scattering parameters by using known standards such as a metal plate and absorber foam. The second calibration step transforms the calibrated scattering parameters of the test object into electric field values so as to use them in the required inverse reconstruction procedure. The transformation is based on comparison of the measured scattering parameters of a known specimen and test object under the same test environment. The imaging algorithm based on the electric field formulation uses the standard iterative distorted Born procedure. For determining the location of the buried object, a novel data interpretation scheme based on analysis of variance (ANOVA) applied to time domain reflection coefficients obtained from various antenna elements is also proposed here. The applicability including performance evaluation of the proposed imaging scheme is tested by retrieving microwave image of various standard objects buried inside sand.

Numerical investigation of polymer rheology in electrohydrodynamic structuring on geometrical dielectric (ESGD) process

Electrohydrodynamic structuring on geometrical dielectric (ESGD) is an interesting approach to fabricate micro-/nanostructures from a rheological polymer which uses an electric field to produce the dielectrowetting force on the three-phase contact line, electrostatic force at the air–polymer interface and electrical attraction between the electrode pair to promote the fluidic polymer deformation. In this paper, the dynamic evolution of polymer rheology is investigated to understand the ESGD process from the viewpoints of the morphology of the air–polymer interface, polymer height, contact angle and the polymeric velocity, which is performed via a proposed numerical model that couples electric field, flow field and level set formulation accompanied with a verification by comparing the numerical simulations with the experimental tests. Furthermore, the driving force is disassembled to study the role of each one, by which the components of driving force can be listed in a sequence from strong to weak: dielectrowetting force, electrostatic force and electrical attraction. Finally, the influence of process parameters on the deformations, consisting of voltage, polymer thickness, dielectric permittivity and thickness of the dielectric layer coated on the template, is studied from the point of view of dielectrowetting force and electrostatic force, which provides a possible approach to enhance the driving force for a high aspect ratio.

Modeling of Forward Gate Leakage Current in MOSHEMT Using Trap-Assisted Tunneling and Poole–Frenkel Emission

Investigation of various forward gate leakage current mechanisms in an AlGaN/GaN metal–oxide–semiconductor high-electron-mobility transistor is done in this paper. During high temperature ( $T>388$ K), the trap-assisted tunneling (TAT) mechanism dominates the gate leakage current at low electric field for a range of gate biases from 0 to 0.2 V, whereas the Poole–Frenkel emission is the major component during medium and high electric field. During low temperature ( $T K), TAT alone is dominant and consistent throughout the whole range of electric field. A formulation of vertical electric field, across the oxide and barrier, is framed by incorporating oxide/barrier interface density of states. Then a physics-based compact analytical model for the TAT mechanism is developed and along with the existing PFE model, the forward gate leakage current is calculated. The results of the developed model in different regions of forward gate characteristics are in good agreement with the experimental results available in the literature.

The Application of Kiuttu’s Formulation to Study Coaxial Flux Compression Generators

A class of flux compression generators (FCGs) is based on the compression of the cross-sectional area of a coaxial geometry where the current flows along the outer conductor and returns through the inner conductor. This compression causes an increase in current since magnetic flux must be conserved. Kiuttu’s inductive electric-field formulation is a powerful tool for the conceptual design of coaxial FCGs. The usefulness of this formulation is demonstrated in this paper for a simplified geometry using a finite-element partial differential equation solver (FlexPDE) for calculation of the inductive electric field. A time-varying applied current or a moving surface creates the nonconservative electric field. Losses due to diffusion of magnetic flux into conducting surfaces can also be accounted for and modeled in this setting. This analytical–computational approach serves as an important step in validating the magnetohydrodynamic (MHD) portion of the complex multiphysics parallel Lawrence Livermore code, Arbitrary Lagrangian-Eulerian (ALE3D). The nonintuitive boundary conditions involved in solving the otherwise straightforward partial differential equations are described in detail and illustrated in a simple model. The physical parameters used in the simulations are not based on a specific design.

A New Low-Frequency Stable Potential Formulation for the Finite-Element Simulation of Electromagnetic Fields

The finite-element formulation proposed in this paper is for the frequency domain and covers the entire range from static/stationary fields to wave propagation. It does not involve any frequency-dependent thresholds, leads to complex symmetric system matrices and unique solutions, and applies to the most general structures, comprising both lossy and lossless regions. Compared with non-stabilized methods, e.g., the electric field formulation, the price to be paid is one extra scalar potential in the lossy region. Numerical examples demonstrate that the proposed method is well-suited for direct and iterative solvers, and remains stable in the static case.

