Development and extensions of stable and efficient alternating-direction-implicit / locally one-dimensional FDTD methods - Tay, Wei Choon (2014)
Recently, there has been increasing interest in the development of unconditionally stable finite-difference time-domain (FDTD) methods in electromagnetics that are not constrained by the Courant-Friedrich-Lewy (CFL) condition. This thesis presents the development and extensions of stable and efficient alternating-direction-implicit (ADI) or locally one-dimensional (LOD) FDTD methods. From Maxwell's equations (which are hyperbolic partial differential equations), the efficient fundamental ADI (FADI) and LOD (FLOD) FDTD methods are formulated into simpler, more concise, and more efficient form of the ADI- and LOD-FDTD methods. The boundary conditions are first investigated on the FADI- and FLOD-FDTD methods. For closed region simulation, the perfect magnetic conductor (PMC) and perfect electric conductor (PEC) are derived regardless of the implicit updating of electric or magnetic fields. Next, for unbounded region simulation, the Mur absorbing boundary condition (ABC) is incorporated into the FADI- and FLOD-FDTD methods using consistent implementation and a novel implementation with lower reflection coefficient. For even better absorption, the perfectly matched layers (PMLs) have been incorporated into the FADI-FDTD method. Further, the FADI-FDTD method with complex frequency shifted convolutional PML (CFS-CPML) has been incorporated into the graphics processor units (GPUs) to exploit data parallelism for higher efficiency. To demonstrate the usefulness of the FADI-FDTD method with CFS-CPML, a practical microstrip low-pass filter is presented. A high computational power is attained while preserving a good agreement with the Yee-FDTD method. Based on the FDTD technique used for solving Maxwell's equation in electromagnetics, it can be extended to solve the heat transfer equation (which is a parabolic partial differential equation) in thermodynamics. The thesis next proposes the stable and efficient FADI and FLOD methods for heat transfer. In inhomogeneous media, the two-dimensional (2-D) Douglas-Gunn (DG) ADI method for the heat transfer equation takes into account both Laplacian and gradient terms. The potential instability of the conventional DG-ADI method caused by the gradient terms is alleviated. Subsequently, the proposed stabilized DG-ADI method is cast into the (stabilized) Peaceman-Rachford (PR) ADI method in compact form. A stable and efficient FADI method is then formulated. The FADI method is further extended into the heat transfer equation with convection heat flux due to fluid motion. To achieve higher efficiency, the FADI method has been incorporated into the GPU for efficient thermal simulation of integrated circuits (ICs) with microchannel cooling. For the three-dimensional (3-D) DG-ADI method in the homogeneous media, its implementation is highly complex and requires a considerable number of memory variables. To overcome these complications, the conventional DG-ADI method is formulated into the efficient ADI method with single operator and heat generation input on the right-hand-side (RHS) of the first procedure. The proposed method has substantially less RHS update coefficients and field variables; as well as lower number of memory variables than the conventional DG-ADI method. Thus, it achieves higher efficiency with reduced memory indexing and arithmetic operations. As the current 3-D DG-ADI method is still conditionally stable within inhomogeneous media, two stable 3-D FLOD methods are proposed for solving the heat transfer equation. Stability analysis by means of analyzing the eigenvalues of the amplification matrix is provided to substantiate the stability of the FLOD method. Further, the relative maximum error of the FLOD method for heat transfer is examined. It exhibits good trade-off between accuracy and efficiency. To show the effectiveness of the proposed stabilized FLOD method, the heat distribution of the closely resembled Alpha 21364 processor chip is presented and analyzed. By using the FDTD technique for solving the heat transfer equation, this technique can be extended to solve the Schrödinger equation (which is a parabolic partial differential equation with complex variables) in quantum mechanics. A stable and efficient FADI method for Schrödinger equation is proposed. For the tridiagonal system equations of the ADI (Tri-ADI) method, the computation of the complex wave function is rather taxing and time consuming. A novel pentadiagonal system of equations for the ADI (Penta-ADI) method is introduced through the separation of the complex wave function into real and imaginary parts. Subsequently, the Penta-ADI method is formulated into the pentadiagonal efficient fundamental ADI (Penta-FADI) method. Such efficient fundamental scheme has matrix-operator-free RHS, leading to computationally efficient update equations. As the Penta-FADI method involves five stencils on the left-hand-sides (LHS) of the pentadiagonal update equations, special treatments are provided for the implementation of the Dirichlet boundary condition. Further analysis of the Penta-FADI method over Tri-ADI method is also shown. Using the Penta-FADI method, a significantly higher efficiency gain can be achieved.