IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,VOL. 51, NO. 10, OCTOBER 2003 2097 The Transfinite-Element Time-Domain Method Din-Kow Sun, Member, IEEE, Jin-Fa Lee, Member, IEEE, and Zoltan Cendes, Member, IEEE Abstract—This paper presents an efficient time-domain method for computing the propagation of electromagnetic waves in mi- crowave structures. The procedure uses high-order vector bases to achieve high-order accuracy in space, the Newmark’s method to provide unconditional stability in time, and the transfinite-element method to truncate the waveguide ports. The resulting system ma- trix is real, symmetric, positive–definite, and can be solved by using the highly efficient multilevel preconditioned conjugate gradient algorithm. Since the method allows large time steps and nonuni- form grids, the computational complexity for problems with ir- regular geometries is superior to the finite-difference time-domain method. Index Terms—Hierarchical vector bases, late-time instability, Newmark’s scheme, scattering matrix, time-domain Maxwell’s equations, transfinite-element method, vector finite-element method. I. INTRODUCTION A LTHOUGH the finite-difference time-domain (FDTD) method is widely used to model electromagnetic-wave propagation in microwave structures, it has three major draw- backs. First, FDTD employs a low-order approximation in space that requires at least ten cells per wavelength to achieve acceptable accuracy. Second, FDTD’s stability is tied to grid size through the Courant condition and imposes small time steps in structures involving fine details. Third, FDTD is not compliant with interface conditions in materials varying in both permittivity and permeability. Even newer conformal FDTD methods are not compliant because two grids are required [1]. These drawbacks necessitate the use of fine grids and many time steps and, thus, result in long solution times with large complex problems. While FDTD has been very widely used in the past, finite- element time-domain (FETD) methods have also been developed in recent years [7]–[9]. FETD is an outgrowth of advances in finite-element frequency-domain (FEFD) methods [2]–[6]. In [7], an unconditionally stable FETD method is presented that circumvents the major drawbacks of FDTD. Advantages of FETD include the use of high-order vector basis functions to achieve high accuracy, unconditional stability to allow the time step to be taken independent of the mesh size, and the use of a single mesh that easily conforms to material interfaces. FETD provides more accurate solutions for a given mesh size, it allows the time step to depend on the rise/fall time of the Manuscript received February 27, 2003; revised April 30, 2003. D.-K. Sun and Z. Cendes are with the Ansoft Corporation, Pittsburgh, PA 15219 USA. J.-F. Lee is with the Department of Electrical Engineering, The Ohio State University, Akron, OH 43210 USA. Digital Object Identifier 10.1109/TMTT.2003.817457 input signal rather than the mesh size, and it adapts the mesh to fine details without requiring the mesh to be fine elsewhere. This paper improves FETD in several ways. First, we rewrite the time-domain vector-wave equation in scaled time and space coordinates, making the new equation applicable to any dimen- sional scale. We then discretize the equation with the high-order hierarchical vector bases developed in [6]. These hierarchical basis functions bestow a hierarchical structure on the system matrix; this allows the system equation to be solved by using the efficient preconditioned conjugate gradient algorithm. Next, we employ the transfinite-element method [10] to couple the input and output waveguide ports to the three-dimensional (3-D) re- gion. In the past, the absorbing boundary condition (ABC) [11] was applied at port boundaries. However, the ABC approach has limited success with multimode scattering because both the port impedance and mode patterns change with frequency. Thus, the ABC approach is limited to less dispersive cases such as mi- crostrip problems with single-mode propagation. The transfi- nite-element method reduces the field unknowns on the ports to the coefficients of the modal expansion. It is proven in [12] that the transfinite-element method produces unitary scattering ma- trices. Thus, the method yields more accurate solutions than the ABC approach. Finally, Newmark’s method is used to derive an unconditionally stable second-order finite-difference equation in time. II. FORMULATION A. Scaled Time-Domain Vector Wave Equation Maxwell’s equations yield the following time-domain vector-wave equation in source-free regions: in (1) Here, is the electric field and , , and are the con- ductivity, relative permittivity, and relative permeability, respec- tively. The boundary conditions to be imposed on (1) are on on (2) Here, denotes the outward unit normal of the surface. Let be the highest angular frequency of interest. Scaling time as , (1) becomes (3) 0018-9480/03$17.00 © 2003 IEEE