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This dissertation develops surface integral equations using constraint-based Helmholtz decompositions for electromagnetic modeling. This new approach is applied to the electric field integral equation (EFIE), and it incorporates a Helmholtz decomposition (HD) of the current. For this reason, the new formulation is referred to as the EFIE-hd. The HD of the current is accomplished herein via appropriate surface integral constraints, and leads to a stable linear system. This strategy provides accurate solutions for the electric and magnetic fields at both high and low frequencies, it allows for the use of a locally corrected Nyström (LCN) discretization method for the resulting formulation, it is compatible with the local global solution framework, and it can be used with non-conformal meshes. To address large-scale and complex electromagnetic problems, an overlapped localizing local-global (OL-LOGOS) factorization is used to factorize the system matrix obtained from an LCN discretization of the augmented EFIE (AEFIE). The OL-LOGOS algorithm provides good asymptotic performance and error control when used with the AEFIE. This application is used to demonstrate the importance of using a well-conditioned formulation to obtain efficient performance from the factorization algorithm.

Understanding, predicting, and controlling electromagnetic field interactions on and between complex RF platforms requires high fidelity computational electromagnetic (CEM) simulation. The primary CEM tool within NASA is GEMINI, an integral equation based method-of-moments (MoM) code for frequency domain electromagnetic modeling. However, GEMINI is currently limited in the size and complexity of problems that can be effectively handled. To extend GEMINI’S CEM capabilities beyond those currently available, primary research is devoted to integrating the MFDlib library developed at the University of Kentucky with GEMINI for efficient filling, factorization, and solution of large electromagnetic problems formulated using integral equation methods. A secondary research project involves the hybrid parallelization of GEMINI for the efficient speedup of the impedance matrix filling process. This thesis discusses the research, development, and testing of the secondary research project on the High Performance Computing DLX Linux supercomputer cluster. Initial testing of GEMINI’s existing MPI parallelization establishes the benchmark for speedup and reveals performance issues subsequently solved by the NASA CEM Lab. Implementation of hybrid parallelization incorporates GEMINI’s existing course level MPI parallelization with Open MP fine level parallel threading. Simple and nested Open MP threading are compared. Final testing documents the improvements realized by hybrid parallelization.

As the convergence of multiple functions in a single electronic device drives current electronic trends, the need for increasing integration density is becoming more emphasized than in the past. To keep up with the industrial need and realize the new system integration law, three-dimensional (3-D) integration called System-on-Package (SoP) is becoming necessary. However, the commercialization of 3-D integration should overcome several technical barriers, one of which is the difficulty for the electrical design of interconnections. The 3-D interconnection design is difficult because of the modeling challenge of electrical coupling from the complicated structures of a large number of interconnections. In addition, mixed-signal design requires broadband modeling, which covers a large frequency spectrum for integrated microsystems. By using currently available methods, the electrical modeling of 3-D interconnections can be a very challenging task. This dissertation proposes a new method for constructing a broadband model of a large number of 3-D interconnections. The basic idea to address the many interconnections is using modal basis functions that capture electrical effects in interconnections. Since the use of global modal basis functions alleviates the need for discretization process of the interconnection structure, the computational cost is reduced considerably. The resultant interconnection model is a RLGC model that describes the broadband electrical behavior including losses and couplings. The smaller number of basis functions makes the interconnection model simpler, and therefore allows the generation of network parameters at reduced computational cost. Focusing on the modeling of bonding wires in stacked ICs and through-silicon via (TSV) interconnections, this research validates the interconnection modeling approach using several examples from 3-D full-wave EM simulation results.

