Surface Integral Equation Method Research Articles

A Domain Decomposition Method for Surface Integral Equation Method in Solving Multitarget Problems

A Domain Decomposition Method for Surface Integral Equation Method in Solving Multitarget Problems

An Efficient and Effective Preconditioner for the Discontinuous Galerkin Domain Decomposition Method of Surface Integral Equation

This paper proposes a new preconditioning technique based on a restricted additive Schwarz (RAS) approach to improve the efficiency of the discontinuous Galerkin surface integral equation method. The RAS preconditioning is implemented efficiently using an oct-tree structure derived from the multilevel fast multipole algorithm (MLFMA) and is constructed using near-field matrices associated with boxes at the finest level. Compared to existing domain decomposition preconditioning methods based on subdomain block matrices, the proposed RAS preconditioning significantly reduces computation time and memory requirements, while providing scalable convergence for iterative solutions. Numerical experiments are presented to demonstrate the performance of the proposed cost-effective preconditioning technique.

Accurate and Robust Surface MLFMA Solution for Scattering From a Large Plasma Object With Large Negative Permittivity

We present an accurate and robust surface integral equation (SIE) method for electromagnetic analysis of large-scale plasma media with large negative permittivity. To accelerate the matrix-vector multiplication (MVM) in the iterative solution of the matrix equation, the multilevel fast multipole algorithm (MLFMA) is generally preferred. Traditional implementation of MLFMA will lead to inaccurate results or poor convergence due to the numerical instability arising from the large negative permittivity. To tackle this problem, we analyze the formula of MLFMA in detail and propose a simple and efficient truncation strategy to improve its robustness. A practical and elegant truncation bound is derived based on the ratios of matrix elements between an ordinary permittivity background medium and the negative permittivity plasma. Applying the proposed truncation strategy, the performance of several commonly used SIE formulations is investigated comprehensively. Numerical results demonstrate that the proposed method can avoid numerical instability while maintaining high efficiency for large-scale plasma media with large negative permittivity.

A High-Order-Accurate 3D Surface Integral Equation Solver for Uniaxial Anisotropic Media

This paper introduces a high-order accurate surface integral equation method for solving 3D electromagnetic scattering for dielectric objects with uniaxially anisotropic permittivity tensors. The N-Müller formulation is leveraged resulting in a second-kind integral formulation, and a finite-difference-based approach is used to deal with the strongly singular terms resulting from the dyadic Green’s functions for uniaxially anisotropic media while maintaining the high-order accuracy of the discretization strategy. The integral operators are discretized via a Nyström-collocation approach, which represents the unknown surface densities in terms of Chebyshev polynomials on curvilinear quadrilateral surface patches. The convergence is investigated for various geometries, including a sphere, cube, a complicated NURBS geometry imported from a 3D CAD modeler software, and a nanophotonic silicon waveguide, and results are compared against a commercial finite element solver. To the best of our knowledge, this is the first demonstration of high-order accuracy for objects with uniaxially anisotropic materials using surface integral equations.

An Efficient and Error-Controllable Angular Sweep Algorithm for Electromagnetic Wave Scattering and its Analysis

The angular sweep of electromagnetic wave scattering is formulated as a matrix equation with multiple right-hand sides (RHSs). Although the low-rank approximation of an RHS matrix is a popular choice for reducing the computational costs of multiple RHSs, only a small amount of research has been conducted to explore how this approximation impacts the solution quality. Furthermore, there has not been sufficient research on the quality of the solution as a function of the accuracy of the iterative solver. We present an error analysis of the approximated solution considering both the reduced number of RHSs and the tolerance of the iterative solver. Based on the error analysis, a new angular sweep algorithm is proposed with fine-tuned tolerances of the iterative solver for individual singular vectors. The different tolerances for each singular vector increase the efficiency of the proposed algorithm. Another benefit of the proposed algorithm is that the error can be bounded by a user-defined global tolerance. In addition, a variant of the generalized conjugate residual method for multiple RHSs is introduced to accelerate iterative solvers. Finally, numerical validation is conducted with three examples in which the discontinuous Galerkin surface integral equation method is applied. The experiments support two conclusions: tight upper and lower bounds of the solution error exist, and fine-tuning the tolerances reduces the computational costs.

