In order to provide a systematic and structured presentation of this work, the remaining contents of the thesis are organized as follows.
The basic theory which both hybrid methods based on will be described in Chapter 2. A brief review of the finite-difference time-domain method will be presented firstly, including the traditional iterative FDTD equations and its matrix form, equations involving both the electric and magnetic fields, as well as those with
single field only. Time-reversal technique for FDTD will also be described. The time-domain method based on Delaunay-Vononoi modeling of power-ground planes will then discussed, which shows how a time marching scheme similar to FDTD can be established from the efficient power-ground plane model. The last part of chapter 2 shows the key component of both hybrid methods, i.e., model order reduction with Krylov subspace methods. With these methods modes of systems can be extracted effectively for the reconstruction of late-time responses.
Chapter 3 presents the first hybrid method which efficiently evaluates the late-time response by combining finite-difference time-domain method and Lanczos algorithm. Technique that hybridizing FDTD and Lanczos Algorithm by introducing time-reversal FDTD equations is firstly described, followed by numerical examples with a homogeneous PEC cavity and an inhomogeneous PEC cavity with different excitations. Convergence and complexity is then discussed with verification from the simulation results.
The other hybrid method that reconstructs the late-time response for Delaunay-Vononoi modeling of power-ground planes in time-domain with Krylov subspace method is presented in Chapter 4. After describing the hybridizing
technique that applies Krylov subspace method on the Delaunay-Vononoi modeling of power-ground planes in time-domain, several examples inclusive of power-ground planes of a simple geometry and a more realistic power-ground plane is demonstrated.
The convergence and complexity for this method is also analyzed and verified with simulation results.
The thesis is then concluded with a summary of this work. A few suggestions for the future works will also be provided.
13
2 2
Theory
HE hybrid methods presented in this work is based on the basic theory described in this chapter. The finite-difference time-domain (FDTD) method will be briefly reviewed first. The traditional iterative FDTD equations and its matrix form, equations involving both the electric and magnetic fields and those with single field only, and time-reversal technique for FDTD will be described. Based on the Delaunay-Vononoi modeling of power-ground planes provided in [22], a time marching scheme similar to FDTD is developed. The key component of the proposed hybrid methods, model order reduction with Krylov subspace methods, with which the modes of a system are extracted effectively for the reconstruction of late-time responses, will be discussed in the last part of this chapter.
T
x
Fig. 2.1 Yee’s cell [1] for descretization of Maxwell equation for the finite-difference
time-domain method in three-dimension.
2.1 Finite-Difference Time-Domain Method
The derivation of finite-difference time-domain equations in source free regions begins with source free Maxwell equations.
t
2.1.1 Time Marching in TDTD
In source-free region with lossless isotropic media, (2.1) can be discretized with
the Yee’s cell [1] setting shown in Fig. 2.1, where the simplified coordinate notation
(
i ,,j k)
denotes the point located at(
iΔx, jΔy,kΔz)
, integer multiples i, j, and k ofthe space descretization Δx, Δy, and Δz, respectively. If the descretization in time is denoted as Δt with integer multiples n, the field values of the x-component of the electric field locate at ⎟
⎠ field values of the y-component of the magnetic field locate at ⎟
⎠
H . Notations of other field values can be
defined in a similar manner. Applying central difference method for both the differentiation in time and space, the iterative finite-difference time-domain equations can be derived as
( )
( )
light in free space. At time step n, the magnetic field is first updated to the next half time step by (2.2a) with the electric field at current time step and magnetic field at previous half time step; the updated magnetic field at next half time step and the electric field at current time step is used to update the electric field to the next time step by (2.2b). After updating both fields, the FDTD iteration in a time step is completed. Time step is than moved from n to n+1. Repeatedly, the desired time responses of both fields can be evaluated.
2.1.2 FDTD Equations in Matrix Form
Let
E
n andH
n be the column vectors formed by the electric and magnetic field values at time step n in (2.2), respectively. The FDTD update equation can be cast into matrix form as( )
r nn
n
H t μ D E
H
+21 = 0 −21 − ⋅Δ ⋅ −1⋅ ⋅0
η
cη
(2.3a)and
( )
1 0 211 c − +
+ = n + ⋅Δ ⋅ r ⋅ T ⋅ n
n
E t ε D H
E η
, (2.3b)where D denotes the discrete curl operator. Substitute (2.3a) and the (n-1)-th time step of (2.3b) into the n-th time step of (2.3b), the FDTD update equation in matrix
form can be written as
where the identity matrix is denoted by I.
Eliminating the equations involving magnetic fields in (2.4) yields an update equation with electric field only.
( )
2 1 1 1On the other hand, an update equation with only magnetic field concerned can also be obtained.
These two matrix equations with only a single field in (2.5) play an important role in the next chapter, where Lanczos algorithm is applied to the symmetric matrix inside the bracket. With the hybridizing technique which will be detailed later, direct evaluation of (2.5) is not required. Existing code of FDTD solvers can be reused with the introducing of additional time-reversal FDTD equations.
2.1.3 Time-Reversal FDTD Equations
The time-reversal technique in FDTD method has been proposed for the numerical synthesis of a microwave structure [20]. By slightly rearranging (2.3), the update equation can be operated backward in time as
n r
n
n
H c t μ D E
H
−21 = 0 +21 + ⋅Δ ⋅ −1⋅ ⋅0
η
η
(2.6a)and
2 1 0 1
1 − −
− = n− ⋅Δ ⋅ r ⋅ T⋅ n
n
E c t ε D H
E η
. (2.6b)The time-reversal FDTD iterations proceed in a similar manner as ordinate FDTD. At time step n, the magnetic field is first updated to the previous half time step by (2.6a) with the electric field at current time step and magnetic field at next half time step;
the updated magnetic field at previous half time step and the electric field at current time step is then used to update the electric field to the previous time step by (2.6b).
After updating both fields, another iteration begins with the time step moving from n to n−1.
For the hybrid method presented in the following chapter, the time-reversal FDTD equations will only activate after the model order reduction procedure begins.
As long as the model order reduction procedure converges rapidly, the computation overhead increased by introducing the time-reversal FDTD equation and the numerical error cumulates in the backward iteration in time [20] can be minimized.