Surface Integral Equation Approach Research Articles

Deterministic–stochastic modeling of transcranial magnetic stimulation featuring the use of method of moments and stochastic collocation

This work examines the influence of the human brain tissue parameters’ uncertainty and the stimulation coil positioning variations within the framework of transcranial magnetic stimulation (TMS). A combination of deterministic modeling and the stochastic collocation method (SCM) is used for the assessment of the input parameter uncertainties’ effects on several outputs of interest: the induced electric field and the related electric current density in the homogeneous human brain model, as well as the maximum values of electric field, magnetic flux density and the induced electric current density. The deterministic model features the formulation based on the surface integral equation approach whose numerical solution is carried out using the method of moments. The SCM shows satisfactory convergence in the assessment of the stochastic mean, variance and standard deviation for the output parameters of interest. Confidence intervals are computed, and the impact of input permittivity, conductivity and coil positioning is assessed. It is found that due to a non-symmetric nature of the utilized brain model the highest electric field variance is shifted from the expected location directly under the coil geometric center.

A Discontinuous Galerkin Combined Field Integral Equation Formulation for Electromagnetic Modeling of Piecewise Homogeneous Objects of Arbitrary Shape

We present a novel discontinuous Galerkin (DG) surface integral equation approach, based on the electric and magnetic current combined field integral equations (JMCFIE), for the electromagnetic analysis of arbitrarily shaped piecewise homogeneous objects. In the proposed scheme, nonoverlapping boundary surfaces and interfaces between materials can be handled independently, without any continuity requirement through multimaterial junctions and tear lines between surfaces in contact. The use of nonconformal meshes provides improved flexibility for computer-aided-design (CAD) prototyping and tessellation. The proposed formulation can readily address nonconformal multimaterial junctions, where three or more material regions meet. The continuity of the electric surface current across the junction contours is enforced by the combination of the boundary conditions implicit in the JMCFIE formulation and the weakly imposed interior penalty (IP) between the contacting surfaces within each region. This completely avoids the cumbersome junction problem, which no longer requires any special treatment. Numerical experiments are included to validate the accuracy and demonstrate the great versatility of the proposed JMCFIE-DG formulation for the management and solution of complex composite objects with junctions.

A Discontinuous Galerkin Surface Integral Equation Method for Scattering From Multiscale Homogeneous Objects

A discontinuous Galerkin (DG) surface integral equation approach is proposed for scattering from homogeneous dielectric objects. The formulation of DG for homogeneous bodies is derived from the combined tangential field integral equation. The differences of DG for penetrable and nonpenetrable objects are presented by numerical experiments to demonstrate the numerical mechanism of DG. Numerical experiments demonstrate the great advantages of our presented formulation of DG for homogeneous objects in efficiency, flexibility, and scalability. A series numerical results are presented to show the capability of the presented DG solution for homogeneous bodies, especially for multiscale homogeneous bodies.

Characteristic Mode Analysis of Plasmonic Nanoantennas

The theory of characteristic modes (TCMs) based on the surface integral equation (SIE) formalism is presented for the analysis of plasmonic nanostructures. With TCM, excitation-independent characteristic solutions of the underlying electromagnetic field problem are obtained. Both single and multiple particle nanostructures are analyzed using the conventional SIE approach and the TCM formalism. By comparing these results, the modes that are responsible of the observed scattering and absorption characteristics can be identified. These solutions can be further utilized in the design and tuning nanoantenna configurations for specific application purposes.

Internally Combined Volume-Surface Integral Equation for a 3-D Electromagnetic Scattering Analysis of High-Contrast Media

An internally combined volume surface integral equation (ICVSIE) for analyzing three-dimensional (3-D) electromagnetic scattering from heterogeneous high-contrast (HC) dielectric objects is proposed. Iteratively solved (generalized) volume integral equations [(g)VIEs] often converge slowly when applied to the analysis of electromagnetic interactions involving dielectric objects with permittivities much higher than those of the surrounding medium [HC breakdown] or discretized using mesh elements much smaller than the wavelength [low-frequency (LF) breakdown]. The proposed ICVSIE hybridizes a VIE and a surface integral equation (SIE) in a manner that eliminates both breakdown phenomena. Just as in previous hybrid VIE–SIE approaches, the ICVSIE invokes Huygens surface equivalence principle to transform the original problem into coupled equivalent exterior and interior scenarios. In the exterior scenario, surface currents radiate in the original surrounding medium; in the interior problem, surface and volume polarization currents radiate in a medium with permittivity closer to that of the object. By judiciously combining integral equations pertinent to both scenarios, the ICVSIE system is assembled. Numerical results show that the proposed ICVSIE is both HC and LF stable when applied to the analysis of 3-D scattering problems.