A Hybrid Volume-Surface Integral Equation Method for Rapid Electromagnetic Simulations in MRI.

We developed a hybrid volume surface integral equation (VSIE) method based on domain decomposition to perform fast and accurate magnetic resonance imaging (MRI) simulations that include both remote and local conductive elements. We separated the conductive surfaces present in MRI setups into two domains and optimized electromagnetic (EM) modeling for each case. Specifically, interactions between the body and EM waves originating from local radiofrequency (RF) coils were modeled with the precorrected fast Fourier transform, whereas the interactions with remote conductive surfaces (RF shield, scanner bore) were modeled with a novel cross tensor train-based algorithm. We compared the hybrid-VSIE with other VSIE methods for realistic MRI simulation setups. The hybrid-VSIE was the only practical method for simulation using 1 mm voxel isotropic resolution (VIR). For 2 mm VIR, our method could be solved at least 23 times faster and required 760 times lower memory than traditional VSIE methods. The hybrid-VSIE demonstrated a marked improvement in terms of convergence times of the numerical EM simulation compared to traditional approaches in multiple realistic MRI scenarios. The efficiency of the novel hybrid-VSIE method could enable rapid simulations of complex and comprehensive MRI setups.

Simulation of realistic speckle fields by using surface integral equation and multi-level fast multipole method

This report presents a speckle simulator which calculates rigorously speckle fields with full randomization from rough surfaces. To achieve this, surface integral equation (SIE) method accelerated via multilevel fast multipole method (MLFMM) is implemented. Among several linear formulations to combine the electric and magnetic field integral equations, we demonstrate that one is most suitable for dielectric rough surfaces and one for metallic rough surfaces. As examples, silver and silicon rough surfaces are simulated at a wavelength of 500 nm. To investigate speckles from extended areas, silicon rough surfaces with a size of 30×30 µm2 and two million unknowns are calculated. Based on the resultant speckle fields, speckle contrast, angular speckle-correlation and transverse autocorrelation function of intensity in the van Cittert-Zernike zone are further calculated and known relations versus surface roughness are manifested, indicating that sufficient statistic surface information is encoded in the speckle fields. The SIE-MLFMM method with the suitable formulation proves to be an efficient rigorous simulation tool for relatively large rough surface problems and would enable more profound numerical studies for speckle involved optical metrologies.

Surface Integral Equation With Multibranch RWG Basis Functions for Electromagnetic Scattering From Dielectric Objects

A surface integral equation method with multibranch Rao–Wilton–Glisson (MB-RWG) basis functions for scattering from dielectric objects is proposed based on the electric and magnetic current combined field integral equations (JMCFIE). The MB-RWGs are defined at the boundaries between dense and coarse meshes and can keep the current continuity. Thus, the JMCFIE with MB-RWGs allows different surfaces of the dielectric models to be independently discretized by triangles with different sizes. Compared with the conventional JMCFIE, the JMCFIE with MB-RWGs increases the flexibility of mesh generation, which is attractive for multiscale dielectric models. Furthermore, the iterative convergence of JMCFIE with MB-RWGs is similar to that of conventional JMCFIE. Numerical examples are presented to demonstrate the advantages of the proposed method.

