Conformal PML-FDTD Schemes for Electromagnetic Field Simulations: A Dynamic Stability Study

Transcription

1 902 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 6, JUNE 2001 Conformal PML-FDTD Schemes for Electromagnetic Field Simulations: A Dynamic Stability Study F. L. Teixeira, K.-P. Hwang, W. C. Chew, and J.-M. Jin Abstract We present a study on the dynamic stability of the perfectly matched layer (PML) absorbing boundary condition for finite-difference time-domain (FDTD) simulations of electromagnetic radiation and scattering problems in body-conformal orthogonal grids. This work extends a previous dynamic stability analysis of Cartesian, cylindrical and spherical PMLs to the case of a conformal PML. It is shown that the conformal PML defined over surface terminations with positive local radii of curvature (concave surfaces as viewed from inside the computational domain) is dynamically stable, while the conformal PML defined over surface terminations with a negative local radius (convex surfaces as viewed from inside the computational domain) is dynamically unstable. Numerical results illustrate the analysis. Index Terms Absorbing boundary condition, curvilinear grids, dynamic stability, FDTD methods. I. INTRODUCTION THE application of the perfectly matched layer (PML) absorbing boundary condition to the truncation of the computational domain in the finite-difference time-domain (FDTD) and finite-element methods (FEM) for electromagnetic field computations has become very popular [1] [18]. This is mainly because of its excellent numerical efficiency and ease of implementation. Applications in optics [19], acoustics [20], quantum mechanics [21], and elastodynamics [22] have also been recently reported. Many studies have focused on the application of the PML in Cartesian grids. In several instances, however, the FDTD method is used in conjunction with non-cartesian grids. For example, a number of body-conformal FDTD methods have been proposed to model more general geometries in curvilinear coordinates. Using the complex coordinate stretching approach [2], the PML absorbing boundary condition has been extended to cylindrical and spherical coordinates [11] [14] and to general orthogonal curvilinear coordinates (conformal PML) [15]. The use of a hyperbolic grid generation technique in conjunction Manuscript received December 21, 1999; revised November 27, This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under MURI Grant F F. L. Teixeira was with Research Laboratory of Electronics, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA USA. He is now with ElectroScience Laboratory and Department of Electrical Engineering, The Ohio State University, Columbus, OH USA ( teixeira@ee.eng.ohio-state.edu). K.-P. Hwang, W. C. Chew, and J.-M. Jin are with the Center for Computational Electromagnetics, Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL USA. Publisher Item Identifier S X(01) with the conformal PML to achieve an FDTD scheme for complex geometries in a compact grid is described in [16]. Moreover, a study of the dynamic stability of the Cartesian, cylindrical, and spherical PMLs was presented in [17]. In this work, we extend the analysis in [17] to study the dynamic stability of the conformal PML for electromagnetic applications. We show that the local geometry of the termination plays a fundamental role in determining the behavior of the conformal PML. More specifically, we show that the conformal PML defined over surface terminations with positive local radii of curvature (concave surfaces as viewed from inside the computational domain) is dynamically stable, while the conformal PML defined over surface terminations with a negative local radius (convex surfaces) is dynamically unstable. The conclusions have impact on the design of PMLs for numerical simulations, as well as their use as a physical basis (blue-prints) for artificially engineered materials over curved surfaces. The use of absorbing materials inside the computational domain is of interest if the current tendency toward global modeling and the requirement of interfacing heterogeneous simulations environments is considered [23]. Numerical simulation results illustrate the main conclusions. II. ANALYSIS Two basic formulations for the PML exist in the frequency domain. The first one is the complex-space formulation [1] [4], [11], [13], in which added degree of freedom modify Maxwell s equations through the introduction of complex stretching variables. This can be shown to be equivalent to an analytic continuation of Maxwell s equations to a complex coordinate spatial domain [11], hence the name complex-space. In the second class of formulations, Maxwell s equations retain their form and the PML is obtained through the insertion of artificial anisotropic tensors for the permittivity and permeability within the PML layer [5] [7], [10]. In the spectral analysis of the PML in global coordinate systems (Cartesian, cylindrical, and spherical) [17], one can directly use the (global) ordinary closed-form solutions of Maxwell s equations to study the behavior of the field solutions inside the PML. This is because the PML solutions, in the complex-space PML formulation, are just an analytic continuation of the ordinary Maxwell s solutions to a spatial domain of complex variables. Such analytical continuation is carried out in the normal coordinate to the grid surface termination (in the case of corner regions, two or more coordinates are analytically continued simultaneously). The complex-space coordinates are X/01$ IEEE