Mode analysis of second-harmonic generation in plasmonic nanostructures

Using a surface integral equation approach based on the tangential Poggio–Miller–Chang–Harrington–Wu–Tsai formulation, we present a full wave analysis of the resonant modes of 3D plasmonic nanostructures. This method, combined with the evaluation of second-harmonic generation, is then used to obtain a better understanding of their nonlinear response. The second-harmonic generation associated with the fundamental dipolar modes of three distinct nanostructures (gold nanosphere, nanorod, and coupled nanoparticles) is computed in the same formalism and compared with the other computed modes, revealing the physical nature of the second-harmonic modes. The proposed approach provides a direct relationship between the fundamental and second-harmonic modes in complex plasmonic systems and paves the way for an optimal design of double resonant nanostructures with efficient nonlinear conversion. In particular, we show that the efficiency of second-harmonic generation can be dramatically increased when the modes with the appropriate symmetry are matched with the second-harmonic frequency.

Analysis of Transcranial Magnetic Stimulation Based on the Surface Integral Equation Formulation.

The aim of this paper is to provide a rigorous model and, hence, a more accurate description of the transcranial magnetic stimulation (TMS) induced fields and currents, respectively, by taking into account the inductive and capacitive effects, as well as the propagation effects, often being neglected when using quasi-static approximation. The formulation is based on the surface integral equation (SIE) approach. The model of a lossy homogeneous brain has been derived from the equivalence theorem and using the appropriate boundary conditions for the electric field. The numerical solution of the SIE has been carried out using the method of moments. Numerical results for the induced electric field, electric current density, and the magnetic flux density distribution inside the human brain, presented for three typical TMS coils, are in a good agreement with some previous analysis as well as to the results obtained by analytical approach. The future work should be related to the development of a more detailed geometrical model of the human brain that will take into account complex cortical columnar structures, as well as some additional brain tissues. To the best of authors knowledge, similar approach in modeling TMS has not been previously reported, albeit integral equation methods are seeing a revival in computational electromagnetics community.

Double-Higher-Order Large-Domain Volume/Surface Integral Equation Method for Analysis of Composite Wire-Plate-Dielectric Antennas and Scatterers

A novel double-higher-order large-domain Galerkin-type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for analysis of composite dielectric and metallic radiation/scattering structures combining the volume integral equation (VIE) approach for dielectric parts and the surface integral equation (SIE) approach for metallic parts of the structure. The technique employs Lagrange-type interpolation generalized hexahedra and quadrilaterals of arbitrary geometrical-mapping orders for the approximation of geometry and hierarchical divergence-conforming polynomial vector basis functions of arbitrary expansion orders for the approximation of currents within the elements. The double-higher-order VSIE (VIE-SIE) method is extensively validated and evaluated against the analytical solutions and the numerical results obtained by alternative higher order methods.

MLFMA-FFT Parallel Algorithm for the Solution of Extremely Large Problems in Electromagnetics

An efficient parallel implementation of the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) has been successfully used to solve an electromagnetic problem involving one billion of unknowns, which indeed becomes the largest problem solved with the surface integral-equation approach up to now. In this paper, we present a deep review of this challenging execution, focusing on the details of the parallel implementation step by step, with the aim of describing the different stages of the parallel algorithm and analyzing its overall parallel performance.

Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function

A rigorous surface integral equation approach is proposed to study the spontaneous emission of a quantum emitter embedded in a multilayered plasmonic structure with the presence of arbitrarily shaped metallic nanoscatterers. With the aid of the Fermi's golden rule, the spontaneous emission of the emitter can be calculated from the local density of states, which can be further expressed by the imaginary part of the dyadic Green's function of the whole electromagnetic system. To obtain this Green's function numerically, a surface integral equation is established taking into account the scattering from the metallic nanoscatterers. Particularly, the modeling of the planar multilayered structure is simplified by applying the layered medium Green's function to reduce the computational domain and hence the memory requirement. Regarding the evaluation of Sommerfeld integrals in the layered medium Green's function, the discrete complex image method is adopted to accelerate the evaluation process. This work offers an accurate and efficient simulation tool for analyzing complex multilayered plasmonic system, which is commonly encountered in the design of optical elements and devices.

