Maxwell's Equations In Frequency Domain 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.

2.5-Dimensional modeling of EM logging-while-drilling tool in anisotropic medium on a Lebedev grid

A 2.5D finite-difference (FD) algorithm for the modeling of the electromagnetic (EM) logging-while-drilling (LWD) tool in anisotropic media is presented. The FD algorithm is based on the Lebedev grid, which allows for the discretization of the frequency-domain Maxwell's equations in the anisotropic media in 2.5D scenarios without interpolation. This leads to a system of linear equations that is solved using the multifrontal direct solver which enables the simulation of multi-sources at nearly the cost of simulating a single source for each frequency. In addition, near-optimal quadrature derived from an optimized integration path in the complex plane is employed to implement the fast inverse Fourier Transform (IFT). The algorithm is then validated by both analytic and 3D solutions. Numerical results show that two Lebedev subgrid sets are sufficient for TI medium, which is common in geosteering environments. The number of quadrature points is greatly reduced by using the near-optimal quadrature method.

Stratigraphic identification with airborne electromagnetic methods at the Hanford Site, Washington

Stratigraphic units can influence the fate and transport of subsurface contaminants within groundwater. Units having coarse-grained sediments act as preferential flow pathways, and therefore can accelerate the transport of contaminants to reach human and ecological receptors. At legacy waste sites, detailed knowledge of subsurface stratigraphy can be used for effective monitoring and remediation planning to help minimize risk to human health and the environment. Airborne electromagnetic (AEM) methods can non-invasively provide information on kilometer-scale or larger subsurface stratigraphic features and fill informational gaps in directly sampled data from sparsely located boreholes. In this paper, we present inversion results of a 412 line-km frequency-domain AEM survey to delineate subsurface stratigraphic features at the Hanford Site, located in southeastern Washington State. The inversion was performed using a massively parallel 3D electromagnetic modeling and inversion code, where the modeling is based on solving frequency-domain Maxwell's equations using an unstructured-mesh finite-element method and the inversion employs a Gauss-Newton optimization scheme. The results are compared to an underlying geologic framework model (GFM), built by interpolating contact depths of stratigraphic units interpreted from site borehole datasets. In areas with good borehole coverage, the inversion results show a good match with the GFM to a depth of about 60 m. Outside of these areas, the inversion results exhibit inconsistencies from the assumptions made to create the GFM, demonstrating that the AEM survey results can be used to improve the understanding of the geological conceptual model.

RF wave propagation simulation for ICRF antenna in EAST

ABSTRACT The propagation performance of radiofrequency (RF) waves excited by ICRF antenna in plasma has a great influence on the coupling effect. In this paper, the propagation characteristics of fast wave (FW) and slow wave (SW) under tokamak experimental condition are calculated, and the influence of angular frequency w 0, antenna power spectrum k// and tokamak total magnetic field B 0 is illustrated based on the typical operating parameters in EAST. By applying perfect matched layer (PML) boundary processing method to simulate single-pass absorption and using finite element method (FEM) to solve Maxwell equations in frequency domain, the propagation and coupling simulation of RF waves in cold plasma for EAST ICRF antenna in B-window were carried out to obtain the polarization characteristics of fields and the relationship between the coupling impedance and the gap length from antenna to the FW cut-off layer. The analysis results in this paper provide useful reference and guidance for the study of propagation coupling of ICRF waves in EAST and other magnetic confined fusion devices.

Simulation acceleration for transmittance of electromagnetic waves in 2D slit arrays using deep learning

When designing new optical devices, many simulations must be conducted to determine the optimal design parameters. Therefore, fast and accurate simulations are essential for designing optical devices. In this work, we introduce a deep learning approach that accelerates a simulator solving frequency-domain Maxwell equations. Our model achieves high accuracy while predicting transmittance per wavelength in 2D slit arrays under certain conditions to achieve 160,000 times faster results than the simulator. We generated a dataset using an open-source simulator and compared its performance with those of other machine learning models. Additionally, we propose a new loss function and performance evaluation method for creating better performance models with multiple regression outputs from one input source. We observed that using a loss function that adds binary cross-entropy loss, which predicts whether the differential of the transmittance is positive or negative at wavelengths adjacent to the root mean-squared error of the transmittance value, is more effective for predicting variations in multiple regression outputs. The simulation results show that a four-layer convolutional neural network model demonstrates the best accuracy (R2 score: 0.86). The overall approach presented here is expected to be useful for simulating and designing optical devices.

