Comparison of integral equation and physical scale modeling of the electromagnetic responses of models with large conductivity contrasts

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. G169-G177 ◽  
Author(s):  
Colin G. Farquharson ◽  
Ken Duckworth ◽  
Douglas W. Oldenburg
Keyword(s):  
Integral Equation ◽  
Electric Field ◽  
Tangential Component ◽  
Basis Functions ◽  
Electromagnetic Modeling ◽  
Equation Solution ◽  
Scale Modeling ◽  
Electromagnetic Responses ◽  
Physical Scale ◽  
A Cell

A comparison is made between the results from two different approaches to modeling geophysical electromagnetic responses: a numerical approach based upon the electric-field integral equation and the physical scale modeling approach. The particular implementation of the integral-equation solution was developed recently, and the comparison presented here is essentially a test of this new formulation. The implementation approximates the region of anomalous conductivity by a mesh of uniform cuboidal cells and approximates the total electric field within a cell by a linear combination of bilinear edge-element basis functions. These basis functions give a representation of the electric field that is divergence free but not curl free within a cell, and whose tangential component is continuous between cells. The charge density (which arises from the discontinuity of the normal com-ponent of the electric field across interfaces between cells of different conductivities and between cells and the background) is incorporated in a similar manner to integral equation solutions to dc resistivity modeling. The scenarios considered for the comparison comprise a graphite cube of [Formula: see text] conductivity and 14-cm length in free space and in brine ([Formula: see text] conductivity). The transmitter and receiver were small horizontal loops; measurements and computations were made for various fixed transmitter-receiver separations and various heights of the transmitter-receiver pair for frequencies ranging from [Formula: see text]. The agreement between the numerical results from the integral-equation implementation and the measurements from the physical scale modeling was very good, contributing to the verification of this particular implementation of the integral-equation solution to electromagnetic modeling.

Advanced three-dimensional electromagnetic modeling using a nested integral equation approach

2021 ◽  
Author(s):  
Chaojian Chen ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Region Of Interest ◽  
Electromagnetic Modeling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Uniform Grids ◽  
The Matrix ◽  
Integral Equation Approach ◽  
Matrix Vector

Summary Most of the existing three-dimensional (3-D) electromagnetic (EM) modeling solvers based on the integral equation (IE) method exploit fast Fourier transform (FFT) to accelerate the matrix-vector multiplications. This in turn requires a laterally-uniform discretization of the modeling domain. However, there is often a need for multi-scale modeling and inversion, for instance, to properly account for the effects of non-uniform distant structures, and at the same time, to accurately model the effects from local anomalies. In such scenarios, the usage of laterally-uniform grids leads to excessive computational loads, both in terms of memory and time. To alleviate this problem, we developed an efficient 3-D EM modeling tool based on a multi-nested IE approach. Within this approach, the IE modeling is first performed at a large domain and on a (laterally-uniform) coarse grid, and then the results are refined in the region of interest by performing modeling at a smaller domain and on a (laterally-uniform) denser grid. At the latter stage, the modeling results obtained at the previous stage are exploited. The lateral uniformity of the grids at each stage allows us to keep using the FFT for the matrix-vector multiplications. An important novelty of the paper is a development of a “rim domain” concept which further improves the performance of the multi-nested IE approach. We verify the developed tool on both idealized and realistic 3-D conductivity models, and demonstrate its efficiency and accuracy.

scholarly journals Fast Integral Equation Solution of Scattering of Multiscale Objects by Domain Decomposition Method with Mixed Basis Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ran Zhao ◽  
Hua Peng Zhao ◽  
Zai-ping Nie ◽  
Jun Hu
Keyword(s):  
Integral Equation ◽  
Domain Decomposition ◽  
Mesh Generation ◽  
Electromagnetic Scattering ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Basis Functions ◽  
Equation Solution ◽  
Nonoverlapping Domain Decomposition ◽  
Well Posed

Nonconformal nonoverlapping domain decomposition method (DDM) with mixed basis functions is presented to realize fast integral equation solution of electromagnetic scattering of multiscale objects. The original multiscale objects are decomposed into several closed subdomains. The higher order hierarchical vector basis functions are used in the electrically large smooth subdomains to significantly reduce the number of unknowns, while traditional Rao-Wilton-Glisson basis functions are used for subdomains with tiny structures. A well-posed matrix is successfully derived by the present DDM. Besides, the nonconformal property of DDM allows flexible mesh generation for complicated objects. Numerical results are presented to validate the proposed method and illustrate its advantages.