L'équation intégrale du champ électrique (EFIE) est très utilisée pour résoudre des problèmes de diffusion d'ondes électromagnétiques grâce à la méthode aux éléments de frontière (BEM). En domaine fréquentiel, les systèmes matriciels émergeant de la BEM souffrent, entre autres, de deux problèmes de mauvais conditionnement : l'augmentation du nombre d'inconnues et la diminution de la fréquence entrainent l'accroissement du nombre de conditionnement. En conséquence, les solveurs itératifs requièrent plus d'itérations pour converger vers la solution, voire ne convergent pas du tout. En domaine temporel, ces problèmes sont également présents, en plus de l'instabilité DC qui entraine une solution erronée en fin de simulation. La discrétisation en temps est obtenue grâce à une quadrature de convolution basée sur les méthodes de Runge-Kutta implicites.Dans cette thèse, diverses formulations d'équations intégrales utilisant notamment des conditions d'impédance aux frontières (IBC) sont étudiées et préconditionnées. Dans une première partie en domaine fréquentiel, l'IBC-EFIE est stabilisée pour les basses fréquences et les maillages denses grâce aux projecteurs quasi-Helmholtz et à un préconditionnement de type Calderón. Puis une nouvelle forme d'IBC est introduite, ce qui permet la construction d'un préconditionneur multiplicatif. Dans la seconde partie en domaine temporel, l'EFIE est d'abord régularisée pour le cas d'un conducteur électrique parfait (PEC), la rendant stable pour les pas de temps larges et immunisée à l'instabilité DC. Enfin, unerésolution efficace de l'IBC-EFIE est recherchée, avant de stabiliser l'équation pour les pas de temps larges et les maillages denses. / The Electric Field Integral Equation (EFIE) is widely used to solve wave scattering problems in electromagnetics using the Boundary Element Method (BEM). In frequency domain, the linear systems stemming from the BEM suffer, amongst others, from two ill-conditioning problems: the low frequency breakdown and the dense mesh breakdown. Consequently, the iterative solvers require more iterations to converge to the solution, or they do not converge at all in the worst cases. These breakdowns are also present in time domain, in addition to the DC instability which causes the solution to be completely wrong in the late time steps of the simulations. The time discretization is achieved using a convolution quadrature based on Implicit Runge-Kutta (IRK) methods, which yields a system that is solved by Marching-On-in-Time (MOT). In this thesis, several integral equations formulations, involving Impedance Boundary Conditions (IBC) for most of them, are derived and subsequently preconditioned. In a first part dedicated to the frequency domain, the IBC-EFIE is stabilized for the low frequency and dense meshes by leveraging the quasi-Helmholtz projectors and a Calderón-like preconditioning. Then, a new IBC is introduced to enable the development of a multiplicative preconditioner for the new IBC-EFIE. In the second part on time domain,the EFIE is regularized for the Perfect Electric Conductor (PEC) case, to make it stable in the large time step regime and immune to the DC instability. Finally, the solution of the time domain IBC-EFIE is investigated by developing an efficient solution scheme and by stabilizing the equation for large time steps and dense meshes.

With the rapid increase in available compute power and memory, and bolstered by the advent of efficient formulations and algorithms, the role of 3D full-wave computational methods for accurate modelling of complex electromagnetic (EM) structures has gained in significance. The range of problems includes Radar Cross Section (RCS) computation, analysis and design of antennas and passive microwave circuits, bio-medical non-invasive detection and therapeutics, energy harvesting etc. Further, with the rapid advances in technology trends like System-in-Package (SiP) and System-on-Chip (SoC), the fidelity of chip-to-chip communication and package-board electrical performance parameters like signal integrity (SI), power integrity (PI), electromagnetic interference (EMI) are becoming increasingly critical. Rising pin-counts to satisfy functionality requirements and decreasing layer-counts to maintain cost-effectiveness necessitates 3D full wave electromagnetic solution for accurate system modelling. Method of Moments (MoM) is one such widely used computational technique to solve a 3D electromagnetic problem with full-wave accuracy. Due to lesser number of mesh elements or discretization on the geometry, MoM has an advantage of a smaller matrix size. However, due to Green's Function interactions, the MoM matrix is dense and its solution presents a time and memory challenge. The thesis focuses on formulation and development of novel techniques that aid in fast MoM based electromagnetic solutions. With the recent paradigm shift in computer hardware architectures transitioning from single-core microprocessors to multi-core systems, it is of prime importance to parallelize the serial electromagnetic formulations in order to leverage maximum computational benefits. Therefore, the thesis explores the possibilities to expedite an electromagnetic simulation by scalable parallelization of near-linear complexity algorithms like Fast Multipole Method (FMM) on a multi-core platform. Secondly, with the best of parallelization strategies in place and near-linear complexity algorithms in use, the solution time of a complex EM problem can still be exceedingly large due to over-meshing of the geometry to achieve a desired level of accuracy. Hence, the thesis focuses on judicious placement of mesh elements on the geometry to capture the physics of the problem without compromising on accuracy- a technique called Adaptive Mesh Refinement. This facilitates a reduction in the number of solution variables or degrees of freedom in the system and hence the solution time. For multi-scale structures as encountered in chip-package-board systems, the MoM formulation breaks down for parts of the geometry having dimensions much smaller as compared to the operating wavelength. This phenomenon is popularly known as low-frequency breakdown or low-frequency instability. It results in an ill-conditioned MoM system matrix, and hence higher iteration count to converge when solved using an iterative solver framework. This consequently increases the solution time of simulation. The thesis thus proposes novel formulations to improve the spectral properties of the system matrix for real-world complex conductor and dielectric structures and hence form well-conditioned systems. This reduces the iteration count considerably for convergence and thus results in faster solution. Finally, minor changes in the geometrical design layouts can adversely affect the time-to-market of a commodity or a product. This is because the intermediate design variants, in spite of having similarities between them are treated as separate entities and therefore have to follow the conventional model-mesh-solve workflow for their analysis. This is a missed opportunity especially for design variant problems involving near-identical characteristics when the information from the previous design variant could have been used to expedite the simulation of the present design iteration. A similar problem occurs in the broadband simulation of an electromagnetic structure. The solution at a particular frequency can be expedited manifold if the matrix information from a frequency in its neighbourhood is used, provided the electrical characteristics remain nearly similar. The thesis introduces methods to re-use the subspace or Eigen-space information of a matrix from a previous design or frequency to solve the next incremental problem faster.

