A look at some challenging problems in computational electromagnetics

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.

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.

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

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.

Various methods are available at the disposal for computing the solution of electromagnetic problems expeditiously. Electromagnetic scattering problems can be classified into two broader categories named as surface and volume problems. These problems can be solved using differential and integral equations. Differential equations based techniques use Finite Element Method (FEM) and Finite Difference Time Domain (FDTD) methods and major integral formulation based technique is Method of Moment (MOM). Any set of linear equations for a given system can be solved using direct or iterative methods. Direct solvers confront many computational complexities and use large amount of memory while iterative solvers reduce computational complexities and use less amount of memory. In this research work some integral equations based solver schemes such ass Finite Element Method (FEM), Finite Difference Time Domain Method (FDTD) and Method of Moments (MOM) for surface and volume scatters are discussed and results are compared. The comparison is with respect to complexity and memory requirements for different solution schemes.

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.

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.

A regularized fast multipole method of moments for rapid calculation of three-dimensional time-harmonic electromagnetic scattering from complex targets

A recommended practice for computational electromagnetics (CEM) applied to the modeling and simulation of antennas (IEEE P2816) is currently being developed by the IEEE Antennas and Propagation Standards Committee (APS/SC). Four commonly adopted CEM solvers to simulate antenna problems are considered in this document: integral equations and method of moments (MoM), finite element method (FEM), finite difference time domain (FDTD) method and transmission line matrix (TLM) method. A broadband biconical antenna was also proposed and included as a benchmark problem that was tackled as an international interlaboratory comparison with twenty sets of results obtained using the different solvers. Overall a good agreement was observed with some deviating results, especially at the lower end of the frequency band. Herein this biconical antenna is numerically analyzed and the results compared with the analytical solution in an aim to explain the deviating results.

In this paper, an easy method is proposed to reduce the computational complexity of matrix-vector multiplication that is required in solving integral equations using method of moments. This method approximates the Green’s function using a Fourier series. It has the advantage that its formulation separates the observation point from the source point effortlessly and regardless of the Green’s function. Since the fast multipole method uses this idea, the two methods are discussed and compared theoretically and numerically by solving the electromagnetic wave scattering from perfect conductor of three-dimensional basic canonical shape (sphere). The results showed that the proposed method is accurate, and for large number of unknowns in the problem, it has a computational complexity less than the method of moments and over the conventional fast multipole method. Hence, the generality and simplicity of this method compare with the conventional fast multipole method and the efficiency compare with the method of moments, which will be achieved.

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.

Surface-integral-equation (SIE) approaches are presented for the accurate solution of different problems in computational electromagnetics. First, an efficient message passing interface (MPI)/OpenMP parallel implementation of the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) is presented for the solution of large-scale conducting bodies. Second, looking for the extension of these rigorous approaches to new demanded areas, the SIE method is successfully applied to the solution of left-handed metamaterials and plasmonic nanostructures.

Though computational molecular dynamics is an effective tool for nano-scale phenomenon analysis, computational costs associated with its computer simulation are extremely high. There are two major computational steps associated with computer simulation of dynamics of molecular structures. They are calculation of interatomic forces and formation and solution of the equations of motion. Currently, these two computational steps are treated separately in most commonly-used methods. For example, Fast Multipole Method (FMM) and Cell Multipole Method (CMM) have been used for calculation of interatomic forces, and Cartesian Coordinate Method (CCM) and Internal Coordinate Molecular Dynamics Method (ICMD) have been created for the formation and solution of equations of motion of an atomistic molecular system. In this paper, a new procedure is presented through a proper integration between multibody molecular algorithms (MMA) and fast multipole methods to improve computational efficiency for computer simulation of the dynamical behaviors of multibody molecular structures in polymers and biopolymers. For the computational costs associated with interatomic forces, a fast multipole method is used to calculate the interatomic forces due to the potentials. For the computational costs associated with formation and solution of equations of motion, a multibody molecular algorithm developed by the author in his previous work will be utilized to integrate with fast multipole methods. The algorithm significantly improves computational efficiency when comparing with its counterpart procedures. The fast multipole method begins by scaling all atoms into a box with coordinate ranges to ensure numerical stability of subsequent operations. The parent box is then divided into half in the direction of each Cartesian axis and each child box is then subdivided to form a computational family tree. The flow of calculations is carried out along the tree structure with five passes. The fast multipole method has been improved and modified to achieve better effectiveness and higher efficiency since it was created. The multibody molecular algorithm starts with numbering subsets, forming bond graph, and developing three computing passes along the tree structure of an atomistic molecular system. Computing data flows in the fast multipole method and the multibody molecular algorithm will properly line up with the parent-child recursive relationship along the configuration of the tree structure due to linear recursive natures of both fast multipole method and multibody molecular algorithm. Then the time spent on the recursive simulation passes in the fast multipole method for computing forces may overlap with the time spent on the three recursive computational passes in the multibody molecular algorithm for forming and solving equations of motion.

The fast multipole method (FMM) is developed to analyse electromagnetic scattering using higher-order hierarchical Legendre basis functions, which have been proposed recently and make the impedance matrix, from method of moments, a low condition number. In this method, the point-to-point based-on-sample quadrature is used because the basis functions are defined on large surface patches of scatterers. The numerical implementations of matrix–vector multiply by FMM are particularly presented. These numerical results show that the proposed second- or third-order FMM (high-order FMM) is accurate and efficient.

Standard numerical models in acoustics rely on the isentropic Helmholtz equation. Its derivation assumes adiabatic and reversible, i.e., dissipation-free, wave propagation. Sound waves in fluids are, however, subject to viscous and thermal losses. These losses originate from viscous friction and heat conduction, leading to the formation of acoustic boundary layers. Considering these effects becomes significant in setups with acoustic cavities of similar dimension as the boundary layers. Recently, boundary element methods (BEM) accounting for the viscothermal dissipation have been proposed. These methods limit the discretization to the surface of the fluid domain and require significantly fewer degrees of freedom than comparable finite element models. However, the BEM coefficient matrices are fully populated, resulting in high computational costs and storage requirements. This study develops a new BEM formulation for acoustics in viscothermal fluids that uses the fast multipole method - a low-rank approximation technique based on a hierarchical subdivision of the computational domain - to alleviate this shortcoming. It is shown that the fast viscothermal BEM improves the algorithmic complexity over the conventional formulation, reducing both the memory requirements and solution time. The results indicate good scalability of the method making it feasible for larger applications such as acoustic metamaterials.