David 1). Davidson
John 1. Volakis
Dept.E&E Engineering University of Steiiencosch Steilenbosch 7600, South Africa (t27) 21 BO8 4458 (t27) 21 BO8 4981 (Fax)
[email protected](e-mail)
Rad. Lab., EECS Dept. University of Michigan Ann Arbor, MI 48 109-2 122
(734) 647-1797 (734) 647-2 106 (Fax) volakisQumich.edu(email)
Foreword by the Editors The FDTD method has become one of the workhorses of computational electromagnetics. Although it originated during the 1960s, a number of theoretical issues had to be addressed before it achieved widespread acceptance in our community. Furthermore, it was only by the late 1980s that computer hardware had progressed sufficiently to satisfy the method’s voracious appetite for memory and CPU time. The PC revolution that followed during the 1990s has brought the method to desktop computers.
Despite the great popularity of the FDTD, the method has some problems. From a user’s viewpoint, the fine meshes required to control numerical dispersion art: at best often undesirable, and at worst can render the solution of elrxtromagnetically large problems impractical. One technique for addressing this is to increase the order of the finite-difference operators, which is the topic of this paper. This is the first of a two-part paper, concentrating on theoretical aspects; in the next issue, .various applications will be presented.
Higher-Order Finite-Difference Schemes for Electromagnetic Radiation, Scattering, and Penetration, Part I: Theory Stavros V. Georgakopoulos*, Craig R. Birtcher*, Constantine A. galanis*, and Rosemary A. Renauf ‘Department of Electrical Engineering Telecommunication Research Center, Arizona State University Tempe, AZ 85287-7206 USA
‘Department of Mathematics, Arizona State University Tempe, AZ 85287-1804 USA
Abstract Higher-order schemes for the Finite-DifferenceTime-Domain (FDTD) Method are presented, in particular, a second-order-intime, fourth-order-in-space method: FDTD(2,4). This method is compared to the original Yee FDTD scheme. One-dimensional update equations are presented, and the characteristics of the FDTD(2,4) scheme are investigated. Theoretical results for numerical stability and dispersion are presented, with numerical results for the latter, as well. The use of the perfectly matched layer for the FDTD(2,4) scheme is discussed, and numerical results are shown. Applications follow in the second part of this two-part paper. Keywords: FDTD methods; electromagnetic radiation; electromagnetic scattering; numerical stability
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/E€€ Antenna’s and Propagation Magazine, Voi. 44, No. 1, February 2002
1. Introduction
T
he technological advancements of the last few decades have triggercd new cngineering problems and challenges. With the clock speed of all electronic equipment increasing, classical engineering analysis tools have become obsolete. Furthermore, as communication systems operate at higher frequencies, the antenna elements become smaller, whereas the platforms they operate on (e.g., helicopter airframes) become electrically larger. These problems yield large computational domains, and they require significant amounts of computational resources, such as memory and execution time. Traditional finite methods (FDTD and FEM) are second-order accurate, thereby restricting the size of the domains that can be handled efficiently. The subject of this paper is the investigation o f methods that can solve electrically large domains very accurately. Electrically large problems are particularly challenging, due to their computermemory and processing-time requirements. The classical FDTD method, as proposed by Yee [l], is second-order accurate in both space and time (FDTD(2,2)), thereby requiring many grid points per wavelength to accurately model wave propagation. The FDTD method, as with all finite methods, suffers from dispersion errors, meaning that the phase velocity in the FDTD grid is not the same as the phase velocity in the continuous physical domain. In order to reduce dispersion errors, the discretization of a structure should become finer, and therefore a smaller cell size should be used. On the other hand, finer discretizations yield larger computational spaces, thereby causing the two issues referred to above (computational time and resources) to become even more restrictive. Consequently, mesh refinement is not an efficient solution, and sometimes not even possible. Numerous attempts have been made during the development of FDTD to minimize phase errors [2-61 One of the most promising approaches for reducing dispersion errors is based on higherorder accurate schemes [7-111. The second-order accuracy in both time and space of the standard FDTD scheme implicitly imposes strict limitations on the size of problems that can be handled efficiently. On the contrary, higher-order schemes exhibit low dispersion errors, and can utilize coarser cell sizes than those needed by second-order schemes to achieve satisfactory levels of accuracy. In addition, coarser meshes yield smaller computational spaces, reduced computational times, and require fewer computational resources. Thus, the implementation of higher-order schemes into the FDTD algorithm will permit the efficient analysis of electrically larger problems.