Full-Wave Characterization of Rough Terrain Surface Scattering for Forward-Looking Radar Applications

In characterizing ground surface clutter as relevant to forward-looking radar applications, a finite-difference time-domain (FDTD)-based solver is proposed to study dielectric surface scattering at low depression angles. The solver's effectiveness and accuracy are carefully evaluated for one-dimensional surfaces by comparing Monte Carlo scattering results to those from a surface integral equation (SIE) approach for various surface parameters and incidence angles. It is demonstrated that satisfactory results can be attained at near-grazing angles for most surface parameters of interest with a relatively small simulation domain size, independent of the incidence angle. Subsequently, FDTD simulations of two-dimensional terrain surfaces are featured, along with a demonstration of the effects of ground clutter on target imaging generated by the time-reversal technique. By providing a practical full-wave simulation framework for the emulation of forward-looking radar operation and imaging, this study is intended to facilitate ongoing investigations into the detectability of discrete ground targets in the presence of distributed variable ground clutter in the near-grazing regime.

Novel postcompression technique in the matrix decomposition algorithm for the analysis of electromagnetic problems

In order to efficiently analyze an electromagnetic scattering problem with the surface integral equation approach, the matrix decomposition algorithm (MDA) is used to accelerate the matrix‐vector multiplication when the corresponding matrix equation is solved by a Krylov‐subspace iterative method. Although the MDA is more efficient than the direct solution, this paper presents a novel recompression technique to further reduce computation time and storage memory. The technique applies singular value decomposition (SVD) to the matrices of MDA. Using the novel recompression technique, a sparser representation of the impedance matrix is obtained, and a more efficient matrix‐vector multiplication is implemented. The modified MDA is comparable with the multilevel matrix decomposition algorithm (MLMDA) and the matrix decomposition algorithm‐singular value decomposition (MDA‐SVD) in terms of computation time and memory requirement. Remarkably, the new formulation can reduce the computational time and memory significantly, with excellent accuracy.

Efficient Analyzing EM Scattering of Objects Above a Lossy Half-Space by the Combined MLQR/MLSSM

A large dense complex linear system is obtained when solving an electromagnetic scattering of arbitrary shaped object located above a lossy half-space with the surface integral equation approach. To analyze the large dense complex linear system efficiently, the multilevel QR (MLQR) is used to accelerate the matrix-vector multiplication when the corresponding matrix equation is solved by a Krylov-subspace iterative method. Although the MLQR is more efficient than the direct solution, this paper presents a novel recompression technique to further reduce computation time and storage memory. The technique applies the multilevel simply sparse method (MLSSM) to the matrices of MLQR. Using the MLSSM, a sparser representation of the impedance matrix is obtained, and a more efficient matrix-vector multiplication is implemented. The combined MLQR/MLSSM is comparable to the MLQR and the adaptive cross-approximation/singular value decomposition (ACA-SVD) in terms of computation time and memory requirement. Remarkably, the new formulation can reduce the computational time and memory by about one order of magnitude, with excellent accuracy.

Formulation and applications of an integral-equation approach for solving scattering problems involving an object consisting of a set of piecewise homogeneous material regions

The paper presents the formulation and selected applications of the surface integral equation approach for finding pressure, displacement, and traction fields in a complex object consisting of a set of piece-wise homogeneous regions characterized by different Lame' material parameters. Representative applications of the approach are presented, which involve finding pressure field distribution inside human inner ear treated as an inclusion embedded in the surrounding inhomogeneous material. (the embedding region is treated with volumetric integral equations with suitable coupling to the inclusion). The method uses a set of coupled elastodynamics integral equations for two unknown fields: displacement and traction at each interface, which allow us to find displacement and traction field distribution inside a general complex object composed of sub-regions with different material parameters. The matrix elements of strongly singular kernels appearing in the integral equations were converted, through suitable i...

