Finally, we simulate a 3-D parallel plate excited by a current source. The mesh details are shown in Fig. 5.5, and it involves 350 tetrahedral elements and 544 edges. The current source is launched along the green line shown in Fig. 5.5. Its expression is J = ˆz2(t − t0) exp −(t − t0)2/τ2 with τ = 1 s and t0 = 4τ . The matrix-free time-
domain method requires the time step to be less than 2.4 × 10−11 s to guarantee stability. This renders an estimated total CPU time 2.0104 × 108 s to finish the simulation. It’s impossible to run such a long time to obtain the solution. For convenience, we can find out the voltage drop between the two PEC plates analytically since the input frequency is very low. In that case, the parallel plate can be viewed
0 0.5 1 1.5 2 x 10−8 −1 −0.5 0 0.5 1x 10 −9 Time (s)
Electric field (V/m) Point 1 (Proposed) Point 2 (Proposed) Point 1 (Analytical) Point 2 (Analytical)
Fig. 5.4. Simulation of a 3D domain with a tetrahedral mesh: electric field at observation points.
as a capacitor of capacitance C = 5.9027 pF, thus the voltage can be calculated as −τC2 exp (−(t − t0)2/τ2) V. On the other hand, only null space contributes to the
solution, and all the unstable eigenmodes should be removed. This results in a much larger time step that is 0.01 s for the proposed unconditionally stable matrix-free time-domain method. Therefore, it only takes 30.7393 s to finish the simulation. In Fig. 5.6, the voltage simulated from the proposed method in comparison with analytical solution is plotted as a function of time. Obviously, the simulated result agrees very well with the reference result, validating the accuracy of the proposed method.
5.4 Conclusion
In this chapter, we develop an unconditionally stable matrix-free time-domain method for analyzing general electromagnetic problems discretized into arbitrarily shaped unstructured meshes. This method does not require the solution of a system
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 X (m) Y (m) Z
Fig. 5.5. Simulation of a parallel plate: Mesh details.
Fig. 5.6. Simulation of a parallel plate: Voltage drop between the two plates compared with analytical solution.
matrix, no matter which element shape is used for space discretization. Furthermore, this property is achieved irrespective of the time step used to perform the time domain
simulation. As a result, the time step can be solely determined by accuracy regardless of space step. Numerical experiments have validated the accuracy and efficiency of the proposed new method.
6. FAST EXPLICIT AND UNCONDITIONALLY STABLE
FDTD METHOD
6.1 Introduction
Among so many time-domain methods, the finite-difference time-domain (FDTD) method is one of the most popular methods for electromagnetic analysis. This is mainly because of its simplicity and optimal computational complexity at each time step. However, as the matrix-free time-domain methods developed in previous chap- ters naturally reduce to the FDTD method in orthogonal grid, the time step of a conventional FDTD [1, 2] is also restricted by space step for stability, as dictated by the Courant-Friedrich-Levy (CFL) condition. If the space step of a given problem is determined solely from an accuracy point of view, the time step required by stability has a good correlation with the time step determined by accuracy. However, if the problem involves fine features relative to working wavelength, the time step required by stability can be orders of magnitude smaller than that required by accuracy. As a result, a large number of time steps must be simulated to finish one simulation, which is time consuming.
To overcome the aforementioned barrier, researchers have developed implicit un- conditionally stable FDTD methods, such as the alternating-direction implicit (ADI) method [3, 4], the Crank-Nicolson (CN) method [5], the CN-based split step (SS) scheme [6], the pseudo-spectral time-domain (PSTD method) [7], the locally one- dimensional (LOD) FDTD [8, 9], the Laguerre FDTD method [10, 11], the associated Hermite (AH) type FDTD [12], a series of fundamental schemes [13] and many oth- ers, but the advantage of the conventional FDTD is sacrificed in avoiding a matrix solution. When the problem size is large, the implicit unconditionally stable FDTD methods become inefficient. Research has also been pursued to address the time
step problem in the original explicit time-domain methods [14–16]. In [17, 18], the source of instability is identified, and subsequently eradicated from the underlying numerical system to make an explicit FDTD unconditionally stable. It is shown that the source of instability is the eigenmodes of the discretized curl-curl operator whose eigenvalues are the largest. These eigenvalues are higher than what can be stably simulated by the given time step. To find these unstable modes, in [18], a partial solution of a global eigenvalue solution is computed. In general, only a small set of the largest eigenpairs of the system matrix need to be found, and the system matrix is also sparse. The same idea is also applied to the matrix-free time-domain method in Chap. 5 to solve time step problem. However, the computational overhead of the resultant scheme may still be too high to tolerate when the matrix size is large.
The time step required for a stable explicit simulation is limited by the largest eigenvalue of the system matrix. However, the finer the space step, the larger the largest eigenvalues of the system matrix. Therefore, there should exist a relationship between the fine cells present in a space discretization and the unstable modes that cannot be stably simulated by the given time step. We do not have to perform a brute-force eigenvalue solution to identify the unstable modes. Instead, we can utilize the relationship between the fine cells and the unstable modes to develop a more efficient explicit and unconditionally stable method. Along this line of thought, in this work, we first develop a new patch-based single-grid FDTD formulation. Using this formulation, we identify the theoretical relationship between fine cells and the largest eigenmodes of the underlying system matrix. We prove that once there exists a difference between the time step required by stability and the time step determined by accuracy, i.e., a difference between the fine-cell size and the regular-cell size, the largest eigenmodes of the original system matrix can be extracted from fine cells. The larger the contrast ratio between the two time steps, the more accurate the eigenmodes extracted in this way. Based on this theoretical finding, we propose an efficient algorithm to find the unstable modes directly from fine cells, and subsequently deduct these unstable modes from the numerical system to achieve an explicit time
Fig. 6.1. Illustration of a patch-based discretization of Faraday’s law.
marching with unconditional stability. The essential idea of this work can also be applied to other time-domain methods.