2 TEIXEIRA et al.: CONFORMAL PML-FDTD SCHEMES FOR ELECTROMAGNETIC FIELD 903 obtained through the following simple transformation in the original (real) coordinates [11], [13] [15]: where for or (normal coordinates in the Cartesian, cylindrical, and spherical systems). The variables are the so-called complex stretching variables [2]. We note that because depends on the coordinate only, the above transformation leads to continuous complex-space coordinates, for any (bounded) in the interval of integration. In the case of the anisotropic PML formulation in these coordinate systems, closed-form solutions are also available because the field solutions of the anisotropic PML formulation and the complex-space PML formulation are related to each other through simple relations [14]. Alternatively, one may directly study the behavior of the solutions of the anisotropic PML formulation in these coordinate systems through an examination of the spectral properties of the corresponding PML constitutive anisotropic tensors, and by noting that the anisotropic PML formulation preserves the form of Maxwell s equations [17]. In contrast to the Cartesian, cylindrical, and spherical PMLs, however, the conformal PML [15], [16] is defined through a local coordinate system attached to each point of the termination surface. As a result, global methods of analysis are, in general, not applicable. In this case, the anisotropic PML formulation is the most direct route for the analysis of the field behavior inside the PML layer. Since the anisotropic PML is uniquely defined through its constitutive tensors, the analysis may simply focus on the analytical properties of such tensors. The anisotropic conformal PML constitutive tensors are written as and, with [15], [16] (1) surface. The equation defines the mesh termination surface. The Cartesian, cylindrical, and spherical PML constitutive tensors are special cases of (2). The tensor is frequency-dependent because the variables, and are frequency-dependent. If the Fourier inversion contour for is carried out along the real axis, the primitive causality condition on, i.e., for, implies that must be analytic (holomorphic) in the upper half-plane of the complex plane (zeros are also forbidden on the upper half-plane [17]: for a more detailed discussion of the connection between causality and the analytical properties of the constitutive tensors, see [25] [29]). However, when is not analytic in the upper half-plane, causality can still be preserved provided that the Fourier inversion contour is taken above any singularities [25]. In this case, the medium will behave as an active medium and its response will not be dynamically stable anymore. The definition of a dynamically stable system adopted here is the same as one used in [17], viz., a system such that all of its eigenmodes approach zero as (asymptotically stable) or remain bounded as (marginally stable). It is important to stress that the our analysis here is focused on the spectral characteristics of the PML equations. The dynamic stability criterion derived from this spectral analysis discussed here is distinct from some other stability criteria, such as the numerical stability criterion resulting from the field-splitting of the modified Maxwell s equations in the time domain PML [18]. This is discussed elsewhere [17]. The PML constitutive tensor in (2) contains factors of the form where or. The variable is the local normal variable to the mesh termination, after the complex stretching (i.e., analytic continued to the upper-half plane) is applied to the local normal coordinate [generalization of (1)]: (3) (2) where is the complex stretching variable along the normal coordinate to the mesh termination surface, and and are the stretched and nonstretched local metric coefficients, respectively. The unit vectors and are tangential to at along the principal lines of curvature, and is the unit vector that is outwardly normal to at this point. In terms of the local coordinates,we write, where is the position vector. These unit vectors define the Darboux Dupin local frame on the surface [24]. In such an orthogonal curvilinear coordinate system, the metric coefficients are given by, and, where and are the principal radii of curvature at the point in. Moreover,. The conformal PML is therefore defined over parallel surfaces to the mesh termination where and [2], [15]. For positive (concave termination as viewed from inside the computational domain), the zeros of the equations will introduce poles for restricted to the lower half of the complex plane only, since its solutions have a negative imaginary part. As a result, the PML constitutive tensors on a concave termination have their analyticity preserved over the entire upper half-plane. In the case when one or both radii of curvature are negative, however, poles will eventually appear in the upper half-plane, (4) (5)

