Total Field Solution Research Articles

3D vector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids

Unstructured tetrahedral grids with local refinement facilitate the use of total-field solution approaches to geophysical electromagnetic (EM) forward problems. These approaches, when combined with the vector finite-element (FE) method and with refinement near transmitters and receivers, can give accurate solutions and can easily handle realistic models with complex geometry and topography. We have applied this approach to 3D forward modeling for fixed- and moving-loop configurations. MUMPS, a direct solver, was used to solve the linear system of equations generated by FE analysis. A direct solver is particularly suited to the moving-loop configuration for which the right side is different for every transmitter loop, but for which the coefficient matrix is unchanged. Therefore, the coefficient matrix need only be factorized once, and then the system can be solved efficiently for all different right sides. We compared our results with several typical scenarios from the literature: a conductive brick in a homogeneous half-space and a complex conductor at a vertical contact both for fixed-loop configurations, and a homogeneous half-space for a moving-loop configuration. We also evaluated results for the massive sulfide ore deposit of the Ovoid Zone at Voisey’s Bay, Labrador, Canada, for which we considered fixed- and moving-loop configurations. This model also provides an illustration of the complex vortex current systems that are generated by time-domain EM methods within highly conductive ore bodies in a resistive host.

Numerical modeling of the 2D time-domain transient electromagnetic secondary field of the line source of the current excitation

To effectively minimize the electromagnetic field response in the total field solution, we propose a numerical modeling method for the two-dimensional (2D) timedomain transient electromagnetic secondary field of the line source based on the DuFort-Frankel finite-difference method. In the proposed method, we included the treatment of the earth-air boundary conductivity, calculated the normalized partial derivative of the induced electromotive force (Emf), and determined the forward time step. By extending upward the earth-air interface to the air grid nodes and the zero-value boundary conditions, not only we have a method that is more efficient but also simpler than the total field solution. We computed and analyzed the homogeneous half-space model and the flat layered model with high precision—the maximum relative error is less than 0.01% between our method and the analytical method—and the solution speed is roughly three times faster than the total-field solution. Lastly, we used the model of a thin body embedded in a homogeneous half-space at different delay times to depict the downward and upward spreading characteristics of the induced eddy current, and the physical interaction processes between the electromagnetic field and the underground low-resistivity body.

A 3-D application of the Bymoment method for electromagnetic scattering

One can find various mesh truncation techniques in the literature for the finite element solution of electromagnetic scattering problems. In this paper, a mesh truncation approach referred to as the three-dimensional bymoment method is considered. This approach uses two z-directed Hertz vector potentials instead of three Cartesian field components to formulate the relevant boundary conditions. The pertaining FEM field solutions are calculated by treating the scattering problem as an interior region problem. The FEM field solutions are then numerically coupled on a second boundary surface that is conformal to the first one and completely enclosed by it. The coupling procedure yields a set of expansion coefficients, with the help of which the total field solution is written as an appropriate linear combination of the interior region FEM field solutions.

Transform technique for planar optical waveguides

A new method for obtaining total field solutions and propagation constants for planar optical waveguiding structures by using a transform technique is presented. The method reduces the two-dimensional Helmholtz equation into a system of linear algebraic equations that can be solved on a computer. The method takes into account arbitrary input excitation and is applicable to arbitrary refractive-index variations in the transverse direction. We illustrate the method by applying it to directional couplers and straight waveguides with graded-index profiles-parabolic and exponential profiles. The results are in good agreement with analytical results.

Time-explicit simulation of wave interaction in optical waveguide crossings at large angles

The time-explicit finite-difference time-domain method is used to simulate wave interaction in optical waveguide crossings at large angles. The wave propagation at the intersecting structure is simulated by time stepping the discretized form of the Maxwell's time dependent curl equations. The power distribution characteristics of the intersections are obtained by extracting the guided-mode amplitudes from these simulated total field data. A physical picture of power flow in the intersection is also obtained from the total field solution; this provides insights into the switching behavior and the origin of the radiations.

Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields

A method for obtaining total field solutions from the Helmholtz equation for optical waveguiding structures is presented. The method is based on a conversion of the Helmholtz equation into a matrix total differential equation and direct solution of the resulting equation by means of standard numerical methods. The principle of the method is discussed in detail, and numerical results are presented for planar guiding structures. The method has been extended to three-dimensional guiding structures. Comparisons with the propagating-beam method show that the present method yields better accuracy and is more efficient numerically for planar as well as for three-dimensional waveguiding structures. Further, unlike the propagating-beam method, the present method needs no inherent approximation and hence can be used to obtain any desired accuracy, if the increase in computation time is not a consideration.