An integral equation formulation with global series expansion for resonant wireless power transfer

2017 ◽  
Vol 36 (5) ◽  
pp. 1474-1487
Author(s):  
Sándor Bilicz ◽  
József Pávó ◽  
Szabolcs Gyimóthy ◽  
Zsolt Badics
Keyword(s):  
Integral Equation ◽  
Series Expansion ◽  
Computational Cost ◽  
Wireless Power Transfer ◽  
Basis Functions ◽  
Power Transfer ◽  
Electromagnetic Modeling ◽  
Wireless Power ◽  
Content Type ◽  
Integral Equation Formulation

Purpose The electromagnetic modeling of inductively coupled, resonant wireless power transfer (WPT) is dealt with. This paper aims to present a numerically efficient simulation method. Design/methodology/approach Recently, integral equation formulations have been proposed, using piecewise constant basis functions for the series expansion of the current along the coil wire. In the present work, this scheme is improved by introducing global basis functions. Findings The use of global basis functions provides a stronger numerical stability and a better control over the convergence of the simulation; moreover, the associated computational cost is lower than for the previous schemes. These advantages are demonstrated in numerical examples, with special attention to the achievable efficiency of the power transfer. Practical implications The method can be efficiently used, e.g., in the optimal design of resonant WPT systems. Originality/value The presented computation scheme is original in the sense that global series expansion has not been previously applied to the numerical simulation of resonant WPT.

scholarly journals Simple and Efficient Computational Method to Analyze Cylindrical Plasmonic Nanoantennas

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Karlo Costa ◽  
Victor Dmitriev
Keyword(s):  
Integral Equation ◽  
Electric Field ◽  
Method Of Moments ◽  
Surface Impedance ◽  
Near Infrared ◽  
Basis Functions ◽  
Computational Method ◽  
Good Precision ◽  
Cylindrical Structures ◽  
Plasmonic Nanoantennas

We present in this work a simple and efficient technique to analyze cylindrical plasmonic nanoantennas. In this method, we take into account only longitudinal current inside cylindrical structures and use 1D integral equation for the electric field with a given surface impedance of metal. The solution of this integral equation is obtained by the Method of Moments with sinusoidal basis functions. Some examples of calculations of nanoantennas with different geometries and sources are presented and compared with the commercial software Comsol 3D simulations. The results show that the proposed technique provides a good precision in the near-infrared and lower optical frequencies 100–400 THz.

A robust integral equation solution for electromagnetic scattering by a thin plate in conductive media

Geophysics ◽  
1991 ◽  
Vol 56 (8) ◽  
pp. 1140-1152 ◽  
Author(s):  
P. W. Walker ◽  
G. F. West
Keyword(s):  
Magnetic Field ◽  
Integral Equation ◽  
Electric Field ◽  
Thin Plate ◽  
Current Density ◽  
Scalar Potential ◽  
The Other ◽  
Em Scattering ◽  
Equation Solution ◽  
Host Medium

An integral equation solution for electromagnetic (EM) scattering by a thin plate robustly models scattering in either perfectly resistive, very resistive, or conducting host media. Because the solution is not restricted to modeling certain ranges of host conductivity, it can be used to model scattering over the large ranges in conductivity encountered in geophysics. The solution is developed around a pair of coupled integral equations for the scattering distributions on the plate. In one equation, the scattering distribution is the scalar potential set up by the scattered charge distribution. In the other, it is the component of the scattered magnetic field perpendicular to the plate. The equations are solved numerically using the Galerkin method with simple polynomial basis functions. To find the fields scattered by the conductor, the scattered current density is first calculated from the scalar potential and the magnetic field. The scattered fields can then be found by integrating over the scattered current density. To test the solution, we model horizontal loop EM responses with our solution and compare the results with those from two established integral equation solutions. One of these solutions models pure induction and is used to test our solution when the host is perfectly resistive. Agreement with this solution is very good. Comparisons with the other solution, an electric field integral equation, tests our solution when the host medium is conductive. Agreement with the latter solution is good where induction is not too strong: i.e., where the electric‐field solution is known to work well. Our solution therefore can accurately model EM scattering by a plate in a host medium with any conductivity.

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