Modeling deformed transmission lines for continuous strain sensing applications

Transmission lines are essential components in various signal and power distribution systems. In addition to their main use as connecting elements, transmission lines can also be employed as continuous sensors for the measurement and detection of external influences such as mechanical strains and deformations. The measuring principle is based on deformation-induced changes of the characteristic impedance. Reflections of an injected test signal at resulting impedance mismatches can be used to infer applied deformations. To determine the effect of deformations on the characteristic impedance, we develop a numerical framework that allows us to solve Maxwell’s equations for any desired transmission-line geometry over a wide frequency range. The proposed framework utilizes a staggered finite-difference Yee method on non-uniform grids to efficiently solve a set of decoupled partial differential equations that we derive from the frequency domain Maxwell equations. To test our framework, we compare simulation results with analytical predictions and corresponding experimental data. Our results suggest that the proposed numerical framework is able to capture experimentally observed deformation effects and may therefore be used in transmission-line-based deformation and strain sensing applications. Furthermore, our framework can also be utilized to simulate and study the electromagnetic properties of complex arrangements of conductor, insulator, and shielding materials.

Modeling of 2D Photonic Crystal based 1x4 All Optical Splitter

Objectives: In this paper a 1x4 optical beam splitter with a power splitting ratio of 80:20, based on 2D photonic crystal has been modeled for TE-polarized light. Methods: Designing of prototypes was done on the OptiFDTD software. Further, the simulations were performed and post-simulation the results were analyzed, this included the transmission spectra of the design which showed that the power was divided in a ratio of 80:20 that is 80 % and 20% of the transmitted signal. Also, the photonic band gaps were calculated using the Plane Wave Expansion (PWE) band solver. Findings: This prototype comprises of 2D Hexagonal lattice where the elliptical Si-rods are lined up in the air. The size of the wafer used for designing the prototype was taken to be 21µm*15µm.This particular design is a H-shaped splitter Finite Difference Time Domain (FDTD) and Plane Wave Expansion (PWE) methods are the mostly used mathematical analysis methods that are used for analyzing and calculating the band gap configuration and calculating the Photonic Band Gap of the photonic crystals. The Plane Wave Expansion method calculates the Eigen frequencies and dispersal characteristics of photonic crystals, with the help of Maxwell equations in frequency domain. The output power at output 1 and 3 was 80% and at the output 2 and 4 was calculated to be 20% of the transmitted signal. With the tolerance as 0.001, three band gaps were calculated and the values of their ban gaps were found to be 0.235179, 341851; 0.459574, 0.575928; 0.616383, 0.628378. Novelty/Improvement: This structure is designed in 1x4 dimensions, further the structure can be designed in various dimensions like 1x8, 1x16, 1x32 etc., depending on different applications. Keywords: Optical Splitter, Photonic Crystals, FDTD Simulation, PWE Band Solver

PETGEM: A parallel code for 3D CSEM forward modeling using edge finite elements

We present the capabilities and results of the Parallel Edge-based Tool for Geophysical Electromagnetic modeling (PETGEM), as well as the physical and numerical foundations upon which it has been developed. PETGEM is an open-source and distributed parallel Python code for fast and highly accurate modeling of 3D marine controlled-source electromagnetic (3D CSEM) problems. We employ the Nédélec Edge Finite Element Method (EFEM) which offers a good trade-off between accuracy and number of degrees of freedom, while naturally supporting unstructured tetrahedral meshes. We have particularised this new modeling tool to the 3D CSEM problem for infinitesimal point dipoles asumming arbitrarily isotropic media for low-frequencies approximations. In order to avoid source-singularities, PETGEM solves the frequency-domain Maxwell's equations of the secondary electric field, and the primary electric field is calculated analytically for homogeneous background media. We assess the PETGEM accuracy using classical tests with known analytical solutions as well as recent published data of real life geological scenarios. This assessment proves that this new modeling tool reproduces expected accurate solutions in the former tests, and its flexibility on realistic 3D electromagnetic problems. Furthermore, an automatic mesh adaptation strategy for a given frequency and specific source position is presented. We also include a scalability study based on fundamental metrics for high-performance computing (HPC) architectures.

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

We put forward the idea of using a Block Low-Rank (BLR) multifrontal direct solver to efficiently solve the linear systems of equations arising from a finite-difference discretization of the frequency-domain Maxwell equations for 3-D electromagnetic (EM) problems. The solver uses a low-rank representation for the off-diagonal blocks of the intermediate dense matrices arising in the multifrontal method to reduce the computational load. A numerical threshold, the so-called BLR threshold, controlling the accuracy of low-rank representations was optimized by balancing errors in the computed EM fields against savings in floating point operations (flops). Simulations were carried out over large-scale 3-D resistivity models representing typical scenarios for marine controlled-source EM surveys, and in particular the SEG SEAM model which contains an irregular salt body. The flop count, size of factor matrices and elapsed run time for matrix factorization are reduced dramatically by using BLR representations and can go down to, respectively, 10, 30 and 40 per cent of their full-rank values for our largest system with N = 20.6 million unknowns. The reductions are almost independent of the number of MPI tasks and threads at least up to 90 × 10 = 900 cores. The BLR savings increase for larger systems, which reduces the factorization flop complexity from O(N2) for the full-rank solver to O(Nm) with m = 1.4–1.6. The BLR savings are significantly larger for deep-water environments that exclude the highly resistive air layer from the computational domain. A study in a scenario where simulations are required at multiple source locations shows that the BLR solver can become competitive in comparison to iterative solvers as an engine for 3-D controlled-source electromagnetic Gauss–Newton inversion that requires forward modelling for a few thousand right-hand sides.