3 904 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 6, JUNE 2001 Fig. 1. Mutual interaction mechanism affected by a hypothetical convex PML. Fig. 2. Transversal discontinuity of the complex stretching variables in an inner rectangular corner. because or, is now negative. In such a case, the PML will behave as an active media with constitutive tensors having poles on the upper half-plane. The resultant field solutions are expected to exhibit an exponential growth in time. This will be illustrated by the numerical simulation results in the next section. Note that this active behavior is local because it depends on the local radii of curvature. This local behavior (which, nevertheless, eventually contaminates the numerical solution in the whole domain) will also be illustrated in the next section. For curvatures with a very large radius (in absolute value), the instability tends to become weaker, since the unstable poles are pushed near to the real axis. In the limit, the complex metric factors become (asymptotically) real,. Such a limit corresponds to the Cartesian PML. The analysis indicates that the application of the conformal PML formulation is restricted to concave surface terminations (when viewed from the inside of the computational domain). It is interesting to note that electromagnetic waves incident upon a concave or planar (half-space) PML region cannot return back to the interior domain through a direct path, as it would be possible in a convex surface termination. Note also that if the PML were applicable to convex terminations, then the interior solution could be affected by the PML itself. For example, mutual interactions between different portions of the scatterer could be eventually attenuated by a hypothetical convex PML region. In this case, the solution of the PML-FDTD scheme on the interior domain would be different from the true open space solution. This is illustrated in Fig. 1. The fact that the convex PML is unstable precludes these situations. In light of the discussion in the previous paragraph, a question naturally arises: if the use of a conformal PML is dynamically unstable on convex surfaces, then what happens in the situations like that of a PML constructed over a rectangular, inner domain, e.g., an hypothetical PML coating a box located inside the computational domain, as depicted in Fig. 2? At first, this would seem to correspond to PMLs defined over the planar surfaces comprising each face and corner regions of the box, i.e., to Cartesian PML s, which are dynamically stable. On the other hand, its corner regions correspond to the limit case of a convex PML with a negative infinite (again, when viewed from the inside of the computational domain) curvature at the edges or corners, and, as discussed before, this would lead to dynamic instability. The answer to this apparent dilemma is that the Cartesian PML constructions over such (inner) edge or corner regions of Fig. 2 are fundamentally different from the usual PML corner regions constructed over the outer termination of a Cartesian grid. In the case of the hypothetical PML depicted in Fig. 2, the inner rectangular surface does not correspond to a separable geometry (i.e., the corresponding boundary value problem is not separable, as opposed to the outer rectangular case). As such, it is not possible to construct a true PML over its corners using the Cartesian coordinate system because the complex stretching variables [2],, would not be functions of the corresponding normal coordinates only ( function of only, ). The fact that is a function of only is a crucial property of the PML in any coordinate system because it ultimately guarantees the continuity of the complex-space coordinates. The continuity of the complex-space coordinates is important because it preserves the continuity of the fields in the complex-space PML formulation, and hence the boundary conditions (including the perfect matching condition). If a nonmatched construction is nevertheless imposed (i.e., with, or etc.), we conjecture that instabilities are expected because of fictitious artificial sources of energy created at the regions of transversal discontinuity of the complex stretching variables, as indicated in Fig. 2. As opposed to normal discontinuities, transversal discontinuities on cause discontinuous complex-space coordinates. To see that, we may use the very definition of the complex-space coordinates given by (1) with substituted by Clearly, the function is a discontinuous function of if is a discontinuous function of (necessary in the case of the geometry such of Fig. 2). Note that normal discontinuities (6)

