Fast surface integral equation formulations for large-scale conductors, metamaterials, and plasmonic problems

Many problems in Computational Electromagnetics (CEM) and Computational Fluid Dynamics (CFD) require millions of grid cells to properly represent the physics. Advances in parallel computing, both in hardware and software, make it feasible to solve such large scale simulations in a timely manner. The objective of this paper is to describe some of the recent work at Rockwell in performing parallel computing for providing quick-turnaround solutions to problems in CFD, CEM and other computational disciplines.

In a recent invited paper in the IEEE Antennas and Propagation Magazine, some of the challenging problems in computational electromagnetics were presented. One of the objectives of this note is to simply point out that challenging to one may be simple to another. This is demonstrated through an example cited in that article. The example chosen is a Vivaldi antenna array. What we discuss here also applies to the other examples presented in that article, but we have chosen the Vivaldi antenna array to help us make our point. It is shown in this short article that a higher-order basis using a surface integral equation a la a PMCHWT (Poggio-Miller-Chu-Harrington-Wu-Tsai) method-of-moments formulation may still be the best weapon that one have in today's arsenal to deal with challenging complex electromagnetic analysis problems. Here, we have used the commercially available code WIPL-D to carry out all the computations using laptop/desktop systems. The second objective of this paper is to present an out-of-core solver. The goal is to demonstrate that an out-of-core 32-bit-system-based solver can be as efficient as a 64-bit in-core solver. This is quite contrary to the popular belief that an out-of-core solver is generally much slower than an in-core solver. This can be significant, as the difference in the cost of a 32-bit system can be 1/30 of a 64-bit system of similar capabilities using current computer architectures. For the 32-bit system, we consider a Pentium 4 system, whereas for the 64-bit system, we consider an Itanium 2 system for comparison. The out-of-core solver can go beyond the 2 GB limitation for a 32-bit system and can be run on ordinary laptop/desktop; hence, we can simultaneously have a much lower hardware investment while better performance for a sophisticated and powerful electromagnetic solver. The system resources and the CPU times are also outlined.

The surface integral equation method is one of the most popular numerical methods for the electromagnetic analysis of composite metallic and dielectric structures. In this paper, the authors seek the "optimal" surface integral equation formulations for closed metallic, homogeneous and piecewise homogeneous dielectric and composite objects by studying the properties of the integral equations. The goal is to develop formulations in which all unknown quantities are well tested in the Galerkin method and the discretized matrix equation has good spectral properties leading to rapidly converging iterative solutions. The results show that by deriving the equations so that they map the currents onto itself and are directly tested in the same space as the unknowns are expanded can significantly improve conditioning of the matrix equation.

Conference of 42nd Annual Review of Progress in Quantitative Nondestructive Evaluation, QNDE 2015, Incorporating the 6th European-American Workshop on Reliability of NDE ; Conference Date: 26 July 2015 Through 31 July 2015; Conference Code:119404

Maxwell's equations are valid from the atomic to intergalactic length scales. Hence, features at various spatial or temporal scales may all be significant in electromagnetic (EM) problems of interest and may affect the results of multiscale modeling and simulation in computational electromagnetics (CEM). Solving electromagnetic problems that involve multiscale structures efficiently is challenging and is of growing interest in CEM. In recent decades, various approaches have been proposed for addressing the multiscale issue of electrically large problems. These methods can be roughly categorized into three types: the “geometry‐based,” “algebra‐based,” and “physics‐based” approaches. In this chapter, we introduce these three representative approaches for modeling multiscale problems in CEM: (1) the discontinuous Galerkin integral equation method (DGIE), which is a general domain decomposition framework for dealing with structures with mesh nonconformity; (2) the nested dissection accelerated H‐LU‐based fast direct solver for sparse matrices, which is used to construct effective preconditioners; and (3) the resistive sheet boundary condition‐based hybrid finite element‐boundary integral‐multilevel fast multipole algorithm (FE–BI–MLFMA), which can simulate electromagnetic field interactions with microwave‐absorbing honeycomb structures efficiently. After we present the formulation for each approach, we present examples to demonstrate its accuracy, capability, and performance.

Computation of scattering by large, complex, composite targets is one of most challenging problems in computational electromagnetics (CEM). A formulation of hybrid FE/BI/MLFMA is presented for this kind of problem. The numerical performance of the hybrid FE/BI/MLFMA is investigated by computation of a conducting sphere coated with anisotropic material and a conducting brick with deep material-filled cavity.

Electromagnetic wave scattering from large and complex bodies is currently the most challenging problem in computational electromagnetics. There is an increasing need for more efficient algorithms with reduced computational complexity and memory requirements. In this work we solve the problem of electromagnetic wave scattering involving three-dimensional, homogeneous, arbitrarily shaped dielectric objects. The fast multipole method (FMM) is used along with the algebraic multigrid (AMG) method, that is employed as a preconditioner, in order to accelerate the convergence rate of the Krylov iterations. Our experimental results suggest much faster convergence compared to the non preconditioned FMM, and hence significant reduction to the overall computation time.