Windowed Green Function MoM for Second-Kind Surface Integral Equation Formulations of Layered Media Electromagnetic Scattering Problems

This paper presents a second-kind surface integral equation method for the numerical solution of frequency-domain electromagnetic scattering problems by locally perturbed layered media in three spatial dimensions. Unlike standard approaches, the proposed methodology does not involve the use of layer Green functions. It instead leverages an indirect M\"uller formulation in terms of free-space Green functions that entails integration over the entire unbounded penetrable boundary. The integral equation domain is effectively reduced to a small-area surface by means of the windowed Green function method, which exhibits high-order convergence as the size of the truncated surface increases. The resulting (second-kind) windowed integral equation is then numerically solved by means of the standard Galerkin method of moments (MoM) using RWG basis functions. The methodology is validated by comparison with Mie-series and Sommerfeld-integral exact solutions as well as against a layer Green function-based MoM. Challenging examples including realistic structures relevant to the design of plasmonic solar cells and all-dielectric metasurfaces, demonstrate the applicability, efficiency, and accuracy of the proposed methodology.

Effects of the coupling of dielectric spherical particles on signatures in infrared microspectroscopy

Infrared microspectroscopy is a powerful tool in the analysis of biological samples. However, strong electromagnetic scattering may occur since the wavelength of the incident radiation and the samples may be of comparable size. Based on the Mie theory of single spheres, correction algorithms have been developed to retrieve pure absorbance spectra. Studies of the scattering characteristics of samples of different types, obtained by microspectroscopy, have been performed. However, the detailed, microscopic effects of the coupling of the samples on signatures in spectra, obtained by infrared microspectroscopy, are still not clear. The aim of this paper is to investigate how the coupling of spherical samples influences the spectra. Applying the surface integral equation (SIE) method, we simulate small dielectric spheres, arranged as double-spheres or small arrays of spheres. We find that the coupling of the spheres hardly influences the broad oscillations observed in infrared spectra (the Mie wiggles) unless the radii of the spheres are different or the angle between the direction of the incident radiation and the normal of the plane where the spheres are located is large. Sharp resonance features in the spectra (the Mie ripples) are affected by the coupling of the spheres and this effect depends on the polarization of the incident wave. Experiments are performed to verify our conclusions.

Electromagnetic response and optical properties of anisotropic CuSbS2 nanoparticles

We investigate the electromagnetic response of anisotropic (non-spherical) copper antimony disulfide ( C u S b S 2 ) nanoparticles and layers embedded with them using computational methods. To this end, we calculate the scattering and absorption efficiencies of oblate spheroidal C u S b S 2 nanoparticles using the surface integral equation method. We find strong dependence of the response depending on the anisotropy of the spheroids and their orientation with respect to the electric field polarization of incoming radiation. Thin spheroids display a sharp plasmonic resonance in the ultraviolet, which is observed only for the electric field polarization along the short axis. Fano resonances that appear in the near infrared (NIR) blueshift when the short axis length is reduced, and they can be either strongly suppressed or enhanced depending on the relative orientation of the spheroid. We further investigate the optical response of thin layers containing C u S b S 2 spheroids at a low volume fraction using a Monte Carlo method. We find that the response of these layers can be considerably modified by changing the short axis length and the orientation of particles within the layer with respect to polarization. Our results demonstrate the potential of anisotropic dielectric particles for polarization-dependent-response applications such as solar devices and NIR sensors.

Simulation of Realistic Speckle Fields by Using Surface Integral Equation and Fast Multipole Method

Simulation of Realistic Speckle Fields by Using Surface Integral Equation and Fast Multipole Method

AIMx: An Extended Adaptive Integral Method for the Fast Electromagnetic Modeling of Complex Structures

Surface integral equation (SIE) methods are of great interest for the efficient electromagnetic modeling of various devices, from integrated circuits to antenna arrays. Existing acceleration algorithms for SIEs, such as the adaptive integral method (AIM), enable the fast approximation of interactions between well-separated mesh elements. Nearby interactions involve the singularity of the kernel, and must instead be computed accurately with direct integration at each frequency of interest, which can be computationally expensive. We propose a novel algorithm for reducing the cost-per-frequency of near-region computations for both homogeneous and layered background media. In the proposed extended AIM (AIMx), the SIE operators are decomposed into a frequency-independent term containing the singularity of the kernel, and a nonsingular frequency-dependent term. Direct integration is only required for the frequency-independent term, and can be reused at each frequency, leading to significantly faster frequency sweeps. The frequency-dependent term is captured with good accuracy via fast Fourier transform (FFT)-based acceleration even in the near region, as confirmed with an error analysis. The accuracy and efficiency of the proposed method are demonstrated through numerical examples drawn from several applications, and CPU times are significantly reduced by factors ranging from three to 16.

