<p> The thesis presents an efficient self-adjoint approach to the S-parameter sensitivity analysis based on full-wave electromagnetic (EM) time-domain simulations with two commonly used numerical techniques: the finite-difference time-domain (FDTD) method and the transmission-line matrix (TLM) method. Without any additional simulations, we extract the response gradient with respect to all the design variables making use of the full-wave solution already generated by the system analysis. It allows the computation of the S-parameter derivatives as an independent post-process with negligible overhead. The sole requirement is the ability of the solver to export the field solution at user-defined points. Most in-house and commercial solvers have this ability, which makes our approach readily applicable to practical design problems.</p> <p> In the TLM-based self-adjoint techniques, we propose an algorithm to convert the electrical and magnetic field solutions into TLM voltages. The TLM-based discrete adjoint variable method (AVM) is originally developed to use incident and reflected voltages as the state variables. Our conversion algorithm makes the TLM-AVM method applicable to all time-domain commercial solvers, FDTD simulators included, with comparable accuracy and less memory overhead. Our approach is illustrated through waveguide examples using a TLM-based commercial simulator.</p> <p> Currently, our TLM-based self-adjoint approach is limited to loss-free homogeneous problems. However, our FDTD-based self-adjoint approach is valid for lossy inhomogeneous cases as well. The FDTD-based self-adjoint technique needs only the E-field values as the state variables. In order to make it also applicable to a TLM-based solver, whose mesh grid is displaced from the FDTD grid, we interpolate the E-field solution from the TLM mesh to that on the FDTD mesh. Our FDTD-based approach is validated through the response derivatives computation with respect to both shape and constitutive parameters in waveguide and antenna structures. The response derivatives can be used not only to guide a gradient-based optimizer, but also to provide a sufficient good initial guess for the solution of nonlinear inverse problems.</p> <p> Suggestions for further research are provided.</p> / Thesis / Master of Applied Science (MASc)

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.