Full-Vectorial Analysis of Optical Waveguide Discontinuities Using Denman–Beavers Iterative Scheme

An accurate and efficient full-vectorial numerical approach for analyzing three-dimensional optical waveguide discontinuities is proposed, where the resulting square root of the characteristic matrix from the field-based operators is approximated by the Denman-Beavers iterative scheme with a branch-cut rotation in the complex plane. Both the magnetic and electric field formulations are rigorously derived along with the robust perfectly matched layer absorbing boundary conditions. Numerical results from a typical buried waveguide, a channel waveguide, and a rib waveguide show that the present scheme is more accurate and stable than the widely used Padé approximation, while the computational load is comparative.

Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method

We present a robust and scalable solver for time-harmonic Maxwell's equations for problems with large conductivity contrasts, wide range of frequencies, stretched grids and locally refined meshes. The solver is part of the fully distributed adaptive 3-D electromagnetic modelling scheme which employs the finite element method and unstructured non-conforming hexahedral meshes for spatial discretization using the open-source software deal.II. We use the complex-valued electric field formulation and split it into two real-valued equations for which we utilize an optimal block-diagonal pre-conditioner. Application of this pre-conditioner requires the solution of two smaller real-valued symmetric problems. We solve them by using either a direct solver or the conjugate gradient method pre-conditioned with the recently introduced auxiliary space technique. The auxiliary space pre-conditioner reformulates the original problem in form of several simpler ones, which are then solved using highly efficient algebraic multigrid methods. In this paper, we consider the magnetotelluric case and verify our numerical scheme by using COMMEMI 3-D models. Afterwards, we run a series of numerical experiments and demonstrate that the solver converges in a small number of iterations for a wide frequency range and variable problem sizes. The number of iterations is independent of the problem size, but exhibits a mild dependency on frequency. To test the stability of the method on locally refined meshes, we have implemented a residual-based a posteriori error estimator and compared it with uniform mesh refinement for problems up to 200 million unknowns. We test the scalability of the most time consuming parts of our code and show that they fulfill the strong scaling assumption as long as each MPI process possesses enough degrees of freedom to alleviate communication overburden. Finally, we refer back to a direct solver-based pre-conditioner and analyse its complexity in time. The results show that for multiple right-hand sides the direct solver-based pre-conditioner can still be faster for problems of medium size. On the other hand, it also shows non-linear growth in memory, whereas the auxiliary space method increases only linearly.

Basis Functions With Divergence Constraints for the Finite Element Method

Basis functions for solving partial differential equations of vector fields using the finite element method are presented. The basis functions are a combination of cubic Hermite splines and second-order Lagrange interpolation polynomials and allow the divergence to be set as a constraint. The basis functions are tested on 3-D resonant cavities and there are no spurious modes. There is good agreement with analytical solutions in cases where they exist and with calculations using edge elements in other cases. The method is extended to solve problems with singularities at edges and corners. For perfect conductors, this includes mesh refinement in the neighborhood of the edge or corner. For dielectrics, constraints are derived so that the flux of the field is zero through a closed surface that contains the edge or corner. The method is used to solve problems using the electric and magnetic field formulations. Through a change of variables, the method is applied to problems in cylindrical coordinates.

Existence and approximation of a mixed formulation for thin film magnetization problems in superconductivity

We recall a recently introduced mixed formulation of thin film magnetization problems for type-II superconductors written in terms of two variables, the electric field and the magnetization function, see [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. A finite element approximation, [Formula: see text], based on this mixed formulation, involving the lowest-order Raviart–Thomas element for approximating the electric field, was also introduced in [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. Here h, τ are the spatial and temporal discretization parameters, and [Formula: see text] with p-1 the value of power in the current–voltage relation characterizing the superconducting material. In this paper, we establish well-posedness of [Formula: see text], and prove convergence of the unique solution of [Formula: see text] to a solution of the power law model ( Q r), for a fixed r > 1, as h, τ → 0. In addition, we prove convergence of a solution of ( Q r) to a solution of the critical state model (Q), as r → 1. Hence, we prove existence of solutions to ( Q r), for a fixed r > 1, and (Q). Finally, numerical experiments are presented.