Recent years have seen a spectacular increase in our capability to model, simulate the performance of, and design complex electromagnetic systems. Much progress has been made in enhancing the available numerical techniques, viz., the method of moments (MoM), the finite-element method (FEM), and the finite-difference time-domain (FDTD) or its variants. Great strides have recently been made in enlarging the scope of MoM via the use of the fast multipole method (FMM), which has made it feasible for us to solve problems that require the handling of 106 degrees of freedom, or even higher, and distributed processing has enabled the FDTD to handle upward of 109 degrees of freedom on a moderate-size computing platform. Despite this recent progress, many practical computational electromagnetic (CEM) modeling problems of interest present formidable challenges, and the search for numerically efficient techniques to solve large problems involving complex structures continues unabated. The objectives of this paper are to identify some of these challenging problems encountered by the author during the last five years, and to present the results of application of a technique called CBFM - developed at the EMC Laboratory at Penn State - that has been found useful for addressing them.

The analysis of electromagnetic diffraction phenomena in arbitrarily linear media is a challenging problem in computational electromagnetics either with numerical, analytical, semi-analytical and asymptotic techniques.

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.

Fast Fourier Transform (FFT) is an important part of many applications, such as in wireless communication based on OFDM (Orthogonal Frequency Division Multiplexing). With Cloud Radio Access Networks, implementing FFTs on multiprocessor clusters is a challenging task. For instance, supporting the Long Term Evolution (LTE) protocol requires processing 100 independent FFTs (with sizes ranging from 128 to 2048 points) in 66.7 μs. In this work, seven native FFT candidate implementations are compared. The considered implementation environments are: OpenMP (Open Multi-Processing) on 1 core, MPI (Message Passing Interface) on 1 core, 2 cores, and 3 cores, Hybrid OpenMP+MPI on 1 core and 3 cores, and MPI on an heterogeneous platform composed of Xeon-Phi and 3 cores. The reported experimental results show that the latter method meets the latency requirements of LTE. It is shown that the OpenMP and MPI paradigms running only on MICs (Many Integrated Cores) cannot benefit fully from the computing capability of many-core architectures. The heterogeneous combination of Xeon+MICs provides a better performance.

In this communication, a novel message passing interface (MPI) parallel algorithm for nodal discontinuous Galerkin time-domain (NDGTD) method has been developed. A unified MPI + MPI technique has been introduced for extreme parallelism on a large-scale computer cluster. Through the data transmission between CPU nodes using MPI persistent nonblocking two-side communication and the direct data connection between processors in the same node via MPI shared memory windows, a two-layered parallel architecture is implemented to minimize the communication. To further accelerate the solution of the multiscale problems, the local time stepping (LTS) technique has been employed in the NDGTD method. A fast time step estimation method has been presented in this communication. With high overlap between the information transmission and the data calculation, the proposed MPI + MPI scheme overcomes the degradation of the parallel efficiency of the pure MPI technique in the scenario of the LTS technique and the large-scale CPU cores. Up to 94% parallel efficiency in 6400 CPU cores is achieved for the average single-core loading about 1700 finite elements, and 18 times acceleration for time step estimation can be obtained with the fourth-order basis function. Three practical complex examples are given to demonstrate a good performance of the proposed method.

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.

Recently, scattering from 3D composite conducting and dielectric object receives much attention. Traditionally, FEM-BI method is applied to solve the problem. But absorption boundary condition (ABC) is needed in FEM-BI method. In this paper, we use two combinations of surface integral equation to solve the RCS of 3D composite conducting and dielectric object. Fast multipole method (FMM) is used. The problem is formulated in terms of a set of coupled integral equations involving equivalent electric and magnetic surface currents based on the equivalence theorem. The conducting structures and the dielectric materials are modeled by planar triangular patches, RWG basis and Galerkin method are used. The fast multipole method is applied to accelerate the computation of matrix-vector multiplication. The computational complexity and storage requirement is O(N/sup 1.5/), respectively. Numerical results are given for various structures and compared with other available data. The numerical results show that the present method has satisfying accuracy.

A key quantity in the design of plasmonic antennas and metasurfaces, as well as metamaterials, is the electrodynamic polarizability of a single scattering building block. In particular, in the current merging of plasmonics and metamaterials, subwavelength scatterers are judged by their ability to present a large, generally anisotropic electric and magnetic polarizability, as well as a bi-anisotropic magnetoelectric polarizability. This bi-anisotropic response, whereby a magnetic dipole is induced through electric driving, and vice versa, is strongly linked to the optical activity and chiral response of plasmonic metamolecules. We present two distinct methods to retrieve the polarizibility tensor from electrodynamic simulations. As a basis for both, we use the surface integral equation (SIE) method to solve for the scattering response of arbitrary objects exactly. In the first retrieval method, we project scattered fields onto vector spherical harmonics with the aid of an exact discrete spherical harmonic Fourier transform on the unit sphere. In the second, we take the effective current distributions generated by SIE as a basis to calculate dipole moments. We verify that the first approach holds for scatterers of any size, while the second is only approximately correct for small scatterers. We present benchmark calculations, revisiting the zero-forward scattering paradox of Kerker et al (1983 J. Opt. Soc. Am. 73 765–7) and Alù and Engheta (2010 J. Nanophoton. 4 041590), relevant in dielectric scattering cancelation and sensor cloaking designs. Finally, we report the polarizability tensor of split rings, and show that split rings will strongly influence the emission of dipolar single emitters. In the context of plasmon-enhanced emission, split rings can imbue their large magnetic dipole moment on the emission of simple electric dipole emitters. We present a split ring antenna array design that is capable of converting the emission of a single linear dipole emitter in forward and backward beams of directional emission of opposite handedness. This design can, for instance, find application in the spin angular momentum encoding of quantum information.