Compact higher-order finite-difference schemes have also been discussed in the literature. Lele [ 121 examined compact finitedifference schemes with improved spectral resolution. Some optimized compact finite-volume stencils were developed, analyzed, and implemented for linear wave propagation phenomena in [13]. A comparison of higher-order schemes for free-space propagation was presented in [ 14, 151. Also, fourth-order difference operators, both explicit and compact (implicit), for initial boundary-value problems, have been discussed in [16]. Liu [17] presented a Fourier analysis of the dispersive, dissipative, and isotropy errors of various spatial and time stencils, applied to Maxwell’s equations on multi-dimensional grids. Furthermore, a family of higher-order finite-difference methods with good spectral resolution was described in [18]. These schemes were a generalization of the standard compact (Pade) schemes, discussed by Lele [ 121.
IEEE Antenna,’s and Propagation Magazine, Vol. 44, No. 1, February 2002
Another difficulty of higher-order FD methods relates to boundary conditions and modeling of discontinuities, and this still remains a research topic. Most of the research that has appeared in the literature related to higher-order schemes has examined limited types of problems, such as one- and two-dimensional problems with certain types of metal and/or dielectric discontinuities [ 191. Three-dimensional analysis has been restricted to free-space simulations. One of the biggest challenges in the design of higher-order schemes is the formulation of not only accurate, but also stable, boundary conditions. Many different methods have been proposed in [20-231. However, many higher-order implementations that were stable for scalar (one-equation) one-dimensional problems were later proven unstable for systems of one-dimensional equations. Also, most of the papers have discussed and addressed boundary conditions that are on the outside of the domain, and not in the interior of the computational space. However, most practical engineering problems involve treatment of boundary conditions inside the domain. The issue of boundary conditions becomes even more complex and challenging in the context of three-dimensional analysis. It is the purpose of this paper to investigate low-dispersion methods that can accurately, as well as efficiently, solve electrically large problems. Specifically, a second-order-in-time and fourth-order-in-space FDTD(2,4) will be used throughout this paper. The main objective is to develop a three-dimensional (3D) code that incorporates FDTD(2,4). This code will be applied in “open-space” electrically large problems, Le., antenna analysis, EM penetration, shielding-effectiveness studies, etc. In order to solve such unbounded-domain problems, absorbing boundary conditions (ABCs) need to be formulated and implemented. Furthermore, boundary conditions have to be formulated in a stable and accurate way.
Here, the theory of FDTD is presented for both second- and fourth-order accurate schemes. Their different characteristics, such as dispersion and stability, are described and compared. The accuracy of the different FDTD schemes is initially examined through numerical experiments in one-dimensional (1D) domains. Then, the implementation of absorbing boundary conditions, such as the. perfectly matched layer (PML), is carried out in the context of a fourth-order-accurate FDTD method. Applications of higher-order FDTD schemes to practical engineering problems are presented in the second part of this paper, to be published in the next (April, 2002) issue of the IEEE Antennas and Propagation Magazine.
2. FDTD Schemes FDTD methods attempt to solve the differential form of Maxwell’s equations, and therefore their implementation consists of finite-difference (FD) approximations to spatial and time derivatives. Consequently, the accuracy of FDTD solutions depends on the accuracy of the finite-difference schemes that are used. Finite-difference stencils appeared initially in the field of mathematics, and numerous schemes have been examined, each exhibiting their own advantages and disadvantages. In this section, two schemes are considered: the standard second-order (both in time and space) finite-difference scheme (FDTD(2,2)), and the second-order-in-time and fourth-ordcr-in-space scheme (FDTD(2,4)). The characteristics of the two schemes are presented and compared.
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+
-1
e
Ax 2
Ax 2
istics, and is directly related to the numerical dispersion. Another important characteristic of a scherne is its stability criterion, which defines the largest time step that can be used in order for the scheme to be stable. In the following subsections, the stability and the dispersion of the FDTD(2,2) and FDTD(2,4) schemes are described and compared.
