Surface Integral Equation Research Articles

Tucker-FMM-FFT -Accelerated SIE Solver for Large-Scale Electromagnetic Analysis

A tensor decomposition methodology is combined with the fast multipole method-fast Fourier transform (FMM-FFT) technique to accelerate the surface integral equation (SIE) solvers. The proposed methodology leverages Tucker and hierarchical Tucker (H-Tucker) decompositions to compress the three-dimensional (3D) arrays storing the far-fields and five-dimensional (5D) arrays storing the translation operator samples, respectively. The compressed tensors are then used in the matrix-vector and element-wise products in aggregation/disaggregation and translation stages. By doing so, all stages of the FMM-FFT are performed via the Tucker-compressed tensors. The resulting Tucker-FMM-FFT-accelerated SIE simulator is far more memory and CPU efficient than the traditional FMM-FFT-accelerated SIE simulators. Preliminary results show that the Tucker-Fmm - Fftacceleration technique requires 12x less memory and 17x less CPU time compared to the traditional FMM-FFT acceleration technique for the electromagnetic (EM) scattering analysis of a frequency-selective surface.

Nested Pseudoskeleton Approximation Algorithm for Generating H 2 -Representation of Electrically Large Surface Integral Operators

In this paper, we develop a fast nested pseudoskeleton approximation algorithm to generate a low-rank $\mathcal{H}^{2}$ -matrix to represent electrically large surface integral equations (SIEs). The algorithm only uses $O (N\log N)$ entries of the original dense SIE matrix of size $N$ to generate the $\mathcal{H}^{2}$ -representation. It also provides a closed-form expression of the cluster bases and coupling matrices using original matrix entries. The resultant $\mathcal{H}^{2}$ -matrix is then directly solved for electrically large scattering analysis. Numerical experiments have demonstrated the accuracy and efficiency of the proposed algorithm.

A Multitrace Surface Integral Equation Method for PEC/Dielectric Composite Objects

The multitrace domain decomposition surface integral equation (MT-DD-SIE) originally developed to analyze electromagnetic scattering from dielectric composite objects is extended to efficiently account for perfect electrically conducting (PEC) bodies. This is achieved by adopting Robin transmission conditions (RTCs) to PEC surfaces. These PEC-RTCs, which are the only governing equations for a PEC body, are used to “couple” it to the dielectric bodies. Upon discretization, PEC-RTCs produce a well-conditioned matrix block and, therefore, does not negatively affect the convergence of the iterative solution of the MT-DD-SIE matrix equation. The resulting method is significantly faster and has a smaller memory footprint than the traditional globally coupled contact-region-modeling method that uses the coupled electric field and Poggio-Miller-Chang-Harrington-Wu-Tsai SIEs in analyzing electromagnetic scattering from composite PEC/dielectric objects. This is demonstrated by numerical examples involving electrically large scatterers.

An Accurate Integral Equation Formulation to Scattering of Periodic Grating Structures

Broadband wave transmission grating structures have potentials in many engineering applications. Theoretical predictions of the transmission spectra and near field characteristics of such structures are crucial for the understanding and design of wave functional gratings. However, accurate theoretical modeling of the wave interactions with such periodic grating structures turns out to be difficult, especially at near grazing incidence angles. In this paper, an efficient and accurate method to solve the scattering of periodic grating structures is proposed based upon integral equation formulations utilizing periodic lattice Green's functions. The method of moments is used to discretize the integral equation with the pulse basis function and the point matching scheme. The imaginary wavenumber extraction technique and the integral transformation approach are combined to efficiently and accurately evaluate the period Green's function. Meanwhile, an over-determined testing scheme on the integral equation is devised to overcome the intrinsic internal resonance of surface integral equations. Such strategy has significantly improved the numerical accuracy of the proposed approach. Calculation results are compared with Comsol simulations for various scenarios, and the proposed approach is found to be predicting physically sound results when Comsol fails at certain frequencies and incident angles. The proposed method is applicable to grating periodic structures of various shapes and it provides a useful reference and tool for researchers and designers.

Light scattering of Laguerre-Gaussian vortex beams by arbitrarily shaped chiral particles.

Laguerre-Gaussian (LG) beams with vortex phase possess a handedness, which would produce chiroptical interactions with chiral matter and may be used to probe structural chirality of matter. In this paper, we numerically investigate the light scattering of LG vortex beams by chiral particles. Using the vector potential method, the electric and magnetic field components of the incident LG vortex beams are derived. The method of moments (MoM) based on surface integral equations (SIEs) is applied to solve the scattering problems involving arbitrarily shaped chiral particles. The numerical results for the differential scattering cross sections (DSCSs) of several selected chiral particles illuminated by LG vortex beams are presented and analyzed. In particular, we show how the DSCSs depend on the chiral parameter of the particles and on the parameters describing the incident LG vortex beams, including the topological charge, the state of circular polarization, and the beam waist. This research may provide useful insights into the interaction of vortex beams with chiral particles and its further applications.