Multilevel fast multipole method based on a potential formulation for 3D electromagnetic scattering problems

A combination of the multilevel fast multipole method (MLFMM) and boundary element method (BEM) can solve large scale photonics problems of arbitrary geometry. Here, MLFMM-BEM algorithm based on a scalar and vector potential formulation, instead of the more conventional electric and magnetic field formulations, is described. The method can deal with multiple lossy or lossless dielectric objects of arbitrary geometry, be they nested, in contact, or dispersed. Several examples are used to demonstrate that this method is able to efficiently handle 3D photonic scatterers involving large numbers of unknowns. Absorption, scattering, and extinction efficiencies of gold nanoparticle spheres, calculated by the MLFMM, are compared with Mie's theory. MLFMM calculations of the bistatic radar cross section (RCS) of a gold sphere near the plasmon resonance and of a silica coated gold sphere are also compared with Mie theory predictions. Finally, the bistatic RCS of a nanoparticle gold-silver heterodimer calculated with MLFMM is compared with unmodified BEM calculations.

Electric field formulation for thin film magnetization problems

We derive a variational formulation for thin film magnetization problems for type-II superconductors written in terms of two variables: the electric field and the magnetization function. A numerical method, based on this formulation, makes it possible to accurately compute all variables of interest, including the electric field, for any value of the power in the power law current–voltage relation characterizing the superconducting material. For high power values we obtain a good approximation to the critical state model solution. Numerical simulation results are presented for simply and multiply connected films, and also for an inhomogeneous film.

A residual-based a posteriori error estimator for the hp-finite element method for Maxwellʼs equations

We present a residual-based a posteriori error estimator for Maxwellʼs equations in the electric field formulation in three space-dimensions. The error estimator is constructed in terms of the residual of the considered problem and we prove upper and lower bounds in the energy norm, which are explicit in the mesh size and the polynomial degree of the ansatz space.

An hp‐Efficient Residual‐Based A Posteriori Error Estimator for Maxwell's Equations

AbstractWe present a residual‐based a posteriori error estimator for Maxwell's equations in the electric field formulation. The error estimator is formulated in terms of the residual of the considered problem and we state upper and lower bounds in terms of the energy error of the computed solution for the estimator. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analytical formulation of electric field and dielectrophoretic force for moving dielectrophoresis using Fourier series

Electrokinetics manipulation and separation of living cells employing microfluidic devices require good knowledge of the strength and distribution of electric field in such devices. AC dielectrophoresis is performed by generating non-uniform electric field using microsize electrodes. Among the several applications of dielectrophoretic phenomenon, this present study considers the recently introduced phenomenon of moving dielectrophoresis. An analytical solution using Fourier series is presented for the electric field distribution and dielectrophoretic force generated inside a microchannel. The potential at the upper part of the microchannel has been found by solving the governing equation of the electric potential with specific boundary conditions. The solutions for the electric field and dielectrophoretic force show excellent agreement with the numerical results. Microdevices were fabricated and experiments were carried out with living cells confirming and validating the analytical solutions.

Theoretical Estimation of Scattering Waves in Transverse Section of Upper Body for On-Body Wireless Communications

This paper attempts to analyze theoretically the propagation characteristics in the transverse section of upper body to support on-body wireless communications. The analytical estimation assumes that the human body is structured as a lossy-dielectric circular cylinder with infinite length that consists of the 2/3-muscle equivalent uniform tissue. Each scattering electric field formulation inside and outside of the cylinder is derived for scattering characteristics in the propagation environment including the human body when the source current has the vertical direction to the cylinder surface or the horizontal direction to the cylinder axis. In order to confirm the reliability of the formulation, total electric field distributions at 2.45GHz are compared with the results by the finite-difference time-domain (FDTD) method. In each current direction, general scattering characteristics and the influence on the total propagation are estimated. Furthermore, from scattering and total electric field intensities evaluated with the variations of operating frequency, radius of the human body, and distance between a source and the human body, propagation characteristics are investigated to assist in the design of a device for on-body propagation channel with the upper body.

Improved Full-Vector Finite-Difference Complex Mode Solver for Optical Waveguides of Circular Symmetry

An improved full-vector finite-difference (FD) complex mode solver for circularly symmetric optical waveguides is developed and validated. The formulations are derived from Taylor expansion of transverse electric and magnetic fields and match of boundary conditions at radial index discontinuities. Calculation of the guided and leaky modes of step index fibers, Bragg fibers, and surface plasmon polaritons structures shows significant improvement in terms of accuracy and rate of convergence without increase of computational efforts, in comparison with the conventional FD scheme based on the average-index approximation. It is demonstrated that the performance of the new scheme is robust for high-order modes and waveguides with high index contrast. Contrary to the conventional FD solvers that claim the magnetic field formulation is more rigorous than the electric field formulation, the improved solver yields practically identical results for both formulations.