Explicit and unconditionally stable finite difference time domain methods for general electromagnetic analysis
Explicit FDTD method is one of the most popular methods for time domain analysis because it does not need to solve a matrix. It has gained a wide-spread popularity not only in electromagnetic simulations but also for photonic, thermal, biological, aerodynamic, and many other applications. However, the time step of the FDTD method is constrained by the smallest space step to ensure the stability of a time-domain simulation. When the problem being analyzed involves fine features relative to working wavelengths, a tremendous number of time steps are required to finish one simulation, which is computationally expensive. So far, there is no explicit unconditionally stable FDTD but all unconditionally stable FDTD methods are implicit, which require a matrix solution. This problem is solved by the proposed work, where we develop explicit and unconditionally stable FDTD methods. In the second chapter of this thesis, we find the root cause of the instability associated with an explicit time marching, based on which we create the first explicit FDTD method that is unconditionally stable. This method is stable for an arbitrarily large time step irrespective of space step, and accurate for a time step solely determined by sampling accuracy. Besides, it keeps the matrix-free property of the original FDTD method, and hence no matrix solution is required. In the third chapter, we identify the root cause of the instability for the analysis of general lossy problems where both dielectrics and conductors can be lossy and in-homogeneous. Based on the root cause analysis, we develop an efficient algorithm to eradicate the root cause of instability, and afterwards achieve unconditional stability in an explicit FDTD-based simulation of general lossy problems. In chapter 4 , we develop a preprocessing and processing method and an alternative method for making an explicit FDTD unconditionally stable. In this method, we bypass the preprocessing algorithms developed in chapter 2 for finding the stable eigenmodes for the given time step. Instead, we up-front deduct the source of instability from the system matrix (discretized curl curl operator) resulting from the space discretization, and directly perform the explicit FDTD marching on the updated system matrix. In the fifth chapter, we extend the method of chapter 3 to the analysis of general lossy problems. During researching explicit and unconditionally stable FDTD methods, we also find a simple approach to make an implicit FDTD unconditionally stable. This approach only requires the change of one time instant in the original FDTD algorithm to achieve the unconditional stability, which is both theoretically proved and numerically verified. This approach is presented in Chapter 6. The success of the unconditionally stable methods developed in this work also enables us to discover the low-frequency breakdown problem in time-domain FDTD simulations. In Chapter 7, we elaborate the breakdown problem identified and a rigorous solution to the problem. In Chapter 8, we use our proposed methods with modification to simulate practical large interconnect structures. In chapter 9, we draw conclusions and discuss the future work on our present proposed methods.
Jiao, Purdue University.
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