Electromagnetic Scattering and Emission From Large Rough Surfaces With Multiple Elevations Using the MLSD-SMCG Method

Electromagnetic scattering and emission from 1-D rough surfaces with multiple elevations are studied using full-wave simulations. Both the root-mean-square (rms) heights and the surface length are large compared to the wavelength. A novel multilevel steepest decent-sparse matrix canonical grid (MLSD-SMCG) method is proposed to address limitations in the original SMCG. The uniform Nystrom method and neighborhood impedance boundary condition (NIBC) are also incorporated in solving the dual surface integral equations (SIEs) of the method of moments (MoM). Simulation results are illustrated at L-band for soil and ocean surfaces. The surface rms heights and lengths are up to 1.43 and 243.8 m corresponding to 6 and 1024 wavelengths at 1.26 GHz, respectively. For ocean surfaces, the wind speeds up to 20 m/s are considered, and the entire spectrum is included to capture all relevant surface length scales. Numerical results indicate the proposed approach is computationally efficient and accurate. Energy conservation checks in simulations are at $10^{-4}$ for ocean scattering and emission. Also, the effects of wind-driven roughness on ocean emissivity are further investigated using the proposed approach in terms of wind speed and observation angle for both polarizations.

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.

Compression of Far-Fields in the Fast Multipole Method via Tucker Decomposition

Tucker decomposition is proposed to reduce the memory requirement of the far-fields in the fast multipole method (FMM)-accelerated surface integral equation simulators. It is particularly used to compress the far-fields of FMM groups, which are stored in three-dimensional (3-D) arrays (or tensors). The compressed tensors are then used to perform fast tensor-vector multiplications during the aggregation and disaggregation stages of the FMM. For many practical scenarios, the proposed Tucker decomposition yields a significant reduction in the far-fields' memory requirement while dramatically accelerating the aggregation and disaggregation stages. For the electromagnetic scattering analysis of a 30{\lambda}-diameter sphere, it reduces the memory requirement of the far-fields more than 87% while it expedites the aggregation and disaggregation stages by a factor of 15.8 and 15.2, respectively, where {\lambda} is the wavelength in free space.

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.

Compressing H2 Matrices for Translationally Invariant Kernels

H2 matrices provide compressed representations of the matrices obtained when discretizing surface and volume integral equations. The memory costs associated with storing H2 matrices for static and low-frequency applications are O(N). However, when the H2 representation is constructed using sparse samples of the underlying matrix, the translation matrices in the H2 representation do not preserve any translational invariance present in the underlying kernel. In some cases, this can result in an H2 representation with relatively large memory requirements. This paper outlines a method to compress an existing H2 matrix by constructing a translationally invariant H2 matrix from it. Numerical examples demonstrate that the resulting representation can provide significant memory savings.

Efficient Jacobian Matrix Determination for H2 Representations of Nonlinear Electrostatic Surface Integral Equations

In this paper, a nonlinear electrostatic surface integral equation is presented that is suitable for predicting corrosion-related fields. Nonlinear behavior arises due to electrochemical reactions at polarized surfaces. Hierarchical H2 matrices are used to compress the discretized integral equation for the fast solution of large problems. A technique based on randomized linear algebra is discussed for the efficient computation of the Jacobian matrix required at each iteration of a nonlinear solution.

Variant Characteristic Mode Equations Using Different Power Operators for Material Bodies

This paper is concerned with the electromagnetic theory of characteristic modes for general homogeneous material bodies using surface integral equations (SIEs), which are computationally more efficient than using volume integral equations (VIEs). However, the generalized eigenvalue equations (GEEs) based on SIEs are prone to generate spurious or extra modes, unlike their counterparts based on VIEs. In this work, several variant SIE-based GEEs are proposed to eliminate the spurious/extra modes and improve the numerical performances. These variant GEEs share a common point that the reactive power operator related to the material domain should be removed from the left-hand sides of GEEs by explicitly enforcing the boundary condition, because it seems sensitive to incur unwanted modes. These variant GEEs are diverse in choosing the active power operator related to the material domain on the right-hand sides to suit different medium parameters, which even can be dropped directly for lossless cases. Numerical results show that these variant GEEs do not generate spurious/extra resonant modes in observed frequency range, and at least one of the variants would have acceptable numerical accuracy.

Scattering by thin shells in fluids: Fast solver and experimental validation.

A numerical model facilitating fast analysis of acoustic scattering by thin shells immersed in fluids is presented. The shell is simulated by an effective boundary condition, in which the inertial properties of the shell are taken into account while the elastic ones are neglected. The problem reduces to a hypersingular surface integral equation, which is solved using the boundary element method accelerated by the multilevel nonuniform grid approach. The validity of the numerical method and the adopted physical approximations were tested by comparison with a water tank experiment on acoustic scattering from cylindrical metallic shells filled with water.