4 TEIXEIRA et al.: CONFORMAL PML-FDTD SCHEMES FOR ELECTROMAGNETIC FIELD 905 Fig. 4. Field distribution after 100 time steps, illustrating the launching of the pulse inside the computational domain. Fig. 3. Grid generated using a hyperbolic grid generator. A convex region is present to test the stability of the PML as a function of its geometry. The PML region comprises the ten most external cells. in for do not affect the continuity of because the integration in (6) is carried over the normal variable itself. III. NUMERICAL RESULTS We have constructed a two-dimensional (2-D) conformal FDTD scheme based on the formulation discussed in [30]. A hyperbolic grid generation technique was employed to ensure that the grid is orthogonal on the PML region, as required by the conformal PML formulation [15]. Details on this grid generation technique can be found in [16]. The anisotropic version of the conformal PML [15] is used to truncate the FDTD domain. Fig. 3 shows the computational grid used in our numerical evaluations. It contains a convex region (as viewed form inside the computational domain) which extends to grid termination, so that the conformal PML is defined over a surface which includes a convex part. The scatterer is 4-m long and perfectly conducting. The grid has 35 cells in the radial direction, with the ten most external ones comprising the conformal PML, and 90 cells in the azimuthal direction. The source is an Hertzian dipole located at the left portion of the grid, as illustrated in Fig. 4, and the source pulse is a modulated Gaussian with central frequency MHz. The envelope of the modulated Gaussian is specified by, where, and MHz. The average grid density is 15 points per wavelength. Fig. 5 shows the amplitude field distribution after 1100 time steps. The asymmetry of the field behavior in the concave and convex portions of the PML boundary is evident. While the field is attenuated in the concave portions of the PML and the reflections suppressed, a strong instability builds up in the convex portion (active region) as soon as the incident wave reaches this portion of the domain. This instability eventually contaminates the whole solution. To avoid such instability when using a general body-conformal orthogonal grid, we may distort the grid so that the PML is introduced over a strictly concave termination surface. This is Fig. 5. Field distribution after 1100 time steps, after the passage of the pulse. The instability coming from the convex PML region is clearly visible. Fig. 6. Second grid generated using a hyperbolic grid generator. Now the grid is distorted toward its ends so that the PML (ten most external cells) may be introduced over a strictly concave surface. illustrated in Fig. 6, where the same scatterer as in Fig. 3 is considered, but now the computational grid is constructed such that its ten outermost cells (PML region) comprise a system of concave surfaces (curves). Figs. 7 and 8 depict the amplitude field

5 906 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 6, JUNE 2001 that a conformal PML defined over a convex termination surface (when viewed from inside the computational domain) leads to dynamically unstable solutions. Numerical results illustrate the analysis. REFERENCES Fig. 7. Field distribution after 100 time steps, illustrating the launching of the pulse inside the new computational domain. Fig. 8. Field distribution after 1100 time steps, after the passage of the pulse. The field is absorbed by the conformal PML and the dynamic instability depicted in Fig. 5 is not present anymore. evolution, which show no dynamic instability coming from the PML. This example also serves to verify that the instability depicted in Fig. 5 is actually coming from the PML itself and is not from any spurious property [31] [33] of the spatial discretization scheme employed. Through our numerical experiments, we have also observed another form of instability, which appears in a very late time regime and which seems not to be associated with the local geometry of the termination. Because we use an unsplit scheme, this instability is also of a different nature from the known weak instability caused by the field splitting in the PML formulation [18], [34]. Since this other observed instability is much weaker than the dynamical instability studied in this paper, in principle it does not hamper the FDTD solution for transient problems. However, it may be a factor when dealing with FDTD for problems which require longer integration times (e.g., to achieve a steady state regime). We are currently investigating the origin of this instability and the best filtering schemes to supress them. IV. CONCLUSION We have analyzed the dynamic stability of the conformal PML. 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Teixeira received the B.S. and M.S. degrees from the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Brazil, in 1991 and 1995, respectively, and the Ph.D. degree from the University of Illinois, Urbana, in 1999, all in electrical engineering. From 1992 to 1994, he was with the Military Institute of Engineering and the Brazilian Army Research and Development Center (IPD-CTEx). From 1994 to 1996, he was with the Satellite Transmission Department of Embratel S. A. (MCI/Worldcom), Rio de Janeiro. From 1996 to 1999, he was a Research Assistant with the Center for Computational Electromagnetics, University of Illinois. From 1999 to 2000, he was a Postdoctoral Associate at the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge. Since 2000, he has been an Assistant Professor with the Department of Electrical Engineering and Electro- Science Laboratory (ESL), The Ohio State University, Columbus. His current research interests include wave propagation and scattering modeling for communication, sensing, and device applications. He has published over 20 journal articles and 30 conference papers. Dr. Teixeira is a member of Phi Kappa Phi. He was the recipient of a CAPES Fellowship for He was awarded the Raj Mittra Outstanding Research Award from the University of Illinois, a IEEE MTT-S Fellowship Award, and paper awards at the 1999 USNC/URSI Meeting (Boulder, CO) and at the 1999 IEEE AP-S International Symposium (Orlando, FL). He was the Technical Program Coordinator for the Progress in Electromagnetics Research Symposium, Cambridge, MA, in K.