A stable finite element solution for two-dimensional magnetotelluric modelling

We report herein on a finite element algorithm for 2-D magnetotelluric modelling which solves directly for secondary variations in the field parallel to strike, plus the subsequent vertical and transverse auxiliary fields, for both transverse electric and transverse magnetic modes. The governing Helmholtz equations for the secondary fields along strike are the same as those for total field algorithms with the addition of source terms involving the primary fields and the conductivity difference between the body and the host. Our approach has overcome a difficulty with numerical accuracy at low frequencies observed in total field solutions with 32-bit arithmetic for the transverse magnetic mode especially, but also for the transverse electric mode. Matrix ill-conditioning, which affects total field solutions, increases with the number of element rows with the square of the maximum element aspect ratio and with the inverse of frequency. In the secondary formulation, the field along strike and the auxiliary fields do not need to be extracted in the face of an approximately computed primary field which increasingly dominates the total field solution towards low frequencies. In addition to low-frequency stability, the absolute accuracy of our algorithm is verified by comparison with the TM and the TE mode analytic responses of a segmented overburden model.

A variable power singular element for analysis of fracture mechanics problems

A variable power displacement formulated singular element is developed. The stress and displacement fields within the element are expressed in terms of singular and regular components. The singular components of these fields are shown to provide a more accurate representation of the singularity than the total field solutions. The problem of a crack terminating at a bi-material interface is solved for two extreme ratios of singular element radius to crack length. The solution shows that accurate results can be achieved for either case.

Comparison of two leading uniform theories of edge diffraction with the exact uniform asymptotic solution

The diffraction of an arbitrary cylindrical wave by a half‐plane has been treated by Rahmat‐Samii and Mittra who used a spectral domain approach. In this paper, their exact solution for the total field is expressed in terms of a new integral representation. For large wave number k, two rigorous procedures are described for the exact uniform asymptotic expansion of the total field solution. The uniform expansions obtained are valid in the entire space, including transition regions around the shadow boundaries. The final results are compared with the formulations of two leading uniform theories of edge diffraction, namely, the uniform asymptotic theory and the uniform theory of diffraction. Some unique observations and conclusions are made in relating the two theories.

Theory of Brillouin scattering from opaque media

The general theory of Brillouin scattering from the surfaces of opaque media is developed using a total field solution approach. Acoustical modes appropriate to a stress-free surface are found from solutions to the acoustical-wave equation and boundary conditions for a finite isotropic medium. Two light-scattering mechanisms, namely, the surface corrugation and bulk elasto-optical effects, were analyzed by deriving optical fields which satisfy both the acousto-optically driven wave equation and the electromagnetic boundary conditions. The special case for a metal of scattering by acoustically created conductivity fluctuations was also investigated in a similar way. The Brillouin spectrum was obtained by summing the scattering from the individual acoustical modes over the density of phonon states. Excellent agreement was obtained with experiment for Brillouin scattering from the metals gallium (liquid) and aluminum (solid).

Theory of non phase-matched second harmonic generation of surface plasmons

Non phase-matched harmonic generation of surface plasmons has been analysed using total field solutions which satisfy the nonlinear current driven Maxwell's Equations and electromagnetic boundary conditions. The results are compared with a recent calculation by Mills.

On the role of the geometrical optics field in aperture diffraction

The asymptotic solution of a high-frequency electromagnetic field transmitted through a finite aperture is studied. In applying Keller's geometrical theory of diffraction (GTD), a basic yet unanswered question for an observation point in the lit region is: "Should the geometrical optics field on a direct incident ray be included in the total field solution?" By studying a test problem and utilizing the newly developed uniform asymptotic theory (UAT), we have deduced simple and explicit rules for the role of the geometrical optics field and the regions of validity for GTD in a general aperture diffraction. The rules reveal that the success of GTD in treating aperture problems in the literature depends critically on the assumption that both the source and the observation points are infinitely far away from the aperture. Had either point been a finite distance away, Keller's GTD, in general, would fail and UAT must be used. The paper also demonstrates a physical phenomenon: the diffusion of the incident field as the observation point moves away from the aperture.

The perturbation of alternating geomagnetic fields by discontinuities with high conductivity contrasts

The perturbation of alternating geomagnetic fields by conductivity discontinuities is considered. A numerical method is used to solve the two-dimensional induction problem. Models in which the conductivity contrast between conductive regions is great, such as between oceanic and continental regions, are considered. The perturbation field distributions for models with higher conductivity contrasts are compared with lower conductivity contrast models using a method recently developed by Jones (1972). Total field solutions are obtained for several models by using a very large grid as well as a folding-in technique. Results from the two methods for the various models are compared. It is found that the folding-in technique offers an alternative method for handling higher conductivity contrasts.