Butterfly Factorization Via Randomized Matrix-Vector Multiplications

This paper presents an adaptive randomized algorithm for computing the butterfly factorization of a $m\times n$ matrix with $m\approx n$ provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The resulting factorization is composed of $O(\log n)$ sparse factors, each containing $O(n)$ nonzero entries. The factorization can be attained using $O(n^{3/2}\log n)$ computation and $O(n\log n)$ memory resources. The proposed algorithm applies to matrices with strong and weak admissibility conditions arising from surface integral equation solvers with a rigorous error bound, and is implemented in parallel.

Formulation of Single-Source Surface Integral Equation for Electromagnetic Analysis of 2D Partially Connected Penetrable Objects

This paper presents a new single-source surface integral equation (SS-SIE) to model the partially contacted penetrable objects. In the proposed formulation, the surface electric and magnetic fields on all interior boundaries are first eliminated through combining integral solutions inside each object. Then, by enforcing the surface electric fields in the original and equivalent configurations are equal to each other, an equivalent model with only electric current density on the outermost boundaries is derived. Compared with other SIEs, like the PMCHWT formulation, all unknowns are residing on the outermost boundaries in the proposed formulation and therefore, less count of unknowns can be obtained. Finally, three numerical examples are carried out to validate the effectiveness of the proposed SS-SIE.

Wigner–Smith Time Delay Matrix for Electromagnetics: Computational Aspects for Radiation and Scattering Analysis

The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative, and enables the synthesis of modes that experience well-defined group delays when interacting with the system. The elements of the WS time delay matrix for surface scatterers and antennas comprise renormalized energy-like volume integrals involving electric and magnetic fields that arise when exciting the system via its ports. Here, direct and indirect methods for computing the WS time delay matrix are presented. The direct method evaluates the energy-like volume integrals using surface integral operators that act on the incident electric fields and current densities for all excitations characterizing the scattering matrix. The indirect method accomplishes the same task by computing scattering parameters and their frequency derivatives. Both methods are computationally efficient and readily integrated into existing surface integral equation codes. The proposed techniques facilitate the evaluation of frequency derivatives of antenna impedances, antenna patterns, and scatterer radar cross sections in terms of renormalized field energies derived from a single frequency characterization of the system.

Surface Integral Equations for Low-Frequency Simulation in Well Logging Applications

Borehole and fracture in well-logging applications are usually simulated by the finite-element method (FEM) or volume integral equation (VIE). In this work, we propose the surface integral equation (SIE) to simulate them together. Specifically, the borehole is treated by the Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) formulation, whereas the fracture is handled by the thin dielectric sheet (TDS)-based SIE, i.e., TDS-SIE. The notorious low-frequency breakdown of PMCHWT is cured by using loop and tree basis functions for discretization. The hybridization of the loop-tree enhanced PMCHWT and TDS-SIE may be referred to as LT-TDS, which is reported for the first time. Moreover, we extend LT-TDS into the case where the background is a layered medium (LM), leading to the more universal and powerful solver entitled LM-LT-TDS. We validate the proposed (LM-)LT-TDS by simulating several low-frequency examples and comparing with the reference results obtained by FEM. Of particular concern is a practical well-logging case where both the borehole and the tilted fracture straddle a three-layer medium.

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 new DBEM for solving crack problems in arbitrary dissimilar materials

In this paper, a new and efficient Dual Boundary Element Method (DBEM) named Dual Boundary-Interface Integral Equation Method (DBIIEM) is presented to solve crack problems in objects composed of arbitrary number of different materials. The conventional DBEM has been widely used in crack analysis and proved to be one of the most effective BEM for solving crack problems, however it cannot be directly employed to solve more complicated multi-medium crack problems. The presented DBIIEM can remedy this deficiency by dually collocating the displacement boundary-interface integral equation on one of the multi-medium crack surface and traction boundary-interface integral equation on the other. The newly derived displacement and traction boundary-interface integral equations are naturally fit for solving multi-medium elasticity problems, and independent each other, therefore they can be employed together in multi-medium crack analysis. The established DBIIEM inherits the advantage of single -region formulation of DBEM for solving more complicated crack problems. Complex stress intensity factors which characterize the oscillatory singularity of crack tips located at the interface crack are evaluated. Several numerical examples are given to verify correctness of the presented method.

Fast solver for uncertainty EM scattering problems by the perturbed-based MLFMA

A novel fast algorithm is proposed to analyze the EM scattering from electrically large conducting targets with varying geometrical shapes. Firstly, the target surface can be constructed with several control points by using the non-uniform rational B-spline surface modeling method. More specifically, the surface integral equation can be rewritten in terms of these control points, which contains the geometrical coordinate information. Moreover, it should be noted that the derivatives of both the L and K operates are needed to describe the perturbation of electric currents. Therefore, a perturbed multilevel fast multipole algorithm (P-MLFMA) is proposed to accelerate the matrix vector multiplication. At last, numerical results are given to verify the accuracy and efficiency of the proposed P-MLFMA. It can be seen that higher computational efficiency can be obtained when compared with both the traditional Monte Carlo method and the Perturbation Approach for MoM.