Time-Domain Techniques for Transient Scattering from Dielectric Bodies and Sea Surface Governed by Jonswap's Sea Spectra

A time-domain surface integral equation (TDCFIE) approach is utilized to calculate the transient scattering from arbitrarily shaped, three-dimensional dielectric bodies. In conjunction with the marching-on-in-time (MOT) method, the TDCFIE-MOT method is used to derive explicit expressions for the present-time current as a function of the incident field. Sample results showing various geometries are presented and are compared with other numerical techniques. Finally, by an incident Gaussian plane tapered wave, transient scattering from the sea surface governed by Jonswap's sea spectra is computed.

Searching for Electrostatic Resonances in Metamaterials Using Surface Integral Equation Approach

In this paper, the possibilities of flnding out surface plasmons in negative-permittivity metamaterial scatterers, using surface integral equations, are discussed. Electrostatic resonances manifest themselves as very high values of the polarizability for a certain (negative) value of the permittivity. Another way to locate the position of the resonances is to calculate the eigenvalues of the source-free integral equation operator for the static potential on the surface of the particle. DOI: 10.2529/PIERS060906073935

A Coupled PEC-TDS Surface Integral Equation Approach for Electromagnetic Scattering and Radiation From Composite Metallic and Thin Dielectric Objects

The recently proposed surface electric field integral equation for thin dielectric sheets is combined with that for conductors to handle objects consisting of both dielectric and conducting materials. This leads to coupled conducting-dielectric surface integral equations which can easily be solved by the method of moments. This approach can handle both scattering and radiation problems such as coated objects, radome enclosed antennas and scatterers, and planar structures. It is observed that the normal electric field in a thin dielectric layer is indispensable when this dielectric layer resides in the vicinity of a conducting object. To validate this approach, couple numerical examples are examined. The results by this approach show agreement with other methods while it economizes on computation resource. It can be used to obtain rapid solutions in the initial phase of the design process. More importantly, it indicates the feasibility for microwave planar circuit applications

Discretization of Hybrid VSIE Using Mixed Mesh Elements With Zeroth-Order Galerkin Basis Functions

The hybrid volume and surface integral equation approach is applied to solve electromagnetic scattering and radiation problems involving conducting and/or dielectric objects. To flexibly and accurately model the complex structures and reduce the number of unknowns, mixed mesh scheme is developed to discretize the object. In this scheme, the triangles and quadrangles are used to discretize the conducting part of the object, and the tetrahedrons, hexahedrons, prisms and pyramids are used to model the dielectric volumes of the scatterer. Numerical results showed the solution accuracies from the mixed element meshes are of the same level compared with the single element meshes, but uses much less number of unknowns. This leads to flexibility for mesh generating and reduces the use of computing resources.

Method of Moments Solution for a Printed Patch/Slot Antenna on a Thin Finite Dielectric Substrate Using the Volume Integral Equation

In this paper, a volume integral equation (VIE)-based modeling method suitable for a patch or slot antenna on a thin finite dielectric substrate is developed and tested. Two new key features of the method are the use of proper dielectric basis functions and proper VIE conditioning, close to the metal surface, where the surface boundary condition of the zero tangential E -component must be extended into adjacent tetrahedra. The extended boundary condition is the exact result for the piecewise-constant dielectric basis functions. The latter operation allows one to achieve a good accuracy with one layer of tetrahedra for a thin dielectric substrate and thereby greatly reduces computational cost. The use of low-order basis functions also implies the use of low-order integration schemes and faster filling of the impedance matrix. For some common patch/slot antennas, the VIE-based modeling approach is found to give an error of about 1% or less in the resonant frequency for one-layer tetrahedral meshes with a relatively small number of unknowns. This error is obtained by comparison with fine finite-element method (FEM) simulations, or with measurements, or with the analytical mode matching approach. Hence it is competitive with both the method of moments surface integral equation approach and with the FEM approach for the printed antennas on thin dielectric substrates.

Efficient analysis of electromagnetic scattering and radiation from patches on finite, arbitrarily curved, grounded substrates

A precorrected‐fast Fourier transform (FFT) accelerated surface integral equation approach formulated using the homogeneous medium Green's function is presented for the analysis of patch arrays on finite, arbitrarily shaped, grounded substrate. The integral equation is solved by the method of moments, and the precorrected‐FFT method is applied to reduce the memory requirement and computational complexity of the solution procedure. The memory required for this algorithm is O(N1.5), and the computational complexity is NiterN1.5log N, where N is the number of unknowns and Niter is the iteration number. Numerical results are presented to demonstrate the accuracy and capability of the method.