Efficient Electromagnetic Modeling of Multidomain Planar Layered Medium by Surface Integral Equation

A novel electromagnetic modeling method for multidomain planar layered medium (MD-PLM) by surface integral equation (SIE) is proposed. The MD-PLM refers to the whole space that is separated into multiple PLM volumes, e.g., the infinite-sized PLM background with inserted finite PLM objects or multilayered parallel-plate structures. In the conventional SIE method for this scenario, only the exterior domain is considered as PLM that is modeled by layered medium Green's functions (LMGFs), while the interior domain is treated by the multidomain homogeneous SIE. However, in this work, the surface equivalence formulation for this special scenario is reestablished by the LMGFs. Then, both the exterior and interior domains are considered as PLM that is modeled by the corresponding LMGFs and SIE in each subdomain. As a result, only the boundary of each sub-PLM domain is required to be discretized rather than the interfaces of each interior medium, which is different from the way in conventional SIE. Consequently, compared with the conventional SIEs, the proposed method can reduce a large quantity of unknowns and significantly increase the efficiency by reducing the memory consumption. By some numerical examples, the performance of this method is evaluated and compared with the conventional methods. The validity and efficiency of the newly proposed method are also well demonstrated in the examples.

Massively Parallel Discontinuous Galerkin Surface Integral Equation Method for Solving Large-Scale Electromagnetic Scattering Problems

In this communication, we present flexible and efficient solutions of large-scale electromagnetic scattering problems using the parallel discontinuous Galerkin (DG) surface integral equation (SIE) method. A simplified formulation of the conventional DG is used by removing the interior penalty term, thereby avoiding computational burden of the contour integral in the conventional DG. It is demonstrated numerically that the simplification causes convergence deficiency and changes the accuracy convergence rate of the solution for objects with sharp corners. The former is overcome by constructing preconditioners using the near-field matrix of the whole region, and the latter is proved to have a neglectable influence on the accuracy of the final solutions as long as normal mesh density requirement is satisfied. The well-scaling ternary parallelization approach of the multilevel fast multipole algorithm (MLFMA) is incorporated into the DG domain decomposition framework to reduce the complexity of the solution and strength its capability for large-scale problems, with the distribution of the near-field-related calculation adjusted to adapt the nonuniform meshes for a better workload balance. Numerical results are included to validate the accuracy and demonstrate the scalability and versatility of the proposed method. In addition, we demonstrated the effectiveness of our algorithm by solving a high-definition complicated aircraft model with inlets and exhausts involving over one billion unknowns.

An Accelerated Surface Integral Equation Method for the Electromagnetic Modeling of Dielectric and Lossy Objects of Arbitrary Conductivity

Surface integral equation (SIE) methods are of great interest for the numerical solution of Maxwell's equations in the presence of homogeneous objects. However, existing SIE algorithms have limitations, either in terms of scalability, frequency range, or material properties. We present a scalable SIE algorithm based on the generalized impedance boundary condition, which can efficiently handle, in a unified manner, both dielectrics and conductors over a wide range of conductivity, size, and frequency. We devise an efficient strategy for the iterative solution of the resulting equations, with efficient preconditioners and an object-specific use of the adaptive integral method (AIM). With a rigorous error analysis, we demonstrate that the AIM can be applied over a wide range of frequencies and conductivities. Several numerical examples, drawn from different applications, demonstrate the accuracy and efficiency of the proposed algorithm.