G0501101 マイクロ波加熱の撹拌系内熱対流に及ぼす影響について

In this paper microwave heating a liquid rotated and stirred by paddle wings is investigated numerically. In order to feed electromagnetic energy into this rotating system directly, we proposed to use the rotating metal shaft as a coaxial antenna. A set of multiphysics equations representing this system was solved by using finite-element method. In the actual computation, commercial software, COMSOL Multiphysics, was utilized. This software enables us to simultaneously solve frequency domain Maxwell equations and time-dependent Navier-Stokes and heat-transfer equations. The flow and temperature fields were extensively visualized as a result of this investigation.

Oblique incidence of semi-guided waves on rectangular slab waveguide discontinuities: A vectorial QUEP solver

The incidence of thin-film-guided, in-plane unguided waves at oblique angles on straight discontinuities of dielectric slab waveguides, an early problem of integrated optics, is being re-considered. The 3-D frequency domain Maxwell equations reduce to a parametrized inhomogeneous vectorial problem on a 2-D computational domain, with transparent-influx boundary conditions. We propose a rigorous vectorial solver based on simultaneous expansions into polarized local slab eigenmodes along the two orthogonal cross section coordinates (quadridirectional eigenmode propagation QUEP). The quasi-analytical scheme is applicable to configurations with — in principle — arbitrary cross section geometries. Examples for a high-contrast facet of an asymmetric slab waveguide, for the lateral excitation of a channel waveguide, and for a step discontinuity between slab waveguides of different thicknesses are discussed.

A Multidomain Pseudospectral Mode Solver for Optical Waveguide Analysis

We propose a pseudospectral mode solver for optical waveguide mode analysis formulated by the frequency-domain Maxwell equations. Special attention is paid upon identifying the required boundary operator for the formulation and the relationships between the derived operator and the physical boundary conditions. These theoretical results are adopted into a Legendre pseudospectral multidomain computational framework to compute the propagation characteristics of optical waveguides. Numerical experiments are conducted, and the expected spectral convergence of the scheme is observed for smooth problems and for problems having field jumps at material interfaces. For dielectric waveguides with sharp corners, the spectral convergence is lost due to the singular nature of fields at the corner. Nevertheless, compared with other methods, the present formulation remains as an efficient approach to obtain waveguide modes.

Light propagation in nonuniform plasmonic subwavelength waveguide arrays

We study light propagation in nanoscale periodic structures composed of dielectric and metal in the visible range. We demonstrate that diffraction curves of nonuniform waveguide arrays can be tailored by varying the geometric and dielectric features of the waveguides. The results obtained from a proper formulation of coupled mode theory for nonuniform arrays are validated through numerical solution of Maxwell equations in frequency domain.

Broad Band Modeling of a Superconducting Magnetic Energy Storage (SMES) Coil

In this paper, we compute with accuracy the impedance parameters of a superconducting magnetic energy storage (SMES) system. We do this by using a three-dimensional integral formulation of the full set of frequency domain Maxwell's equations in the quasistatic limit. The impedance parameter allows the construction of a time domain equivalent circuit to be used for the simulation of the overall SMES system. The numerical model is based on a volume integral formulation where the unknown is the total current density J, expressed as the sum of its solenoidal and non-solenoidal components. This separation allows to avoid the ill-conditioning of the relevant stiffness matrix at low frequencies, being essential for developing a numerical model accurately working where the SMES resonances are located. The model is applied to the case of an ideal model coil consisting of an 18-layers solenoid.

Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation

In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1) (1999) 204–225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements. The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.

A High-Order Accurate Method for Frequency Domain Maxwell Equations with Discontinuous Coefficients

Maxwell equations contain a dielectric coefficient ? that describes the particular media. For homogeneous materials the dielectric coefficient is constant. There is a jump in this coefficient across the interface between differing media. This discontinuity can significantly reduce the order of accuracy of the numerical scheme. We present an analysis and implementation of a fourth order accurate algorithm for the solution of Maxwell equations with an interface between two media and so the dielectric coefficient is discontinuous. We approximate the discontinuous function by a continuous one either locally or in the entire domain. We study the one-dimensional system in frequency space. We only consider schemes that can be implemented for multidimensional problems both in the frequency and time domains.