### Computational Electromagnetics

The Electromagnetics Research Group staff are experienced at formulating, implementing, and applying advanced Computational Electromagnetics (CEM) methods to challenging problems in the general areas of scattering, antenna design and characterization, microwave cavity design and characterization, and microwave circuits. Our styaff have experience in all major techniques of CEM including the finite element method, finite difference-time domain method, the method of moments, and high frequency asymptotic methods.

Recent research projects in CEM have focused on several different topics. One involved the design and characterization of complex antenna apertures using a hybrid finite element-boundary integral methods where the antennas were flush-mounted in planar or cylindrical structures. A second on-going research thrust involves the characterization of electromagnetic fields in microwave cavities with either dielectric or plasma loads. A third effort involves the development of a highly efficient integral equation-based solver for very large geometries. Another effort involves the characterization of ferrite-loaded structures using a hybrid finite element-boundary integral method. All of these efforts are on-going and being performed with the collaboration of colleagues at MSU, in other academic institutions, industrial partners, and in close cooperation with Federal sponsors.

### Fast and Accurate Integral Equation Solvers for Mixed-scale Electromagnetic Simulation

In the electromagnetic simulation of realistic structures, the spatial representation of the domain being analyzed depends not only on the frequency of interest but also on the need to capture possible fine geometric features. Such mixed scales cause havoc in standard integral equation based solvers on three fronts; (i) discretized integral equations become poorly conditioned as the size of the element becomes smaller, (ii) the function spaces used do not optimally represent the underlying physics, and (iii) the overall computational burden is exceedingly large. This largely limits the applicability of the existing methods. The proposed project seeks to develop a demonstrably unified, robust and accurate solution methodology that is well conditioned over a wide range of frequencies and, at the same time, has the flexibility to handle complicated (and possibly near singular) geometries. This is achieved by (i) developing a well conditioned integral equation scheme (that are Fredholm equations of the second kind) with provable bounds on convergence rates and accuracy to solve for electromagnetic quantities over a large range of spatial frequencies; (ii) enlarging the approximation space used for representing the unknown quantity so as to include the local physics; (iii) designing a scheme that permits seamless interplay between a variety of basis functions to model the unknown quantities to be used with the above integral equation scheme; (iv) deriving error bounds and convergence estimates on these schemes to demonstrate clear and easy user control over the error, and (iv) developing a domain decomposition framework so that these schemes can be integrated seamlessly with classical integral equation and finite element methods to solve electrically large problems. The educational objective is to develop a publicly available set of tutorials/teaching modules based on this research.

### PACE-Parallel Accelerated Cartesian Expansions with Application to Molecular Dynamics

Computation of pairwise potential functions is crucial, albeit computationally expensive, to simulating the underlying physics in many fields. To mitigate this cost, fast and approximate potential computation methods have been developed for several potential functions; for example, particle-mesh methods, Fast Fourier Transforms, Fast Multipole Method (FMM), and limiting computation to neighborhoods. These methods differ in efficiency, accuracy, and applicability. Recent work by one of the PIs provides the foundation for the development of unified, robust, accurate and parallel methods for fast computation of non-oscillatory potentials using the Accelerated Cartesian Expansion framework.

A two pronged approach undertaken herein involves the development of (i) translation operators to enable FMM based computation for different pairwise potentials, including Yukawa, Lennard Jones, Gauss, Morse, and Buckingham potentials, and (ii) parallel framework for computing individual and multiple potentials simultaneously. These techniques are to be applied to a set of practical systems involving the Poisson, diffusion, retarded and Helmholtz (sub-wavelength), and Klein-Gordon equations, and to computing van-der Waals (in mesoscopic systems). The underlying methodology requires that only translation operators change from potential to potential, and provides a mathematically exact formulation for traversal up and down the FMM tree. The unifying treatment for computing multiple potentials simplifies parallel code development, especially with regard to scalability.

### Parallel Transient Solvers for Multiscale Electromagnetics Simulation

This proposal seeks to answer a growing engineering need: the development of robust computationally efficient methods to analyze transient radiation and scattering from electrically large multiscale objects. The proposed work can be categorized into two interrelated areas: (i) building parallel transient potential evaluators for computing interactions between random non-uniform source/observer pairs wherein separation between two points ranges from a millionth to a thousand of the minimum wavelength; (ii) development of parallel time domain higher-order integral equation solvers that include these potential integrators. The four-fold objectives of this proposal are as follows: (i) rigorous methods that can be integrated with the plane wave time domain (PWTD) algorithm to extend its applicability to the quasi-static regime; (ii) windowed operators that will morph PWTD with beams; (iii) parallel, multiscale, fast potential evaluators that include the above developments; and (iv) integration of these into time domain integral equation solvers. To realize these objectives, advances will be made on two fronts: (i) numerical methods to effect these operations with a proper understanding of error bounds and the means to control them; and (ii) parallel algorithms that are provably scalable.

The design and analysis of realistic devices is the holy grail of any computational endeavor. The same is true of Maxwell solvers. As Maxwell's equations form the foundation to a wide array of modern technology, methods developed to efficiently and accurately solve these equations can have wide ranging impact. To date, simulation tools have been complementary to, but have not supplanted experiments. The principal challenge has been bottlenecks posed by complex structural topologies with fine features, embedded in electrically large structures. Our goal-to enable the analysis of field deployable systems-will be realized by making advances in both the underlying numerics and parallel algorithms. These, in turn, will enable transition of this technology from tens of processors to thousands and tens of thousands of processors. Methods developed will yield a robust, accurate, and adaptable code that can be widely adopted in multiple domains in electromagnetics, acoustics, plasma dynamics, etc.