Figure 1. The computational molecule of a central secondorder finite-difference stencil.
2.1 The FDTD(2,2) Scheme In 1966, Kane Yee presented a set of finite-difference equations for the system of Maxwell's curl equations for lossless materials [l]. Yee introduced the following notation for a function of space and time on,a Cartesian grid: F ( x , y , z , t ) =F(ihx,jAy,ICAz,nAt)=F"(i,j,k), (1) where Ax, Ay , and Az are the grid sizes in the x, y , and z directions, respectively; At is the time increment; and i, j , k, and n are integers. Yee used central finite-difference expressions for the space and time derivatives, which are second-order accurate in space and time, respectively:
a~ (i, j , k )
-=
at
F"+"~(i, j , IC)At
(i, j , IC)
+
(At ').
2.3.1 Numerical Stability Finite-difference schemes rcquire that the time increment, A t , have a specific bound relative to the spatial discretizations, A x , Ay , and Az . This bound is necessary to prevent numerical instability, which is an undesirabls feature of explicit differentialequation schemes. Numerical ins1ability can cause the computed results to spuriously increase without limit as time-stepping progresses, thereby yielding meaningkss solutions.
An investigation of the numerical stability of Yee's algorithm (FDTD(2,2)) was initially presented in [24], and detailed afterwards in [25]. The stability analysis is based upon finding the Fourier numerical-wave modes in the grid for both the electric and magnetic field components, and requiring that each Fourier mode be stable for arbitrary angles of propagation through the mesh. Enforcing the stability in the three-dimensional FDTD(2,2) algorithm provides the following constraint on the algorithm's time step, relative to the spatial-grid increments:
(3)
Finite-difference schemes are commonly visualized using so-called computational molecules. The computational molecule corresponding to the second-order central finite difference of Equations (2) or (3) is illustrated in Figure l .
where c = l/Gis the speed of light in the homogeneous material being modeled. For uniform grids with equal cell sizes in all dimensions ( A x = Ay = Az = A ) , Equation (5) reduces to A
2.2 The FDTD(2,4) Scheme More-accurate approximations to derivatives are provided by higher-order finite-difference schemes. Here a second-order-intime and fourth-order-in-space scheme (FDTD(2,4)) is described. In this scheme, the positions of the electric- and magnetic-field components remain the same as those of the FDTD(2,2) scheme. The FDTD(2,4) stencil uses central finite differences, fourth-orderaccurate in space and second-order-accurate in time: 8~ (i, j , k ) 9 F" (i + 1/,j , k ) - F" (i - %, j , k ) ax 8 Ax 1 F" (i + 3/2, j , k ) - F" (i - 3/2, j , k ) __ +O(Ax4). 24 Ax -==-
(4)
The stability criterion for the FDTD(2,4) scheme was derived by Fang [9], who based his formulation on the Von Neumann condition, suitable for systems of equations and multi-step schemes. This analysis requires that the eigenvalues of the amplification matrix be bounded. However, this is only a necessary but not sufficient condition for stability. A necessary and sufficient condition for stability is that the norm of the amplification matrix is bounded. Therefore, after a stability constraint has been obtained by the eigenvalue analysis, it should be verified that the norm of the amplification matrix remains bounded under the same constraint. The stability constraint for the FDTD(2,4) scheme derived by Fang is
The computational molecule of a central fourth-order finite-difference stencil corresponding to Equation (4) is shown in Figure 2.
2.3 Characteristics of FDTD Schemes Any FDTD scheme exhibits its own strengths and weaknesses. The order of accuracy of a scheme dominates its character136
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&
2
2
3& 2
Figure 2. The computational molecule of a central fourth-order finite-difference stencil. /€€€Antenna's and Propagation Magazine, Vol. 44, No. 1, February 2002
(7)
It is apparent that the stability criterion of FDTD(2,4) is more constraining than that of FDTD(2,2). For uniform grids with equal cell sizes in all dimensions ( A x = Ay = Az = A), Equation (7) becomes 6 A 7 c43
At