Electromagnetic Scattering Analysis of SHDB Objects Using Surface Integral Equation Method

A surface integral equation (SIE) method is applied in order to analyze electromagnetic scattering by bounded arbitrarily shaped three-dimensional objects with the SHDB boundary condition. SHDB is a generalization of SH (Soft-and-Hard) and DB boundary conditions (at the DB boundary, the normal components of the D and B flux densities vanish). The SHDB boundary condition is a general linear boundary condition that contains two scalar equations that involve both the tangential and normal components of the electromagnetic fields. The multiplication of these scalar equations with two orthogonal vectors transforms them into a vector form that can be combined with the tangential field integral equations. The resulting equations are discretized and converted to a matrix equation with standard method of moments (MoM). As an example of use of the method, we investigate scattering by an SHDB circular disk and demonstrate that the SHDB boundary allows for an efficient way to control the polarization of the wave that is reflected from the surface. We also discuss perspectives into different levels of materialization and realization of SHDB boundaries.

A Fast Macromodeling Approach to Efficiently Simulate Inhomogeneous Electromagnetic Surfaces

The full-wave simulation of complex electromagnetic surfaces such as reflectarrays and metasurfaces is a challenging problem. In this paper, we present a macromodeling approach to efficiently simulate complex electromagnetic surfaces composed of PEC traces, possibly with fine features, on a finite-sized multilayer dielectric substrate. In our approach, we enclose each element of the structure with a fictitious surface. By applying the equivalence principle on each surface, we derive a macromodel for each element of the array. This macromodel consists of a linear operator that relates the equivalent electric and magnetic current densities introduced on the fictitious surface. Mutual coupling between the elements of the structure is captured by the equivalent current densities in a fully accurate way. The crux of the proposed technique is to solve for equivalent current densities on the fictitious surface instead of directly solving for the actual current densities on the original scatterer. When simulating complex surfaces, this approach leads to fewer unknowns and better conditioning. We also propose a rigorous acceleration algorithm based on the fast Fourier transform to simulate electrically large surfaces. Numerical results demonstrate that the proposed approach is significantly faster and requires less memory than commercial solvers based on the surface integral equation method, while giving accurate results.

Surface Integral Equation Method for Soft-and-Hard/DB Boundary Condition

A surface integral equation (SIE) method is developed to analyze electromagnetic scattering by 3-D objects with soft-and-hard/DB (SHDB) boundary conditions. The SHDB boundary condition is a generalization of the soft-and-hard (SH) and DB boundary conditions that associate the normal and tangential field components on the boundary. In the developed method, the SHDB boundary condition is expressed in vector form that allows combining it with the tangential field integral equations. The obtained equations can be discretized with the standard method of moments (MoM) using the Rao–Wilton–Glisson (RWG) functions. Different combinations of the integral equations and boundary conditions are derived, and their numerical performance is studied and compared. It is demonstrated with numerical experiments that a much more stable system is obtained by considering the boundary conditions as extra equations, rather than integrating them into the SIEs. The solutions of the proposed nonsquare integral equation are verified with the physical optics approximations.

Hybrid Surface Electric Field Volume Magnetic Field Integral Equations for Electromagnetic Analysis of Heterogeneous Dielectric Bodies With Embedded Electrically Conducting Structures

In this article, hybrid surface electric field volume magnetic field integral equations (SEVMIE) for electromagnetic analysis of heterogeneous dielectric bodies with embedded perfect electrical conductor (PEC) structures are presented. These are then solved using a quasi-exact vector moment method employing mixed order edge-based divergence conforming basis functions to expand the current on the PEC surface and node-based linear Lagrangian basis functions to expand the magnetic field within the dielectric region. For an incident plane wave scattered by the combined structure, this procedure results in a linear matrix equation for the surface currents and the volume magnetic fields. The method is demonstrated for three spherical test cases and the results compared to those obtained by a semi-analytic Mie series expansion. The performance is compared to that of the well-known volume surface integral equation (VSIE) method and found to provide more accurate results with approximately one-third the number of degrees of freedom. The method is then demonstrated on the scattering cross section calculation of a heterogeneous composite structure. This new method enables the efficient and accurate analysis of composite structures with large dielectric regions.