The objective of this research work is to develop an efficient methodology for chip-package cosimulation. In the traditional design flow, the integrated circuit (IC) is first designed followed by the package design. The disadvantage of the conventional sequential design flow is that if there are problems with signal and power integrity after the integration of the IC and the package, it is expensive and time consuming to go back and change the IC layout for a different input/output (IO) pad assignment. To overcome this limitation, a concurrent design flow, where both the IC and the package are designed together, has been recommended by researchers to obtain a fast design closure. The techniques from this research work will enable multiscale cosimulation of the chip and the package making the concurrent design flow paradigm possible. Traditional time-domain techniques, such as the finite-difference time-domain method, are limited by the Courant condition and are not suitable for chip-package cosimulation. The Courant condition gives an upper bound on the time step that can be used to obtain stable simulation results. The smaller the mesh dimension the smaller is the Courant time step. In the case of chip-package cosimulation the on-chip structures require a fine mesh, which can make the time step prohibitively small. An unconditionally stable scheme using Laguerre polynomials has been recommended for chip-package cosimulation. Prior limitations in this method have been overcome in this research work. The enhanced transient simulation scheme using Laguerre polynomials has been named SLeEC, which stands for simulation using Laguerre equivalent circuit. A full-wave EM simulator has been developed using the SLeEC methodology. A scheme for efficient use of full-wave solver for chip-package cosimulation has been proposed. Simulation of the entire chip-package structure using a full-wave solver could be a memory and time-intensive operation. A more efficient way is to separate the chip-package structure into the chip, the package signal-delivery network, and the package power-delivery network; use a full-wave solver to simulate each of these smaller subblocks and integrate them together in the following step, before a final simulation is done on the integrated network. Examples have been presented that illustrate the technique.

As the current electronic trend is toward integrating multiple functions in a single electronic device, there is a clear need for increasing integration density which is becoming more emphasized than in the past. To meet the industrial need and realize the new system-integration law [1], three-dimensional (3-D) integration is becoming necessary. 3-D integration of multiple functional IC chip/package modules requires co-simulation of the chip and the package to evaluate the performance of the system accurately. Due to large scale differences in the physical dimensions of chip-package structures, the chip-package co-simulation in time-domain using the conventional FDTD scheme is challenging because of Courant-Friedrich-Levy (CFL) condition that limits the time step. Laguerre-FDTD has been proposed to overcome the limitations on the time step. To enhance performance and applicability, SLeEC methodology [2] has been proposed based on the Laguerre-FDTD method. However, the SLeEC method still has limitations to solve practical 3-D integration problems. This dissertation proposes further improvements of the Laguerre-FDTD and SLeEC method to address practical problems in 3-D interconnects and 3-D integration. A method that increases the accuracy in the conversion of the solutions from Laguerre-domain to time-domain is demonstrated. A methodology that enables the Laguerre-FDTD simulation for any length of time, which was challenging in prior work, is proposed. Therefore, the analysis of the low-frequency response can be performed from the time-domain simulation for a long time period. An efficient method to analyze frequency-domain response using time-domain simulations is introduced. Finally, to model practical structures, it is crucial to model dispersive materials. A Laguerre-FDTD formulation for frequency-dependent dispersive materials is derived in this dissertation and has been implemented.

In this thesis, we first propose a new scalar auxiliary variable (SAV) approach for general dissipative nonlinear systems. This new approach is half computational cost of the original SAV approach, can be extended to high order unconditionally energy stable backward differentiation formula (BDF) schemes and not restricted to the gradient flow structure. Rigorous error estimates for this new SAV approach are conducted for the Allen-Cahn and Cahn-Hilliard type equations from the BDF1 to the BDF5 schemes in a unified form. As an application of this new approach, we construct high order unconditionally stable, fully discrete schemes for the incompressible Navier-Stokes equation with periodic boundary condition. The corresponding error estimates for the fully discrete schemes are also reported. Secondly, by combining the new SAV approach with functional transformation, we propose a new method to construct high-order, linear, positivity/bound preserving and unconditionally energy stable schemes for general dissipative systems whose solutions are positivity/bound preserving. We apply this new method to second order equations: the Allen-Cahn equation with logarithm potential, the Poisson-Nernst-Planck equation and the Keller-Segel equations and fourth order equations: the thin film equation and the Cahn-Hilliard equation with logarithm potential. Ample numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.