Estimation of the Radar Cross Section of large inhomogeneous scattering objects such as composite aircrafts, ships and biological bodies at high frequencies has posed large computational challenge. The detection of scattering from wake vortex leading to detection and possible identification of low observable aircrafts also demand the development of computationally efficient and rigorous numerical techniques. Amongst the various methods deployed in Computational Electromagnetics, the Method of Moments predicts the electromagnetic characteristics accurately. Method of Moments is a rigorous method, combined with an array of modeling techniques such as triangular patch, cubical cell and tetrahedral modeling. Method of Moments has become an accurate technique for solving electromagnetic problems from complex shaped homogeneous and inhomogeneous objects. One of the drawbacks of Method of Moments is the fact that it results into a dense matrix, the inversion of which is a computationally complex both in terms of physical memory and compute power. This has been the prime reason for the Method of Moments hitherto remaining as a low frequency method. With recent advances in supercomputing, it is possible to extend the range of Method of Moments for Radar Cross Section computation of aircraft like structures and radiation characteristic of antennas mounted on complex shaped bodies at realistic frequencies of practical interest. This thesis is a contribution in this direction. The main focus of this thesis is development of parallel Method of Moments solvers, applied to solve real world electromagnetic wave scattering and radiation problems from inhomogeneous objects. While the methods developed in this thesis are applicable to a variety of problems in Computational Electromagnetics as shown by illustrative examples, in specific, it has been applied to compute the Radar Cross Section enhancement due to acoustic disturbances and flow inhomogeneities from the wake vortex of an aircraft, thus exploring the possibility of detecting stealth aircraft. Illustrative examples also include the analysis of antenna mounted on an aircraft. In this thesis, first the RWG basis functions have been used in Method of Moments procedure, for solving scattering problems from complex conducting structures such as aircraft and antenna(s) mounted on airborne vehicles, of electrically large size of about 45 and 0.76 million unknowns. Next, the solver using SWG basis functions with tetrahedral and pulse basis functions with cubical modeling have been developed to solve scattering from 3D inhomogeneous bodies. The developed codes are validated by computing the Radar Cross Section of spherical homogeneous and inhomogeneous layered scatterers, lossy dielectric cylinder with region wise inhomogeneity and high contrast dielectric objects. Aerodynamic flow solver ANSYS FLUENT, based on Finite Volume Method is used to solve inviscid compressible flow problem around the aircraft. The gradients of pressure/density are converted to dielectric constant variation in the wake region by using empirical relation and interpolation techniques. Then the Radar Cross Section is computed from the flow inhomogeneities in the vicinity of a model aircraft and beyond (wake zone) using the developed parallel Volume Surface Integral Equation using Method of Moments and investigated more rigorously. Radar Cross Section enhancement is demonstrated in the presence of the flow inhomogeneities and detectability is discussed. The Bragg scattering that occurs when electromagnetic and acoustic waves interact is also discussed and the results are interpreted in this light. The possibility of using the scattering from wake vortex to detect low visible aircraft is discussed. This thesis also explores the possibility of observing the Bragg scattering phenomenon from the acoustic disturbances, caused by the wake vortex. The latter sets the direction for use of radars for target identification and beyond target detection. The codes are parallelized using the ScaLAPACK and BiCG iterative method on shared and distributed memory machines, and tested on variety of High Performance Computing platforms such as Blue Gene/L (22.4TF), Tyrone cluster, CSIR-4PI HP Proliant 3000 BL460c (360TF) and CRAY XC40 machines. The parallelization speedup and efficiency of all the codes has also been shown.