-P. Hwang received the B.S. degree in electronic engineering from Seoul National University, Seoul, Korea, in 1993, and the M.S. degree in electrical engineering from The Ohio State University, Columbus, in He is currently working toward the Ph.D. degree in electrical engineering at the University of Illinois at Urbana-Champaign, Urbana. His current research interests include numerical partial differential equation theory, time-domain Maxwell solver algorithms, electrical packaging, optical interconnects, and electromagnetic compatibility. W. C. Chew was born on June 9, 1953, in Malaysia. He received the B.S., M.S., Engineer s, and Ph.D. degrees, all in electrical engineering, from the Massachusetts Institute of Technology, Cambridge, in 1976, 1978, and 1980, respectively. His recent research interest has been in the area of wave propagation, scattering, inverse scattering, and fast algorithms related to scattering, inhomogenous media for geophysical subsurface sensing, and nondestructive testing applications. Previously, he has also analyzed electrochemical effects and dielectric properties of composite materials, microwave and optical waveguides, and microstrip antennas. From 1981 to 1985, he was with Schlumberger-Doll Research, Ridgefield, CT, where he was a program leader and later a department manager. From 1985 to 1990, he was an Associate Professor with the University of Illinois, where he is currently a Professor and teaches graduate courses in Waves and Fields in Inhomogenous Media, and Theory of Microwave and Optical Waveguides and supervises a graduate research program. He has authored a book titled Waves and Fields in Inhomogenous Media (Piscataway, NJ: IEEE Press, 1995), published over 200 scientific journals, and presented over 270 conference papers. From 1989 to 1993, he was the Associate Director of the Advanced Construction Technology Center at the University of Illinois. Presently, he is Director of the Center for Computational Electromagnetics and the Electromagnetics Laboratory at the same university. Dr. Chew is a member of Eta Kappa Nu, Tau Beta Pi, URSI Commissions B and F, and an active member of the Society of Exploration Geophysics. He was an NSF Presidential Young Investigator for He was also an AdCom member of the IEEE Geoscience and Remote Sensing Society and is an Associate Editor of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (1984 present), the Journal of Electromagnetic Waves and Applications (1996 present), and Microwave Optical Letters (196 present). He was also an Associate Editor with the International Journal of Imaging Systems and Technology ( ) and has been a Guest Editor of Radio Science (1986), the International Journal of Imaging Systems and Technology (1989), and Electromagnetics (1995). He is the winner of the 2000 IEEE Graduate Teaching Award and is a Founder Professor, College of Engineering at the University of Illinois.He has been listed many times in the List of Excellent Instructors on campus. J.-M. Jin (S 87 M 89 SM 94) received the B.S. and M.S. degrees in applied physics from Nanjing University, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in He is an Associate Professor of Electrical and Computer Engineering and Associate Director of the Center for Computational Electromagnetics at the University of Illinois at Urbana-Champaign. He has authored or co-authored over 100 papers in refereed journals and several book chapters. He has also authored The Finite Element Methods in Electromagnetics (New York: Wiley, 1993) and Electromagnetic Analysis and Design in Magnetic Resonance Imaging (Boca Raton, FL: CRC Press, 1998) and co-authored Computation of Special Functions (New York: Wiley, 1996). He currently serves as an Associate Editor of Radio Science and is also on the Editorial Board for Electromagnetics Journal and Microwave and Optical Technology Letters. Dr. Jin is a member of Commission B of USNC/URSI, Tau Beta Pi, and the International Society for Magnetic Resonance in Medicine. He was a recipient of the 1994 National Science Foundation Young Investigator Award and the 1995 Office of Naval Research Young Investigator Award. He also received the 1997 Xerox Junior Research Award and the 2000 Xerox Senior Research Award presented by the College of Engineering, University of Illinois at Urbana-Champaign, and was appointed as the first Henry Magnuski Outstanding Young Scholar in the Department of Electrical and Computer Engineering in He was a Distinguished Visiting Professor in the Air Force Research Laboratory in He served as an Associate Editor of the IEEE Transactions on Antennas and Propagation ( ). He was the symposium co-chairman and technical program chairman of the Annual Review of Progress in Applied Computational Electromagnetics in 1997 and 1998, respectively. He has been listed many times in the University of Illinois at Urbana-Champaign s List of Excellent Instructors.